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Transfer Function Measurement with Sweeps

SWEN MÜLLER, AES member,

Institut für Technische Akustik, RWTH, 52056 Aachen, Germany

AND

PAULO MASSARANI

Laboratório de Ensaios Acústicos (LAENA), Instituto Nacional de Metrologia, Calibraçãoe Qualidade Industrial (INMETRO), Av. Nossa Senhora das Graças, 50 – Xerém/Duque deCaxias – RJ, Brazil

Compared to using pseudo noise signals, transfer function measurements using sweeps asexcitation signal show significantly higher immunity against distortion and time variance.Capturing binaural room impulse responses for high-quality auralization purposes requiresa signal-to-noise of >90 dB that is unattainable with MLS-measurements due toloudspeaker non-linearity, but fairly easy to reach with sweeps due to the possibility ofcompletely rejecting harmonic distortion. Before investigating the differences and practicalproblems of measurements with MLS and sweeps and arguing why sweeps are thepreferable choice for the majority of measurement jobs, the existing methods to obtaintransfer functions are reviewed. The perennial need to use pre-emphasized excitationsignals in acoustical measurements will be addressed. A method to create sweeps witharbitrary spectral contents, but constant or prescribed frequency-dependent temporalenvelope is presented. The possibility of simultaneously analysing transfer function andharmonics is investigated.

1 INTRODUCTION

Measuring transfer functions and their associated impulse responses is one of the mostimportant daily tasks in all areas of acoustics. The technique is practically neededeverywhere. A loudspeaker developer will check the frequency response of a new prototypemany times before releasing it for production. As the on-axis response does not sufficientlycharacterize a loudspeaker, a full set of polar data requiring many measurements is needed.

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In room acoustics, the impulse response plays a central role, as many acoustical parametersrelated to the perceived transmission quality can be derived from it. The room transferfunction obtained by Fourier-transforming the room impulse response may be useful todetect modes at low frequencies. In building acoustics, the frequency dependent insulationagainst noise from outside or other rooms is a common concern. In vibro-acoustics, thepropagation of sound waves in materials and radiation from their surface is a vast field ofsimulation and verification by measurements with shakers. Profiling by detection ofreflections (sonar, radar) is another area closely linked to the measurement of impulseresponses.

While most of these measurement tasks do not require an exorbitant dynamic range, thesituation is different when it comes to acquiring room impulse responses (RIR) for use inconvolutions with dry audio material. Due to the wide dynamic range of our auditorysystem and the logarithmic relation between SPL and perceived loudness, anyabnormalities in the reverberant tail of a RIR are easily recognizable. This hold especiallywhen speech with its long intermediate pauses is used for convolution and when theauralization results are monitored with headphones, as required for “virtual reality” basedon binaural responses. With digital recording technology of today offering signal-to noise-ratios in excess of 110 dB, it does not seem too daring to demand an S/N ratio for acquiredRIRs that is at least equivalent to the 16 bit CD standard.

This work has been fueled by the constant disappointment about MLS-based measurementsof RIRs. Even under optimal conditions, with supposed absence of time variance, very littlebackground noise, appropriate pre-emphasis and an arbitrary number of synchronousaverages, it seems impossible to achieve a dynamic range superior to that of, say, ananalogue tape recorder. In any measurement using noise as excitation signal, distortion(mainly induced by the loudspeaker) is spreading out over the whole period of therecovered impulse response. The ensuing noise level can be reduced using longer excitationsignals, but never be secluded entirely. Distortion can be reduced using lower volume. Butthis leads to more background noise contaminating the results. Hence, a compromise levelhas to be arduously adjusted for every measurement site, often leaving the powercapabilities of the driving amplifier and the speaker largely unexploited.

In contrast, using sweeps as excitation signal showed to relieve to a great extent from theselimitations. Using a sweep somewhat longer than the RIR to be recovered allows excludingall harmonic distortion products, practically leaving only background noise as limitation forthe achievable S/N ratio. The sweep can thus be fed with considerable more power to thespeaker without introducing artifacts in the acquired RIR. Moreover, in unecoic conditions,the distortion can be classified into single harmonics related to the fundamental, allowingfor a simultaneous measurement of transfer function and frequency-dependent distortion.This possibility already anticipated by Griesinger [16] and described by Farina [15] will befurther examined in chapter 6.

Sweep-based measurements are also considerable less vulnerable against the deleteriouseffects of time variance. For this reason, they are sometimes the only option in long-distance outdoor measurements under windy weather conditions or measurements ofanalogue recording gear.

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2 EXISTING METHODS

Quite a number of different ways to measure transfer functions have evolved in the pastcentury. Common to all of them is that an excitation signal (stimulus) that contains all thefrequencies of interest is used to feed the device under test (DUT). The response of theDUT is captured and in some way compared with the original signal. Of course, there isalways a certain amount of noise reducing the certainty of a measurement. Therefore, it isdesirable to use excitation signals with high energy so to achieve sufficient signal-to-noiseratio in the whole frequency range of interest. Using gating techniques to suppress noiseand unwanted reflections further improves the S/N. In practice, there is also always acertain amount of non-linearity, and time-variances are commonplace as well in acousticalmeasurements. We will see that the different measurement methods react quite differentlyto these kinds of disturbances.

2.1 The Level Recorder

One of the oldest methods to bring a transfer function onto paper already involved sweepsas excitation signal. The DUT’s response to a sweep generated by an analogue generator isrectified and smoothed by a low-pass filter. The resulting voltage is input to a differentialamplifier whose other input is the voltage derived from a precision potentiometer that ismechanically linked to the writing pen. The differential amplifier’s output controls thewriting pen that is swept over a sheet of paper with the appropriate scaling printed on it.The potentiometer may be either linear or logarithmic to produce amplitude or dB readingson the paper. Obviously, this method does not need any digital circuitry and used to be thestandard in frequency response testing for many years. Still today, the famous old B&Klevel writers can be seen in many laboratories and due to their robustness, might continueso in the future. The excitation signal being used is a logarithmic sweep, this means that thefrequency increases by a fixed factor per time unit (for example, doubles every second). Asthe paper is moved with constant speed under the writing pen, the frequency scaling on thepaper is correspondingly logarithmic. The spectrum of such a logarithmic sweep declinesby 3 dB/octave. Every octave shares the same energy, but this energy has to be spread outover an increasing bandwidth, therefore the magnitude of each frequency componentdecreases. We will later see that this excitation signal that is already in use for such a longtime has some unique properties that keeps it attractive for use in the digital word of today.One of these advantageous properties is that the spectral distribution is often quite welladapted to the ambient noise, resulting in a good S/N ratio also at the critical low end of thefrequency scale. While the level recorder cannot really suppress neither noise norreflections, a smoothing effect is obtained by reducing the velocity of the writing pen. Theripple in a frequency response caused by a reflection as well as irregular movement inducedby noise can be “flattened out” by this simple means. If the spectral details to be revealedare blurred too much by the reduced responsiveness of the writing pen, reducing the sweeprate helps to reestablish the desired spectral resolution. This way, measurement length andmeasurement certainty can be compromised, just as with the more modern methods basedon digital signal processing.

The evident shortcomings of level recorders are that they do not show phase informationand the produced spectra reside on a sheet of paper instead of being written to a hard disc

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for further processing. Clearly, the “horizontal” accuracy of displayed frequencies cannotmatch the precision offered by digital solutions deploying quartz-based clocking of AD-and DA converters. On the vertical scale, the resolution of the dB or amplitude readings isrestricted due to the discrete nature of the servo potentiometer that is composed of discreteprecision-resistors.

2.2 Time Delay Spectrometry (TDS)

TDS is another method to derive transfer functions with the help of sweeps. Devised byHeyser [1-4] especially for the measurement of loudspeakers, it is also applicable for roomacoustic measurements or any other LTI system in general. The principal functionality of aTDS analyzer is shown in Fig. 1. The analyzer features a generator that produces both aswept sine and simultaneously a phase-locked swept cosine. The sine is fed to the LUT(loudspeaker under test) and its captured response is multiplied separately by both theoriginal sine (to get the transfer function’s real part) and the 90° phase-shifted cosine (to getthe imaginary part). The multiplier outputs are filtered by a low-pass with fixed cut offfrequency. The multipliers work similar to the mixers used in the intermediate frequencystages of HF receivers (superhet principle), producing sum and difference of the inputfrequencies. The sum terms of both multiplier outputs have to be rejected by the low-passfilters, whereas the difference terms may pass, depending on their frequency. If both thegenerated and the captured frequency are almost equal, the output difference frequency willbe very low and thus not be attenuated by the low-pass filters. As the sound that travelsfrom the loudspeaker under test to the microphone arrives with a delay, its momentaryfrequency will be lower than the current generator signal. This causes a higher mixer outputdifference frequency that, depending on the cut-off frequency, will be attenuated by thelow-pass filters. For this reason, the generated signal has to be “time-delayed” by anamount equivalent to the distance between loudspeaker and microphone before beingmultiplied with the LUT’s response. This way, the difference frequency will be near DC. Incontrast, reflections will always take a longer way than the direct sound and thus arrivewith a lower instantaneous frequency, causing higher frequency components in themultiplier output that will be attenuated by the low-pass filters. By proper selection of thesweep rate and the low-pass filter cut-off frequency, simulated quasi-free-field-measurements are possible with TDS. Not only the disturbing reflections will be attenuated,but also distortion products (that arrive with a higher instantaneous frequency and alsocause high mixer output frequencies that will be suppressed by the filters) and noise (thelarge part in the frequency range above the low pass filter cut off frequency).

The controlled suppression of reflections is the motivation why TDS analyzers utilize alinear sweep (df/dt = const) as the excitation signal. The frequency difference betweenincoming direct sound and reflection will thereby stay constant over the whole sweeprange, maintaining the attenuation of each reflection frequency-independent. If alogarithmic sweep was used instead, the low pass filter would have to increase their cut offfrequency by a constant factor per time to avoid a narrowing of the imposed equivalenttime window (the low pass filter’s impact indeed is similar to the windowing of impulseresponses in the FFT methods described later). On the other hand, the higher frequencycomponents of a typical loudspeaker impulse response will decay faster than the lowerones. Thus, a narrowing of the window at higher frequencies (corresponding to “adaptive

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windowing” proposed by Rife to process impulse responses) should even be desirable formany measurement sceneries. It would increase the signal-to-noise ratio at high frequencieswithout corrupting the impulse response more than at low frequencies.

There are a number of drawbacks associated with TDS measurements. The most serious isthe fact that TDS uses linear sweeps and hence a white excitation spectrum. In mostmeasurement setups, this will lead to poor signal-to-noise ratio at low frequencies. If thewhole audio range from 20 Hz to 20 kHz is swept through in 1 second, then the subwooferrange up to 100 Hz will only receive energy within 4 ms. This most often is insufficient in afrequency region where the output of a loudspeaker decreases while ambient noiseincreases. To overcome the bad spectral energy distribution, the sweep has to be made verylong or the measurement splitted into two ranges, for example one below and one above500 Hz. Both methods extend the measurement time far beyond of what would be neededphysically to perform a measurement of the particular spectral resolution.

Another problem is ripple that occurs at low frequencies. As previously mentioned, themultiplier produce sum and difference terms of the “time-delayed” excitation signal and theincoming response. At higher momentary frequencies, the sum is sufficiently high to beattenuated by the output low-pass filter. But at the low end of the sweep range, when thesum is around or lower than the low pass cut off frequency, “beating” will appear in therecovered magnitude response. To remedy this, the sweep can be made very long and thelow-pass cut-off frequency decreased by the same factor. The better method, however, is torepeat the measurement with a “mirrored” setup, that is, exciting the DUT with a cosineinstead of a sine and treat the captured signal as depicted by the dotted lines in Fig. 1. Thereal part of the complex result of this second measurement is added to the real part obtainedby the previous measurement, while the imaginary part is subtracted. The effect of thisoperation is that the sum terms of the mixer output will be cancelled, as shown in [5], [7],[8]. As a consequence, thanks to the absence of the interfering sum terms over the wholesweep range, the low pass filters following the multiplier stages may be omitted. In fact,they have to be omitted if a full room impulse response shall be recovered, a case in whichobviously the low pass filter impact of attenuating reflections is not desired. Indeed, roomacoustic measurements that always involve acquiring lengthy impulse responses are onlyfeasible with the double excitation method. If, however, a loudspeaker is the object ofinterest, it is worth it to keep the low pass filters inserted in order to reject reflections, noiseand harmonics.

Even with the sum-term-canceling double excitation method, some ripple might still appearat the very beginning and at the end of the sweep frequency range, due to the sudden onsetof the linear sweep. By system theory, this switched sine corresponds to a corrugatedspectrum near the initial frequency (see Fig. 15). A common way to circumvent thisproblem is to let the excitation sweep start well below the lowest frequency of interest. Thismight entail starting the sweep at “negative” frequencies which in practice means starting atthe corresponding positive frequency, then lowering down to 0 Hz and from thereincreasing the frequency normally [7]. A better possibility would be to formulate theexcitation sweep in the spectral domain to create a signal that does not suffer spectralleakage (see Fig. 16), as will be demonstrated in chapter 5.2.

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Of course, the necessity to use the double excitation method to recover a full IR furtherextends the time needed to complete a TDS-measurement. On the other hand, FFT or MLSbased methods using periodic stimuli in practice also require emitting the excitation signaltwice to recover the periodic impulse response. With these methods, the DUT’s periodicresponse is captured and processed only in the second run after stabilizing. In contrast, theTDS double excitation method makes use of both passes and this attributes it an(additional) advantage of 3 dB in S/N ratio over MLS, given the same excitation length(both linear sweep and MLS have a white spectrum). With the absence of the output low-pass filters, the spectral resolution of a TDS measurement is just as high as with periodicexcitation of same period length. So a TDS double excitation measurement requires justabout the same time as an MLS measurement to achieve the same spectral resolution. Thiscontrasts to some TDS-MLS comparisons in the literature in which a certain TDS low-passfilter frequency is assumed and a correspondent frequency resolution calculated, in anattempt to proof that an exorbitant sweep length would be required to achieve the resolutionof an unwindowed MLS measurement. But of course, the TDS low-pass filter’s impact isequivalent to the application of a window to the retrieved impulse response, and anywindowing reduces the spectral resolution. The possibility to perform simulated free fieldmeasurements and the associated smoothing effect occurring when attenuating reflectionswith either method is normally very welcome for loudspeaker measurements (at least in thehigher octaves).

However, the relation between TDS sweep rate, low-pass cut-off frequency, and achievedattenuation of a delayed reflection is not evident immediately, albeit not too difficult tocalculate [6]. But it is more intuitive to inspect the full impulse response as derived by MLSand FFT measurements (or dual excitation TDS) and position a window whose right lobeends just in front of the first annoying reflection. This way, all subsequent reflections aremuted entirely. In contrast, even after arduous adjustment of the cut-off frequency, TDS isnot capable of complete reflection suppression due to the limited steepness of the low-passfilters (let alone a high order digital FIR version is used). Their smoothing effect on theretrieved transfer functions is not very well defined, while windowing offers a well-explained [50] compromise of main lobe broadening and side lobe suppression.

Using sweeps purely in conjunction with FFT analysis, without multipliers to produceintermediate results, obviates many of the problems inherent to TDS, especially insufficientenergy at low frequencies and long measurement cycles. However, some advantages ofTDS measurements over measurements with noise such as MLS should not be ignored. Thementioned higher achievable signal/noise ratio in a full double-excitation TDSmeasurement can be augmented further when taking advantage of the low crest factor ofonly 3 dB inherent to a swept sine. In practice, MLS have a crest factor of at least 8 dB, aswill be revealed later. TDS measurements should also offer higher tolerance against timevariance and better rejection of harmonic distortion that can be filtered out along with noiseand reflections.

2.3 Dual-Channel FFT-Analysis

A review of transfer function measurement methods would be incomplete withoutmentioning dual-channel FFT analysis. It is as old as the first FFT analyzers and has passed

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through a certain revival in the past years due to the omnipresence of stereo sound boards inPCs, although it neither boasts speed nor precision. The basic principle of dual-channelanalysis is to divide the output spectrum of the DUT through that of the input signal, as inall FFT-based transfer function measurements. However, the noise source traditionally usedin dual-channel analyzers is non-deterministic and hence its spectrum is not known inadvance. This inflicts capturing and processing both the input and the outputsimultaneously.

While this already seems disadvantageous in comparison to all other techniques that onlyrequire one channel, the main drawback of traditional dual-channel FFT is the nature of thesignal source commonly employed. It usually has a white (or pink or whatever) spectrumwhen averaged over a long time, but a single snapshot of the noise signal has a verycorrugated spectrum that suffers from deep magnitude dips. Thus, a dual channel analyzeralways has to average over many single measurements before being able to present areliable result. As in almost any analysis interval, the S/N ratio is insufficient at somefrequencies, those should be excluded from the averaging process as due to the division ofboth channels, gross errors that impair the display frequency response might occur. Thedetection of lacking S/N is mostly done by means of the coherence function [40], [17]. Dueto the necessity to average many blocks of data to achieve a consistent display, theresponsiveness of dual channel analysis is very poor and makes is unattractive foradjustment purposes.

In acoustic measurements, the precise delay of the acoustical transmission path has to beknown, as the direct signal will be delayed by exactly this amount of time so thatcorrespondent chunks of the excitation signal are analyzed (see Fig. 1). Moreover, in orderto avoid leakage effects, the signal blocks submitted to the FFT analysis have to bewindowed. This is a considerable source of error as delayed components are attenuatedmore than the direct sound.

The dual channel analysis could be improved considerably by generating the impulseresponse of every single measurement by inverse FFT. Windowing the impulse responseoffers the crucial freedom to control the amount of reflections entering into the result and tomute the noise outside the windowed interval, thus speeding up the convergence process.Not every dual channel analyzer deploys this helpful feature that is also not included in Fig.2.

There is one well-known application for dual-channel analysis that no other measurementtechnology offers: The possibility to measure sound systems unobtrusively during aperformance, using the program material itself as excitation signal. However, music with itserratic spectral distribution is usually a much worse excitation signal than uncorrelatednoise sources and requires even longer averaging periods to achieve a reliable result, ifpossible anyway. Thus, when the unobtrusiveness is not needed, it is advisable to use anoise generator as source. Far better results in far shorter time, however, can be achievedwith the FFT techniques using custom-tailored deterministic excitation signals.

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2.4 Stepped Sine

Probably the most time consuming method to acquire a transfer function is by exciting thestep by step DUT with pure tones of increasing frequency. The DUT’s response to thissteady-state excitation can be either analyzed by filtering and rectifying the fundamental, orby performing an FFT and retrieving the fundamental from the spectrum. The latter methodrequires the use of a sine that is exactly periodic within the bounds of one FFT block-lengthto avoid spectral leakage. In practice, this can only be realized generating the sine digitallyand emitting it via a DA converter that is synchronized to the capturing AD converter. As abig plus, the FFT method allows for the complete suppression of all other frequencies andthus is the preferable method over analysis in the time domain, involving band-pass filterswith restricted selectivity and precision.

After each single measurement, the excitation sweep’s frequency is raised by a valueaccording to the desired spectral resolution. In acoustic measurements, the frequency willusually be incremented by multiplying the previous value with a fixed factor to obtain alogarithmic spacing. Clearly, the spectral resolution in stepped sine measurements is muchlower at high frequencies compared to what could be achieved using a broadband excitationsignal with FFT analysis. But this is not necessary a disadvantage as the frequency-linearresolution of FFT-spectra yields often unnecessary fine frequency steps in the HF regionwhile sometimes lacking information in the LF region, provided that the time interval usedfor the FFT is too small. The possible logarithmic spacing of the stepped sinemeasurements results in much smaller data records than those obtained by FFT, but this isnot a serious advantage in the age of gigabyte hard disks.

The biggest advantage of the stepped-sine method is the enormous signal-to-noise ratio thatcan be realized in a single measurement. All energy is concentrated at a single frequency,and the feeding sine wave has a low crest factor of only 3 dB, allowing to establish a highcontinuous level. Thus, the measurement certainty and repeatability for a single frequencycan be made very high compared to broadband excitation, especially when using thesynchronous FFT technique.

That’s why despite of the considerable amount of time needed for the complete evaluationof a transfer function, this method is still popular for precision measurement and calibrationof electronic equipment or acoustic transducers such as microphones. Stepped sines are alsothe established method when it comes to precise distortion measurements. Every harmoniccan be picked up easily and with high precision from the FFT spectra.

However, when solely the transfer function is of interest, the stepped sine method iseverything but elegant. Only part of the energy emitted by the DUT can be used for theanalysis, as after each switching to a new frequency, a certain time has to be waited untilthe DUT settles to a steady state. Especially when high-Q resonances (corresponding tolarge IRs) exist, this settling time must be made very long to reduce errors to a negligiblelevel. On the other hand, when the system is very noisy and many synchronous averageshave to be executed to achieve an acceptable measurement certainty, the settling time playsonly a minor role [42].

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In acoustic measurements with pure tones, gating out reflections is only possible when thedifference in flying time between direct sound and reflection is bigger than the analysisinterval. This is a clear disadvantage compared to the methods that recover impulseresponses.

The unmatched accuracy achievable with single tone measurements also has to be put indoubt. It should be clear that the same accuracy could be achieved with broad-bandmeasurements in lesser time. While the stepped-sine measurements delivers singlefrequency magnitudes with very high signal-to-noise ratio, a broad-band measurementyields many values in the particular frequency interval. Each of them clearly has a lessercertainty, but by performing a spectral smoothing over a width corresponding to thefrequency increment used in the pure tone measurements, the random noise shoulddiminish to comparable values. This, however, assumes that the harmonic distortionproducts can be excluded entirely from the broad-band measurement, a condition that canonly be fulfilled by sweep measurements, as will be revealed later.

2.5 Impulses

Using an impulse as excitation signal is the natural way to obtain an impulse response andalso the most straight-forward approach to perform FFT based transfer functionmeasurements. The impulse can be created by analogue means or preferably sent out by aDA-converter and amplified. It feeds the DUT whose response is captured by themicrophone, amplified and digitized by an AD converter (Fig. 3). As the name implies, thiscaptured response already is the desired impulse response (IR), provided that a dirac stylepulse with its linear frequency response has been used. To increase the signal-to-noise ratio,the pulse can be repeated periodically and the responses of each period added. This leads tothe periodic impulse response (PIR) that is practically equal to the non-periodic IR if it isshorter than the measurement period (in practice: if the IR has vanished in the noise floorbefore the end of the period). As is well known, such synchronous averaging leads to areduction of uncorrelated noise by 3 dB relative to the IR per doubling of the number ofaverages.

The IR may optionally be shifted to the left (or the PIR shifted in a cyclic fashion) tocompensate the delay introduced by the flying time between loudspeaker and microphonein an acoustical measurement. Windowing then mutes unwanted reflections and increasesthe signal/noise ratio.

The IR can then be transformed into the transfer function by FFT. To increase the precisionof the measurement considerably, the result should be multiplied with a reference spectrum.This reference spectrum is obtained by linking output and input of the measurement systemand inverting the measured transfer function. Applying this technique (independently of thekind of excitation signal) offers the crucial freedom to pre-emphasize the excitation signalto adapt it to the spectral contribution of background noise. This pre-emphasis willautomatically be removed from the resulting transfer function by applying the obtainedreference spectrum in all subsequent measurements.

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Impulses are a simple and not too bad choice when the measurement is purely electrical (noacoustic path in the measurement chain), and the measurement should be as fast as possible.However, they require a low noise floor of the DUT to achieve reasonable measurementcertainty. But when measuring low noise audio equipment, this requirement is easilyfulfilled. Despite of their far-from-optimal S/N performance, impulses can even be useful inacoustics. In an anecoic chamber where ambient noise is typically very low at highfrequencies, tweeters can be measured with reasonable S/N ratio. Due to their shortness,pulses can be fed with very high voltage without having to consider the danger ofoverheating the voice coil. Care however has to be taken not to cause excursion into thenon-linear range of the speaker (although this will hardly be provoked with a very narrowpulse [35]), as this will make the amplitude smaller then expected and hence lead to anapparent loss of sensitivity [20]. Generally, all distortion in a pulse measurement occurssimultaneously with the impulse response and hence cannot be separated from it. Toincrease the S/N ratio of a tweeter measurement, the impulses can be repeated and averagedin quite short intervals, as the impulse response to be recovered is very short and therequired linear frequency resolution fairly low.

Impulse testing doesn’t allow identifying distortion, but is pretty immune against thedetrimental effects of time variance that frequently haunt MLS or noise-based outdoormeasurements. It is simple, doesn’t require sophisticated signal processing and works verywell for some measurement tasks. That’s why it has been a popular method quite a while[35], [36]. When amplifier power is available in abundance, the increase of S/N ratio ofmeasurements using excitation signals stretched out in time compared to single pulsemeasurement is not as big as one might expect, because a loudspeaker can generally be fedwith pulses of very high voltage. The S/N gain decreases further in favor of pulse testingwhen considering the distortion products bedraggling impulse responses obtained fromnoise measurements, such as MLS.

2.6 Maximum length sequences (MLS)

MLS are binary sequences that can be generated very easily with an N-staged shift registerand an XOR-gate (with up to four inputs) connected with a shift register in such a way thatall possible 2N states, minus the case “all 0”, are run through [22]. This can beaccomplished by hardware with very few simple TTL-ICs or by software with less than 20lines of assembly code.

In the times when MLS grew popular, the possibility to create the sequences by hardwarerelieved from memory constraints. In the 80s, the maximum memory an 8088 based IBMPC deployed was 640 KB. To circumvent the need to store the excitation signal allowed forlarger data arrays to capture and process the DUT’s response. Today, this advantage hastotally vanished and it is more cost-effective and flexible to create MLS by software andoutput them from memory via the DA converter of a measurement system.

As the case “all zeros” is excluded from the sequence, the length of an MLS is 2N-1.Though missing one value to have an exact FFT block length, MLS have some uniqueproperties that make them suited for transfer function measurements. Their auto-correlationcomes close to a dirac pulse, indicating a white spectrum. Repeated periodically as a pulse

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train, all frequency components have indeed exactly the same amplitude, meaning theirspectrum is perfectly white. Compared to a pulse of same amplitude, much more energycan be fed to the DUT as the excitation signal is now stretched out over the wholemeasurement period. This means increased S/N ratio.

Normally, an MLS is not output as a pulse train, as this would mean feeding very littlepower to the subsequent DUT. Instead of this, the output of a hardware-MLS generator isusually kept constant between two clock pulses. This first order hold function leads to asinc(x) aperture loss that reaches almost 4 dB at fS/2 and therefore must be compensated. Incontrast, when the MLS is output by an oversampling audio DA converter, as is standardtoday, the spectrum will be flat up to the digital filter’s cut-off frequency. In the case ofcheaper codecs, a noticeable ripple might be introduced over the whole passband. Thesefrequency-linear undulations are always present to a certain extent in oversampling audioconverters. They originate from the linear-phase FIR anti-alias filters. These usually tradepassband ripple against stop band attenuation by means of the Parks-McClellan algorithm[57]. Furthermore, they are practically always designed as half-band filter that halve therequired calculation power, but exhibit an attenuation of merely 6.02 dB at fS/2. That’s whya small invalid aliasing region always exists near the Nyquist frequency when measuringwith audio converters. The anti-alias filter also induce a hefty overshoot of the output MLSwhich means that they cannot be fed with full level. Chapter 3.2 will clarify the issue.

Excitation signals with white spectrum allow the use of the cross-correlation to retrieve asystem’s impulse response. While normally, a cross correlation is most efficientlyperformed in the spectral domain by complex-conjugate multiplication, the well-known fastHadamard transform (FHT) can perform this task for MLS without leaving the timedomain. We will omit the theory behind entirely, as it has been thoroughly explained manytimes, for example in [22], [19], [23], [24], [17]. The butterfly algorithm employed in theFHT only uses additions and subtractions and can operate on the integer data delivered bythe AD converter. In former times, this meant calculation times that were much shorter thanthat of an FFT of similar length, but today this difference has shrunken a lot. Modernprocessors as those of the Pentium II family are able to perform floating-pointmultiplications, additions and subtractions as fast as the respective integer operations.

Again back in the 80s, the time-saving property of the Hadamard transform was verywelcome, as the calculation of a long broad-band room impulse response still consumedmany seconds. The advantage became especially prominent when just the IR, but not itsassociated transfer function was of interest. This, for example, holds for the evaluation ofreverberation times by backward integration of room impulse responses [43]. In thesecases, just one FHT is required to transform the MLS-response captured by the microphoneinto the desired PIR. And this FHT is faster than an FFT. In contrast, using arbitrary noisesignals or sweeps as stimulus requires at least one FFT and one IFFT to retrieve the IR.However, the processing times are not anything to worry about anymore, as with thepowerful processors of today, the transformations can be performed much faster than realtime. For example, processing an MLS of degree 18 (period length of six seconds at44.1 kHz sampling rate, a typical length for broad-band measurements in quite reverberantambiences) completes in only 138 ms on a Pentium III/500, using a 32-Bit-integer Radix-4MMX-FHT partitioned into sub-chunks accommodated to the sizes of 1st and 2nd level

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cache. This encompasses the permutation needed before and after the butterfly algorithmand a peak search. A real-valued FFT [50] for the same length, also using nested sub-chunks that can be processed entirely in the caches, terminates in roughly double the time(280 ms), still being a lot less than the measurement period. So regardless of themeasurement principle, today it is possible to transmit the excitation signal continuouslyand to complete calculation and display updating within every period, even for two or moreinput channels. Shorter measurement periods even achieve a higher real-time score as theyare handled entirely in the caches. Of course, the number of operations per output samplealso decreases slightly by the ratio of the degree (e.g. an FFT of degree 12 only needs twothirds of the operations per output sample as one of degree 18).

In an MLS based measurement, the FHT is the first signal-processing step after digitizationby the AD converter (Fig. 4). The resulting impulse response can be shifted in a cyclicfashion and windowed, as with simple impulse testing. If the transfer function is what isbeing looked for, an additional FFT has to be carried out. But as MLS have a length of 2n-1,one sample has to be inserted to patch the impulse response to full 2n length. Whilecomputationally, this is a trivial action, care has to be exercised where to place this sample.It must be in a region where the impulse response has decayed to near zero to avoid grosserrors. When using a window, the sample can be placed in the muted area. The acquiredtransfer function again can and should be corrected by multiplication with a referencespectrum obtained previously by a self-response measurement.

MLS measurements have proved quite popular in acoustics, but have a lot of drawbacks.Among high vulnerability against distortion and time variance (these will be compareddirectly to sweep measurements in chapter 3), the most lamentable property of MLS is theirwhite spectrum. As will be advocated further in chapter 4, a non-white spectrum isdesirable for almost all acoustical measurements. This requirement can be achieved bycoloring the MLS with an appropriate emphasis. Clearly, the MLS will loose its binarycharacter by pre-filtering. Thus, this technique is only viable if the pre-filtered MLS isoutput by a true DA converter, not just a one bit switching stage as used in some old-fashioned hardware-based MLS analyzers. The latter ones are restricted to analogue post-filters to emphasize the MLS, but these don’t offer the versatility of FIR filters, such aslinear phase or compensation of the measurement system’s self-response [27].

Creating an emphasized MLS can be done most efficiently by means of the inverse fastHadamard transform (IFHT) [25], [27]. The IFHT simply consists in time inverting theimpulse response of the desired emphasis filter (curtailed to 2n-1 samples), applying anormal FHT on the inverted IR and then time inverting the result again. This will yield anMLS periodically convolved with the emphasis filter. Due to the periodicity, every discretefrequency component of the former MLS can be influenced independently in bothamplitude and phase.

When an emphasized instead of a pure MLS is being used as stimulus, obviously theimpulse response obtained by FHT will consist of the IR of the DUT convolved with the IRof the emphasis filter. For acoustical measurements, it is meaningful to give the excitationsignal a strong bass boost of maybe 20 or 30 dB, as will be illustrated later. In this case, therecovered IR may become much broader than the one of the DUT alone. This broadening

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often constraints windowing, especially when reflections that shall be muted are in closeproximity of the main peak. In these cases, applying a window “pre-comp” to the non-equalized impulse response will noticeably attenuate the low frequency energy spread outin time. Thus, it is better to perform the windowing “post-comp”, meaning aftertransforming the uncorrected impulse response into the spectral domain, multiplying it withthe inverse emphasis frequency response, and eventually back-transforming it into the timedomain. This will yield the true impulse response of the DUT alone that can now bewindowed with lesser low frequency energy loss.

If the transfer function is the desired result of the measurement, another final FFT will haveto be performed after the windowing of the compensated impulse response (Fig. 5). So thenumber of transformations is totaling to 1 FHT and 3 FFTs when measuring transferfunctions with pre-emphasized MLS and applying “post-comp”-windowing of the impulseresponse.

2.7 Periodic Signals of length 2N

Examining thoroughly the MLS measurement setup in Fig. 5 reveals that actually, the useof MLS and the application of the FHT are pretty superfluous. If only the excitation signalhad 2N samples instead of the odd 2N-1 of an MLS, the DUT’s response could betransformed directly to the spectral domain, omitting the FHT. There, it could simply bemultiplied by the reference spectrum (the inverse of the product of the excitation signal’sspectrum and the measurement system’s frequency response). As this multiplication is acomplex operation, not only the magnitude, but also the phases are thereby corrected toyield the true complex transfer function of the DUT, regardless of the excitation signal’snature. Performing an IFFT on this compensated spectrum will produce the correspondenttrue impulse response of the DUT (Fig. 6).

When comparing this technique (FFT, compensation, IFFT) to the FHT, it becomes clearthat it is far more powerful and flexible, allowing the use of arbitrary signals of length 2N.The only restriction is that the excitation signal should have enough signal energy over thewhole frequency range of interest.

The FHT, being a cross-correlation algorithm, is able to merely reshuffle the phases of aspecial class of excitation signals, namely MLS. Its operation is “pulse-compressing” theMLS by means of convolution with the correspondent “matched filter”. In contrast, the FFTapproach compensates phase and magnitude of any excitation signal, be that noise, sweepsor even short chunks of music. This operation is sometimes referred to as “mismatchedfiltering”, although in contrast to “matched filters”, it is not restricted to white excitationsignals. The only obvious restriction is that the excitation signal must have enough signalenergy over the whole frequency range of interest to avoid noisy parts in the obtainedtransfer function.

Clearly, performing two FFTs consumes more processing time than one single FHT. Butwith today’s powerful microprocessors, this disadvantage is insignificant. As we have seen,using pre-emphasized MLS and “post-comp” windowing (Fig. 5) even leads to slightlylonger calculation times than using arbitrary 2N-signals.

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The technique resembles the good old dual channel FFT analysis, but differs from it in thatthe excitation signal is known in advance. Hence, its spectrum needs to be calculated onlyonce and can be used in all subsequent measurements. This obviates the need of a secondchannel, or if present anyway, allows to analyze two inputs simultaneously. Anaccompanying benefit is the fact that the achievable precision of a “single channel FFTanalyzer” outperforms that of any dual channel analyzer. With the latter, any difference inthe frequency response of the two input channels will be reflected in the DUT’s measuredfrequency response. Of course, manufacturers of pricey dual channel analyzers endeavor tomake these differences as small as possible. However, creating a reference file by replacingthe DUT with a wire makes the excitation signal pass over exactly the same stages andguarantees higher precision. Even with consumer equipment, a certainty of 1/1000 dB orbetter can be achieved without big effort in purely electrical measurements. The referencevoltage sources included in modern AD and DA converters are stable enough to permit thisfor a certain while (when slow drift due to heat-up occurs, the reference measurement canquickly be repeated). However, a source of error to always be aware of are the impedancesof the analogue input and output stages. The DA output impedance should be as low aspossible to prevent a drop of the generated voltage when connecting the DUT, whereas theinput impedance should be sufficiently high to avoid influencing the DUT’s output voltage.Clearly, these conditions are rarely fulfilled when using simple sound-boards withoutbuffering amplifiers.

The even bigger asset of the “single channel analysis” is the use of a deterministic signal incontrast to the uncorrelated noise sources normally used in dual channel FFT analyzers. Asstated before, the latter have a steady spectrum when averaged over a long time, but asingle snapshot of the noise signal suffers from deep magnitude dips. Thus, a dual channelanalyzer always has to average over many single measurements before being able topresent a reliable result. In contrast, the deterministic stimulus used in the method of Fig. 6can be custom tailored by defining an arbitrary magnitude spectrum free of dips, adapted tothe prevailing noise floor, to accomplish a frequency-independent S/N ratio. According tothe desired signal type, the corresponding phase spectrum can then be constructed in quitedifferent manners.

A noise signal can be generated easily by setting its phases to random values. Theexcitation signal is obtained by IFFT. Repeated periodically, it will have exactly themagnitude spectrum previously defined (for example, flat or pink). Noise signals havesimilar properties as MLS, especially concerning their vulnerability against distortion andtime variance. Some people refer to noise signals as “multi sine signals”, but of course, justany non-pure tone is a multi sine signal.

As the predefined spectrum of a noise signal is only valid for periodic repetition, themeasurement cannot be started immediately after turning on the stimulus. At least a timecorrespondent to the length of the impulse response has to be awaited for the DUT responseto stabilize. As the length of the IR is not always known in advance, it is practical to simplyeject two periods of the excitation signal. Signal acquisition only starts in the second run,just as with MLS measurements. Similarly, only half of the emitted energy is used foranalysis in a single-shot measurement. On the other hand, the periodicity again allowsmanipulating every frequency component completely independently. For example, single

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frequencies can be selectively muted or enhanced to reduce or improve signal energy inparticular frequency bands.

2.8 Non periodic sweeps

Instead of randomizing the phases to obtain a noise signal with the desired spectral shape,the phase spectrum can also be adjusted to yield an increasing group delay. The IFFT willthen reveal a sweep instead of a noise signal. For several reasons, sweeps are a far betterchoice for transfer function measurements than noise sequences. First of all, in contrast tothe latter, the spectrum of a non-repeated single sweep is almost identical compared to itsperiodic repetition. This means that it is not necessary to emit the excitation signal twice toestablish the expected spectrum. The sweep has to be sent out only once and the DUT’sresponse can immediately be captured and processed. Thus, the measurement durationhalves, maintaining the same spectral resolution and signal-to-noise ratio as in ameasurement with the stimulus periodically repeated. The minor remaining differences inthe spectrum of the repeated and the non-periodic sweep do not matter, as they are reflectedand later cancelled by the reference measurement that also uses the non-repeated sweep.

The other enormous advantage of a sweep measurement is the fact that the harmonicdistortion components can be entirely kept away from the acquired impulse response. Theseappear at negative times where they can be separated completely from the actual impulseresponse. Thus, the impulse response remains untouched from distortion energy. Incontrast, measurements using noise as stimulus unavoidably lead to distribution of thedistortion products over the whole period. The reason for the distortion rejecting propertycan be explained easily with a small example: Consider a sweep that glides through 100 Hzafter 100 ms and reaches 200 Hz at 200 ms. To compress this excitation signal to a diracpulse, the reference spectrum needs to have a correspondent group delay of -100 ms at100 Hz and -200 ms at 200 Hz. When the momentary frequency is 100 Hz and the DUTproduces second order harmonics, a 200 Hz component with the same delay as the 100 Hzfundamental will be present in the DUT’s response. This 200 Hz component will then betreated with the -200 ms group delay of the reference spectrum at 200 Hz and hence appearat -100 ms after the deconvolution process. Likewise, higher-order harmonics will appear ateven more negative times.

In order to actually place the distortion products at negative times of the acquired impulseresponse, a linear deconvolution suited for non-periodic signals would be necessary.Instead of this, it is also possible to maintain the normal FFT operation and referencemultiplication as used in measurements with periodic stimuli, provided that either theexcitation sweep is considerably longer than the DUT’s impulse response or zeros areinserted to stuff the DUT’s sweep response to double the length. In both cases, thedistortion products will appear at the end of the recovered impulse response where they canbe interpreted as bearer of negative arrival times. They can be chopped off withoutcorrupting the actual impulse response, as in the first case, the latter has already decayedinto the noise floor, and in the second case, any causal information cannot reside in thisregion.

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However, both cases mean that an FFT block length longer than required for the finalspectral resolution has to be employed as a first signal-processing step. For the goal tocapture high-quality room impulse responses advocated in this paper, it is even advisable touse sweeps that are considerably longer than the impulse response. This is the best means toincrease the signal-to-noise ratio and decrease the influence of time variance.

The data-acquiring period in non-periodic measurements has to be made sufficiently largeso to capture all delayed components. This means that the sweep always has to besomewhat shorter than the capturing period and the subsequent FFT length. In roomacoustic measurements, it is very beneficent that the reverberation times at the highestfrequencies are usually much shorter than the ones encountered at low frequencies. Thus,the sweep has to be shortened only by a time correspondent to the reverberation at highfrequencies, provided that the entire sweep is long enough to avoid that low frequencyreverberation is stumbling behind the high frequency components.

3 A COMPARSION BETWEEN SWEEP AND MLS-BASEDMEASUREMENTS

3.1 Measurement Duration

In all measurements, the AD capture period has to be at least as long as the impulseresponse itself (in practice: the time until the response has gone under in noise) to avoiderrors. This is obvious for the measurement with a non-periodic pulse. All its energy isemitted at the very beginning, and the AD converter simply has to collect samples until theimpulse response has decayed. In case of a non-periodic sweep being used as excitationsignal, the capture period has to be a little bit longer, but in general not much. This isthanks to the sweep’s nice property to start with the low frequencies. With normal DUTssuch as loudspeakers, the largest signal delays will occur at the low frequencies. Thus,while sweeping through the high frequencies, there should be sufficient time to catch thedelayed low components. For loudspeaker measurements, the decay for the highestfrequencies is usually so short that the AD capturing can be stopped almost immediatelyafter the excitation signal swept through (provided that the sweep is considerably longerthan the delay at low frequencies, and of course observing the flying time betweenloudspeaker and microphone). In room acoustic measurements, the gap of silence followingthe emission of the sweep usually has to be as long as the reverberation at the highestfrequencies.

In the case of periodic excitation, the period length and AD capture time has to be no longerthan the impulse response length. Using a shorter length would lead to time aliasing, that is,folding back the tail of the IR that is crossing the period’s end to its beginning where it isadded to the start of the IR. Depending on the amount of energy folded back, this createsmore ore less tolerable errors. Compared to non-periodic pulse or sweep-measurements, theneed to emit the excitation signal twice means longer measurement durations than required.Moreover, only half of the total energy fed to the DUT is used for the analysis.

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3.2 Crest factor

The crest factor is the relation of Peak-to-RMS voltage of a signal, here expressed in dB.Assuming that either the measurement system or the DUT are limited by a distinct voltagelevel, the peak value of any considered excitation signal has to be normalized to this valueto extract the maximum possible energy in a measurement. The RMS level will then belower according to the crest factor. Thus, the crest factor imparts how much energy is lostwhen employing a certain excitation signal, compared to the ideal case of a stimulus whoseRMS voltage equals the peak value (crest = 0 dB). For this reason, it has become a kind ofsport among signal theorists to devise excitation signals with the lowest possible crestfactor.

The assumption that a certain voltage level is defining the upper limit in a measurement ismostly true for purely electrical measurements (for instance, audios gear such as EQs,mixers, etc). In acoustic measurements, it is only valid when the driving amplifier is theweak link in the chain. Even then, most amplifiers equipped with a traditional line-transformer based power supply can reproduce surge peaks with 2 or 3 dB higher level thancontinuous power.

If the danger of overheating of loudspeaker voice coils is the primary restriction, the totalenergy going out to the loudspeaker is more important than the crest factor. However, veryhigh crest factors should always be avoided, as single high-level peaks might causedistortion.

At first glimpse, a bipolar MLS produced by a 1st order hold output seems to be the idealexcitation signal in the sense of extracting maximal energy. The peak value equals its RMSvalue. However, the resulting ideal crest factor of 0 dB cannot be exploited in practice. Assoon as an MLS goes out to the real world and passes through a filter, the rectangularwaveform can change considerably. Especially the steep anti-aliasing filters used in theoversampler stages of audio DA converters cause dramatic overshoot. In order to avoiddrastic distortion caused by clipping filter stages, MLS must therefore be fed to the DAconverter with a level at least 5 to 8 dB below full scale, depending on the anti-aliasingfilter characteristics. This means that MLS cannot be ejected distortion-free with the sameenergy as a (swept) sine that features a crest factor of merely 3.01 dB.

But even if the MLS is produced by a hardware generator, it will not retain its favorablecrest factor for a long time. Power amplifiers are always equipped with an input low-passfilter to reject radio interference and to avoid transient intermodulation induced by highslew rates of the audio signal - precisely what is prevailing in profusion in an unfilteredMLS. A typical input filter would be a 2nd order Butterworth low-pass with a cut offfrequency of maybe 40 kHz. The overshoot produced by such a filter is much moremoderate than that of a steep anti-aliasing filter (Fig. 7), but still merits consideration.

There are cases in which not the voltage at the measurement system’s output, but at someinternal nodes of the DUT is restricting the driving level. For example, if a resonance withhigh gain is encountered in a chain of equalizer stages, the excitation signal will most likelyfirst be clipped at the output of that stage. In these cases, s sweep has to be fed with a level

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lower by the amount of gain at the resonance frequency of that specific filter. In contrast tothis, filtered MLS tend to take a gaussian amplitude distribution, with 1% of the amplitudesreaching a level more than 11 dB above the RMS value [17]. In most loudspeaker and roomacoustical measurements, even with the presence of moderate resonances, a sweep can stillbe fed with a higher RMS level than an MLS. In practice, distortion already occursgradually with MLS long before the clip level of the driving amplifier is reached, as section3.5 will examine more thoroughly.

3.3 Noise Rejection

Any measurement principle using excitation signals with equal length, spectral distributionand total energy will lead to exactly the same amount of noise rejection, if the entire periodof the unwindowed impulse response is considered. For every frequency, the S/N ratiosolely depends on the energy ratio of the DUT’s response to that of extraneous noisecaptured in the measurement period. The difference between the various measurementmethods lies merely in the way how the noise is distributed over the period of the recoveredIR. Clearly, using the same spectral distribution in an excitation signal requires the sameinverted coloration in the deconvolution process. That’s why the magnitude of a noisesource will not vary when changing the stimulus type. The phases, however, will turn outvery different. Still, some kinds of noise sources will appear similarly in all measurements,as their general character is not altered by manipulating the phases. Monofrequent noise,such as hum, is an example. Likewise, uncorrelated noise (for instance, air condition) willstill appear as noise. Any other disturbance, however, will be reproduced quite differently,depending on the type of stimulus. Short, impulsive noise sources, such as clicks and pops,will be again transformed into noise when using noise as a stimulus. In contrast, they willbecome audible as time-inverted sweeps in a sweep measurement. Usually, steady noise isconsidered to be more unobtrusive than other error signals. However, the time-invertedsweeps that give the impulse response tail a bizarre melodic touch do not sound toodisturbing as long as their level is low. Generally, if any loud noise suddenly happens toappear, the measurement should simply be repeated, or, when averaging, the specific periodshould be discarded from the synchronous averaging process.

3.4 Time Variance

Time variances tend to haunt measurements whenever they are performed over a longdistance outside, when synchronous averaging is performed over a long period or when theDUT itself is not very time-invariant. The first case often holds for measurements instadiums or at open-air sites under windy weather conditions. The second is an issue whenvery low signal-to-noise ratios force using many hundreds or even thousands ofsynchronous averages. In such long periods, a slight temperature drift or movement of theair can thwart the averaging process. Finally, an example for DUTs that are not time-invariant themselves is any kind of analogue recording gear.

It is well known that periodic noise sequences in general and MLS in special are extremelyvulnerable to even slight time variance. A considerable amount of theoretical work hasalready been executed to explain these effects in detail [29, 30, 34]. While the complicate

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equation framework in these publications looks threatening, it is neither easy to comparethe effects of time variance in practice, as these tend to have an erratic and unpredictablenature. It is likely that two outdoor measurements performed in series are affected quitedifferently from wind gusts. Only a simulation allows circumventing this “time variance ofthe time variance”. Fig. 8 shows a small example. A noise sequence and a sweep, both withwhite flat spectrum, have been submitted to a slight sinusoidal time variance of±0.5 samples. To simulate this jitter, the signals have first been oversampled by a factor256 (without filtering, simply inserting 255 consecutive zeros after each sample). Then theexact arrival times of the samples have been shifted in the range of ±128 according to thesinusoidal jitter curve. The resulting disturbance of the base band spectra is negligible atlow frequencies, but then increases extremely for the jittered noise spectrum. In contrast,the jittered sweep spectrum only displays a minor corrugation at the high end that couldeasily be removed by applying gentle smoothing.

Another expressive test is the frequency response measurement of an analogue tape deck, akind of machine that always suffers from wow and flutter to some extent. As the magnetictape material tends to saturate much earlier at higher frequencies, an emphasis of 24 dB atlow frequencies has been applied to both the MLS and the sweep used in this experiment.Additionally, the sweep’s envelope has been tailored to decrease by 12 dB above 5 kHz,but maintaining the initial coloration. Section 5.4 will reveal how this is accomplished.Both stimuli were normalized to contain identical energy and were recorded with the sameinput level. As the results in Fig. 9 show, the measured frequency response using MLS asexcitation becomes pretty noisy above 500 Hz. The one using a sweep as stimulus alsocontains some time-variance induced contamination, but to a much lesser extent. Besidesthe deleterious effects of time variance, distortion certainly also plays an important role inpolluting the recovered frequency responses of an analogue recorder. To minimizeintermodulation products, the recording level has been adjusted to about 20 dB below thetape’s saturation limit in this trial.

Even when measuring systems that are virtually free of time variance, using pseudo noiseas excitation is disadvantageous when adjustments (volume, EQ or other) are made withinthe measurement period. In this case, gross errors occur in the displayed frequencyresponse. Sweep and impulse measurements do not show this annoying reaction to timevariance and are thus more pleasant to use for adjustment work monitored by repetitivemeasurements.

3.5 Distortion

Hardware audio engineers endeavor a lot to optimize the dynamic range of their circuitswhich ideally should match that of our auditory system with values up to 130 dB. Yet, thesignal-to-noise ratio in acoustic measurements seems to have been somewhat neglected byacousticians in the past. At relatively calm sites, it is less the background noise that limitsthe quality of the acquired impulse responses, but very often the distortion produced by theemployed loudspeaker. In any measurement using noise as excitation signal, thesedistortion products will be distributed as noise over the entire impulse response period. Thereason for this is that the distortion products of a stimulus with (pseudo-) random phasesalso have more or less random phases, and the deconvolution process again involves

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random phases that will eventually produce an error spectrum with random phases,corresponding to a randomly distributed noise signal. As this error signal is correlated withthe excitation signal, synchronous averaging does not improve the situation.

While it is true that the noise level diminishes relative to the impulse response peak valuewhen choosing a longer sequence, it can hardly be reduced to an acceptable level. Roomacoustic measurements already involve lengthy sequences. For example, increasing thelength of an MLS of degree 18 by a factor of 128 would theoretically reduce the noise levelfrom a typical value of –65 dB to –86 dB. While it would even be feasible to process anMLS of degree 25 (length: 12 minutes, 41 seconds at 44.1 kHz sampling rate) on acomputer generously equipped with memory, it would not work out to bring the noise leveldown this way, because when using such long sequences, the vulnerability against timevariance becomes predominant.

Instead of reducing the influence of distortion products by spreading them out over an everincreasing measurement period, it is far more beneficent to simply exclude them totallyfrom the recovered impulse response. This can be accomplished easily by using sweeps asexcitation signal, as explained in section 2.8. The great improvement that can be realized inroom acoustical measurements by replacing MLS with sweeps is demonstrated in Fig. 10.The room impulse response of a reverberant chamber has been measured here, using eitheran MLS and a sweep of degree 20, both with exactly the same energy and the same pre-emphasis of 20 dB at low frequencies. In favor for the MLS, the volume has been carefullyadjusted so to yield minimum contamination of the impulse response. Indeed, this findingof the optimum level by trial and error is crucial in MLS measurements, as too much powerleads to excessive distortion, visible by an increasing noise floor with lumpy structure,whereas a low level leads to background noise mainly impairing the measurement.

After this adjustment, 1, 10 and 100 synchronous averages have been performed with bothexcitation signals. It is clearly visible that even when using no averages, the impulseresponse decays into a noise level that is already by 5 dB lower in the sweep measurement.The accumulated distortion products reside at the end of the measurement period. As thelength of the excitation signal has been chosen to be almost 8 times bigger than thereverberation time, they can easily be segregated from the actual impulse response.Executing 10 synchronous averages reduces the noise floor by the expected 10 dB whenusing a sweep. In contrast, only a minor decrease of the noise floor is noticeable when anMLS serves as stimulus, showing that the correlated intermodulation products now prevailin the recovered impulse response. Performing 100 synchronous averages does not ensueany improvement in the MLS measurement, whereas a further decrease is recognizable inthe sweep measurement, albeit somewhat less than the expected 10 dB. Time variance by asmall temperature drift might come into play here, after all 100 averages take almost 40minutes to complete.

In this experiment, the output level has been optimized for use with the MLS. The level ofthe emitted sweep could have been raised by 15 dB without causing amplifier clipping, andindeed the noise floor dropped exactly by this value when doing so. Thus, a signal-to-noiseratio of 100 dB could have been reached with just 10 averages of the sweep. In contrast, it

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is impossible to achieve this high quality by an MLS measurement, regardless of the leveland the number of averages.

The loudspeaker used in that setup was a coaxial 12” PA woofer/tweeter combinationoptimized for high-efficiency. This kind of speaker certainly produces more distortion thanhigh-fidelity types with soft suspension and long voice coils, but even with the latter ones, asignal-to-noise ratio of 90 dB or better proves unfeasible in MLS-measurements.

Apart from acoustical measurements, there are more measurement situations in which therequired linearity for MLS measurements is not fulfilled. This especially holds for signalpaths including psycho-acoustic coders. An obvious example are cellular phones that usevery high compression to achieve low bit rates. Preliminary experiments showed that shortexcitation signals produce unpredictable and erratic results, regardless of whether MLS orsweeps were used. Extending the length to degree 18 at 44.1 kHz sampling rate deliveredmore or less reliable results with a sweep (linear between 100 Hz to 7 kHz), whereas themeasurement with an MLS of same length and coloration produced a very rugged curvethat hardly allows recognizing the transfer function (Fig. 11). However, it must be admittedthat these results only hold when the impulse response is transformed to the spectraldomain without prior processing. Applying a narrow window with a width of only 80 msaround the main peak relieved the situation to a great extent (Fig. 12). Nonetheless,surprising differences appear above 2 kHz between the measurements with MLS andsweeps. The encoder seems to produce different results for broad-band noise and signalsthat appear almost monofrequent in a short term analysis.

Another example that certainly appeals more to the audio community than low-fidelitytelephone-quality encoding is the popular MPEG 1 / layer 3 compression. Fig. 13 shows acoder’s transfer function for the common rate of 128 kbit/s, as measured with MLS andsweep. The advantage for the sweep becomes overwhelming in this application. In fact, thebroad-band noise the coder has to deal with when an MLS is used as excitation signalpresents the worst case for psycho-acoustic coding. All frequency bands contain energy, sonone falls below the masking level which would allow to omit it. Consequently, all bandshave to be submitted to a fairly course quantization to achieve the required bit rate,resulting in distortion that disturbs the MLS measurement significantly. In contrast, thesweep glides through only a couple of bands per analysis interval, allowing to quantizethese with high resolution while discarding the others that contain no signal energy.

To summarize, when measuring data-compressing coders, sweeps bear the advantage ofconsiderably reduced signal complexity compared to noise sequences. This makes codingthem an easy task. While in “natural” measurements using sweeps, the distortion productscan be isolated and rejected by windowing, the advantage in measuring signal pathsinvolving coders seems to lie more in the fact that the generation of distortion is simplyprevented.

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4 PREEMPHASIS

In almost any acoustical measurement, it is not advisable to use an excitation signal withwhite spectral contents. When, for instance, measuring a loudspeaker in an unecoicchamber, two effects account for a heavy loss of S/N ratio at low frequencies: The loss ofsensitivity (with 12 or 24 dB per octave) below the (lowest) resonance of the bass cabinet,and the increase of ambient noise in this frequency region due to the wall’s decreasinginsulation against outside noise. So, in order to track the speaker’s low frequency roll-off(if possible at all, see [54], [55]) without the corrupting effects of low S/N, a strongemphasis of more than, say 20 dB is required to establish reasonable measurementcertainty.

This also results in a better contribution of the power fed to a multi-way loudspeaker. Thewoofer typically endures much higher power than the tweeter. But when using a whiteexcitation signal, the tweeter has to bear the brunt of the excitation signal’s energy. A dometweeter can be overheated and damaged with as little as a few watts, and this limit caneasily be trespassed by any power amplifier. So also from this point of view, an emphasisof lower frequencies is highly desirable.

A third reason to mention is less of physical but of social nature: When measuring “on site”(concert hall, stadium etc…), an excitation signal with bass boost will sound warmer andmore pleasant than white noise, and hence is more acceptable by other people present.Moreover, a strong increase of power in the low band will not correspond to the impressionof much increased loudness due to the decreased sensitivity of our hearing sense in thisfrequency region. While this may sound little scientific, everybody who alreadyparticipated in measurement sessions in public knows that the maximum applicable volumeis dictated by the persons congregating in the venue, not by the loudspeakers nor theamplifiers (see also [16]). When setting-up a PA for large-scale sound reinforcement,dozens of technicians not related to the audio area (light etc..) are also in charge. Forinstance, riggers climbing on trusses near a loudspeaker cluster definitely appreciateunobtrusive excitation signals. In these cases, using white noise as stimulus even bears apotential health risk and security problem. An MLS accidentally fed with full level to apowerful sound system may cause hearing damage or even lead to accidents of startledpersonnel. This holds especially in the vicinity of horn-loaded 2” drivers. Using excitationsignals with decent LF-emphasis relaxes the situation to some extent. In particular, longsweeps can be interrupted before reaching the excruciating mid-frequencies.

4.1 Equalizing loudspeakers for room acoustical measurements

Acquiring room impulse responses is one of the ever-repeating tasks in room and buildingacoustics. All typical parameters that describe the acoustical properties of a room (or, to bemore precise, the acoustical transmission path between two points in the room, using asource and a receiver with distinct directivity) such as reverberation, clarity, definition,center time, STI and many others can be derived form it. A closer look on the impulseresponse can help to identify acoustical problems such as unwanted reflections orinsufficient ratio between direct sound and reverberation. Visually examining the

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associated room transfer function (obtained by FFT) can reveal disturbing room modes or,of course, tonal misbalance of a sound reinforcement system.

Another very interesting application is the creation of “virtual reality” by convolution ofdry audio material with binaural room impulse responses captured by a dummy head. Wewill later see that only sweeps are applicable to fulfill this task with sufficient dynamicrange.

Up to today, capturing RIRs for reverberation time measurements is occasionally doneusing non-electro-acoustic impulsive sources. Firing a pistol (delicate action especially inchurches) or stinging balloons are common means. While achieving high levels in somefrequency bands, these methods have a very poor repeatability and produce unpredictablespectra. The low frequency energy contents usually is scant, especially for pistols due totheir small dimension. Even the omnidirectionality is not at all guaranteed [16]. The onlybet to avoid these severe drawbacks is the use of an electro-acoustical system, and hencebring a loudspeaker into play.

Obviously, when using a loudspeaker without any further precautions, the acquired roomtransfer functions will be colored by the loudspeaker’s frequency response. This isparticularly a problem when the RIR shall be used for auralization. To worsen things, thefrequency response is direction-dependent. For room acoustical measurements, the ISO3382 prescribes the use of a loudspeaker “as omni-directional” as possible (a condition thatin practice hardly can be satisfied over, say, 2 kHz).

No loudspeaker will be able to produce a frequency-independent acoustical output. This isnot a dramatic problem in reverberation time measurements in octaves or third of octaves,as long as the deviation within these bands doesn’t get too high. Alas the coloring of theRTFs by the loudspeaker’s frequency response is highly undesirable for auralizationpurposes. In these cases, it’s mandatory to use a pre-emphasis to remove the coloring. Ofcourse, this equalization could also be done by post-processing the RTFs with the inverseof the speaker’s response, but this wouldn’t improve the poor S/N ratio at frequencyregions where the acoustical output of the loudspeaker is low. That’s why it isadvantageous to pre-filter the excitation signal to allow for a frequency-independent poweroutput.

A loudspeaker equalizing is but one component of pre-emphasis that should be applied inroom acoustical measurements. Additionally, the measurement can be enhanced to a greatextent by adapting the emitted power spectrum to the ambient noise spectrum. And in mostof the cases, this background noise tends to be much higher in the lower frequency regions.Thus, in order to achieve an S/N-ratio that is almost independent of frequency, there shouldbe an extra pre-emphasis that tracks the background noise spectrum.

While a frequency independent S/N-ratio certainly is desirable for room acoustical analysis(especially for the extraction of reverberation times in filtered bands), it may be argued thatin order to minimize audible noise, RIRs acquired for auralization purposes better had anoise floor that is approximately the inverse of our audition sensitivity at low levels. Noiseshaping to reduce the perceived noise in recordings with a fixed quantization (for instance,

24

the 16 bits of a CD) attempts the same goal. A very expert introduction to this area is givenin [48]. To achieve a noise level that is particularly low in the spectral regions of high earsensitivity, an emphasis equal to such a sensitivity curve (e.g. E-curve) would have to beapplied to the excitation signal. On the other hand, giving an extra boost in the midfrequency region of highest ear sensitivity will lead to particularly annoying excitationsignals. Anyway, the question which emphasis is best suited is a multifaceted problem andmay be answered differently for every measurement scenario.

A conundrum is how to handle measurements for acoustic power equalization ofmeasurement loudspeakers. Usually, the acoustical power response of the speaker isobtained by magnitude averaging many transfer functions measured in the diffuse field of areverberant chamber. A correction with the inverse of the reverberation times with10 log {1/T(f)} converts the diffuse field sound pressure spectrum into a spectrumproportional to the acoustical power.

After this, a smoothing with 1/6 or 1/3 octave is indispensable to obtain a non-corrugatedcurve. But this smoothing will not be sufficient for low frequencies, as can be seen in thetop right curve in Fig. 14. In this range, the chamber transfer function consists only in a fewsingle high-Q modes. It’s a good idea to replace these needles by a slope with thetheoretical decrease of the speaker’s sensitivity (12 dB/oct. for closed box or 24 dB/oct. forvented systems).

The problem now is how to deal with the phase and the associated group delay. Evidently,the phase of a reverberant chamber’s transfer function cannot be used. A feasiblecompromise is to combine the acoustic power magnitude response (as obtained in thereverberant chamber) with a free-field phase response (as obtained by measuring thespeaker’s sensitivity on-axis). Obviously, merging the amplitude of one measurement withthe phase of another leads to an artificial spectrum that will also correspond to a syntheticimpulse response with some artifacts. However, this seems to be a viable way to minimizeamplitude and phase distortion in room acoustical measurements.

It is interesting to note here that if a sweep shall be created, only the target’s magnituderesponse will influence the excitation signal. Having no influence on the creation of thesweep, the measurement loudspeaker’s phase can thus only be equalized by post-processing(that is, applying the inverse of this phase to the reference file). Thus, for general roomacoustical measurements, the signal processing will always consist of a combination of pre-and post-processing.

5 SWEEP SYNTHESIS

Sweeps can be created either directly in the time domain or indirectly in the frequencydomain. In the latter case, their magnitude and group delay are synthesized and the sweep isobtained via IFFT of this artificial spectrum. The formulas given here do not adhere to thetypical mathematical standards, but their form is suitable for direct implementation insoftware.

25

5.1 Construction in the time domain

Sweeps can be synthesized easily in the time domain by increasing the phase step that isadded to the argument of a sine expression after each calculation of an output sample. Alinear sweep as used in TDS has a fixed value added to the phase increment:

)sin()( ϕAtx =

ϕϕϕ ∆+=

ϕϕϕ Inc+∆=∆

(1)

In contrast, a logarithmic sweep is generated by multiplying the phase increment with afixed factor after each new output sample calculation. So the last line in equation (1) simplychanges to:

ϕϕϕ Mul⋅∆=∆ (2)

The value of ϕ for the first sample is 0 while the start value of ϕ∆ corresponds to thedesired start frequency of the sweep:

SSTARTSTART ff⋅=∆ πϕ 2 (3)

The factor ϕInc used for the generation of a linear sweep depends on the start- and stopfrequency, the sampling rate fS and the number of samples N to be generated:

Nf

ffInc

S

STARTSTOP

⋅−

⋅= πϕ 2 (4)

In contrast, the factor ϕMul necessary to create a logarithmic sweep is calculated by:

N

ffld STARTSTOP

Mul

)(

2

−

=ϕ (5)

While sweeps generated in the time domain have a perfect envelope and thus the same idealcrest factor as a sine wave (3,02 dB), their spectrum is not exactly what is expected. Thesudden switch-on at the beginning and switch-of at the end are responsible for unwantedripple at the extremities of the excitation spectrum, as can be seen in Fig. 15. Half-windowscan be used to soothe the impact of switching, but do not entirely suppress it. Normally,these irregularities will have no effect on the recovered frequency response when correctingthem with a reference spectrum derived by inversion of the excitation spectrum (asobtained by a reference measurement with output connected to input). If, however, thedeconvolution is simply done with the time-inverted and amplitude-shaped stimulus, asproposed in [15], or if the correction is not feasible, as with TDS, then errors can beexpected near the start and end frequency of the sweep.

26

5.2 Construction in the frequency domain

Constructing the sweep in the spectral domain avoids these problems. The synthesis can bedone by defining magnitude and group delay of an FFT-spectrum, calculating real- andimaginary part from them and finally transforming the artificial sweep spectrum into thetime domain by IFFT. The group delay, while sometimes not easily interpretable forcomplex signals, is a well defined function for swept sines, describing exactly at whichtime each momentary frequency occurs. For sweeps, the group delay display looks like atime-frequency distribution with the vertical and horizontal axis interchanged (albeitlacking the third dimension, containing magnitude information).

If a constant temporal envelope of the sweep is desired (guaranteed naturally by theconstruction in the time domain), magnitude and group delay have to be in a certain relationto each other. In the case of a linear sweep, the magnitude spectrum has to be white and inthe case of a logarithmic sweep, must be pink with an inclination of –3 dB per octave. Theassociated group delay for the linear sweep can then simply be set by:

kff GG ⋅+= )0()( ττ (6)

with k being:

N

fk GSG )0()2/( ττ −

= (7)

The group delay of a logarithmic sweep is slightly more complicated:

)()( fldBAfG ⋅+=τ (8)

with

)()()(

)()(STARTSTARTG

STARTEND

STARTGENDG fldBfAffld

ffB ⋅−=

−−

= τττ

(9)

Normally, fSTART will be the first frequency bin in the discrete FFT-spectrum, while fEND

equals to fS/2. Of course, )( STARTG fτ and )( ENDG fτ have to be restricted to values that fit

into the time interval obtained after IFFT. The phase is calculated from the group delay byintegration. Phase and magnitude can then be converted to real and imaginary parts by theusual sin/cosine expressions.

As stated before, creating a sweep in the time domain will cause some dirt effects in thespectral domain. Likewise, synthesizing a sweep in the spectral domain will cause someoddities of the resulting time signal. First of all, it is important that the phase resulting fromthe integration of the constructed group delay reaches exactly 0° or 180° at fS/2. Thiscondition generally has to be fulfilled for every spectrum of a real time signal. It can beachieved easily by subtracting values from the phase spectrum that decrease linearly withfrequency until exactly offsetting the former end phase:

27

ENDSf

fff ϕϕϕ ⋅−=

2/)()( (10)

This is equivalent to adding a minor constant group delay in the range of ± 0,5 samplesover the whole frequency range.

Even satisfying this condition, the sweep will not be confined exactly to the values givenby )( STARTG fτ and )( ENDG fτ , but spread out further in both directions. This is a directconsequence of the desired magnitude spectrum in which the oscillations that would occurwith abrupt sweep start and stop points are precisely not present. Because of thebroadening, the group delay for the lowest frequency bin should not be set to zero, but alittle bit higher. This way, the sweep’s first half-wave has more time to evolve. However, itwill always start with a value higher than zero, while the remaining part left of the startingpoint folds back to “negative times” at the end of the period. There, it can “smear” into thehigh-frequency tail of the sweep if the group delay chosen for fS/2 is too close to the lengthof the FFT time interval. A safe way to avoid contaminating the running-out tail by lowfrequency components is to simply choose an FFT block length that is at least double thedesired sweep length.

To force the desired sweep start and end point to zero, fade operations of the first half-waveand the tail are indispensable to avoid switching noise. Doing so, it is clear that a deviationfrom the desired magnitude spectrum occurs. But it can be kept insignificant by choosingsufficiently narrow parts at the very beginning and end of the sweep. The resulting spectraand time signals for both the linear and logarithmic sweeps are shown in Fig. 16. Thehorizontal platform below 20 Hz for the logarithmic sweep has been introduced deliberatelyto avoid too much subsonic signal energy. Both sweeps cover the full frequency range fromDC (included) to fS/2.

The ripple introduced by the fading operations (half-cosine windows are most suitable) caneasily be kept under 0.1 dB and should not cause any concern, as its impact onmeasurement results is cancelled by performing and applying the reference measurement.However, in cases when an exact amplitude compensation is not feasible (for example, in aTDS device) or not intended, the iterative method depicted in Fig. 17 allows to completelyreject the ripple, establishing exactly the desired magnitude spectrum while maintaining thesweep confined to the desired length. The rather primitive iteration consists ofconsecutively performing the fade in/out-operation, transforming the time signal to thespectral domain, replacing the slightly corrugated magnitude spectrum with the targetmagnitude while maintaining the phases, and eventually back-transforming the manipulatedspectrum into the time domain. Before imposing the fade-in/out windows another time, theresiduals outside the sweep interval are examined. If their peak value is below the LSB ofthe sweep’s intended final quantization, the windowing is omitted and the iteration ends.Usually, the process converges rapidly and even a sweep quantized with 24 bits is availableafter only approximately 15 iterations. However, the perfect magnitude response is traded-off by a light distortion of the phase spectrum. Hence, the derivative group delay will beslightly warped, but the effect is almost unperceivable and restricted to the very narrow

28

frequency strips around DC and fS/2. The iteration works best with broad-band sweepscovering the full frequency range between 0 Hz and fS/2.

5.3 Sweeps with arbitrary magnitude spectrum and constant temporalenvelope

So far we’ve been restricted to linear and logarithmic sweeps. If their magnitude spectrawas altered from the dictated white or pink energy contribution to something different, it isclear that their temporal envelope would not be constant any more. This would entail anincreasing crest factor and hence an energy drop, given a fixed peak value as maximumamplitude. Nevertheless, it would be very attractive to use sweeps with just an arbitraryenergy contribution without loosing the advantage of low crest factor. This can be achievedsurprisingly easy by simply making the group delay grow proportionally to the power ofthe desired excitation spectrum. Generally, the energy of a sweep in a particular frequencyregion can be controlled by either the amplitude or the sweep rate at that frequency. Asteeply increasing group delay means that the corresponding frequency region is stretchedout in time substantially, with the momentary frequency rising only slowly. This way,much energy is packed into the respective spectral section.

The group delay for the arbitrary-magnitude, constant-envelope sweep can be constructedstarting with )( STARTG fτ for the first frequency and then increasing bin by bin:

2)()()( fHCdfff GG ⋅+−= ττ (11)

with C being the sweep length divided by the excitation spectrum’s energy:

∑=

−=

2/

0

2)(

)()(Sf

f

STARTGENDG

fH

ffC

ττ

(12)

The process is illustrated by an example in Fig. 19. The sweep under construction shallserve to equalize a loudspeaker and feature an additional low-frequency boost. The desiredexcitation signal spectrum originates from the inverted loudspeaker response. It is furtheremphasized with a first order low-shelf filter with 10 dB gain and high-pass filtered at30 Hz to avoid an excess of infrasonic energy being fed to the loudspeaker by the newequalizing excitation signal. The constructed group delay shows a relative steep inclinationup to 0,5 seconds, and the resulting time signal reveals that the frequency in this rangeincreases only gradually from the start value. Thus, the sweep will contain a lot of energy inthat frequency region. From 80 Hz to 6 kHz, the group delay increases by merely 200 ms,so the whole mid range is swept through in this short time. Above 6 kHz, the group delayagain inclines due to the increased desired magnitude and the frequency rises moremoderately, thereby extending the HF part of the sweep.

It is noticeable that the sweep’s amplitude is not entirely constant, as would be desirable toachieve the ideal crest factor. At the beginning and at approximately 650 ms, a slight

29

overshoot cannot be denied. This is another imperfection caused by the sweep’s syntheticformulation in the frequency domain. To keep these disturbances small, a slight smoothingof the magnitude spectrum helps. Doing so, the sweep’s crest factor can normally be keptbelow 4 dB, leading to an energy loss of less than 1 dB. Of course, the iterative methoddescribed in 5.2 can be combined with this algorithm to reduce deviations from the desiredmagnitude response near the band limits. Doing so, the crest factor even decreases slightly.

5.4 Sweeps with arbitrary magnitude spectrum and prescribed temporalenvelope

Is the constant envelope sweep with freely definable magnitude spectrum the optimumexcitation signal for room acoustic measurements? Well, if the amplifier power should bethe restricting limit of the measurement equipment, one would say yes. The almost constantenvelope of the excitation signal allows drawing the amplifier’s maximum powerthroughout the whole measurement, thereby pumping the maximum possible energy intothe RUT (room under test) in the distinct time interval. However, lacking amplifier poweris rarely a point of concern today (let alone battery powered portable equipment). It is muchmore likely that the power handling capabilities of the deployed loudspeakers have to beconsidered carefully to avoid damaging them. In the case of a multi-way system, forexample a dodecahedron composed of coaxial speakers, perhaps supported by a powerfulsubwoofer, each way has its own power limit. A constant-envelope sweep must have itspower adjusted to the weakest link of the equipment, mostly the tweeters. This would leavethe power handling capability of the woofer (which exceeds that of the tweeters often by afactor of 10 or more) for the most part idle. On the other hand, an extra boost is oftendesirable precisely in the low frequency region to overcome the increasing ambient noisefloor.

It becomes clear that in the case of loudspeaker impasses, the momentary sweep powershould be controllable according to the frequency just being swept through. This can beaccomplished by controlling the amplitude of the sweep in a frequency-dependent fashion.To do so, only a minor modification of the sweep creation process revealed in section 5.3 isnecessary. As depicted in Fig. 19, the trick is to first divide the target spectrum by the“desired envelope” spectrum. The resulting spectrum is the base for the group delaysynthesis, using the same formula as in the “constant envelope” case. The synthesizedspectrum indeed corresponds to a sweep with constant envelope, but with reduced energy atlower frequencies, according to the inverse “desired envelope” spectrum. After creating thereal/imaginary pair, the sweep spectrum will now be multiplied with the desired envelopespectrum, re-establishing the desired magnitude response. The IFFT will now produce asweep that doesn’t have a constant envelope any more, but faithfully features thefrequency-dependent amplitude course imposed by the “desired envelope” spectrum.

Two degrees of freedom are offered here: An arbitrary spectral distribution along with anarbitrary definition of the frequency-dependent envelope is transformed into a swept sinewave warped suitably in amplitude and time.

It must be admitted that this approach doesn’t entirely remedy the danger of overheatingsensitive tweeters. Compared to a “constant envelope” sweep with identical energy

30

contents, a “desired envelope” sweep only leads to stretching out in time the same energyfed to the tweeter, resulting in the same heat-up if the sweep length is smaller than the timeconstant of the voice coil’s thermal capacity. Room acoustical measurements with longsweeps (many seconds), however, benefit from this expanding. The “desired envelope”sweeps are also advantageous if the tweeter is not only threatened by overheating, but aswell by mechanically caused damage (for example, excessive forces in compressiondrivers). The reduced distortion due to the lesser level may also be motivating, although itis the big asset of sweep measurements that the distortion products can be separated so wellfrom the actual impulse response.

Another application of sweeps with controlled decrease of the envelope at higherfrequencies is the measurement of analogue tape recorders. The envelope can be adapted tothe frequency-dependent saturation curve of the tape, thus making optimum use of thetape’s dynamic range at every frequency. This way, an apparent drop in response atfrequencies where the level gets close to the saturation limit is obviated. In compactcassette tape decks with their narrow and very slowly running tapes, the necessary decreaseof the envelope can reach 20 dB or more, depending on the tape material. The saturationcurve itself can be determined by deliberately feeding a sweep with excessive level thatcauses overload at all frequencies. The distortion products are removed from the impulseresponse, and the remaining spectrum of the main impulse response is a good estimate ofthe frequency-dependent maximal input level for recordings and measurements.

5.5 Dual channel sweeps with speaker equalization and crossoverfunctionality

In many acoustical measurements, multi-way loudspeakers are employed to cover as muchof the audio range as possible. For example, an omnidirectional dodecahedron will usuallyexhibit poor response below approximately 200 Hz, depending on its size. Thus, it isadvantageous to support it with a subwoofer to circumvent the need of excessive pre-emphasis. A normal closed or vented box design with just one chassis will still besufficiently omnidirectional in this frequency range. As all sound cards and somemeasurement systems are equipped with stereo converters, it is rewarding to make use ofthe two channels to include an active crossover functionality into the excitation signal. Thiscan be achieved with some extensions of the sweep generation methods described in thesections 5.3 and 5.4.

First of all, as in any equalization task, it is necessary to measure both speakers at the sameposition to establish valid magnitude, phase and delay relationships. Bearing in mind thespeaker’s power handling capabilities and according to their magnitude responses, anappropriate crossover point can then be selected. At this frequency, the phase and groupdelay difference between both spectra is read out and stored. They will be needed later.Now, after an optional smoothing, both spectra are inverted and treated with a band-pass toconfine them to the desired frequency range to be swept through (see Fig. 20). At this point,a desired additional emphasis is also applied to the stereo spectrum. Should the excitationsignal feature a frequency-controlled envelope, a division by the desired envelope spectrumhas to be executed now. After these preparing steps, the crossover function can be brought

31

into play by multiplying the first channel with a low-pass-filter and the second with thecorresponding high-pass filter.

An optional smoothing effect (with constant width on a linear frequency scale) may beobtained by windowing the impulse responses of the two spectra. For this purpose, both aretransformed to the time domain after deleting their phase. The desired window is appliedand the confined impulse responses transformed back to the frequency domain (see Fig.21). The window should not to be too narrow to avoid too much blurring of the lowfrequency details.

Now the dual-channel sweep can be created by formulation of the already knownrelationship between squared magnitude and group delay increase. But instead of using justone channel, the squared magnitude of both channels has to be summed here to yield thegroup delay growth value:

22

1

)()()( fHCdfffCh

GG ∑=

⋅+−= ττ (13)

Likewise, C is calculated using the sum of both channel’s total energy:

∑∑==

−=

2/

0

22

1

)(

)()(Sf

fCh

STARTGENDG

fH

ffC

ττ

(14)

As the momentary frequency shall be the same for both channels, the group delay synthesishas to be performed only once and the resulting phase can be copied into the secondchannel. If the excitation signal shall feature a frequency-controlled envelope, its dual-channel spectrum has to be multiplied by the envelope spectrum after converting magnitudeand synthesized group delay to the normal real/imaginary part representation.

The IFFT will turn out a dual channel sweep that first glides through the low frequencies inone channel and then through the remaining frequencies in the other channels. The sum ofboth channels will have the desired envelope, while the relation of their amplitude at eachmomentary frequency corresponds to the relation established in the spectral domain.

In the crossover region that cannot be made indefinitely narrow due to the non-repetitivenature of the sweep, both channels interfere. While they are in-phase in the synthesizeddual channel excitation signal, they usually wouldn’t arrive in-phase at the microphonewhen emitted over the two loudspeakers. So a delay/phase correction is necessary to avoidsound pressure level drops at the crossover frequency. That’s why the phase and groupdelay relationship of the speaker responses should be stored previously. This informationcan now be used to shift the signal for the speaker with the smaller group delay to the rightby the difference in arrival times. This can be accomplished best still in the spectral domainby adding the appropriate group delay. The received phases at the crossover point can bebrought into accordance by adding or subtracting a further small group delay to one of thechannels.

32

It is arguable that dual-channel noise signals or sweeps with “two voices” coveringindependently both frequency ranges at the same time would be more advantageous, as theyallow pumping energy to both speakers over the whole measurement period. However, onlya “single-voiced” sweep allows properly excluding the distortion products from therecovered impulse response. Being able to do so, the stereo signal can be fed with muchhigher level, which offsets the disadvantage of restricted emission time in the single ways.

Obviously, the reference spectrum necessary to deconvolute the received excitation signalcannot be created by performing the usual electrical reference measurement. By doing so,the carefully introduced speaker equalization would disappear in the final results. The onlyviable way is to construct the reference spectrum by simulation, as depicted in Fig. 22.First, the previously generated dual channel excitation signal is transformed to the spectraldomain. There, it is multiplied with the speaker response and the self-response of themeasurement system. The latter should include the response of the entire electrical signalpath (converters, power amplifier, microphone pre-amp) and the microphone response,should this one not be sufficiently flat. After these operations, both channels are summed,yielding a simulation of the received sound pressure spectrum at the microphone position(colored by the response of the receiving path). This spectrum is now inverted to negate thegroup delay and to neutralize the chosen pre-emphasis, if any was applied. As the invertedspectrum would lead to excessive boost of the frequencies outside the selected transmissionrange, a multiplication with a band-pass of roughly the same corner frequencies as used inthe pre-processing stages (top right of Fig. 20) is indispensable. To mute the out-of-bandnoise effectively, the order of this band-pass should be somewhat higher than the one usedin the pre-processing. The application of this inevitable band-pass is a critical step, as itmeans that the acoustical measurement results are convoluted with its impulse response.The ensuing effects can be quite disturbing. For example, using a linear-phase band-passwill obviously produce pre-ringing of the recovered room impulse responses. This isundesirable for auralization purposes. In this case, it is more advisable to use minimum-phase IIR-type filters. On the other hand, a linear phase band-pass might induce less errorsin the classical room acoustical parameter evaluations. In any case, the filter order shouldbe as moderate as possible to keep the filter’s impulse response sufficiently narrow.

Generally, these considerations apply to any broad-band room acoustical measurement. Atleast, the active pre-equalization technique presented here allows acquiring room impulseresponses that are free of coloration by the measurement loudspeaker and feature a high,frequency-independent S/N ratio.

6 DISTORTION MEASUREMENT

So far, it has been shown that the harmonic distortion artifacts can be removed entirelyfrom acquired impulse responses when measuring with sweeps. In room acousticalmeasurements, they are usually simply discarded as the loudspeaker is not the object ofinvestigation. In loudspeaker measurements, however, it is very interesting to relate them tothe fundamental to evaluate the frequency-dependent distortion percentage. Indeed, this canbe done separately for every single harmonic, as has already been proposed by Farina [15].To illustrate the technique, Fig. 23 shows the group delay of a logarithmic sweep and its

33

first four harmonics. Obviously, for a given momentary frequency of the fundamental, allits corresponding harmonics have the same group delay. In the example, the sweepfundamental reaches 400 Hz after 2 seconds. Consequently, the 2nd order harmonic traceintersects the horizontal 2 second line at 800 Hz, the 3rd at 1.2 kHz and so on. Nowfocusing on just one specific frequency in the diagram, the harmonic traces intersecting thisvertical line have a lesser group delay then the sweep fundamental, as of course they belongto fundamentals with lower frequencies that have been swept through earlier. Multiplyingtheses curves with the reference spectrum (that is, subtracting the group delay of thefundamental) leads to displacing them to negative times while the fundamental will resideat t=0s, as desired.

Only in the case of a logarithmic sweep, the harmonics will all feature a frequency-independent constant group delay after the application of the reference spectrum. Othergroup delay courses of the sweep could be used, but would require the use of a separatereference spectrum for each harmonic to warp them to straight lines. Moreover, the distancebetween the components of the individual harmonics would not be frequency-independent.Thus, a logarithmic sweep is the preferable excitation signal, equaling the one that alreadyhas been used since such a long time in the venerable level recorder (2.1).

An IFFT of the deconvoluted spectrum yields the actual impulse response at the leftboarder and a couple of “harmonic’s impulse responses” (HIRs) at negative times near theright border, with the 2nd order HIR situated rightmost and the upper order HIRs followingfrom right to left. Their distance between each other can be calculated by:

( )[ ]soctratesweep

ordordldDist BA

ABHIR = (15)

To evaluate the frequency-dependent distortion fraction for every harmonic, the time signalis separated into the fundamental’s impulse response and the single HIRs by windowing.Each of the isolated HIRs is submitted to a separate FFT. The FFT block length usedtherefore can be much shorter than the one used for the initial deconvolution of the sweepresponse, thereby speeding up the whole process. To relate the frequency contents of oneHIR spectrum to the fundamental, a spectral shift operation according to the order of thespecific harmonic has to performed. For example, the spectral components of the 5th orderHIR are shifted horizontally to one fifth of their original frequency. After this operation, theshifted spectrum can be divided by the fundamental’s, yielding the frequency-dependentdistortion fraction. It would seem that a correction of -10 log {order of the harmonic} mustbe applied to the results to compensate for the emphasis of higher frequencies imposed bythe reference spectrum. Suprisingly, as exposed by many trials, this operation has to beomitted to yield the correct results.

Compared to the standard single-tone excitation and analysis with fixed frequencyincrements, this method is many times faster and usually establishes a much higherfrequency resolution, at least in the mid and high frequency range. However, somedisadvantages should not be denied. First of all, the measurement is restricted to anecoicambiences, unless very long sweeps are used. Too much reverberation at the measurementsite would lead to smearing of the distinct harmonic impulse responses into each other by

34

delayed components, thus thwarting their separation. This certainly is an essentialshortcoming, as the strength of impulse response based measurement is precisely the abilityto reject reverberation, provided that the time gap between direct sound and first reflectionis sufficiently large.

Other problems are related to the mandatory use of windows to separate the individualHIRs from each other. Of course, all the usual problems associated with windowing [51]apply here. In particular, the choice of the window type constitutes a compromise betweenmainlobe width (equivalent to the spectral resolution) and sidelobe suppression. To avoidan energy loss and subsequent underestimate of distortion components that are not exactlysituated under the window’s top, a Tukey-style window should be used. However, this kindof window features the rectangular window’s poor sidelobe suppression of just 21 dB. Thiscan lead to filling up with artifacts those frequency regions in which the distortionplummets to very low values. The spectral smoothing caused by any window is constant ona linear frequency scale. On the usual logarithmic display, this means that the spectralresolution gets extremely high at high frequencies, while perhaps lacking details at the lowend of the distortion spectrum. If a higher resolution is desired, the sweep has to be madelonger to space the HIRs further apart, thus allowing to use wider windows.

The window width has to decrease according to equation (15) to separate the higher orderHIRs. This entails a reduced resolution of the corresponding distortion spectrum. However,when compressing it to the right to relate it with the fundamental’s, the resolution becomeshigher than that of lower order HIRs. In practice, the desired resolution of the 2nd orderharmonic at the low end of the loudspeaker’s transmission range dictates the sweep rate andhence the excitation signal length.

Another problem of the fast distortion testing is lacking signal-to-noise ratio, especially forthe higher order harmonics that usually have fairly low levels. While pure-tone testingresults in a lot of energy being packed in either the steady fundamental and its harmonics,the sweep technique disperses each one’s energy continuously over the whole frequencyrange. That’s why the distortion spectra for the higher order harmonics tend to becomepretty noisy, especially at higher frequencies. To alleviate this problem, it is a good practiceto extend the sweep to a length even bigger than necessary to achieve the desired spectralresolution. The windows can then be made narrower than necessary to seclude the singleHIRs, this way rejecting the noise floor that resides between them.

Finally, even when all precautions have been taken to guarantee a high precisionmeasurement, it cannot be denied that sometimes, unexplainable differences between thesteady tone testing and the sweep method occur in some frequency regions. Fig. 25 displaysan example of such a discrepancy. Between 1.3 kHz and 1.7 kHz, the 2nd order harmonictrace acquired with sweep and steady sine testing look fairly different. The reasons forthese occasional divergences are not obvious, maybe different voice coil temperatures playa role. After all, the stepped tone testing lasts a lot longer and hence leads to more voicecoil heat up.

Even in spite of these uncertainties, the sweep based distortion testing is very attractive, asit is so much faster than the conventional pure tone testing. In production testing, it does

35

not only allow occasional spot checks, but facilitates checking 100% of the production,even if the product is cheap mass ware.

7 CONCLUSIONS

FFT techniques using sweeps as excitation signal are the most advantageous choice foralmost every transfer function measurement situation. They allow feeding the DUT withhigh power at little more than 3 dB crest factor and are pretty tolerant against time varianceand distortion. Choosing an adequate sweep length allows complete rejection of theharmonic distortion products. Moreover, these can be classified into single frequency-dependent harmonics, allowing a complete and ultra-fast distortion analysis over the wholefrequency range together with the evaluation of the transfer function. When it comes tocapturing room impulse responses for auralization purposes, there is no alternative to sweepmeasurements: The high dynamic range in excess of 90 dB required for this purpose isunattainable with MLS or noise measurements. But even in standard RT measurements thatdon’t require such a high dynamic range, sweeps are helpful as they allow easily increasingthe dynamic range up to 15 dB compared to MLS-based measurements, using the sameamplifier, loudspeaker, and measurement length. Thus, there is little point in using MLS.Even their asset of saving memory and processing time has almost completely lost itsrelevance with today’s computer technology. From a programmer’s point of view,renouncing to include the MLS generation and Hadamard transform may save preciousdevelopment time when writing a measurement software. While acquiring transferfunctions with MLS may be mathematically elegant, the authors think that using sweeps todo so is elegant from a system theory and signal purity point of view. Measuring withsweeps is also more natural. After all, bats do not emit MLS to do their acoustic profiling.

8 ACKNOLEDGEMENT

The work has been supported by the Brazilian National Council for Scientific andTechnological Development (CNPq). It sponsored a six month’s journey in the laboratoryof acoustics of INMETRO of one of the authors. While the scholarship initially was notexactly intended to develop measurement technology, many of the ideas regarding sweepmeasurements have evolved and been tested in this period, thanks to intense exchange ofideas by the two authors.

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[3] Richard C. Heyser, “Determination of Loudspeaker Signal Arrival Times, Parts I,II& III”, J.AES 1971, pp. 734-743, pp. 829-834, pp. 902, or AES Loudspeakers Anthology,vol. 1–25, pp. 225

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[17] Douglas D. Rife & John Vanderkooy, “Transfer Function Measurement withMaximum-Length Sequences”, J.AES June 1989, pp.419

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[19] H.Alrutz & Manfred R. Schroeder, “A Fast Hadamard Transform Method for theEvaluation of Measurements using Pseudorandom Test Signals”,Proc. 11th ICA, Paris 1983, pp. 235

[20] Chris Dunn, Malcom Omar Hawksford, “Distortion Immunity of MLS-DerivedResponse Measurements”, J.AES May 1993, pp. 314

[21] Douglas D. Rife, “Comments on “Distortion Immunity of MLS D. Rifed ImpulseResponse Measurements” and author’s reply, J.AES June 1994, pp. 490

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[25] Jeffrey Borish, “An Efficient Algorithm for Generating Colored Noise Using aPseudorandom Sequence”, J.AES March 1985, pp. 141

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[33] Johan L. Nielsen, “Improvement of Signal-to-Noise Ratio in Long-TermMeasurements with High-Level Nonstationary Disturbances”, J.AES 1997, vol. 45, pp.1063

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Fig. 1. TDS signal processing. The dotted signal paths are used in the second run of adouble excitation measurement.

Fig. 2. Signal processing steps for 2-channel FFT analysis with program material used asexcitation signal.

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Fig. 3. Signal processing for transfer function measurement with impulses.

Fig. 4: Signal processing stages for transfer function measurements with MLS.

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Fig. 5. Signal processing stages for transfer function measurements with pre-emphasizedMLS.

Fig. 6. Signal processing stages for transfer function measurements with any deterministicsignal.

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Fig. 7. MLS passed through anti-aliasing low-pass of 8x oversampler (left) and 2nd orderButterworth low-pass with corner frequency of 40 kHz (right).

Fig. 8. Artificial sinusoidal time variance of ±0.5 samples imposed on noise signal (above)and sweep (below) and the resulting spectra.

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Fig. 9. Transfer function of analogue tape deck measured with MLS (left) and sweep(right), both with identical coloration and energy.

Fig. 10. Measurement of room impulse response in a reverberant chamber with 1, 10 and100 synchronous averages. Left: with MLS, right: with sweep of identical coloration andenergy. The curves are compressed to 1303 values, each of them representing the maximumof 805 consecutive samples.

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Fig. 11. Transfer function a GSM cellular (fed acoustically by a headphone), received by afixed phone. Left: Measurement with linear MLS of degree 18. Right: Measurement withlinear sweep of same length.

Fig. 12. Same measurement as in Fig. 11, but with a window (Hann, width 80 ms) appliedto the recovered impulse responses.

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Fig. 13. Transfer function of an MPEG3 Coder at 128 kbit/s, measured with linear MLS ofdegree 14 (left) and linear sweep of same length (right).

Fig. 14. Steps to construct the acoustical power spectrum of a loudspeaker

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Fig. 15. Sweeps created in the time domain and their spectral properties

Fig. 16. Sweeps created in the spectral domain by formulating group delay

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Fig. 17. Iteration to construct broad-band sweeps with perfect magnitude response

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Fig. 18. Generation of a sweep with almost constant envelope from an arbitrary magnitudespectrum

Fig. 19. Sweep creation with desired envelope and arbitrary magnitude response.

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Fig. 20. Preprocessing for dual channel sweep with 2 way speaker equalization and activecrossover functionality.

Fig. 21. Further processing of dual channel sweep: Windowing of impulse response, groupdelay synthesizing, inter-channel phase and delay adjust, fade in/out of sweep.

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Fig. 22. Creation of reference file for deconvolution of dual channel sweep emitted by2-way-speaker.

Fig. 23. Group delay of a) fundamental (upper line) and first four harmonics, b) referencefile, c) deconvoluted sweep with harmonics. d) impulse response positions of harmonics.

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Fig. 24. Signal processing stages for evaluation of transfer function and 2nd order harmonicwith logarithmic sweep.

Fig. 25. Comparison of 2nd order harmonic acquired with sweep (left) and traditional puretone testing in 1/24 octave increments (right).