Ch03pulse Modulation

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Chapter 3: Pulse Modulation

7/25/2014©2000, John Wiley & Sons, Inc.

Haykin/Communication Systems, 4th Ed1

Chapter 3

Pulse Modulation






Chapter Outline

Sampling: is basic to all forms of pulse modulation.

Pulse-amplitude modulation (PAM): is the simplest

form of modulation. Quantization: when combined with sampling,

permits to digitize analog signals.

Pulse-code modulation (PCM): is the standard

method used to transmit analog signals by digitalmeans.

Time-division multiplexing: provides for the time

sharing by a common channel.






Chapter Outline (Continued)

Digital multiplexers: combines many slow bit

streams into a single faster stream.

Other forms of PCM: delta modulation (DM) anddifferential PCM (DPCM).

Linear prediction: is a basic form of encoding

analog message signals at low bit rates as in

DPCM. Adaptive forms of DPCM and DM.

The MPEG-1/audio coding standard: is a transpa-

rent, perceptually loss-less compression system for

audio signals.






Sampling ProcessLet g (t ) be a finite-energy band-limited signal with bandwidth W .

The ideal sampled signal g (t ) obtained from g (t ) is

n s s

m s s

n s s

nfT jnT g

mf f G f f GkT t nT g t g

)2exp()(

)()()()()(

where T s is the sampling period and f s=1/T s is the sampling frequency (rate).

From the expression of G( f ), it is clear that sampling in time produces periodicity

in frequency. Also G( f ) can be represented as the discrete-time Fourier transform

(DTFT) of the samples g (kT s). The DTFT may be viewed as a complex Fourier series representation of the period

frequency function G( f ), where g (kT s) play the role of Fourier coefficients.






Sampling Process (Continued) If T s=1/2W (or f s=2W ), then G( f ) takes the form

0

)()(

exp2

)(

mm s s s

n

mf f G f f G f

W

nf j

W

n g f G

The second equation shows that with the choice of f s=2W , the spectrum G( f ) may be

separated from other periods of G( f ), which is essential for the reconstruction

process.

Under the two conditions: G( f )=0 for f W and f s=2W , G( f ) can be written as

W f W

W

nf j

W

n g

W

W f W f GW

f G

n

,exp

22

1

),(2

1)(






Sampling Process (Continued)

This latter expression means that G( f ) is uniquely determined by using the discrete

time Fourier transform (DTFT) of the samples g (n/2W ), taken at a sampling rate

f s=2W.

This is true as long as f s2W.






Figure 3.1

The sampling process. (a) Analog signal.(b) Instantaneously sampled version of

the analog signal.






Figure 3.2(a) Spectrum of a strictly band-limited

signal g (t ). (b) Spectrum of the sampled

version of g (t ) for a sampling period T s =

1/2 W .






Signal Reconstruction In case T s=1/2W , the reconstruction of g (t ) can be performed using inverse FT

n

n

n

W

W

W

W n

nWt W

n g

nWt

nWt

W

n g

df W

nt f j

W W

n g

df W

n

t f jW

nf j

W

n

g W

df ft j f Gt g

t- ),2(csin2

)2(

)2sin(

2

22exp

2

1

2

22expexp22

1

)2exp()()(

where sinc(2Wt -n) is called the interpolation function.






Signal Reconstruction (Cont’d)

As can be seen g (t ) can be reconstructed from its samples g (n/2W ), provided that

the sampling frequency (rate) f s2W .






Sampling TheoremThe sampling theorem for finite-energy band-limited signals can be stated in two equivalent

parts related to the transmitter and receiver of a pulse modulation system:

1. A finite-energy band-limited signal with bandwidth W Hertz , is completely described in

terms of its samples taken at a rate f s=1/T s2W samples per second.

2. A finite-energy band-limited signal with bandwidth W Hertz , may be completely recovered

(reconstructed) from its samples taken at a rate f s=1/T s2W samples per second.

Note: The minimum sampling rate f s=2W is called Nyquist rate and its reciprocal (inverse)

T s=1/2W is called Nyquist interval.






Aliasing Effect

In practice a message signal is not strictly band-limited. This means that the sampling rate

f s=2W may result in an overlapping between the different periods of G( f ) as shown byFigure 3.3.

This overlapping is called aliasing effect .

Aliasing effect can be eliminated by using an anti-aliasing filter prior to sampling and usinga sampling rate slightly higher than Nyquist rate ( f s=2W ). This is shown in Figure 3.4.

g (t ) g (kT s)Anti-aliasing

Filter

Sampler






Figure 3.3(a) Spectrum of a signal. (b) Spectrum of an

undersampled version of the signal exhibiting the aliasingphenomenon.






Figure 3.4(a) Anti-alias filtered

spectrum of aninformation-bearing

signal. (b) Spectrum

of instantaneously

sampled version ofthe signal, assuming

the use of a sampling

rate greater than the

Nyquist rate.

(c ) Magnitude

response of

reconstruction filter.






Example

1. In telephone system, the bandwidth of the voice signal (message signal) is limited

to W =3.1 kHz and the universal sampling frequency is f s = 8kHz > 2W .

2. Determine the Nyquist rates and Nyquist intervals used to sample the following

signals:

x(t )=sinc(200t ), y(t )=sinc2(200t ). To determine the sampling rate of a signal, we have to know its frequency

spectrum. For x(t ) and y(t ), this may be done using Fourier transform.

200200

1)(

f rect f X ,

200 ,0

200 ,200

1200

1

)(

f

f f

f Y






Example (Continued)From X ( f ) and Y ( f ), it is clear that for x(t ) the maximum frequency (bandwidth) W =100 Hz and

for y(t ) the maximum frequency (bandwidth) W =200 Hz

X ( f ) Y ( f )

1/200 1/200

f ( Hz ) f ( Hz )

-100 0 100 -200 0 200

For x(t ), f s=2W =200 Hz and T s=1/ f s=5ms.

For y(t ), f s=2W =400 Hz and T s=1/ f s=2.5ms.






Pulse-Amplitude Modulation

Ideal sampling, as seen before, uses instantaneous Dirac pulses. This cannot berealized (generated) in practice.

A practical way to perform sampling is to use pulses with finite non-zero width

such as rectangular pulse.

The result of sampling with rectangular pulses is pulse-amplitude modulation(PAM). This is shown in Figure 3.5.

The sampling theorem is still valid when a rectangular pulse instead of Dirac pulse

is used for sampling.

The resulting signal is given by a rectangular pulse train but with amplitudes that

are varied in proportion to the corresponding sample values. For a message signal m(t ), PAM signal is expressed as

n s s nT t hnT mt s )()()( ,

where T s is the sampling interval and h(t ) is a rectangular pulse of duration T , given by Figure 3.6a.






Figure 3.5Flat-top samples, representing an analog

signal.






PAM (Continued) From previous (ideal sampling) section, the instantaneous sampled version of m(t )

is given by

n s s nT t nT mt m )()()(

One can show that

n s s nT t hnT mt ht mt s )()()(*)()( , where * denotes the

convolution operation.

In the frequency domain, we have S ( f )= M ( f ) H ( f )=

k s s f H kf f M f )()( , where

S ( f )=FT[ s(t )], M ( f )=FT[m(t )], and H ( f )=FT[h(t )]. FT[.] stands for Fourier transform.

How do we recover m(t ) from PAM signal s(t ) whose Fourier transform S ( f )?

From the expression of S ( f ), it is clear that the spectrum of the sampled PAM s(t ) isa repetition of the spectrum of m(t ) reshaped with the frequency response

H ( f )=T sinc( fT )exp(- j fT ) of the sampling rectangular pulse.






PAM (Continued) If f s>2W , then the output of a reconstruction filter will be M ( f ) H ( f ). This output is

equivalent to passing m(t ) through another low-pass filter of frequency response

H ( f ).

From Figure 3.6 we see that by using rectangular pulses to generate PAM signals,

we have introduced amplitude distortion and phase delay. This effect is calledaperture effect . The amplitude distortion effect can be removed by connecting an

equalizer in cascade with the reconstruction filter. The frequency response of the

equalizer should have a magnitude of the form

)sin()(csin

1

)(

1

fT

f

fT T f H

Usually if the duty cycle T /T s is very small, then no equalization is needed.

The noise performance of PAM systems can never be better than base-band

signal transmission. In practice PAM is used as intermediate step for other form

of modulation or time multiplexing.






Figure 3.6(a) Rectangular pulse h(t ). (b) Spectrum H (f ), made up

of the magnitude |H (f )|, and phase arg[H (f )].






Figure 3.7System for recovering message signal

m(t ) from PAM signal s(t ).






Other Forms of Pulse

ModulationTwo other parameters of a pulse may be used to convey (carry) information. These areduration and position.

The resulting modulation schemes are pulse duration (width) modulation (PDM or PWM)and pulse position modulation (PPM). PDM and PPM waveforms are shown in Figure 3.8.

The generation of PDM and PPM waveforms is performed using the following circuit

Technique for generating PDM and PPM signals

PAMGenerator

Clock

Mono-stable

K

TriangleGnerator






Figure 3.8Illustrating two

different forms of

pulse-time

modulation for the

case of a sinusoidal

modulating wave.

(a) Modulating wave.

(b) Pulse carrier.

(c ) PDM wave.(d ) PPM wave.






Bandwidth-Noise Trade-off

In the context of noise performance, a PPM system is the optimum form of analog

pulse modulation. PPM and FM have similar noise performance.

Both PPM and FM have figure of merit (FOM) proportional to ( BT )2.

In terms of trade-off of increased BT for improved noise performance, the best we

can do with continuous-wave (CW) modulation and analog pulse modulationsystems is to follow a square law.

Can we do better than a square law?

The answer is yes, and the digital pulse modulation is the technique to have better

law for improvement.

Specifically in pulse-code modulation (PCM), the message signal m(t ) isrepresented in discrete form both in time and amplitude.

Digital pulse modulation techniques require sampling and quantization be performed on the base-band signal m(t ).






Quantization Process Quantization is a process that transforms the continuous sample amplitude m(nT s)

of a message signal m(t ) at time t =nT s into a discrete amplitude v(nT s) taken from a

finite set of possible discrete levels.

In our study we assume that the quantization process is memoryless and

instantaneous, which means that the quantization at time t =nT s is not affected by

earlier or later samples of the message signal.

Figure 3.10 shows the input/output characteristics of two types of quantizers.

The relationship between the input and output of the quantizer is represented as

v= g (m).

Quantizers can be of uniform or non-uniform type. In the former type, the levels of

the quantized amplitude are uniformly spaced. However, in the latter type the levelsare not.

Since the quantization is an approximation process, it results into an approximation

error called quantization noise. Figure 3.11 illustrates the quantized signal and the

corresponding error (noise).






Quantization Process (Cont’d) The quantization noise (error) is random and therefore for a specific sample it can

be represented by a random variable Q of sample value q. This sample value o

quantization error is given by: q=m-v.

If the signal m(t ) has an amplitude with dynamic range [-mmax,mmax], then for a

uniform quantization the step size of the quantizer is: =2mmax/ L, where L is the

total number of quantization levels.

According to Figure 3.10, the quantization error Q will have its sample values

bounded by -/2q/2.

If is sufficiently small (which means L is sufficiently large), we can assume that

Q is a uniformly distributed random variable with PDF

otherwise ,0

2/2/ ,/1)(

qq f Q

With this PDF, the mean and variance of Q are zero and12

)(2

22 Q E Q ,

respectively.






Quantization Process (Cont’d) Typically the number of quantization levels L is chosen as power of two, i.e L=2 R,

or equivalently R=log2( L), where R represents the number of bits used to encode

each quantization level. This gives a quantization step size =2mmax/2 R and a

variance of quantization noise R

Q m 22

max

22 2

3

1

12

.

The performance of the quantizer is measured using the output SNR defined as

Rm

Q

mo

m

P P SNR 2

2

max

22

2)(

This expression shows that the SNR at the quantizer output increases exponentiallywith the number of bits R. And since increasing R requires a proportionate increasein the channel (transmission) bandwidth BT , one can see that digital pulse

modulation follows an exponential law for noise performance.






Figure 3.9

Description of a memoryless quantizer.






Figure 3.10Two types of quantization: (a) midtread and (b)

midrise.






Figure 3.11Illustration of the quantization process. (Adapted from

Bennett, 1948, with permission of AT&T.)






Figure 3.12Illustrating the partitioning of the dynamic

range A m A of a message signal

m(t ) into a set of L cells.






Example

Let m(t )= Amcos(2 f mt ).

2/2

mm A P , mmax= Am, and R

mQ A222

23

1

)2(23

3/22/)( 2

22

2

R R

m

mo

A ASNR

In terms of decibels, (SNR)o becomes

10log10(SNR)o = 1.8 + 6 R






Example (Continued)

Table 3.1 Signal-to-(quantization) noise ratio for different number of

quantization levels for sinusoidal modulation.

Number of Levels L Number of bits/sample R

(SNR)o in dB

32 5 31.8

64 6 37.8

128 7 43.8

256 8 49.8






Pulse-Code Modulation

In PCM, a message signal is represented by a sequence of coded pulses, obtainedfrom representing the signal in discrete form in both time an amplitude.

As shown by Figure 3.13 the basic operations performed at the transmitter are:

sampling, quantization, and encoding.

The basic operations in the receiver are regeneration of impaired signals, decodingand reconstruction.

Regeneration also occurs during the route of transmission.

In PCM of voice signals, non-uniform quantization is used to allow smaller

quantization step sizes form smaller amplitudes and larger step sizes for larger

amplitudes so that the (SNR)o remains quasi-constant for all levels of amplitudes. To use non-uniform quantization, the message signal is passed through a

compressor , then a uniform quantization is applied to the compressed signal. Atthe receiver an expander circuit is used to undo the effect of the compressor.






Pulse-Code Modulation (Cont’d)

For the compression, two laws are adopted: the -law in North America and the A-law in Europe.

-law:)1(log

)1(log

e

e mv

A-law:

1

1

log1

)(log1

10 log1

m A

, A

m A

Am ,

Am A

v

e

e

e

The characteristics of these two laws are shown in Figure 3.14. The typical values

used in practice are: =255 and A=87.6.

After quantization the different quantized levels have to be represented in a form

suitable for transmission. This is done via an encoding process.






Figure 3.13The basic elements of a PCM system.






Figure 3.14Compression laws. (a) -law. (b) A-law.







Each discrete level is represented by a code element (symbol). In a binary code,

we have two symbols only, representing two levels of the signal.

The electrical representation of a code is done by assigning a waveform (or a

pulse) to each symbol as shown in Figure 3.15 for the binary case.

Some well-known line codes that can be used for the electrical representation oa binary data stream, are: (a) Unipolar NRZ. (b) Polar NRZ signaling. (c)

Unipolar RZ signaling. (d) Bipolar RZ signaling. (e) Split-phase or Manchester

code. NRZ: non-return to zero. RZ: return to zero.

These lines codes have some interesting spectral properties shown in Figure 3.16.

Differential encoding method is used to encode information in terms of signaltransitions. In practice a transition designates symbol 0, while no transition

designates symbol 1 (see Figure 3.17). The original binary information is

recovered simply by comparing the polarity of adjacent binary symbols to

establish whether or not a transition has occurred.






Figure 3.15Line codes for the

electrical representations

of binary data.

(a) Unipolar NRZsignaling. (b) Polar NRZ

signaling.

(c ) Unipolar RZ signaling.

(d ) Bipolar RZ signaling.(e) Split-phase or

Manchester code.






Figure 3.16aPower spectra of line codes: (a) Unipolar NRZ signal.

The frequency is normalized with respect to the bit rate

1/T b, and the average power is normalized to unity.

Ch 3 l d l i






Figure 3.16bPower spectra of line codes: (b) Polar NRZ signal.



Ch 3 P l M d l i






Figure 3.16cPower spectra of line codes: (c ) Unipolar RZ signal.



Ch t 3 P l M d l ti






Figure 3.16dPower spectra of line codes: (d ) Bipolar RZ signal.



Ch t 3 P l M d l ti






Figure 3.16ePower spectra of line codes: (e) Manchester-encoded signal.

The frequency is normalized with respect to the bit rate 1/T b,

and the average power is normalized to unity.

Ch t 3 P l M d l ti







Regeneration

A regenerative repeater (see Figure 3.18) consists of (1) an equalizer, (2) a timing circuit,

and (3) a decision-making device. The equalizer is used to undo the effect of thetransmission channel to get back the pulses in their original shape before transmission. The

timing circuit is used to recover the clock of the transmitted symbols (pulses), which is then

used in the decision-making process. The function of the decision-making device is to detect

the different pulses based on some threshold information.

The purpose of a regenerative repeater is to clean the PCM signal during its transmission

through a channel.

Ch t 3 P l M d l ti






Noise in PCM Systems

The performance of a PCM system is influenced by two noise sources: (1) channel

noise and (2) quantization noise.

The main effect of channel noise is to introduce bit errors into the received signal.

The presence of this noise can be measured in terms of probability of symbol error

or bit error rate (BER).

The effect of channel noise can be made practically negligible by using high signal

energy-to-noise density ratio through short spacing between regenerative repeaters.

In the absence of channel noise, quantization noise is acting alone. Since

quantization noise is under the designer's control, it can be made negligible by

increasing the number of levels L and selecting a compressor-expander (compander) pair that is matched to the message signal characteristics.







Figure 3.17(a) Original binary data. (b) Differentially

encoded data, assuming reference bit 1.

(c ) Waveform of differentially encoded

data using unipolar NRZ signaling.







Figure 3.18

Block diagram of regenerative repeater.







Time-Division Multiplexing

This technique combines time-domain samples from different message signals

(sampled at the same rate) and transmits them together across the same channel.

The multiplexing is performed using a commutator (switch) as shown in Figure

3.19. At the receiver a decommutator (switch) is used in synchronism with the

commutator to demultiplex the data. TDM system is very sensitive to symbol dispersion, that is, to variation o

amplitude with frequency or lack of proportionality of phase with frequency.

This problem may be solved through equalization of both magnitude and phase.

One of the methods used to synchronize the operations of multiplexing and

demultiplexing is to organize the mutiplexed stream of data as frames with aspecial pattern. The pattern is known to the receiver and can be detected very

easily.







Figure 3.19

Block diagram of TDM system.







Digital Multiplexers

This type of multiplexers is used to combine digital signals at different bit ratessuch as voice, video, audio, and computer data.

There are two groups of digital multiplexers: One group uses low bit-rate data

streams and the other is for high bit-rate data streams.

The first group requires the use of modems.

The second group of digital multiplexers forms a part of the data transmissionservice provided by telecommunication carriers such as AT&T.







Figure 3.20

Conceptual diagram of multiplexing-demultiplexing.







Digital Multiplexers (Continued)

The hierarchy starts at 64 kb/s, which corresponds to PCM representation of voice signal.

One bit stream with bit rate 64 kb/s is called digital signal 0 (DS0).

The first-level hierarchy combines 24 DS0 into one single DS1 at a rate 1.544 Mb/s.

The second-level hierarchy combines 4 DS1 into one single DS2 at a rate 6.312 Mb/s.

The third-level hierarchy combines 7 DS2 into one single DS3 at a rate 44.736 Mb/s. The fourth-level hierarchy combines 6 DS3 into one single DS4 at a rate 274.176 Mb/s.

The five-level hierarchy combines 2 DS4 into one single DS5 at a rate 560.160 Mb/s.

The bit rate produced by each multiplexer is slightly higher than the total rate of the

multiplexed signals because of bit stuffing.

The multiplexed signal must include some form of framing so that its individual

components can be identified at the receiver.







Digital Multiplexers (Continued)

The bit rate of interleaved signals must be locked to a common clock.

The multiplexer has to handle small variations in the bit rates of the incoming digital signal

due to the propagation delay, by using a technique known as bit stuffing. At the

demultiplexer, the stuffed bits must be removed from the multiplexed signal. This requires a

method that can be used to identify the stuffed bits. AT&T M12 multiplexer is designed to combine 4 DS1 bit streams into 1 DS2. The format o

DS2 is given by Figure 3.21. One frame of DS2 consists of 6 48-bit words separated by 6

control bits. The 48-bit words are obtained by multiplexing 12 bits from each incoming DS1.

A DS2 frame consists of 24 control bits. Overall we have 3 types of control bits: F for the

overall frame, M for subframes, and C for stuffing indication.




7/25/2014 ©2000, John Wiley & Sons, Inc.Haykin/Communication Systems, 4th Ed

57

Virtues and Limitations of PCM

The most important advantages of PCM are:

1. Robustness to channel noise and interference.

2. Efficient regeneration of the coded signal along the channel path.

3. Efficient exchange between BT and SNR.

4. Uniform format for different kind of base-band signals.5. Flexible TDM.

6. Secure communication through the use of special modulation schemes o

encryption.

These advantages are obtained at the cost of more complexity and increased BT .

With cost-effective implementations, the cost issue no longer a problem oconcern.

With the availability of wide-band communication channels and the use o

sophisticated data compression techniques, the large bandwidth is not a serious

problem.






58

Delta Modulation (DM)

In DM, the message signal is over-sampled to purposely increase correlation

between adjacent samples.

The DM provides a staircase approximation to the message signal m(t ) as shown in

Figure 3.22.

the difference e[nT s]=m[nT s]-mq[(n-1)T s] is quantized into only two levels . The error e[nT s] is quantized to give

eq= sgn(e[nT s]). The quantity eq is then used to compute the new

quantized level

mq[nT s]=mq[(n-1)T s]+eq[nT s] In DM the quantization levels are represented by two symbols: 0 for - and 1 for

+. In fact the coding process is performed on eq.

The main advantage of DM is its simplicity as shown by Figure 3.23.






59

Delta Modulation (Cont’d)

The transmitter of a DM system (Figure 3.23a) is given by a comparator, a one-bit quantizer,

an accumulator, and an encoder.

The receiver of a DM system (Figure 3.23b) is given by a decoder, an accumulator, and a low-

pass filter.

DM is subject to two types of quantization error: Slope overload distortion and granular noise(see Figure 3.24).

Slope overload distortion is due to the fact that the staircase approximation mq(t ) can't follow

closely the actual curve of the message signal m(t ). In order for mq(t ) to follow closely m(t ), it

is required that

dt

t dm

T s

)(max

be satisfied. Otherwise, step-size is too small for the staircase approximation mq(t ) to follow

m(t ).




p


60

Delta Modulation (Cont’d)

In contrast to slope-overload distortion, granular noise occurs when is too large relative to

the local slope characteristics of m(t ). granular noise is similar to quantization noise in PCM.

It seems that a large is needed for rapid variations of m(t ) to reduce the slope-overload

distortion and a small is needed for slowly varying m(t ) to reduce the granular noise. The

optimum can only be a compromise between the two cases.

To satisfy both cases, an adaptive DM is needed, where the step size can be adjusted in

accordance with the input signal m(t ).




p


61

Figure 3.22Illustration of delta modulation.




p


62

Figure 3.23DM system.

(a) Transmitter.

(b) Receiver.




p


63

Figure 3.24Illustration of the two different forms of

quantization error in delta modulation.




p


64

Delta-Sigma Modulation

The quantizer input of a DM transmitter may be viewed as an approximation to

the derivative of m(t ). This leads to an accumulation of noise in the demodulated

signal.

This drawback can be overcome by integrating m(t ) prior to delta modulation.

The use of integration has the following beneficial effects1. The low frequency content of m(t ) is pre-emphasized.

2. Correlation between adjacent samples of the delta modulator input is increased

which tends to reduce the variance of the error signal at the quantizer input.

3. Design of the receiver is simplified

A DM scheme that includes an integrator at its input is called delta-sigmamodulation (D-M). Figure 3.25 shows two equivalent versions of D-Msystem.

The receiver of D-M system consists simply of a low-pass filter.





65

Figure 3.25Two equivalent versions of delta-sigma modulation system.





66

Linear Prediction

Linear prediction is a signal processing function performed by a finite-duration impulse

response (FIR) discrete-time filter, as shown by Figure 3.26.

Linear prediction consists of estimating the current sample of a signal from a certain

number of previous samples. This is always possible when the signal samples are

correlated.

Linear predictor involves the use of three functional blocks (see Figure 3.26): (1) a set of

delay units, (2) a set of multipliers, and (3) a set of adders.

For a linear predictor, the output is given by

p

k k k n xwn x

1

)()(ˆ ,

where p is the prediction order, wk ' s are the predictor coefficients. Since the linear prediction is an estimation, it results into an error called prediction error

and given by: )(ˆ)()( n xn xne





67

Linear Prediction (Continued)

The objective of designing a linear predictor (FIR filter) is to choose the coefficients

w=(w1, w2, …, w p)t so as to minimize the mean-square error (MSE) criterion

J (w)= E [e2(n)]

The result of minimizing J (w) with respect to w is

xopt x rwR

where R x is a Toeplitz matrix whose entries are given by R x[i,j]= R x[|i-j|]= E [ x(k-i) x(k-j)], 0i,j p-1, wopt is the optimum coefficients vector in the

MSE sense, and r x is a px1 vector given by r x=( R x(1), R x(2), …, R x( p))t .

The corresponding minimum mean square error (MMSE) is given by

x x

t

x x J rR r12

min

It is common in practice to use an adaptive algorithm to perform the linear prediction, as

shown by Figure 3.27. The coefficients w are estimated using a gradient-based algorithm

called linear mean square (LMS) algorithm)()()1( k ek k

xww

where k indicates the iteration number, is the adaptation step size, x=( x(k -1), x(k -2),

…, x(k-p))t , and e(k ) is the prediction error, given by).(ˆ)()( k xk xk e





68

Linear Prediction (Continued)

The LMS algorithm is a stochastic adaptive filtering algorithm, which starting from aninitial vector w(0), seeks to find the minimum point of J (w) by following a zig-zag path. Infact the exact minimum is never found because the algorithm keeps wandering around it

in a random fashion.

The usefulness of the LMS algorithm resides in its tracking property since it can track theminimum even for nonstationary signals such as speech signals, where the optimum

prediction coefficients w vary with time.





69

Figure 3.26Block diagram of a linear prediction filter

of order p.





70

Figure 3.27Block diagram illustrating the linear

adaptive prediction process.





71

Differential PCM (DPCM)

Voice and video signals represented in PCM exhibit high correlation, which means that

PCM signals contain redundant information. The result is an inefficient coding.

By removing the PCM information redundancy a more efficient coded signal may be

obtained. This is done using DPCM.

In DPCM a linear prediction is performed on samples of a message signal m(kT s)=m(k ),

then the prediction error )(ˆ)()( k mk mk e is computed and fed to a quantizer to obtainthe quantized value eq(k )=e(k )+q(k ), as shown by Figure 3.28a. q(k ) is the quantization

error.

The input of the linear predictor of Figure 3.28a is )()()()(ˆ)( k qk mk ek mk m qq ,

which represents a quantized version of the input sample m(k ).

If the prediction is well performed, then the variance of e(k ) will be much smaller than

the variance of m(k ), which results into a smaller number of levels to quantize e(k ).

The receiver as given by Figure 3.28b, consists of a decoder which produces eq(k ), that is

added to the output of a prediction filter identical to the one used in the transmitter. The

result is the quantized message signal mq(k ).




7/25/2014 ©2000, John Wiley & Sons, Inc.Haykin/Communication Systems, 4th Ed 72

Differential PCM (Continued)

DPCM includes DM as a special case, where the prediction filter is a simple delay

element. Simply put, DM is a one-bit version of DPCM.

The problem of slope-overload distortion may also arise in DPCM, whenever the slope of

the message signal changes too rapidly for the prediction filter to track it.

The noise performance of DPCM is measured, as in other digital modulation systems, by

the output signal-to-quantization noise, given by

Q p

Q

E

E

M

Q

M o SNRGSNR )()(

2

2

2

22

2

where222 and,, E Q M are the variances of m(k ), q(k ), and e(k ), respectively.

The factor G p is the processing gain produced by the DPCM quantization scheme. When

G p>1, which is the case most of the time, it represents the gain in SNR obtained by using

DPCM compared to PCM.

The receiver as given by Figure 3.28b, consists of a decoder which produces eq(k ), that is

added to the output of a prediction filter identical to the one used in the transmitter. The

result is the quantized message signal mq(k ).





Figure 3.28DPCM system.(a) Transmitter.

(b) Receiver.





Adaptive DPCM

In PCM, the standard bit rate is 64 kbits/s. The aim of all the variants of PCM is to reduce

the number of bits used in the encoding process by removing redundancies.

Adaptive DPCM (ADPCM) is a scheme that permits the coding of speech (voice) signals

at 32 kbits/s through the combined use of adaptive quantization and adaptive prediction.

Adaptive quantization refers to a quantizer that operates with a time-varying step-size

)(ˆ

)( k k M and adaptive prediction filter refers to a filter with time-varyingcoefficients. is a constant and )(ˆ k M is an estimate of the standard deviation of m(k ).

In ADPCM adaptive quantization can be performed using adaptive quantization with

forward estimation (AQF) or adaptive quantization with backward estimation (AQB).

In ADPCM adaptive prediction can be performed using adaptive prediction with forward

estimation (APF) or adaptive prediction with backward estimation (APB).

AQF and APF use unquantized samples of the input message signal to estimate M and the

predictor coefficients w, respectively.

AQB and APB use quantized samples of the input message signal to estimate M and the

predictor coefficients w, respectively.





Adaptive DPCM (Continued)

Both AQF and APF suffer from the same disadvantages, which are the buffering, the side

(extra) information to be transmitted, and the delay. But by using AQB and APB these

disadvantages are eliminated.

Figure 3.29 shows the AQB scheme and Figure 3.30 shows the APB scheme.





Figure 3.29 Adaptive quantization with backward

estimation (AQB).





Figure 3.30 Adaptive prediction with backward estimation (APB).





Computer Experiment:

Adaptive delta Modulation Adaptive delta modulation (ADM) is a modification of DM, in which the step size is

adapted to the slope (variation) of the message signal.

If successive errors are of opposite polarity, then the delta modulator is operating in the

granular mode; in such a case it is advantageous to use reduced step size.

If successive errors are of the same polarity, then the delta modulator is operating in its

slope-overload mode; in this case, the step size should be increased.

The algorithm used for adaptive DM with step size increase/decrease of 50% is

(k )=|(k -1)|(mq(k )+0.5mq(k -1))/mq(k ), if (k -1)min

(k )=min, if (k -1)min

where if (k ) is the step size at iteration k and mq(k ) is the one-bit quantizer output that is equal to 1.


Fi 3 31




Figure 3.31 Adaptive delta modulation system: (a) Transmitter. (b)

Receiver.


Fi 3 32




Figure 3.32Waveforms resulting from the computer

experiment on delta modulation: (a) Linear deltamodulation. (b) Adaptive delta modulation.





MPEG/Audio Coding System

For speech (voice) signal we have efficient coding schemes such as ADPCM because a

speech production model is available. Unfortunately, nothing similar exists for audio

signal.

MPEG-1/audio coding standard is a lossy compression system that is used for audio

signals. It is capable of achieving transparent, perceptually lossless compression of

stereophonic audio signals at high sampling rate. MPEG stands for Motion Picture Experts Group.

The MPEG-1/audio coding standard achieves such a performance by exploiting two

psycho-acoustic characteristics of the human auditory system: (1) Critical bands and (2)

auditory masking .

Critical Bands: The human auditory system (with bandwidth up to 20 kHz) may be

modeled as a band-pass filter bank, consisting of 25 overlapping bands called critical bands with bandwidths less than 100 Hz for low frequencies up to 5 kHz for high

frequencies.





MPEG/Audio Coding System

(Continued) Auditory masking: A strong signal can mask a weak signal if they are in the same critical

band and the latter lies below a masking threshold (see Figure 3.33).

From Figure 3.33, as long as the quantization noise level lies below the minimum masking

threshold, the quantization noise is inaudible.

The transmitter and receiver of MPEG/audio coding system are given by Figure 3.34a andFigure 3.34b, respectively. In the transmitter, the function of time-to-frequency mapping

network is to decompose the input audio signal into multiple sub-bands with frequency

resolutions closer to the partitions between the critical bands.

The function of the psycho-acoustic model is to analyze the spectral content of the input

audio signal so as to compute the mask level for each sub-band. This will help the

quantizer allocating the appropriate number of bits to each quantized sample.

Chapter 3: Pulse ModulationFigure 3.33Illustrating the definitions of masking threshold and related




Illustrating the definitions of masking threshold and related

parameters. The high-level signal (masker) lies inside the

darker-shaded critical band, hence the masking is more

effective in this band than in the neighboring band shown inlighter shading. (Adapted from Noll (1998) with permission of

the CRC Press.)


Figure 3 34




Figure 3.34MPEG/Audio coding system. (a) Transmitter. (b)

Receiver.





Figure P3.5





Haykin/Communication Systems, 4th Ed 86

Figure P3.22




Figure P3.37

Ch03pulse Modulation

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