My Introduction

• Syed Rizwan-ul-Hasan– Asst. Professor– B.Sc (Industrial Tech), MUET– BS (Electronics / Computer Sc), USA– MS (Computer Engineering), SSUET– (PhD) Computer Engineering ,SSUET– Experience; 27 years, which include 14 years

of field experience in USA, Saudi Arab and Pakistan ; 13 years of academic.

My Introduction …

Research Interest– Multi-Carrier Communication

• MCCDMA(Multi-Code Code Division Multiple Access)• OFDM(Orthogonal Frequency Division Multiplexing)

– Peak Power control

Office contact– srhasan@ssuet.edu.pk– Room no. BT-9– Phone; 4988000-2 / ext 252

Communications Systems

CE-3085TH. SEMESTER

COMPUTER ENGINEERING DEPARTMENT

COURSE OUTLINEIntroduction to Signals & Comm.

• Week 1– Signal & Its Classification– Important Signals– Operations On Signal– Power Signals

• Week 2• Communication System Components• Types of Communication Systems• Application of Communication System– SNR, Noise, Channel Bandwidth, Data rate– Basic Filter Types– Nyquist Theorem, Shannon’s Theorem

FOURIER ANALYSIS

• Week 3– Fourier series for Periodic Signals– Fourier Transform– Parseval’s Theorem– Application of Fourier Transform– Power Spectrum Density Function

• Week 4• Properties of Fourier Transform• Channel & their types with distortions

FOURIER ANALYSIS…

• Week 4 …– Convolution Operation– Impulse Response & T.F of Channels/Filters

AMPLITUDE MODULATION AM

• Week 5– Double Side Band – Suppressed Carrier(DSB-SC)– Modulators & Demodulators of DSB-SC, – DSB Single Band Modulation– Hilbert Transform

• Week 6– Modulators & Demodulators of SSB– Telephone Channels using SSB– Vestigial Sideband VSB & Application– Carrier Acquisition & Phase locked loop PLL

INTRODUCTION TO DIGITAL COMMUNICATION (Week 7 & 8)

• Digital Signal, Digital Communication Component

• Sampling & Quantization• AWGN probability Density Function

BASEBAND DIGITAL COMMUNICATION(Note; Week 9 is Midterm)

• Week 10– Detection of Baseband Signals– Detection 7 likelihood test

• Week 11– Vector representation of Signals– Vector Analysis of Signals & noise

• Week 12– Error probability & Error performance curves– Match filters & Correlation

BAND PASS SIGNALS

• Week 13– ASK, PSK & FSK– Detection of Binary Band Pass Signals – Detection of Multiple Band Pass Signals

• Week 14– Different PSK Implementation– Non Coherent & Coherent Detection of Signals– Orthogonality of Signals

• Week 15– Error performance for different modulation Schemes– M-ary Signaling & Performance

• Week 16 (Revision)

Text and Ref. Books

• Text Books– Modern Digital and Analog Communication

systems, 3rd. Edition by B.P.Lathi– Digital communication fundamental and

Application, 2nd. Edition by Bernard Skalar

• Reference Books– Signals & Systems, 2nd. Edition by Alan V.

Oppenheim and Alan S. Willsky with S. Hamid Nawab

Reference books continue

– Digital Signal Processing, Principles, Algorithms and Application by John G. Proakis and Dimitris G. Manolakis

Marks distributionTotal marks 100

• Theory (20)– Attendance (2)– Quizzes and Assignments (3)– Midterm (15)

• Lab (20)– Attendance (4)– Lab manual (8)– Final test (8)

• Semester Exam (60)

Signals and Systems(Week 1)

• In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying quantity.

• In the physical world, any quantity measurable through time can be taken as a signal.

Signals and Systems ….

• Whereas the systems respond to particular signals by producing other signals or some desired behavior.

• For e.g. In Electrical circuit, voltages and currents as a function of time are signals, and a circuit is itself a system.

• Another example a robot arm is a system, whose movements are the response to control input signal.

Signals and Systems…

• The concept of signals and systems arise in a wide variety of fields.

• Areas – communications, aeronautics and astronautics, circuit design, acoustics, seismology, biomedical engineering, energy generation and distribution systems, chemical process control, and speech processing.

• Graphical rep. of a signal is shown in fig. 1.

Fig.1.Graphical rep. of Signal

Mathematical rep. of a signal

• S(t) = 8t ------------------------------------ (1)• The above equation will given only linear relationship

between signal S(t)and time ‘t’.• S(t) = A Sine(2∏ft + ø) ------------------(2)• However, equation 2 is a sine wave or sinusoidal S(t)

with parameters such as amplitude ‘A’, frequency ‘f’, time ‘t’ and phase angle ‘ø’.

• Hence equation 2 is useful in analysis of for e.g. speech signal, because it gives complete information about the signal compare to equation 1.

Math. Rep of a Signal …

• Sine wave or Sinusoid– Fig. 2, Phase diff of 900,

Cosine wave leads by 900

A, the amplitude, is the peak deviation of the function from its center position.ω or 2∏f , the angular frequency, specifies how many oscillations occur in a unit time interval, in radians per secondɸ, the phase, specifies where in its cycle the oscillation begins at t = 0.The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase.

Math. Rep of a Signal…

• It is the only periodic waveform that has this property. This property leads to its importance in Fourier analysis and makes it acoustically unique.

• Cosine wave said to be sinusoidal, because cosx = sinx(x + ∏/2), which is also a sine wave with a phase difference of ∏/2.

Fig. 3.Occurrence of a Sinusoid

Classification of signalsPeriodic Signal

• A periodic signal repeat itself on a fixed interval of length 2∏ as shown in Fig. 4 . Sinusoidal signals are examples of it.

• S(t) = S(t + T)

Non-periodic Signal

• While non-periodic signal does not repeat itself as shown in Fig. 4.

Fig. 4 Classification of Signals…

Periodic Sig., with two Periods Non-Periodic Signal

(a) A periodic signal with period T 0  

(b) An aperiodic signal

Figure 3

Classification of signals…

Continuous time• Continuous time signal or

analog signal are defined for every value of time and they take on values in the continuous interval ( x, y)

Discrete time• Discrete time signal are

defined only at certain specific values of time. These time instant needs not be equidistant. For e.g. n=0, +1, +2,…..

Classification of Signal…Analog vs. Discrete

• Same as Continuous vs. Discrete

Classification of Signals

Deterministic• Signals whose values at any

instant ‘t’ is known from their analytical or graphical description are called deterministic signals, as shown in Figure 5.

• This type of signal convey no information.

• Example is electronic circuits based on Ohm’s law.

Random• Random means un-

certainty. If the signal or message has un-certainty, it means, there is information as shown in Figure 6.

• Noise signals that perturb information are example of random signals.

Classification of signals…

Random Signal Fig.6Deterministic Signal Fig.5

Classification of Signals…

Causal signals are signals that are zero for all negative time

Anti-causal are signals that are zero for all positive time.

Classification of signal…

• Anti-causal signals have nonzero values in both positive and negative time.

Classification of signals…

A signal fe(t) is said to be even, if it is identical to its time reversed counter part fe(t)=fe(-t)

(a) An even signal

A signal fo(t)is said to be odd, if it is 0 at t=0; fo(t)=-(fo(-t))

(b) An odd signal

Classification of signals…

• Continuous time complex exponential (Fig.7)– If σ is +, then as t increase s(t) is growing

exponential.– Examples; Chain reaction in atomic explosion,

complex chemical reaction etc…– If σ -, then s(t) is a decaying exponential.– Examples; radioactive decay, Responses of RC

circuit etc…

Classification of signals…Fig. 7 Exponential signals

(a) If σ is negative, we have the case of a decaying exponential window

(b) If σ is positive, we have the case of a growing exponential window.

(c) If σ is zero, we have the case of a constant window.

Signal Operations(Time Shifting of a Signal)

Note; Subtracting from the time variable ‘t’ will cause delay (move the signal to the right), while adding advance (to the left).

Signal Operations…Time Scaling

Time scaling compresses or dilates a signal by multiplying the time variable by some quantity. If that quantity is greater than one, the signal becomes narrower and the operation is called compression, while if the quantity is less than one, the signal becomes wider and is called dilation.

Signal operations…Time Reversal

A natural question arises about time scaling is: What happens when the time variable is multiplied by a negative number? The answer to this is time reversal. This operation is the reversal of the time axis, or flipping the signal over the y-axis.

Signal Energy and Power

• In many applications the signals we consider are directly related a physical quantities capturing power and energy in physical system.

)()()( titvtp

)(1

)( 2 tvR

tp

Signal Energy and Power …

• The total energy expended over the time interval is

•

•

• Similarly, for average power

21 ttt

dttvR

dttpt

t

t

t

)(1

)( 22

1

2

1

dttvRtt

tpt

t

)(1

)(

1)(

1

2

2

21

Signal Energy • The previous equations are basic, but they may be

applied to continuous signal for evaluating Energy and Power, with the assumption that R=1.

• We may consider the area under the signal S(t) as a possible measure of its size, because it takes amplitude and duration both.

• This could be a improper measure, due to its large size and its positive and negative value, which cancel each other

Signal Energy…

• This indicates a signal of small size.• This difficulty can be corrected by defining signal size as S2(t),

which is always positive.• This measure is called Signal Energy. For a real valued and

complex signal:-

•

dttsSe )(2

dttsSe2|)(|

Signal Energy…

• The Signal Energy must be finite for it to be meaningful measure of the signal size.

• A necessary condition for the energy to be finite is that the signal amplitude →0 as |t|→∞, otherwise the integral will not converge.

Signal Power

• If the amplitude of a signal does not →0 as |t|→∞, the signal energy is infinite. In such a case a better measure of a signal size would be a, average power Ps , defined for a real and complex signal:-

dttsT

PT

TTs )(

1 2/

2/

2lim

dttsTT

PT

Ts

22/

2/

|)(|1lim

Communications Systems(Week 2)

• Communication means to share information; To transmit or receive information or data.

• Communication may be established between people, Computer to Computer, near distance far distance (Telecommunication)

• The communication system , consist of different components which helps in transmitting or receiving data, between two entities or devices.

Communications Systems…

• As shown in Fig. 8; Communication system components consists of Source, Input transducer , Transmitter, Channel, Receiver, and Output transducer.

• Source originates the message, for e.g. voice, picture or data. If the data is non-Electrical such as voice, then it must be converted to Electrical signal by Transducer known as baseband signal or message signal.

Fig. 8 Communication System

Input Transducer

Transmitter Channel

Noise

Receiver

Output transdu

cer

Input Signal

Input Message

TransmittedSignal

ReceivedSignal

Output Signal

OutputMessage

Communications Systems…

• The transmitter modifies the baseband signal for efficient transmission.

• A transmitter consists of one or more sub-systems, a sampler, a quantizer, a coder and a modulator.

• A channel is a medium, such as coaxial cable, a waveguide, an optical fiber, or a radio link-through which the transmitter output is sent.

Communications Systems…

• The receiver reprocesses the signal from the channel by undoing the signal modification made at the transmitter and the channel

• Finally, the receiver output is fed to the output transducer, which convert back the electrical signal to its original form i.e. the message signal

• The Signal is distorted by Channel and Noise, which are random and unpredictable

Communications Systems…

• That comes from external and internal sources• External sources comes from nearby channel,

lightning, tube light electrical equipment etc…• Internal noise results from thermal motions of

electrons in conductors.• The signal to noise ratio is defined as the ratio

of the signal power to the noise power. The channel distort the signal and the noise accumulates along the path.

Communications Systems…

• The signal strength decreases while the noise level increases with distance from the transmitter.

• Thus SNR is continuously decreasing along the channel and amplification of the noisy signal make no use.

• For good results SNR, supposed to be high. In other words signal value, should be high compare to noise.

Digital Communication System

• Block diagram of CDMA system is shown in next slide. Since the human speech is the analog signal, so it has to be first converted into digital form. This function is performed by the source encoding module.

• After the source information is coded into a digital form, redundancy needs to be added to this digital message or data. This done for error control and power reduction.

Digital Communications Systems

• Block diagram

Source Encode

ModulateMultipleAccess

ChannelEncode

Sp

ee

ch

Transmitter

Source Decode

De-Modulate

MultipleAccess

ChannelDecode

Re

co

nstr

ucte

dS

pe

ech

Receiver

Digital Communication system…

• Thereafter, signal is further transformed to allow access to multiple users. Multiple access by different users means to the sharing of a common resource i.e. RF (Radio Frequency) spectrum.

• The purpose of the modulator is to shift the message or data to the carrier frequency (high frequency), because message or data does not have enough strength to go far distance.

• At the receiver end the purpose of the demodulator is to recover original signal. In other words at the receiver end the reverse operation is performed.

Application of Comm. System

• Telephone Exchange• Wireless Communication

Digital vs. AnalogDigital

• More immune to channel noise and distortion

• Regenerative repeaters for noise free signal.

• Digital hardware implementation is flexible in reconfiguring the hardware simply by changing the program.

• Accuracy

Analog

• Less immune to noise and distortion

• Not possible in analog communication

• Fixed, and need to redesign for new hardware

• Difficult to control the accuracy

Digital vs. Analog

Digital• Coding to yield low error

rate• Efficient in exchange of SNR

for Bandwidth

Analog• Less exchange of SNR for

Bandwidth

SNR, Bandwidth, Data Rate• The fundamental parameters that control the rate and

quality of information transmission are the channel bandwidth and the signal power S.

• The bandwidth (BW) of a channel is the range of frequencies that it can transmit with reasonable fidelity . OR

• Difference between the highest and the lowest frequencies in the specific range of frequencies.

• Example; Voice frequency range of 300hz. To 3300hz. Thus the voice bandwidth (BW) or Pass band is 3000hz wide

SNR, Bandwidth, Data Rate…

• Role of BW, If we want to increase the speed of information transmission by time compression of the signal lets say by a factor of 2.

• The signal can be transmitted in half time.• Frequencies and channel BW must also be doubled.• Thus the rate of information transmission is directly

proportional to channel BW. • The signal power S plays a dual role in information

transmission. Increase S reduced the effect of channel noise and we received accurate data.

SNR, Bandwidth, Data Rate…

• Signal to Noise Ratio (SNR) means, the higher (strength) the value of the signal, compare to Noise, the quality of the signal would be better over a longer distance.

• However, a certain minimum SNR is necessary for communication.

• The second role of signal power ‘S’ is not as obvious, but it is important. The BW and ‘S’ are exchangeable.

SNR, Bandwidth, Data Rate…

• For example if we increase the BW (by adding redundant bits to the message for reliability) the signal power S would reduced and vice versa.

• Another example; telephone channel has limited BW, but requires lot of power.

• So, we have to trade in between BW & S. • Since SNR is proportional to S, Therefore SNR and

BW are exchangeable. • In practice, increasing BW to reduce signal power S

is followed and is rarely vice versa.

Shannon’s and Nyquist Theorem

• The limitation imposed on communication by the channel bandwidth BW and the SNR is highlighted by Shannon’s equation:-

•

• Where ‘C’ is the channel capacity in bits per second. This is the max. number of bits that can be transmitted per second with a probability of error close to zero.

)1(log2 SNRBC

Shannon’s and Nyquist Theorem…

• If there is no noise in the channel, N=0• .• The Nyquist Theorem, also known as the sampling theorem, is

a principle that engineers follow in the digitization of analog signals. For analog-to-digital conversion (ADC) to result in a faithful reproduction of the signal, slices, called samples, of the analog waveform must be taken frequently. The number of samples per second is called the

sampling rate or sampling frequency.

c

Nyquist Theorem…

• Any analog signal, consists of components at various frequencies.

• The simplest case is the sine wave, in which all the signal energy is concentrated at one frequency.

• In practice, analog signals usually have complex waveforms, with components at many frequencies.

• The highest frequency component in an analog signal determines the bandwidth of that signal. The higher the frequency, the greater the bandwidth.

Nyquist Theorem…

• Suppose the highest frequency component, in hertz, for a given analog signal is fmax.

• According to the Nyquist Theorem, the sampling rate must be at least 2fmax, or twice the highest analog frequency component.

• If the sampling rate is less than 2fmax, some of the highest frequency components in the analog input signal will not be correctly represented in the digitized output. When such a digital signal is converted back to analog form by a digital-to-analog converter, false frequency components appear that were not in the original analog signal. This undesirable condition is a form of distortion called aliasing.

Examples of Aliasing• 1. Consider two sinusoidal signals:-

•

• Which are sampled at a rate The corresponding discrete signals are

• •

tts

tts

)50(2cos)(

)10(2cos)(

2

1

hzFs 40

nnns

nnns

2

5cos)

40

50(2cos)(

2cos)

40

10(2cos)(

2

1

Example of Aliasing …

• Result is that after digitizing, the value of signal s2 (5 integer multiple i.e. cos ∏/2)is identical with s1 (cos∏/2) . In other words frequency of signal s2 is alias with the frequency of signal s1.

• So, as per Nyquist criteria, sampling rate Fs >2fmax of the signal components. In this case Fs should be 100hz. i.e. twice of 50hz.

Examples of Aliasing…

• 2. Consider the analog signal:-•

• What is the Nyquist rate for this signal?• Solution; F1,F2,F3 are 25,150,50hz. Respectively.• (Note If the signal for e.g. is cos2∏(10)t, no need to divide it

by 2. Because it is already in the form of 2∏f. In the other case, for e.g. cos 50∏t , in order to make it in the form of 2∏f, we have to divide frequency 50hz by 2.

tttts 100cos300sin1050cos3)(

Aliasing Examples…

• Thus fmax is 150hz, therefore Fs >2fmax =300hz

Basic Filter Types

• A filter is a circuit that is designed to pass a specific band of frequencies, while block all signals outside this band.

• Application include (but certainly not limited to) noise rejection, signal separation, smoothing of digitally generated analog signals, audio signal shaping etc…

• There are four types of filter; low-pass, high-pass, band-pass, and band elimination also known as notch filter.

Basic filter types …

• A low-pass filter is an electronic filter that passes low frequency signals but attenuates (reduces the amplitude of) signals with frequencies higher than the cutoff frequency Fig.1.

• A high-pass filter (HPF) is a device that passes high frequencies and attenuates (i.e., reduces the amplitude of) frequencies lower than its cutoff frequency Fig.1.

Basic filter types…

• A band-pass filter is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range Fig.2.

• A Notch filter is a filter that passes all frequencies except those in a stop band centered on a center frequency Fig.2.

• Notch filters are used to reject unwanted signals for e.g. spikes in sensitive instruments.

Fig.1. Basic Filter types…

Low Pass filter High pass filter

Fig.2.Basic Filter types…

Band pass filter Notch filter

Week 3 and 4

Fourier Series…Fourier Transform…

Time & Frequency domain concept

• Time-domain graph shows how a signal changes over time• In the freq. domain, all the component s (freq. and

amplitude) of a sine wave can be represented by a single vertical line. Similarly, freq. and phase, as shown in Fig. 4

• Frequency domain is a term used to describe the domain for analysis of signals with respect to frequency, rather than time.

• Frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. A frequency-domain representation can also include information on the phase shift.

Time & Frequency domain concept….

• Multiple sine wave called a composite signal, which is used in communication.

• Example• S(t) =1 sine10(10000∏t+ ø)• Where ‘1’ is the amplitude in volts, ’10’ is the no. of

cycles, (angular frequency) ω = 2∏f , so f = 50khz, ‘t’ is the time and ‘ø’ is the phase, in this case it is 0, because sine wave starts at 0 origin.

• Graphical rep. of this equation is shown in fig.4.

Fig. 4 Time & Frequency domain example

10 periods of 1 Volt, 50KHz sine waveAmplitude in the Single Sided DFT in the 50KHz bin is 1.0 V

Time & Frequency domain concept…

• A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called a transform.

• An example is the Fourier transform, which decomposes a function into the sum of a (potentially infinite) number of sine wave frequency components.

• The 'spectrum' of frequency components is the frequency domain representation of the signal. The inverse Fourier transform converts the frequency domain function back to a time function.

Fourier Analysis• Fourier analysis provides us a way to view time

domain signal in frequency domain. Fourier series and Fourier Transform are part of it. The former deal with periodic signal, which can be represented as a sum of sinusoids as shown in fig Fig. 5. Actually, this figure shows the verification of conversion of square wave function into Fourier series. While the latter is related with non-periodic signals. This is shown in Fig.6.

Fourier Analysis …

• Signals like pulse and transient are of practical importance in communications. These signals cannot be analyzed by Fourier series due to mathematical constraints.

• The Fourier Transform decomposes a waveform - basically any real world waveform, into sinusoids. That is, the Fourier Transform gives us another way to represent a waveform.

• The Fourier series is named in honor of Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d Alembert, and Daniel Bernoulli

Fig. 5 Fourier series approximations for a square wave.

Fourier Series

• In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials).

• Therefore, a signal g(t) can be expressed by a trigonometric Fourier series over any interval of duration T0 second as

Trigonometric Fourier Series…

tbtb

tataatg

0201

02010

2sinsin...

...2coscos)(

tnbntnaatgn

n 001

0 sincos)(

Trigonometric Fourier Series…

• Where , for the coefficients a0,an and bn we have

00 /2 T

01

1

01

1

01

0

0

0

0

10

0

sin)(2

,cos)(2

,)(1

Tt

tn

Tt

tn

Tt

t

tdtntgT

b

tdtntgT

a

dttgT

a

Compact Trigonometric Fourier Series

• The trigonometric Fourier series contains sine and cosine of the same frequency. We can combine the two terms in a single term of the same frequency using the trigonometric identity:-

Compact Trigonometric Fourier Series…

)(tan

,

)cos(sincos

1

22

000

n

nn

nnn

nnnn

a

b

baC

where

tnctnbtna

Fourier Transform

• The Fourier transform is a mathematical operation with many applications in physics and engineering that expresses a mathematical function of time as a function of frequency, known as its frequency spectrum; The Fourier Transform is used for non-periodic or Aperiodic signals as shown in Fig.6.

• The function of time is often called the time domain representation, and the frequency spectrum the frequency domain representation. The inverse Fourier transform expresses a frequency domain function in the time domain. Each value of the function is usually expressed as a complex number (called complex amplitude) that can be interpreted as a magnitude and a phase component.

Fourier Transform…Fig. 6;Aperiodic signal

Fourier Transform…

• The representation of non-periodic signals by eternal exponential (everlasting or endless) can be accomplished by a simple limiting process. In other words non-periodic signals can be expressed as a continuous sum (integral) of a signal, which have no duration.

• This is shown in Fig.7a & 7b.

Figure 7a&b; Aperiodic signal and its representation as a continuous sum.

Fourier Transform…

• As shown in figure 7b a new periodic signal gp (t)consisting of signal g(t) repeating itself every T0 sec. This period is made long enough so that there is no overlap between the repeating pulses.

• This new signal gp (t) is a periodic signal and can be represented by Fourier series.

• (Note derivation is given in the B.P. Lathi book)

Fourier Transform…

• The Fourier transform and its inverse of Aperiodic signal in terms of eternal exponential function is given below respectively:-

detgG jwt

)()(

dteGtg jwt

)(2

1)(

Fourier Transform…

• Where G(ω) is the frequency domain representation of the continuous signal g(t), with frequencies lying in the interval (-∞<ω<∞).

• While g(t) is the time domain representation of the G(ω).

Power Spectral Density• Power of the periodic signal is distributed among the

various frequency components.• Power spectral density function (PSD) shows the

strength of the variations(energy) as a function of frequency. In other words, it shows at which frequencies variations are strong and at which frequencies variations are weak.

• The unit of PSD is energy per frequency(width) and you can obtain energy within a specific frequency range by integrating PSD within that frequency range.

Power Spectral Density…

• Computation of PSD is done directly by the method called FFT or computing autocorrelation function and then transforming it.

Power Spectral Density…

• We define the power spectral density (PSD) (ω) as

• Where GT(ω), is signal spectrum and ‘T’ is the time.

dT

G

TS Tg

2|)(|lim)(

Parseval’s Theorem• Signal Energy in time domain can be related to

the signal spectrum:-

dGdtgE tg

22 |)(|2

1)(

Properties of Fourier Transform

• Time-Frequency duality• Symmetry property

– g(t) G(ω)– Then G(t) 2∏g(-ω)

• Scaling property– If g(t) G(ω)– Then g(at)

a

Ga

||

1

Properties of Fourier Transform…

• Reciprocity of signal Duration and its Bandwidth. This suggests that the bandwidth of a signal is inversely proportional to the signal duration or width (in seconds)

• Time shifting property– g(t) G(ω)– g(t-t0)

0)( tjeG

Properties of Fourier Transform…

• Frequency-Shifting Property– g(t) G(ω); )()( 0

0 Getg jwt

Channel & their types with distortions

• The channel is a medium – such as wire, co-axial cable a wave guide, an optical fiber, or a radio link – through which the transmitter output is sent.

• Distortion; spreading or dispersion of the pulse will occur if either the amplitude response or the phase response (linear distortion) or both are not ideal. For example in TDM, pulse spreading causes interference.

Channel & their types with distortions…

• Please note that linear distortion is valid only for small signals. For large amplitudes, non-linearity cannot be ignored for example memory less channel.

Convolution

• The convolution of two signals or functions g(t) and w(t), denoted as follows:-

– g(t) * w(t) =

• Application is in digital filtering

dtwg )()(