Module 2

SPECTRAL ANALYSIS OF COMMUNICATION SIGNAL

Nature and classification of signals

– Signals are physical quantity or variable containing information about the behavior and nature of the phenomenon. For example a picture of a person gives you information regarding whether he is short or tall, fair or black .

The origin of signal is varied and transducers may be required to convert it into an appropriate electrical form, suitable for processing and transmission through the network. Some signals in our daily life are music, speech, and video signals.

Examples of signal include:Electrical signals

– Voltages and currents in a circuitAcoustic signals

– Sound/music over timeMechanical signals

– Velocity of a car over timeVideo signals

– Intensity level of an image or video over time

Classification of signals

Signals can be classified as; – Continuous-time and Discrete-time – Energy and Power – Real and Complex – Periodic and Non-periodic – Analog and Digital – Even and Odd – Deterministic and Random– Causal and Non Causal signals

Continuous time signal and Discrete time signal:A signal is continuous-time, if it is defined for all time,

x(t) and Discrete time if it is defined only at discrete instants of time, x[n].

(a) (b)

(a) Continuous-time signal x(t) (b) Discrete-time signal x[n].

Deterministic and Random Signals : – A deterministic signal is known for all time that there

is no uncertainty with respect to its value i.e. future values can be predicted accurately.

– Random signals exhibit some degree of uncertainty before the signal actually occurs i.e. future values cannot be predicted with complete accuracy.

Periodic and Non-periodic Signals: – A signal can be classified as periodic signal if it

repeats itself after a time interval of T Mathematically satisfies x ( t )= x ( t+T) for all values of t where T is called period of the signal.

A signal which is not periodic is said to be Aperiodic signal. Mathematically it can be defined as a periodic signal with infinite period T = ∞ which signifies that the signal never repeats itself. i.e. x(t) ≠ x(t+T)

Examples of Periodic and Aperiodic signal

a) Square wave with amplitude A = 1 and period T = 0.2s. (b) Rectangular pulse of amplitude A and duration T1.

Even and Odd Signals– A signal is even if x(t)=x(-t) and odd if x(t)=-x(-t)Example: Sin(t) is an odd signal Cos(t) is an even signal. A signal can be even, odd or neither. Any signal can be

written as a combination of an even and odd signal.

Example:Find the even and odd components of the signal

2( ) costx t e t

2

2

( ) cos( )

= cos( )

t

t

x t e t

e t

Even component:2 21

( ) ( cos cos )2

cosh(2 ) cos

t tex t e t e t

t t

Odd component:

2 21( ) ( cos cos ) sinh(2 ) cos

2t t

ox t e t e t t t

Causal and Non Causal SignalsA signal that does not start before t=0 is a causal signal

i.e. x(t)=0, t<0 A signal that starts before t=0 is a non causal signal

x(t)=0 t>0

Energy and Power SignalsA signal is an energy signal if its energy is finite non

zero, i.e.0 <E <∞ while is a power signal if its power is finite non zero, i.e.0 <P <∞.

X(t) is a continuous energy signal if:

X[n] is a discrete energy signal if:

2

0 x t dt

2

0n

x n

X(t) is a continuous power signal if:

X[n] is a discrete power signal if:

• An energy signal has zero power, and a power signal has infinite energy.

21

0 lim2

T

TTx t dt

T

2

10 lim

2 1

N

Nn N

x nN

Periodic signals and random signals are usually power signals while deterministic and aperiodic are usually energy signals.

Example 1:The signal x(t) is given below is energy or power signal. Explain.

x t

3

0 1t

12 2

0

11 1 1 9lim lim 3 lim 9 lim 0

02 2 2 2

T

TT T T TP x t dt dt t

T T T T

Thus this signal is energy signal

Real and Complex Signals

A value of a complex signal x(t) is a complex number

1 2

1 2Where 1 Re Im

x t x t jx t

j x t x t x t x t

The complex conjugate, of the signal x(t) is;

The Magnitude or absolute value

Phase or angle

2 21 2

1

2x t x t x t x t x t

12 1tanx t x t x t

Representation using Basic signals

A fundamental idea in signal processing is to attempt to represent signals in terms of basic signals,

–Complex exponentials:A complex exponential signal is of the form Aejω̥t = A[cos (ωot) +j sin (ωot)] Complex Exponential

Sinusoidal & Exponential Signals– Sinusoids and exponentials are important in signal and

system analysis because they arise naturally in the solutions of the differential equations.

– Sinusoidal Signals can expressed in either of two ways : cyclic frequency form- A sin 2Пfot = A sin(2П/To)t

radian frequency form- A sin ωot

ωo = 2Пfo = 2П/To

To = Time Period of the Sinusoidal Wave

Sinusodial signalx(t) = A sin (2Пfot+ θ)

= A sin (ωot+ θ)

Exponential signal x(t) = Aeat Real Exponential

= Aejω̥t = A[cos (ωot) +j sin (ωot)] Complex Exponential

θ = Phase of sinusoidal wave A = amplitude of a sinusoidal or exponential signal fo = fundamental cyclic frequency of sinusoidal signal ωo = radian frequency

Overview of Fourier Analysis

– Fourier analysis is a tool that changes a time domain signal to a frequency domain signal and vice versa.

– Fourier series is the representation of a signal in the form of linear combination of complex sinusoids

Fourier Series– Every composite periodic signal can be represented

with a series of sine and cosine functions.– The functions are integral harmonics of the

fundamental frequency “f” of the composite signal.– Using the series we can decompose any periodic

signal into its harmonics.

j

eeee

je

je

jjjj

j

j

2sin&

2cos

sinsin& coscos

sincos

sincos

Let the signal x(t) be a periodic signal with period T0.If the following conditions are satisfied

1. x(t) is absolutely integrable over its period

2. The number of maxima and minima of x(t) in each period is finite

3. The number of discontinuous of x(t) in each period is finite then x(t) can be expanded in terms of the complex exponential signal as

dttxT0

0|)(|

– The process of forming a signal X(t) is called synthesis

– while the process of finding the harmonics function xn

of is called Analysis, where n is the harmonic number

– xn = Fourier series coefficients of the signal x(t).

n

tTn

j

nextx 0

2

)(

00

2

0

1( )

nj tT T

nx x t e dtT

– ExampleFind the Fourier coefficient of the signal x(t)=sinwot

– Fourier Transform is a a pair of transform that takes a given signal waveform from time domain to frequency.

– Fourier transform is the extension of Fourier series to periodic and non periodic signals.

– The signal are expressed in terms of complex exponentials of various frequencies.

– The Fundamental period T is considered as infinity in FT

conditions for Fourier Transform:– The signal must be absolutely integrable over a finite

interval of time. – Over a finite interval of time the signal must have

finite number of maxima and minima (orvariations)– Over a finite interval of time, the signal must have

finite number of discontinuities. Also, those discontinuities must be finite.

Convolution Techniques

– When a unit impulse train signal, otherwise called delta function, is fed into time invariant system at time t = 0, the impulse response is h(t).

– A time shift in the input results in a corresponding time shift in the output.

– The impulse function allows us to capture the value of a signal at any point during its existence.

For instance let x(t) and h(t) be two function. The convolution of x and h denoted by x*h is the function on t ≥ 0 given by

The mathematical operation above is called convolution of x(t) and h(t)

The convolution operation is frequently denoted by an asterisk (*)

dthxthtxt

)()()(*)(0

– Convolution technique exploits the superposition principle to model the processes that takes place in a linear system. It also gives an insight into the relationship between the time domain and the frequency domain.

)(*)()()(

)()()(*)(

fHfXthtx

fHfXthtx

– Convolution in time domain becomes multiplication in the frequency domain, while multiplication in the time domain becomes convolution in the frequency domain

Correlation Techniques

– The need to determine the similarity of waveforms is of extreme importance in communication, especially in trying to extract weak signals hidden in noise.

– Correlation techniques are used to measure this similarity.

– If x and h are periodic finite energy waveforms, the cross correlation function becomes

– The reason convolution is preferred to correlation for filtering has to do with how frequency spectra of the two signals interact. Convolving two signals is equivalent to multiplying the frequency spectra of the two signals

dthxthtxt

)()()(*)(0