Chapter 2. Fourier Representation of Signals

and Systems

2.6 Transmission of Signal Through Linear Systems : Convolution Revisited• In a linear system,

– The response of a linear system to a number of excitations applied simultaneously is equal to the sum of the responses of the system when each excitation is applied individually.

• Time Response– Impulse response

• The response of the system to a unit impulse or delta function applied to the input of the system.

– Summing the various infinitesimal responses due to the various input pulses,• Convolution integral • The present value of the response of a linear time-invariant

system is a weighted integral over the past history of the input signal, weighted according to the impulse response of the system

)93.2()()()( dthxty

)94.2()()()( dtxhty

2.6 Transmission of Signal Through Linear Systems : Convolution Revisited• Causality and Stability

– Causality : It does not respond before the excitation is applied

– Stability• The output signal is bounded for all bounded input signals (BIBO)

• An LTI system to be stable– The impulse response h(t) must be absolutely integrable– The necessary and sufficient condition for BIBO stability of a linear

time-invariant system)100.2()(

dtth

dhM

dtxhdtxh

)(

)()()()(

)98.2(0,0)( tth

tMtx allfor )(

)99.2()()()( dtxhty

dhM

)(y(t)

2.6 Transmission of Signal Through Linear Systems : Convolution Revisited• Frequency Response

– Impulse response of linear time-invariant system h(t),– Input and output signal

– By convolution theorem(property 12),

– The Fourier transform of the output is equal to the product of the frequency response of the system and the Fourier transform of the input• The response y(t) of a linear time-invariant system of impulse response

h(t) to an arbitrary input x(t) is obtained by convolving x(t) with h(t), in accordance with Eq. (2.93)

• The convolution of time functions is transformed into the multiplication of their Fourier transforms

)(*)(

)()()(

txth

dtxhty

)109.2()()()( fXfHfY

2.6 Transmission of Signal Through Linear Systems : Convolution Revisited

– In some applications it is preferable to work with the logarithm of H(f)

)110.2()](exp[)()( fjfHfH

)()( fHfH

Amplitude response or magnitude response Phase or phase response

)()( ff

)111.2()()()(ln fjffH

)112.2()(ln)( fHf

)113.2()(log20)( 10' fHf The gain in decible [dB]

)114.2()(69.8)(' ff

2.6 Transmission of Signal Through Linear Systems : Convolution Revisited• Paley-Wiener

Criterion– The frequency-domain

equivalent of the causality requirement

)115.2(1

)(2

df

ff

2.7 Ideal Low-Pass Filters

• Filter– A frequency-selective system that is used to limit the spectrum

of a signal to some specified band of frequencies• The frequency response of an ideal low-pass filter

condition– The amplitude response of the filter is a constant inside the

passband -B≤f ≤B– The phase response varies linearly with frequency inside the

pass band of the filter

)116.2(f ,0

),2exp()( 0

BBfBftj

fH

2.7 Ideal Low-Pass Filters

– Evaluating the inverse Fourier transform of the transfer function of Eq. (2.116)

– We are able to build a causal filter that approximates an ideal low-pass filter, • with the approximation

improving with increasing delay t

B

Bdfttfjth )117.2()](2exp[)( 0

(2.118))]t-c[2B(tsin2 )(

)](2sin[)(

0

0

0

Btt

ttBjth

0for ,1)]t-c[2B(tsin 0 t

2.7 Ideal Low-Pass Filters

– Gibbs phenomenon

2.8 Correlation and Spectral Density : Energy Signals• The autocorrelation function of an energy signal x(t) is defined

as2*( ) ( ) ( ) , (0) ( )x xR x t x t dt R x t dt

2.8 Correlation and Spectral Density : Energy Signals• Energy spectral density

– The energy spectral density is a nonnegative real-valued quantity for all f, even though the signal x(t) may itself be complex valued.

• Wiener-Khitchine Relations for Energy Signals– The autocorrelation function and energy spectral density form a

Fourier-transform pair.

2

x f X f

( ) ( ) exp( 2 )

( ) ( ) exp( 2 )

x x

x x

f R j f d

R f j f df

2.8 Correlation and Spectral Density : Energy Signals• Cross-Correlation of Energy Signals

– The cross-correlation function of the pair

– The energy signals x(t) and y(t) are said to be orthogonal over the entire time domain• If Rxy(0) is zero

– The second cross-correlation function

)139.2()()()( * dttytxRxy

)140.2(0)()( *

dttytx

)141.2()()()( * dttxtyRyx

)142.2()()( * yxxy RR

2.8 Correlation and Spectral Density : Energy Signals

– The respective Fourier transforms of the cross-correlation functions Rxy(τ) and Ryx(τ)

– With the correlation theorem

– The properties of the cross-spectral density1. Unlike the energy spectral density, cross-spectral density is

complex valued in general.2. Ψxy(f)= Ψ*yx(f) from which it follows that, in general, Ψxy(f)≠

Ψyx(f)

)143.2()2exp()()( dfjRf xyxy

)144.2()2exp()()( dfjRf yxyx

)145.2()()()( * fYfXfxy

)146.2()()()( * fXfYfyx

2.9 Power Spectral Density

– The average power of a signal is

• Power signal : • Truncated version of the signal x(t)

• By Rayreigh energy theorem

T

TTdttx

TP )147.2()(

21lim 2

P

(2.148) otherwise ,0),(

2rect)()(

TtTtxTttxtxT

)150.2()(

21lim 2 dffXT

P TT

)149.2()(

21lim 2 dttxT

P TT

)151.2()(

21lim 2 dffXT

P TT

)152.2()(21lim)( 2fXT

fS TTx

)153.2()( dffSP x

Power spectral density

Summary

• Fourier Transform – A fundamental tool for relating the time-domain and frequency-

domain descriptions of a deterministic signal• Inverse relationship

– Time-bandwidth product of a energy signal is a constant• Linear filtering

– Convolution of the input signal with the impulse response of the filter– Multiplication of the Fourier transform of the input signal by the

transfer function of the filter• Correlation

– Autocorrelation : a measure of similarity between a signal and a delayed version of itself

– Cross-correlation : when the measure of similarity involves a pair of different signals

• Spectral Density– The Fourier transform of the autocorrelation function

• Cross-Spectral Density– The Fourier transform of the cross-correlation function