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
