Signals and Systems - Electrical Engineering

10.3 Fourier Series of Discrete-Time Periodic Signals 605

then according to the eigenfunction property of LTI systems, the output is also periodic of periodNwith

Fourier series

y[n]=

N∑− 1

k= 0

X[k]H(ejkω^0 )ejkω^0 n ω 0 =

2 π

N

fundamental frequency

and coefficientsY[k]=X[k]H(ejkω^0 ). That is, it is affected by the frequency response of the system

H(ejkω^0 )=H(z)

∣∣

∣z=ejkω 0

at the harmonic frequencies{kω 0 ,k=0,...,N− 1 }.

Remarks

n Although the input x[n]and the output y[n]of the LTI system are both periodic of the same period, the
Fourier series coefficients of the output are affected by the frequency response H(ejkω^0 )of the system at
each of the harmonic frequencies.
n A similar result is obtained by using the convolution property of the DTFT, so that if X(ejω)is the DTFT
of the periodic signal x[n], then the DTFT of the output y[n]is given by

Y(ejω)=X(ejω)H(ejω)

=

[N− 1

∑

k= 0

2 πX[k]δ(ω− 2 πk/N)

]

H(ejω)

=

N∑− 1

k= 0

2 πX[k]H(ej^2 πk/N)δ(ω− 2 πk/N)

and letting X[k]H(ej^2 πk/N)=Y[k], we get the DTFT of the periodic output y[n].

nExample 10.16

Consider how to implement a crude spectral analyzer for discrete-time signals using MATLAB.

Divide the discrete frequencies [0π] in three bands: [0 0.1π],(0.1π0.6π], and(0.6π π], to obtain

low-pass, band-pass, and high-pass components of the signalx[n]=sign(cos(0.2πn)). Use the

MATLAB functionfir1to design the three filters. Plot the original signal and its components in

the three bands. Verify that the overall filter is an all-pass filter. Obtain the sum of the outputs of

the filters, and explain how it relates to the original signal.

Solution

The script for this example uses several MATLAB functions that facilitate the filtering of signals, and

for which you can get more information by usinghelp.

The DTFT ofx[n] is found using the FFT algorithm as indicated before. The low-pass, band-

pass, and high-pass filters are obtained usingfir1, and the functionfilterallows us to obtain the