Signals and Systems - Electrical Engineering

628 C H A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems

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% Example 10.23---Filtering using convolution and FFT

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clear all; clf

N = 100; n = 0:N - 1;

load laughter

x = 5∗y(1:N)’; x1 = x + 0.3∗(-1).ˆn; % desired signal plus disturbance

h = fir1(40,0.9); [H,w] = freqz(h,1); % low-pass FIR filter design

% filtering using convolution

y = conv(x,h); % convolution of signal and impulse response of FIR

% computing using FFT

M = length(x) + length(h) - 1; % circular and linear convolutions equal

X = fft(x,M);

H = fft(h,M);

Y = X.∗H;

y1 = ifft(Y); % output of filtering n

10.5 What Have We Accomplished? Where Do We Go from Here?....................

In this chapter we have considered the Fourier representation of discrete-time signals and systems.

Just as with the Laplace and the Fourier transforms, in the continuous case there is a large class of

discrete-time signals and impulse responses of systems for which we are able to find their discrete-

time Fourier transform from their Z-transforms. For signals that are not absolutely summable, the

time-frequency duality and other properties of the transform are used to find their DTFTs. Properties

of the DTFT are very similar to those of the Z-transform. Although theoretically useful, the DTFT is

computationally not feasible, due to the continuity of the frequency variable and to the integration

required in the inverse transformation. It is the Fourier series of discrete-time signals that makes

the Fourier representation computationally feasible. In Table 10.1, the DTFT of common signals and

some DTFT properties are given.

The Fourier series coefficients constitute a periodic sequence of the same period as the signal; thus

both are periodic. Moreover, the Fourier series and its coefficients are obtained as sums, and the

frequency used is discretized. Thus, they can be obtained by computer. To take advantage of this,

the spectrum of an aperiodic signal resulting from the DTFT is sampled so that in the time domain

there is a periodic repetition of the original signal. For finite-support signals we can then obtain a

periodic extension that gives the discrete Fourier transform or DFT. The significance of this result is

that we have frequency representations of discrete-time signals that are computed algorithmically.

Table 10.2 displays properties of the discrete Fourier series and of the Discrete Fourier Transform

(DFT). What remains then is to take a look at the algorithm used for those computations or the

fast Fourier transform (FFT). We will do that in Chapter 12, where we will show that this algorithm

efficiently computes the DFT and makes the convolution sum a more feasible procedure.