Discrete Fourier Transform 623

10.4 Discrete Fourier Transform 623

f (Hz)

f (Hz)

(a) 4

FIGURE 10.15

Computation of the

nT

FFT of a periodic

signal using (a) 4 and

(b) 12 periods to

improve the 0 0.02 0.04 0.06 0.08 0.1 0.12 frequency resolution

of the FFT. Notice

that both magnitude

and phase

responses look alike, ( f)| 15 ( but when we use 12 f)

|X

periods these

spectra look sharper

due to the increase in the number of

f (Hz) components added.

f (Hz)

(b)

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

Given x[n] and h[n] of lengths M and K, the linear convolution sum y[n] of length N = M + K − 1 can be found by following these three steps:

Compute DFTs X[k] and H[k] of length L ≥ N for x[n] and h[n].

Multiply them to get Y[k] = X[k]H[k].

Find the inverse DFT of Y[k] of length L to obtain y[n]. Although it seems computationally more expensive than performing the direct computation of the convolution sum, the above approach implemented with the FFT can be shown to be much more efficient.

The above procedure could be implemented by a circular convolution sum in the time domain, although in practice it is not done due to the efficiency of the implementation with FFTs. A circu- lar convolution uses circular rather than linear representation of the signals being convolved. The periodic convolution sum introduced before is a circular convolution of fixed length—the period of the signals being convolved. When we use the DFT to compute the response of an LTI system the length of the circular convolution is given by the possible length of the linear convolution sum. Thus, if the system input is a finite sequence x[n] of length M and the impulse response of the system h[n] has

a length K, then the output y[n] is given by a linear convolution of length M + K − 1. The length L ≥ M + K − 1 of the DFT Y[k] = X[k]H[k] corresponds to a circular convolution of length L of the x[n] and h[n] padded with zeros so that both have length L. In such a case the circular and the linear convolutions coincide.

If x[n] of length M is the input of an LTI system with impulse response h[n] of length K, then

(10.55) where X[k], H[k], and Y[k] are, respectively, DFTs of length L of the input, the impulse response, and the

Y[k] = X[k]H[k] ⇔ y[n] = (x ⊗ L h)[n]

output of the LTI system, and ⊗ L stands for the circular convolution of length L. If L is chosen so that L ≥ M + K − 1, the circular and the linear convolution sums coincide—that is,

(10.56) Remark If we consider the periodic expansions of x[n] and h[n] with period L = M + K − 1, we can use

y[n] = (x ⊗ L h)[n] = (x ∗ h)[n]

their circular representations and implement the circular convolution as shown in Figure 10.16. Since the length of the linear convolution or convolution sum, M + K − 1, coincides with the length of the circular convolution, the two convolutions coincide. Given the efficiency of the FFT algorithm in computing the DFT, the convolution is typically done using the DFT as indicated above.

■ Example 10.22

To illustrate the connection between the circular and the linear convolution, compute using MAT- LAB the circular convolution of a pulse signal x[n] = u[n] − u[n − 21] of length N = 20 with itself for different values of its length. Determine the length for which the circular convolution coincides with the linear convolution of x[n] with itself.