[9.5 DTFT CONNECTION WITH THE](#page-14-0) CTFT

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9.5 DTFT CONNECTION WITH THE CTFT

Consider a continuous-time signal xc(t) (Fig. 9.17a) with the Fourier transform Xc(ω) bandlimited to B Hz (Fig. 9.17b). This signal is sampled with a sampling interval T. The sampling rate is at least equal to the Nyquist rate; that is, T ≤ 1/2B. The sampled signal xc(t) (Fig. 9.17c) can be

Figure 9.17 Connection between the DTFT and the Fourier transform.

expressed as

$$\bar{x}_c(t) = \sum_{n=-\infty}^{\infty} x_c(nT) \,\delta(t - nT)$$

The continuous-time Fourier transform of the foregoing equation yields

$$\overline{X}_c(\omega) = \sum_{n=-\infty}^{\infty} x_c(nT) e^{-jnT\omega}$$

\n(9.40)

In Sec. 8.1 (Fig. 8.1f), we showed that Xc(ω) is Xc(ω)/T repeating periodically with a period ω*^s* = 2π/T, as illustrated in Fig. 9.17d. Let us construct a discrete-time signal x[n] such that its nth sample value is equal to the value of the nth sample of xc(t), as depicted in Fig. 9.17e, that is,

$$x[n] = x_c(nT) \tag{9.41}$$

Now, X(), the DTFT of x[n], is given by

$$X(\Omega) = \sum_{n=-\infty}^{\infty} x[n] e^{-jn\Omega} = \sum_{n=-\infty}^{\infty} x_c(nT) e^{-jn\Omega}$$

Comparison of this equation with Eq. (9.40) shows that letting ωT = in Xc(ω) yields X(), that is,

$$X(\Omega) = X_c(\omega)|_{\omega T = \Omega}$$

Alternately, X() can be obtained from Xc(ω) by replacing ω with /T, that is,

$$X(\Omega) = \overline{X}_c \left(\frac{\Omega}{T}\right) \tag{9.42}$$

Therefore, X() is identical to Xc(ω), frequency-scaled by factor T, as shown in Fig. 9.17f. Thus, ω = 2π/T in Fig. 9.17d corresponds to = 2π in Fig. 9.17f.

9.5-1 Use of DFT and FFT for Numerical Computation of the DTFT

The discrete Fourier transform (DFT), as discussed in Ch. 8, is a tool for computing the samples of the continuous-time Fourier transform (CTFT). Because of the close connection between CTFT and DTFT, as seen in Eq. (9.42), we can also use this same DFT to compute DTFT samples.

In Ch. 8, Eqs. (8.12) and (8.13) relate an N0-point sequence xn to another N0-point sequence Xr. Changing the notation xn to x[n] in these equations, we obtain

$$X_r = \sum_{n=0}^{N_0 - 1} x[n] e^{-j r \Omega_0 n}$$

\n(9.43)

and

$$x[n] = \frac{1}{N_0} \sum_{r=0}^{N_0 - 1} X_r e^{jr\Omega_0 n}$$

\n(9.44)

where ^{0 = 2π} N0 . Comparing Eq. (9.19) with Eq. (9.43), we recognize that Xr is the sample of X() at = r0, that is,

$$X_r = X(r\Omega_0) \qquad \Omega_0 = \frac{2\pi}{N_0}$$

Hence, DFT Eqs. (9.43) and (9.44) can be viewed to relate an N0-point sequence x[n] to the N0-point samples of corresponding X(). We can now use the efficient algorithm FFT (discussed in Ch. 8) to compute Xr from x[n], and vice versa.

If x[n] is not timelimited, we can still find the approximate values of Xr by suitably windowing x[n]. To reduce the error, the window should be tapered and should have sufficient width to satisfy error specifications. In practice, the numerical computation of signals, which are generally non-timelimited, is performed in this manner because of the computational economy of the DFT, especially for signals of long duration.

COMPUTATION OF DISCRETE-TIME FOURIER SERIES (DTFS)

The discrete-time Fourier series (DTFS) equations [(9.3) and (9.4)] are identical to the DFT equations [(8.13) and (8.12)] within a scaling constant N0. If we let x[n] = N0xn and D^r = Xr

886 CHAPTER 9 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS

in Eqs. (9.4) and (9.3), we obtain

$$X_r = \sum_{n=0}^{N_0 - 1} x_n e^{-jr\Omega_0 n} \qquad \text{and} \qquad x_n = \frac{1}{N_0} \sum_{r=0}^{N_0 - 1} X_r e^{jr\Omega_0 n}$$

This is precisely the DFT and IDFT of Eqs. (8.12) and (8.13). For instance, to compute the DTFS for the periodic signal in Fig. 9.2a, we use the values of xn = x[n]/N⁰ as

$$x_n = \begin{cases} \frac{1}{32} & 0 \le n \le 4 \quad \text{and} \quad 28 \le n \le 31\\ 0 & 5 \le n \le 27 \end{cases}$$

Numerical computations in modern digital signal processing are conveniently performed with the discrete Fourier transform, introduced in Sec. 8.5. The DFT computations can be very efficiently executed by using the fast Fourier transform (FFT) algorithm discussed in Sec. 8.6. The DFT is indeed the workhorse of modern digital signal processing. The discrete-time Fourier transform (DTFT) and the inverse discrete-time Fourier transform (IDTFT) can be computed by using the DFT. For an N0-point signal x[n], its DFT yields exactly N⁰ samples of X() at frequency intervals of 2π/N0. We can obtain a larger number of samples of X() by padding a sufficient number of zero-valued samples to x[n]. The N0-point DFT of x[n] gives exact values of the DTFT samples if x[n] has a finite length N0. If the length of x[n] is infinite, we need to use the appropriate window function to truncate x[n].

Because of the convolution property, we can use the DFT to compute the convolution of two signals x[n] and h[n], as discussed in Sec. 8.5. This procedure, known as fast convolution, requires padding both signals by a suitable number of zeros, to make the linear convolution of the two signals identical to the circular (or periodic) convolution of the padded signals. Large blocks of data may be processed by sectioning the data into smaller blocks and processing such smaller blocks in sequence. Such a procedure requires smaller memory and reduces the processing time [1].