DFT for Noncausal or Infinite-Duration Sequence

3.4.3 DFT for Noncausal or Infinite-Duration Sequence

Let us consider the DFT formula (3.4.2):

N −1

for k = 0 : N − 1 This is defined for a causal, finite-duration sequence x[n] so that we can cast

X (k) = DFT N {x[n]} =

x [n]e − j2πkn/N

n=0

it into a computer program without taking much heed of the negative indices. (C language prohibits using the negative indices for array and MATLAB does not accept even zero index.) Then, how can we get the DFT spectrum of noncausal or infinite-duration sequences? The answer is as follows:

156 3 Discrete-Time Fourier Analysis - For a noncausal sequence, append the noncausal part to the end of the sequence

after any necessary zero-padding is done so that the resulting sequence can be causal.

- For an infinite-duration sequence, set the time-duration [0, N − 1] so that the

most significant part of the sequence can be covered. To increase the DFT size is also helpful in making the DFT close to the DTFT. Compare Fig. 3.11(b2)/(c2) with (b3)/(c3). Also compare Fig. 3.12.1(b1) with Fig. 3.12.2(b1).

Example 3.14 DFT of a Non-causal Pulse Sequence Consider a sequence which is depicted in Fig. 3.11(a1). Shifting the noncausal part x[−1] = 1/3 into the end of the sequence yields a causal sequence

where N is the DFT size. This corresponds to one period of the periodic extension ˜x N [n] (see Fig. 3.11(a2) with N = 4 and (a3) with N = 8).

Then we can compute the N -point DFT of x N [n] as

X N (k) = x N [n]e

− j2πkn/N

= e − j2πk0/N

e − j2πk1/N

3 + 3 + e 3 − j2πk(N −1)/N

n=0

1 1 j 2π k/N =

− j2πk/N +e − j2πk(N −1)/N

− j2πk/N

3 3 +e

1 = 3 (1 + cos(2πk/N )) for k = 0 : N − 1

–1 0 1 4 789 n (a1) A noncausal sequence

(a2) Periodic extension with N = 4 (a3) Periodic extension with N = 8

and its one period

and its one period

1 8-point DTFT magnitude

1 1 4-point

DFT magnitude X(Ω)

DFT magnitude

X 4 (k)

X 8 (k)

0 01 23 4 5 6 7 k (b1) The DTFT magnitude spectrum

(b2) The DFT magnitude spectrum

(b3) The DFT magnitude spectrum

– DTFT phase ∠X(Ω) π

8-point DFT phase ∠X 8 (k) (c1) The DTFT phase spectrum

4-point DFT phase ∠X

4 (k)

(c2) The DFT phase spectrum

(c3) The DFT phase spectrum

Fig. 3.11 DTFT and DFT for a noncausal sequence

3.4 Discrete Fourier Transform (DFT) 157 Figure 3.11(b2)/(c2) and (b3)/(c3) show the magnitude/phase of this DFT with

N = 4 and those with N = 8, respectively. Note the following: - The overlapped DTFT spectra in dotted lines, obtained in Example 3.1 and plot-

ted in Figs. 3.1 and 3.11(b1)/(c1), illustrate the fact that the DFT X N (k) are just the samples of the DTFT X (Ω) at Ω = kΩ 0 = 2πk/N for 0 ≤ k ≤ N − 1 as

long as the whole duration of the sequence is covered by [0 : N − 1]. - Figure 3.11(a2) and (a3) differ in the length of zero padding performed before

appending the noncausal part to the end of the sequence. Comparing the corre- sponding DFT spectra in Figure 3.11(b2)/(c2) and (b3)/(c3), we see that longer zero-padding increases the DFT size and thus decreases the digital resolution

frequency Ω 0 = 2π/N so that the DFT looks closer to the DTFT.

Example 3.15 DFT of an Infinite-Duration Sequence Consider a real exponential sequence of infinite duration described by

(E3.15.1) This sequence is shown partially for n = 0 : 7 in Fig. 3.12.1(a1) and for n = 0 : 15

[n] = a n u

s [n] ( |a| < 1 )

in Fig. 3.12.2(a1).

(a) The DTFT of this infinite-duration sequence was obtained in Example 3.2 as

1−ae −jΩ

Thus we get the samples of the DTFT at Ω = kΩ 0 = 2πk/N as

: DTFT magnitude X(Ω) 4 : DFT magnitude X

0 0 4 7 k (a1) A part of an infinite-duration sequence (b1) The 8-point DFT X 8 (k ) and the DTFT X(Ω)

1 x ~ 8 [n]

8-point

N = 8-point samples of DTFT

0 0 π 2π Ω (a2) Periodic extension of x [n ] with period N = 8 (b2) Samples of the DTFT X(Ω) at Ω = 2πk /N

Fig. 3.12.1 Relationship between the 8-point DFT and the DTFT for an infinite-duration sequence

158 3 Discrete-Time Fourier Analysis

: DTFT magnitude X(Ω)

16-point

: DFT magnitude X 16 (k )

1 x [0:15]

DFT

0 0 8 15 k (a1) A part of an infinite-duration sequence

0 0 4 8 15 n

(b1) The 16-point DFT X 16 (k ) and the DTFT X (Ω)

1 ~ x 16-point 16 [n ] N = 16-point samples of DTFT

IDFT 2 π

0 0 8 16 n

0 0 π Ω 2π (a2) Periodic extension of x [ n ] with period N = 16 (b2) Samples of the DTFT X (Ω) at Ω = 2 π k /N

Fig. 3.12.2 Relationship between the 16-point DFT and the DTFT for an infinite-duration sequence

1 − ae − j2πk/N

k with W N =e − j2π/N (E3.15.3)

1 − aW N

(b) We can use the DFT formula (3.4.2) to find the N -point DFT of x[n] for n =

0 : N − 1:

X N −1 (k) = x [n]W nk

n=0

(E3.15.1) N −1

1 − aW N (E3.15.4)

Comparing this with (E3.15.3) reveals the following: - The DFT and the DTFT samples are not exactly the same for an infinite-

duration sequence, while they conform with each other for a finite-duration sequence whose duration can somehow be covered in [0 : N − 1]

- Larger DFT size N will make the DFT closer to the sampled DTFT. This can

also be seen visually by comparing Figs. 3.12.1(b1) and 3.12.2(b1). (c) One might wonder what the time-domain relationship between the DFT X (k)

and the sampled DTFT X (kΩ 0 ) is for the infinite-duration sequence. This

curiosity seduces us to find the IDFT of X (kΩ 0 ) as

3.4 Discrete Fourier Transform (DFT) 159

(3.4.3) 1 N −1

x N [n] = IDFT{X(kΩ 0 )} =

(E3.15.3) 1 N −1

[n] = a u s [n] (E3.15.5) 1−a

N →∞

|a|<1

This corresponds to one period of the periodic extension ˜x N [n] of x[n] with period N and it becomes closer to x[n] as the DFT size N increases. Note the following:

- Just as the sampling of continuous signal x(t ) in the time domain results in the

periodic extension of X (ω) = F{x(t)} in the frequency domain (Eq. (E2.13.3)), so the sampling of continuous spectrum X (ω) or X (Ω) in the frequency domain results in the periodic extension ˜x P (t ) (of x(t ) = F −1 {X(ω)}) or ˜x N [n] (of

x [n] = F −1 {X(Ω)}) in the time domain. - Besides, just as shorter time-domain sampling interval (T ) in the sampling of x (t ) to make x[n] = x(nT ) increases the frequency band on which the CTFT

X (ω) = F{x(t)} is close to the DTFT X(Ω) = F{x[n]}, so narrower frequency- domain sampling interval (ω 0 /Ω 0 ) in the sampling of X (ω)/ X (Ω) to make

X (kω 0 )/ X (kΩ 0 ) expands the time range on which ˜x P (t ) (with P = 2π/ω 0 ) or

˜x N [n] (with N = 2π/Ω 0 ) is close to x(t ) or x[n].

- However short the time-domain sampling interval in the sampling of x(t ) to make x [n] may be, X (Ω) = F{x[n]} for −π ≤ Ω ≤ π cannot be exactly the same as

X (ω) = F{x(t)} for −π/T ≤ ω ≤ π/T due to the frequency-aliasing unless

X (ω) is strictly bandlimited. Likewise, however narrow the frequency-domain sampling interval in the sampling of X (ω)/ X (Ω) to make X (kω 0 )/ X (kΩ 0 ) may

be, the corresponding periodic extension ˜x P (t )/ ˜x N [n] cannot be exactly the same as x(t )/x[n] for one period of length P/N due to the time-aliasing unless

x (t )/x[n] is strictly time-limited.

Remark 3.6 The DFT for Noncausal/Infinite-Duration Sequences The DFT pair (3.4.2) and (3.4.3) can be used to analyze the frequency character- istic of causal, finite-duration sequences. Then, how do we deal with noncausal or infinite-duration sequences?

(1) For a noncausal sequence, append the noncausal part to the end of the sequence after any necessary zero-padding is done so that the resulting sequence can be causal.

(2) For an infinite-duration sequence, set the time-duration [0, N − 1] so that the most significant part of the sequence can be covered. If the duration is shifted,

160 3 Discrete-Time Fourier Analysis apply the time-shifting property to the DFT for obtaining the right phase spec-

trum. You can also increase the DFT size to make the DFT close to the DTFT, which accommodates infinite-duration sequences.