Properties of the Discrete-Time Fourier Transform

3.2 Properties of the Discrete-Time Fourier Transform

As with the continuous-time Fourier transform (CTFT), there are many properties of DTFT that provide us with further insight into the transform and can be used for reducing the computational complexity associated with it. Noting that there are striking similarities with the case of CTFT, we will simply state the properties unless their derivations and interpretations differ from those with the continuous-time case.

3.2.1 Periodicity

Since the DTFT X (Ω) defined by Eq. (3.1.1) is a function of e jΩ , it is always periodic with period 2π in Ω:

(3.2.1) (cf.) The periodicity lets us pay attention to the DTFT only for its one period, say,

X (Ω) = X[e jΩ ] = X(Ω + 2mπ) for any integer m

3.2.2 Linearity

With F{x[n]} = X(Ω) and F{y[n]} = Y (Ω), we have

(3.2.2) which implies that the DTFT of a linear combination of many sequences is the same

ax F [n] + b y[n] ↔ a X(Ω) + b Y (Ω ),

linear combination of the individual DTFTs.

3.2.3 (Conjugate) Symmetry

In general, the DTFT has the time reversal property:

F{ x[−n] } (3.1.1) = x [−n] e − j Ωn −n=m x [m] e − j (−Ω) m

= X(−Ω) x F [−n] ↔ X(− Ω )

n=−∞

m=−∞

3.2 Properties of the Discrete-Time Fourier Transform 139 In case x[n] is a real-valued sequence, we have

n=−∞ (2.2.1a)

=X ∗ (Ω) (complex conjugate of X (Ω)) or equivalently,

x [n] e −(− j) Ωn

n=−∞

Re{X(−Ω)} + j Im{X(−Ω)} = Re{X(Ω)} − j Im{X(Ω)};

(3.2.4) This implies that the magnitude/phase of the DTFT of a real-valued sequence is an

|X(−Ω)|∠X(−Ω) = |X(Ω)|∠ − X(Ω)

even/odd function of frequency Ω. Also in analogy with Eq. (2.5.5) for the CTFT, we have

F even and real-valued x e [n] ↔ Re{X( Ω)} even and real-valued (3.2.5a)

odd and real-valued x o [n] ↔ j Im{X(Ω)} odd and imaginary-valued (3.2.5b)

3.2.4 Time/Frequency Shifting (Real/Complex Translation)

The DTFT has the time-shifting and frequency-shifting properties as x F [n − n

1 ] ↔ X(Ω ) e − j Ωn 1 = X(Ω )∠ − n 1 Ω (3.2.6)

x [n]e jΩ 1 n F ↔ X( Ω − Ω 1 )

3.2.5 Real Convolution

The DTFT has the convolution property

(3.2.8) which can be derived in the same way with Eq. (2.5.11). This is very useful for

y F [n] = x[n] ∗ g[n] ↔ Y (Ω) = X(Ω) G(Ω)

describing the input-output relationship of a discrete-time LTI system with the input x [n], the output y[n], and the impulse response g[n] where G(Ω) = F{g[n]} is called the frequency response of the system.

3.2.6 Complex Convolution (Modulation/Windowing)

In analogy with Eq. (2.5.14) for the CTFT, the DTFT also has the complex convo- lution (modulation) property as

2π ∗ where

y [n] = x[n] m[n] ↔ Y (Ω) =

X (Ω) M (Ω) (3.2.9)

2π ∗ denotes a circular or periodic convolution with period 2π .

140 3 Discrete-Time Fourier Analysis Example 3.9 Effect of Rectangular Windowing on the DTFT of a Cosine Wave

From Eqs. (E3.7.2) and (E3.1.2), we can write the DTFTs of a cosine wave x [n] = cos(Ω 0 n ) and an even rectangular pulse sequence r ′ 2M+1 [n] of duration

2M + 1 as

X (Ω) = DTFT{cos( Ω 0 n )}

i =−∞ (δ(Ω + Ω 0 − 2π i) + δ(Ω − Ω 0 − 2π i)) (E3.9.1) R ′

(E3.7.2)

(E3.1.2) sin(Ω(2M + 1)/2) 2M+1 (Ω) = DTFT{r 2M+1 [n]} =

(E3.9.2)

sin(Ω/2)

We can use the complex convolution property (3.2.9) to find the DTFT of a rectangular-windowed cosine wave y[n] = cos(Ω 0 n )r ′ 2M+1 [n] as

2π ∗ R 2M+1 (Ω) (E3.9.1) 1 ∞

Y (Ω) = DTFT{y[n]} = DTFT{cos( Ω 0 n )r ′

2M+1 [n]} =

X (Ω) 2π

i =−∞ (δ(Ω + Ω 0 − 2π i) + δ(Ω − Ω 0 − 2π i)) ∗ R ′

(R ′ (Ω + Ω 0 − 2π i) + R 2 ′ 2M+1 2M+1 (Ω − Ω 0 − 2π i))

(E3.9.2) 1 0 − 2π i)(2M + 1)/2)

sin((Ω + Ω 0 − 2π i)/2) sin((Ω − Ω 0 − 2π i)(2M + 1)/2)

+ (E3.9.3)

sin((Ω − Ω 0 − 2π i)/2)

which is depicted together with X (Ω) (E3.9.1) and R ′ 2M+1 (Ω) (E3.9.2) in Fig. 3.5. Compared with the spectrum X (Ω) of the cosine wave (Fig. 3.5(b1)), the spectrum Y (Ω) of the rectangular-windowed cosine wave (Fig. 3.5(b3)) has many side lobe ripples (with low amplitude) besides the two main peaks, which is interpreted as the spectral leakage due to the rectangular windowing.

Example 3.10 Impulse Response and Frequency Response of an FIR LPF (Lowpass Filter)

We can use Eq. (E3.5.2) to write the impulse response of an ideal LPF with bandwidth B = π/4 as

(E3.5.2) sin(Bn)

4 4 which has an infinite duration so that it cannot be implemented by an FIR filter.

B=π/4

Thus, to implement it with an FIR filter, we need to truncate the impulse response,

3.2 Properties of the Discrete-Time Fourier Transform 141

x [n] = cos(Ω 0 n)

Time

Frequency domain domain

(a1) A discrete–time cosine signal x [n]

(b1) The DTFT spectrum X (Ω) of x[n]

(a2) A discrete–time rectangular signal r 2M + 1 ' [n] II II –2 π

(b2) The DTFT spectrum R ' 2M + 1 [Ω] of r ' 2M + 1 [n] y [n] = cos(Ω 0 n)r 2M + 1 ' [n]

0 π 2⎯ Ω (a3) A rectangular–windowed cosine signal y [n]

Fig. 3.5 Effect of rectangular windowing on the DTFT spectrum

say, by multiplying the rectangular window

whose DTFT is

(E3.1.2) sin(7Ω/2) W (Ω) =

(E3.10.3)

sin(Ω/2)

Then we obtain the windowed impulse response of an FIR LPF as

(E3.10.4) whose DTFT is

g w [n] = g[n] w[n]

G w (Ω)

G (Ω) ∗ W (Ω) (E3.10.5) complex convolution property 2π

This is the frequency response of the FIR LPF. Figure 3.6 shows the impulse and frequency responses of the ideal and FIR LPFs together with the rectangular window sequence and its DTFT spectrum.

(cf.) Note that the frequency response of the FIR LPF (Fig. 3.6(b3)) has smooth transition in contrast with the sharp transition of the frequency response of the ideal LPF (Fig. 3.6(b1)).

We can run the following program “sig03e10.m” to plot Fig. 3.6 and compare the two DTFTs of the windowed impulse response g 7 [n], one obtained by using the DTFT formula (3.1.1) or the MATLAB routine “ DTFT()” and one obtained by

142 3 Discrete-Time Fourier Analysis

0.25 1 g [n]

Time

Frequency domain domain

G (Ω) B = π/4

–π 0 π/4 π 2π Ω (a1) Ideal LPF impulse response

F (b1) Ideal LPF frequency response

Ω (a2) Rectangular window sequence

(b2) Spectrum of rectangular window 0.25 F 1

–π 0 π 2π Ω (a3) FIR LPF impulse response

(b3) FIR LPF frequency response

Fig. 3.6 Effect of windowing on the spectrum or frequency response

using the circular convolution (E3.10.5) or the MATLAB routine “ conv circular()”.

%sig03e10.m % Fig. 3.6: Windowing Effect on Impulse/Frequency Response of an ideal LPF clear, clf n= [-20: 20]; g= sinc(n/4)/4; N=400; f=[-400:400]/N +1e-10; W=2 ∗ pi ∗ f; global P D D=pi/2; P=2 ∗ pi; G=rectangular wave(W); subplot(321), stem(n,g,’.’), subplot(322), plot(f,G) M=3; wdw= [zeros(1,20-M) ones(1,2 ∗ M+1) zeros(1,20-M)]; Wdw= DTFT(wdw,W,-20); % DTFT Wdw(W) of wdw[n] - Appendix E for DTFT Wdw t= sin(W ∗ (2 ∗ M+1)/2)./sin(W/2); % Eq.(E3.1.2) discrepancy between DTFT and E3 1 2 = norm(Wdw-Wdw t) subplot(323), stem(n,wdw,’.’), subplot(324), plot(f,real(Wdw)) gw= g. ∗ wdw; Gw= DTFT(gw,W,n(1)); % DTFT spectrum Gw(W) of gw[n] Gw 1P= conv circular(G,Wdw,N); discrepancy between circonv and DTFT= norm(Gw(1:N)-Gw 1P)/norm(Gw 1P) subplot(325), stem(n,gw,’.’) subplot(326), plot(f,real(Gw)), hold on plot(f(1:N),real(Gw 1P),’r’)

function z=conv circular(x,y,N) % Circular convolution z(n)= (1/N) sum m=0ˆN-1 x(m) ∗ y(n-m) if nargin<3, N=min(length(x),length(y)); end x=x(1:N); y=y(1:N); y circulated= fliplr(y); for n=1:N

y circulated= [y circulated(N) y circulated(1:N-1)]; z(n)= x ∗ y circulated’/N;

end

3.2 Properties of the Discrete-Time Fourier Transform 143

3.2.7 Differencing and Summation in Time

As with the differentiation and integration property of the CTFT, the DTFT has the following differencing and summation properties:

x [n] − x[n − 1] ↔ (1 − e −jΩ ) X (Ω) (3.2.10)

[m] = x[n] ∗ u

s [n]

m=−∞

(3.2.11) 1−e −jΩ

X (Ω) + π X(0)

i =−∞ (Ω − 2π i)

3.2.8 Frequency Differentiation

By differentiating both sides of the DTFT formula (3.1.1) w.r.t. Ω, we obtain dX (Ω) (3.1.1)

which yields the frequency-differentiation property of the DTFT as

This means that multiplication by n in the time domain results in differentiation w.r.t. Ω and multiplication by j in the frequency domain.

3.2.9 Time and Frequency Scaling

In the continuous-time case, we have Eq. (2.5.21) as

x (at) ↔

|a|

However, it is not so simple to define x[an] because of the following reasons: - If a is not an integer, say, a = 0.5, then x[0.5 n]| n=1 = x[0.5] is indeterminate.

- If a is an integer, say, a = 2, then it does not merely speed up x[n], but takes the even-indexed samples of x[n].

Thus we define a time-scaled version of x[n] as x [n/K ]

x (K ) for n = K r (a multiple of K ) with some integer r

[n] = (3.2.13)

0 elsewhere

144 3 Discrete-Time Fourier Analysis which is obtained by placing (K − 1) zeros between successive samples of x[n].

Then we apply the DTFT formula (3.1.1) to get

X (K ) (Ω) = x (K ) [n]e − j Ωn (3.2.13)

with n=K r

r =−∞

3.2.10 Parseval’s Relation (Rayleigh Theorem)

If x[n] has finite energy and the DTFT X (Ω) = F{x[n]}, then we have

|x[n]| =

|X(Ω)| dΩ (3.2.15)

n=−∞

where |X(Ω)| 2 is called the energy-density spectrum of the signal x[n].