DTFT, DFT, ZT, and FFT 557
d. g(n) = f(n) + f(n − 1) − f(n + 1)e. g(n) = (^) √
f(n)
f. g(n) = |ƒ(n)|
P.5.2 Let the impulse response of an LTI system be h(n). Evaluate in the time and fre-
quency domains if the following impulse responses correspond to stable systems:
a. h(n) = 0.5nu(n)
b. h(n) = 0.5n[u(n − 3) – u(n − 10)]
c. h(n) =1.05nu(n)
d. h(n) = 0.5ncos(n)u(n)
e. h(n) = 1.05nu(n)
f. h(n) = 1/5−nu(n)
g. h(n) = 1.05nu(−n)
P.5.3 Use MATLAB to create a script fi le that returns each of the signals of P.5.2 over the
range 0 ≤ n ≤ 31 and their respective energies.
P.5.4 Let f(n)=[1 2 3 –1 –2 –3 4 5 6] and g(n) = [4 3 2 1 1 2 3 4 5]. Evaluate
a. The linear convolution
b. The circular convolution
P.5.5 Verify that the circular convolution is commutative using the sequences of P.5.4.
P.5.6 Let h(n) = 0.5nu(n) and f(n) = δ(n) + 2 δ(n − 1) − 3 δ(n − 3).
a. Sketch both the sequences by hand.
b. Obtain the plot of the convolution given by f(n) ⊗ h(n).
c. Obtain the autocorrelation plot of the function f(n) with h(n).
d. Obtain the cross-correlation plot of f(n) with h(n).
P.5.7 Let ƒ 1 (n) = δ(n) + 2 δ(n − 1) − 3 δ(n − 3), f 2 (n) = u(n) − u(n − 10) − 3 δ(n − 3), and f 3 (n) =
u(n + 10) − u(n − 10) + δ(n − 3).
a. Determine the sequence length of the following convolutions by hand:
i. g 1 (n) = f 1 (n) ⊗ f 2 (n)
ii. g 2 (n) = f 1 (n) ⊗ f 3 (n)
iii. g 3 (n) = f 2 (n) ⊗ f 3 (n)
b. Determine by hand the fi rst and last nonzero element of part a
c. Determine by hand g 1 (0), g 2 (−1), and g 3 (3)
d. Verify the results of parts a, b, and c by using MATLAB
P.5.8 Let f 1 (n) = 0.5nu(n) and f 2 (n) = 1.5nu(−n). Obtain the infi nite length convolution
sequence g 3 (n) = f 1 (n) ⊗ f 2 (n).
P.5.9 Let the causal LTI system equation be given by g(n) = g(n − 1) + f(n)+ 3f(n − 3),
with g(n) = 0, for n ≤ 0 and f(n) = 0 , for n ≤ 0. Use MATLAB and simulate the out-
put g(n) over the range 0 ≤ n ≤ 10.
P. 5. 1 0 L e t f(n) = sin(πn/5)u(n) + cos(πn/10)u(n) and h(n) = 0.3nu(n − 5). Verify that the cross-
correlation of f(n) with h(n) is identical to the convolution given by f(n) ⊗ h(−n).
P.5.11 Create a script fi le that returns a random sequence f(n) with length 10 and its
convolution with itself 10 times ( f (n) ⊗ f (n) ⊗ f (n) 10 times). Discuss the shape of the
resulting wave and compare it with the standard Gaussian function.