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100 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
P3.11 For each of the linear, shift-invariant systems described by the impulse response,
determine the frequency response function H(e jω ). Plot the magnitude response |H(e jω )|
and the phase response ̸ H(e jω ) over the interval [−π, π].
1. h(n) =(0.9) |n|
2. h(n) =sinc(0.2n)[u(n + 20) − u(n − 20)], where sinc 0 = 1.
3. h(n) =sinc(0.2n)[u(n) − u(n − 40)]
4. h(n) =[(0.5) n +(0.4) n ]u(n)
5. h(n) =(0.5) |n| cos(0.1πn)
P3.12 Let x(n) =A cos(ω 0n + θ 0)beaninput sequence to an LTI system described by the
impulse response h(n). Show that the output sequence y(n) isgiven by
y(n) =A|H(e jω 0
)| cos[ω 0n + θ 0 + ̸ H(e jω 0
)]
P3.13 Let x(n) =3cos (0.5πn +60 ◦ )+2sin(0.3πn) bethe input to each of the systems
described in Problem P3.11. In each case, determine the output sequence y(n).
P3.14 An ideal lowpass filter is described in the frequency domain by
H d (e jω )=
{
1 · e −jαω , |ω| ≤ω c
0, ω c < |ω| ≤π
where ω c is called the cutoff frequency and α is called the phase delay.
1. Determine the ideal impulse response h d (n) using the IDTFT relation (3.2).
2. Determine and plot the truncated impulse response
{
hd (n), 0 ≤ n ≤ N − 1
h(n) =
0, otherwise
for N = 41, α = 20, and ω c =0.5π.
3. Determine and plot the frequency response function H(e jω ), and compare it with the
ideal lowpass filter response H d (e jω ). Comment on your observations.
P3.15 An ideal highpass filter is described in the frequency-domain by
{
H d (e jω 1 · e −jαω , ω c < |ω| ≤π
)=
0, |ω| ≤ω c
where ω c is called the cutoff frequency and α is called the phase delay.
1. Determine the ideal impulse response h d (n) using the IDTFT relation (3.2).
2. Determine and plot the truncated impulse response
{
hd (n), 0 ≤ n ≤ N − 1
h(n) =
0, otherwise
for N = 31, α = 15, and ω c =0.5π.
3. Determine and plot the frequency response function H(e jω ), and compare it with the
ideal highpass filter response H d (e jω ). Comment on your observations
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