Problem 2: A real signal x(t) is sampled at 8 kHz and we only record 256 samples x(n) for n = 0, 1, ...,255. Assume the magnitude of the DFT X(k) has two sharp peaks at k = 15 and k = 85. a) Find the frequencies (in Hz) of these two peaks. b) Compute the frequency resolution and decide if it is possible to separate between the frequencies 465 Hz and 500 Hz? Justify your answer. c) Compare the number of real multiplications used to compute the DFT using the mathematical formula discussed in class and the decimation in frequency FFT. Answers:
Problem 2: A real signal x(t) is sampled at 8 kHz and we only record 256 samples x(n) for n = 0, 1, ...,255. Assume the magnitude of the DFT X(k) has two sharp peaks at k = 15 and k = 85. a) Find the frequencies (in Hz) of these two peaks. b) Compute the frequency resolution and decide if it is possible to separate between the frequencies 465 Hz and 500 Hz? Justify your answer. c) Compare the number of real multiplications used to compute the DFT using the mathematical formula discussed in class and the decimation in frequency FFT. Answers:
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![Problem 2: A real signal x(t) is sampled at 8 kHz and we only record 256 samples x(n) for
n = 0, 1,...,255. Assume the magnitude of the DFT X(k) has two sharp peaks at k = 15
and k = 85.
a) Find the frequencies (in Hz) of these two peaks.
b) Compute the frequency resolution and decide if it is possible to separate between the
frequencies 465 Hz and 500 Hz? Justify your answer.
c) Compare the number of real multiplications used to compute the DFT using the
mathematical formula discussed in class and the decimation in frequency FFT.
Answers:
1| Page
a) f₁ = 468.75 Hz and f₂= 2,656.30 Hz, b) Af= 31.25 Hz, Yes, c) 131072 for DFT
formula and 2048 for FFT.
Problem 3: Using the decimation in frequency FFT, compute the amplitude and phase spectra
of the following sequences:
۲/۲
a) x= [1,0,-1,0] for n = 0, 1, 2, 4
b)x= [j, 0,j, 1] for n = 0, 1, 2, 4
c) x(n)= cos (0.25mm) for n = 0, 1,...,3
d) x(n)= 0.5" for n = 0, 1,...,3
Answers:
Note: You can check your answers using MATLAB fft function to compute the DFT.
A [0 0.4000 0 0.4000]
[0 0 0 0]
b)
A = [0.4472 0.2000 0.4472 0.2000]
[1.1071 1.5708 2.0344 - 1.5708]
c) See question la)
A [0.2500 0.4330
[0-0.9553
d) See question lb)
A = [0.4688 0.2096
[00.4636
0.2500 0.4330]
0 0.9553]
0.1562 0.2096]
0 0.4636]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F161b629b-7d2a-4e7a-943e-cdfd9410a98a%2F3a663f48-a650-4bb4-b483-a8343f4a590c%2F5g6b0cp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 2: A real signal x(t) is sampled at 8 kHz and we only record 256 samples x(n) for
n = 0, 1,...,255. Assume the magnitude of the DFT X(k) has two sharp peaks at k = 15
and k = 85.
a) Find the frequencies (in Hz) of these two peaks.
b) Compute the frequency resolution and decide if it is possible to separate between the
frequencies 465 Hz and 500 Hz? Justify your answer.
c) Compare the number of real multiplications used to compute the DFT using the
mathematical formula discussed in class and the decimation in frequency FFT.
Answers:
1| Page
a) f₁ = 468.75 Hz and f₂= 2,656.30 Hz, b) Af= 31.25 Hz, Yes, c) 131072 for DFT
formula and 2048 for FFT.
Problem 3: Using the decimation in frequency FFT, compute the amplitude and phase spectra
of the following sequences:
۲/۲
a) x= [1,0,-1,0] for n = 0, 1, 2, 4
b)x= [j, 0,j, 1] for n = 0, 1, 2, 4
c) x(n)= cos (0.25mm) for n = 0, 1,...,3
d) x(n)= 0.5" for n = 0, 1,...,3
Answers:
Note: You can check your answers using MATLAB fft function to compute the DFT.
A [0 0.4000 0 0.4000]
[0 0 0 0]
b)
A = [0.4472 0.2000 0.4472 0.2000]
[1.1071 1.5708 2.0344 - 1.5708]
c) See question la)
A [0.2500 0.4330
[0-0.9553
d) See question lb)
A = [0.4688 0.2096
[00.4636
0.2500 0.4330]
0 0.9553]
0.1562 0.2096]
0 0.4636]
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