Solved FALL 25 EC516 Homework 02

Description

Problem 2.1 (DTFT Basics)
A) Consider the complex exponential signal 𝑥[𝑛] = 𝑒'(.*+,-. to answer each of the
following questions:
a) Is the signal 𝑥[𝑛] periodic? If yes, what is its fundamental period?
b) Plot the absolute value of 𝑥[𝑛] as a function of n.
c) Specify a numerical value of 𝑛( such that 𝑥[𝑛(] = −1.
B) Determine the DTFT 𝑋(𝑒’3) of 𝑥[𝑛] = 𝛿[𝑛 − 3] by using 𝑋(𝑒’3)=∑ 𝑥[𝑛]𝑒 9 8’3. .:89
where 𝛿[𝑛] is the discrete-time unit impulse.
C) Calculate the DTFT of each of the following signals using 𝑋(𝑒’3)=∑ 𝑥[𝑛]𝑒 9 8’3. .:89 and
plot |𝑋<𝑒’3=| as a function of 𝜔.
a) 𝑥[𝑛] = 𝑢[𝑛 + 2] − 𝑢[𝑛 − 3] , where 𝑢[𝑛] is the discrete-time unit step.
b) 𝑥[𝑛] = 𝑢[𝑛] − 𝑢[𝑛 − 5] , where 𝑢[𝑛] is the discrete-time unit step.
c) 𝑥[𝑛] = 𝑢[𝑛] − 𝑢[𝑛 − 4] , where 𝑢[𝑛] is the discrete-time unit step.
Problem 2.2 (Analog Sampling)
Suppose is a real-valued speech-signal whose CTFT is and it is known that
for . Let be the output of an analog sampler where T
represents the sampling interval. Answer the following questions about , the DTFT of
, for the specified values of T.
(a) For what values of is guaranteed to be zero if secs. Justify your answer.
(b) For what values of is guaranteed to be zero if secs. Justify your answer.
(c) For what values of is guaranteed to be zero if secs. Justify your answer.
x(t) X ( jω)
| X ( jω)|= 0 |ω |≥10,000π x[n] = x(nT )
X (e jω )
x[n]
ω X (e jω ) T = 0.0001
ω X (e jω ) T = 0.00005
ω X (e jω ) T = 0.00001
Problem 2.3 (Inverse DTFT and DTFT Properties)
A) For each DTFT given below, compute the corresponding signal using 𝑥[𝑛] = H
I- ∫ 𝑋(𝑒’3)𝑒’3. –
8- 𝑑𝜔 and plot |𝑥[𝑛]| as a function of n.
a) 𝑋<𝑒’3= = ∑ 2𝜋𝛿(𝜔 − 0.5𝜋 − 2𝜋𝑘) 9
O:89
b) 𝑋<𝑒’3= = ∑ {𝜋𝛿(𝜔 + 0.5𝜋 − 2𝜋𝑘) + 𝜋𝛿(𝜔 − 0.5𝜋 − 2𝜋𝑘)} 9
O:89
B) Suppose that the input 𝑥[𝑛] to a digital circuit is related to the output 𝑦[𝑛] of that
digital circuit through the following difference equation:
𝑦[𝑛] = 0.5𝑦[𝑛 − 1] + 𝑥[𝑛]
Determine the mathematical relationship between 𝑌(𝑒’3) and 𝑋(𝑒’3).
C) If 𝑥[𝑛] is a real-valued signal with 𝑋<𝑒'(.I,-= = 1 + 𝑗, use 𝑋<𝑒’3= = ∑ 𝑥[𝑛]𝑒 9 8’3. .:89
to determine the value of 𝑋(𝑒8′(.I,-). This result shows that any algorithm for
determining the DTFT of a real-valued signal need not directly calculate the DTFT for
negative frequencies.
Problem 2.4 (Digital Sampling)
Suppose the discrete-time signal 𝑥[𝑛] is a real-valued signal. We perform digital downsampling
by a factor of 3 on 𝑥[𝑛] to obtain the signal 𝑦[𝑛]. The digital downsampling uses a digital antialiasing filter with input 𝑥[𝑛] and output 𝑔[𝑛]. The signal 𝑔[𝑛] then undergoes a sampling rate
change that reduces the sampling rate by a factor of 3.
(a) Draw and label the ideal frequency response of the digital antialiasing that should be
used in the digital sampling process. Justify your answer.
(b) Is the ideal digital antialiasing filter causal? Justify your answer.
(c) Are there any values of 𝜔 for which 𝐺(𝑒’3) is guaranteed to be zero regardless of what
𝑋<𝑒’3= looks like? Justify your answer.
(d) Determine the values of the constants 𝛼, 𝛽, and γ such that
3𝑌<𝑒’3= = 𝐺<𝑒'(38c= + 𝐺<𝑒'(38d= + 𝐺<𝑒'(38e=
Justify your answer.
(e) Is the signal 𝑔[𝑛] guaranteed to be reconstructable from the signal 𝑦[𝑛]? Justify your
answer.
(f) Is the signal 𝑥[𝑛] guaranteed to be reconstructable from the signal 𝑔[𝑛]? Justify your
answer.
Experiential DSP Exercise 02
Part A: Plot and Observe!
1. Place audio.wav from HW02 assignment in your current MATLAB folder (check with
pwd or use the Current Folder pane).
2. Read the audio and sampling rate using the audioread command. (Note that this
command gives you two outputs, x and F.)
3. Following the instructions of Assignment 1, plot the waveform of the audio signal.
4. Observe that the audio signal is roughly periodic. What approximately is the period?
Part B: DTFT in MATLAB
In this part, we want to learn how the fft command in MATLAB works and how to analyze its
output.
1. The fft command takes two inputs. x, which is the signal that you get from the previous
part, and n, which is the number of samples. The fft command computes n uniformly
spaced samples every
I-
. from 𝜔 = 0 to 𝜔 = 2𝜋 of the discrete-time Fourier transform
(DTFT).
2. Set nfft_1 = 512 and nfft_2 = 1024.
3. Get X_1 = fft(x, nfft_1) and Get X_2 = fft(x, nfft_2).
4. Create two stem plots: stem(20*log10(abs(X_1)) and
stem(20*log10(abs(X_2)). Save these figures as figure1 and figure2,
respectively. Note that the x-axis in these figures represents the frequency, which here spans
from 0 to 2𝜋.
5. In each figure, find the middle point and explain how the portion to the left of the middle
point and the portion to the right of the middle point are related to each other.
6. Observe the two saved figures and determine how the samples in figure1 are just a subset
of the samples in figure2. Specify how the index of a sample in figure1 maps to the
corresponding index in figure2.