Solved FALL 25 EC516 Homework 03

Description

Problem 2.1 (Fractional Digital Sampling)
Suppose we are interested in preserving analog frequencies up to 10Khz while sampling a realvalued analog signal 𝑥(𝑡) to produce a digital signal 𝑥[𝑛]. Initially, we use analog sampling with
a sampling frequency of 50 KHz and an analog antialiasing filter (with a flat gain of 1.0 up to
10Khz and a gradual tapering of its gain down to 0.0 at frequencies beyond 25KHz) to produce a
signal 𝑔[𝑛].
(a) By what factor would you downsample the initial analog sampling result 𝑔[𝑛] to
produce the desired signal 𝑥[𝑛]?
(b) Explain how you would use operations of sampling rate change and ideal lowpass digital
filtering to produce 𝑥[𝑛] from 𝑔[𝑛]. You must specify the sampling rate changes that
would be employed and the cutoff frequencies of any filters employed in your proposed
solution.
Problem 2.2 (z-transform Basics)
PART: A
Let 𝑥[𝑛] be a discrete time signal with z-transform 𝑋(𝑧) = ∑ 𝑥[𝑛]𝑧 1 /0 02/1
(a) Show that 𝑥[𝑛 − 𝑛4] has z-transform 𝑧/05𝑋(𝑧)
(b) Show that 𝑥[−𝑛] has z transform 𝑋(𝑧/6)
(c) Show that 𝑥∗[𝑛] has z-transform 𝑋∗(𝑧∗)
(d) Show that 𝑥[𝑛] ∗ ℎ[𝑛] has z-transform 𝑋(𝑧)𝐻(𝑧)
PART B:
Determine the z-transform of each of the following signals and the corresponding region of
convergence in each case, the values of |𝑧| for which 𝑋(𝑧) converges. You may use 𝑋(𝑧) =
∑ 𝑥[𝑛]𝑧 1 /0 02/1 , the Finite Sum Formula and/or the Infinite Sum Formula.
(a) 𝑥[𝑛] = 𝛿[𝑛 − 3]
(b) 𝑥[𝑛] = 𝑢[𝑛] − 𝑢[𝑛 − 5]
(c) 𝑥[𝑛] = (0.25)0𝑢[𝑛]
(d) 𝑥[𝑛] = (0.25)0/6𝑢[𝑛 − 1]
(e) 𝑥[𝑛] = (0.25)0𝑢[𝑛 − 1]
(f) 𝑥[𝑛] = (0.25)0𝑢[𝑛] + (0.5)0𝑢[𝑛]
(g) 𝑥[𝑛] = (0.25)0cos (0.25𝜋𝑛)𝑢[𝑛]
Problem 2.3 (Practical Filter Implementation)
Consider a digital filter with impulse response ℎ[𝑛] which is zero for n<0 and whose input 𝑥[𝑛]
and output 𝑦[𝑛] are related through the following difference equation:
𝑦[𝑛] = 0.25𝑦[𝑛 − 2] + 𝑥[𝑛]
By noting that the impulse response ℎ[𝑛] is the output of the digital filter when the input is 𝛿[𝑛],
use the above difference equation to determine the numerical values of
ℎ[0], ℎ[1], ℎ[2], ℎ[3] & ℎ[1001]. Justify your answers
Problem 2.4
PART A:
Calculate the convolution in each of the following cases and show your work:
a) and
b) and
c) and
d) and
e) and
PART B:
Sketch the phase of the DTFT of . Justify your answer.
PART C:
Sketch the magnitude and the phase of the DTFT of the signal , where
. Show your work.
PART D:
a) Show that if has DTFT , then has DTFT
b) Show that a real and even has a real and even DTFT .
c) Show that a real and odd has an imaginary and odd DTFT .
y[n] = x[n]∗ h[n]
x[n] = u[n]− u[n −5] h[n] = 0.5δ[n −3]
x[n] = n{u[n −1]− u[n −5]} h[n] = 2δ[n +3]
x[n] = u[n]− u[n −5] h[n] = u[n]− u[n −5]
x[n] = u[n]− u[n −5] h[n] = u[n]− u[n −3]
x[n] = u[n]− u[n −5] h[n] = u[n]
x[n] = u[n]− u[n − 4]
r[n] = x[k]x[n − k]
k=−∞
∞
∑
x[n] = u[n]− u[n −6]
x[n] X (e jω ) x[n − n0 ] e
− jω0 X (e jω )
x[n] X (e jω )
x[n] X (e jω )
Experiential DSP Exercise 03
A. Record and Load!
a. Open the MATLAB software.
b. Set these variables: Fs = 16000; nBits = 16; nChannels = 1;
c. Create an audio recorder object using the recObj = audiorecorder (Fs, nBits,
nChannels) command and pass the defined variables in step b to it.
d. Use the record(recObj) command to start recording. In the recording, you need to
say: “Six students went to school”.
e. Write the command pause. With this command, you can stop recording by pressing any
key on your keyboard.
f. Use the stop(recObj) command to stop recording.
g. Get the recorded data using by getaudiodata(recObj) command.
h. Save the recorded audio using the audiowrite (filename, audioData, Fs);
choose filename as: audio.wav.
i. Read the audio and sampling rate using the audioread command. (Note that this
command gives you two outputs, x and F)
B. Digital Upsampling
a. Set U = 4. Create isr_x = upsample (x, U). This upsample Matlab command
is badly named because it does not upsample a signal, rather it merely increases the
sampling rate by a factor of 4 with three zero-valued samples inserted between each pair
of consecutive samples of x. To obtain the digitally upsampled signal we will have to pass
isr_x through an interpolation filter (see steps c and d below).
b. Create a stem plot of x and isr_x in two separate figures. Let’s call the figures “Original”
and “IncreasedSR” for future reference.
c. Load the given non-ideal interplolation filter load (‘filter1.mat’, ‘b’); Pass
the IncreasedSR signal through the loaded filter: upsampled_x = filter (b, 1,
isr_x);
d. Create a stem plot of upsampled_x. Let’s call this figure “Upsampled”.
e. Now comment on the differences between the “Original,” “IncreasedSR,” and “Upsampled”
figures.
C. Digital Downsampling
a. Load the antialiasing filter: load (‘filter2.mat’, ‘b’). Pass the upsampled
signal upsampled_x from Part B above through this digital antialiasing filter:
filtered_upsampled_x = filter (b, 1, upsampled_x).
b. Set D = 4, create downsampled_upsampled_x = downsample
(filtered_upsampled_x, D). The downsample Matlab command is badly
named because it does not downsample, rather it merely decreases the sampling rate
without first applying a digital anti-aliasing filter.
c. Create a stem plot of downsampled_upsampled_x. Is it the same as “Original”?
Explain why there might be differences.