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Material Type: Assignment; Professor: Havlicek; Class: Digital Signal Processing; Subject: ELECTRICAL AND COMPUTER ENGINEERING; University: University of Oklahoma; Term: Unknown 1989;
Typology: Assignments
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1
1
j
−
ω
−
Q3.2 The plots generated by running Program P3_1 are shown below:
Real part of H(e
j ω
ω
π
Amplitude
Imaginary part of H(e
j ω
ω
π
Amplitude
Magnitude Spectrum |H(e
j ω
ω
π
Amplitude
Phase Spectrum arg[H(e
jω
ω
π
Phase in radians
The plots generated by running the modified Program P3_1 are shown below:
Real part of H(e
j ω
ω
π
Amplitude
Imaginary part of H(e
j ω
ω
π
Amplitude
Magnitude Spectrum |H(e
j ω
ω
π
Amplitude
Phase Spectrum arg[H(e
jω
ω
π
Phase in radians
The DTFT is a periodic_ function of ω.
Its period is - 2π
The jump in the phase spectrum is caused by - a branch cut in the arctan function used by
angle in computing the phase. “angle” returns the principal branch of arctan.
The phase spectrum evaluated with the jump removed by the command unwrap is as given
below:
Phase Spectrum arg[H(e
j ω
ω
π
Phase in radians
Q3.4 The required modifications to Program P3_1 to evaluate the given DTFT of Q3.4 are given below:
Magnitude Spectrum |H(e
j ω
ω
π
Amplitude
Phase Spectrum arg[H(e
j ω
ω
π
Phase in radians
The DTFT is a periodic function of ω.
Its period is - 2π
The jump in the phase spectrum is caused by - “angle” returns the principal value of the
arc tangent.
Q3.5 The required modifications to Program P3_1 to plot the phase in degrees are indicated below:
given below:
The parameter controlling the amount of time-shift is - D
Magnitude Spectrum of Original Sequence
ω
π
Amplitude
Magnitude Spectrum of Time-Shifted Sequence
ω
π
Amplitude
Phase Spectrum of Original Sequence
ω
π
Phase in radians
Phase Spectrum of Time-Shifted Sequence
ω
π
Phase in radians
From these plots we make the following observations: As before, of course, the time shift
has no effect on the magnitude spectrum. However, there is a very significant change
to the phase. As before, the time shift adds phase to the DTFT, making the slope of the
phase spectrum steeper. However, in this case with D=5 instead of D=10, the
(negative) increase in the slope is less than it was before.
length for the sequence –
The plots generated by running the modified program are given below:
Magnitude Spectrum of Original Sequence
ω
π
Amplitude
Magnitude Spectrum of Time-Shifted Sequence
ω
π
Amplitude
Phase Spectrum of Original Sequence
ω
π
Phase in radians
Phase Spectrum of Time-Shifted Sequence
ω
π
Phase in radians
Magnitude Spectrum of Original Sequence
ω
π
Amplitude
Magnitude Spectrum of Time-Shifted Sequence
ω
π
Amplitude
Phase Spectrum of Original Sequence
ω
π
Phase in radians
Phase Spectrum of Time-Shifted Sequence
ω
π
Phase in radians
From these plots we make the following observations: Increasing the length makes the
magnitude spectrum more narrow (i.e., makes the signal more “low pass”). It also
makes the phase steeper (i.e., the slope more negative). This is because, if we think of
the sequence as the impulse response of an LTI system, increasing the length adds more
Magnitude Spectrum of Original Sequence
ω
π
Amplitude
Magnitude Spectrum of Frequency-Shifted Sequence
ω
π
Amplitude
Phase Spectrum of Original Sequence
ω
π
Phase in radians
Phase Spectrum of Frequency-Shifted Sequence
ω
π
Phase in radians
From these plots we make the following observations: Both the magnitude and phase
spectra are shifted right by wo, which is given by 0.4π in this case. Note that the
frequency shifted signal was obtained by multiplying the original sequence pointwise
with a complex-valued exponential sequence. Thus, the frequency shifted sequence is
also complex-valued and it’s DTFT does not have conjugate symmetry.
Q3.12 Program P3_3 was run for the following value of the frequency-shift – wo = -0.5π.
The plots generated by running the modified program are given below:
Magnitude Spectrum of Original Sequence
ω
π
Amplitude
Magnitude Spectrum of Frequency-Shifted Sequence
ω
π
Amplitude
Phase Spectrum of Original Sequence
ω
π
Phase in radians
Phase Spectrum of Frequency-Shifted Sequence
ω
π
Phase in radians
From these plots we make the following observations: In this case, the magnitude and
phase spectra are shifted left by π/2 rad. As before, the frequency shifted signal is
complex-valued, so the frequency shift causes a loss of the conjugate symmetry that
was present in the spectrum of the original signal. NOTE: you should keep in mind
that these spectra are all 2π-periodic; we are only displaying the fundamental period.
values of length for the sequence –
The plots generated by running the modified program are given below:
the impulse responses of LTI systems, then a shorter length implies a smaller delay
between the input and output of the system, which corresponds to a phase spectrum
with a negative slope that is less steep. Likewise, a longer length implies a longer
delay between the input and output, which corresponds to a phase spectrum with a
negative slope that is steeper. In both cases shown here, the effect of the frequency
shift is to translate both the magnitude and phase spectrum by wo radians. Whereas the
original sequences are real-valued and hence have conjugate symmetric spectra, the
frequency shifted sequences are complex-valued and do not exhibit any inherent
spectral symmetry.
given below:
Q3.15 The plots generated by running the modified program are given below:
Product of Magnitude Spectra
ω
π
Amplitude
Magnitude Spectrum of Convolved Sequence
ω
π
Amplitude
Sum of Phase Spectra
ω
π
Phase in radians
Phase Spectrum of Convolved Sequence
ω
π
Phase in radians
From these plots we make the following observations: The DTFT magnitude and phase
spectra obtained by performing pointwise multiplication of the two DTFT’s of the
original sequences are identical to those obtained by performing time domain
convolution of the two original sequences; this verifies the convolution property of the
DTFT.
Q3.16 Program P3_4 was run for the following two different sets of sequences of varying lengths –
1 2
n
for 0 ≤ n ≤ 7;
Length of x2 = 4; x2[n]= [0.25 0.25 0.25 0.25]
3 4
n
− for 0 ≤ n ≤ 15;
Length of x2 = 8; x2[n]= [1 3 5 7 9 11 13 15]
The plots generated by running the modified program are given below:
are identical to the magnitude and phase spectra obtained from the DTFT of the time
domain convolution of the two original sequences.
Q3.18 The plots generated by running the modified program are given below:
Magnitude Spectrum of First Sequence
ω
π
Amplitude
Magnitude Spectrum of Second Sequence
ω
π
Amplitude
Magnitude Spectrum of Product Sequence
ω
π
Amplitude
From these plots we make the following observations: The DTFT of the product sequence
y is 1/2π times the convolution of the DTFT’s of the two sequences x1 and x2, as
expected. The low-pass mainlobe of the DTFT of x1 combines with the high-pass
mainlobe of the DTFT of x2 to produce a high-pass mainlobe centered at ±π in the
magnitude spectrum of the product signal. The low-pass mainlobe of the DTFT of x
combines with the low-pass sidelobes of the DTFT of x2 to produce a low-pass smooth
region of relatively lower gain centered at DC in the magnitude spectrum of the product
signal.
Q3.19 Program P3_5 was run for the following two different sets of sequences of varying lengths –
1 2
n
for 0 ≤ n ≤ 7;
Length of x2 = 8; x2[n]= [1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8]
3 4
n
− for 0 ≤ n ≤ 15;
Length of x2 = 16; x2[n]= [1 3 5 7 9 11 13 15 0 0 0 0 0 0 0 0]