Digital Signal Processing Homework Assignment #3

EE3660 Intro. to Digital Signal Processing
Homework Assignment #3: Chap. 5-6

I Paper Assignment (74%)
1. (6%) Determine the system function, magnitude response, and phase response of the following
systems and use the pole-zero pattern to explain the shape of their magnitude response:
(a) y[n] = 1
4 (𝑥𝑥[𝑛𝑛] + 𝑥𝑥[𝑛𝑛 − 1]) − 1
4 (𝑥𝑥[𝑛𝑛 − 2] + 𝑥𝑥[𝑛𝑛 − 3])
(b) y[n] = 𝑥𝑥[𝑛𝑛] − 𝑥𝑥[𝑛𝑛 − 4] + 0.6561𝑦𝑦[𝑛𝑛 − 4]
2. (12%) Consider a periodic signal
x[n] = sin(0.1𝜋𝜋𝜋𝜋) +
1
3 sin(0.3𝜋𝜋𝜋𝜋) +
1
5 sin(0.5𝜋𝜋𝜋𝜋)
For each of the following systems, determine if the system imparts (i) no distortion, (ii) magnitude
distortion, and/or (iii) phase (or delay) distortion.
(a) h[n] = {1𝑛𝑛=0, −2,3, −4,0,4, −3,2, −1}
(b) y[n] = 10𝑥𝑥[𝑛𝑛 − 10]
3. (12%) An economical way to compensate for the droop distortion in S/H DAC is to use an
appropriate digital compensation filter prior to DAC.
(a) Determine the frequency response of such an ideal digital filter 𝐻𝐻𝑟𝑟(𝑒𝑒𝑗𝑗𝑗𝑗) that will perform
an equivalent filtering given by following 𝐻𝐻𝑟𝑟(𝑗𝑗𝑗𝑗)
(b) One low-order FIR filter suggested in Jackson (1996) is
𝐻𝐻𝐹𝐹𝐹𝐹𝐹𝐹(𝑧𝑧) = − 1
16 +
9
8 𝑧𝑧−1 − 1
16 𝑧𝑧−2
Compare the magnitude response of 𝐻𝐻𝐹𝐹𝐹𝐹𝐹𝐹(𝑒𝑒𝑗𝑗𝑗𝑗) with that of 𝐻𝐻𝑟𝑟(𝑒𝑒𝑗𝑗𝑗𝑗) above.
(c) Another low-order IIR filter suggested in Jackson (1996) is
𝐻𝐻𝐼𝐼𝐼𝐼𝐼𝐼(𝑧𝑧) = 9
8 + 𝑧𝑧−1
Compare the magnitude response of 𝐻𝐻𝐼𝐼𝐼𝐼𝐼𝐼(𝑒𝑒𝑗𝑗𝑗𝑗) with that of 𝐻𝐻𝑟𝑟(𝑒𝑒𝑗𝑗𝑗𝑗) above.
4. (12%) Consider the following continuous-time system
𝐻𝐻(s) = 𝑠𝑠4 − 6𝑠𝑠3 + 10𝑠𝑠2 + 2𝑠𝑠 − 15
𝑠𝑠5 + 15𝑠𝑠4 + 100𝑠𝑠3 + 370𝑠𝑠2 + 744𝑠𝑠 + 720
(a) Show that the system H(s) is a nonminimum phase system.
(b) Decompose H(s) into the product of minimum phase component 𝐻𝐻𝑚𝑚𝑚𝑚𝑚𝑚(𝑠𝑠) and an all pass
component 𝐻𝐻𝑎𝑎𝑎𝑎(𝑠𝑠).
(c) Briefly plot the magnitude and phase responses of H(s) and 𝐻𝐻𝑚𝑚𝑚𝑚𝑚𝑚(𝑠𝑠) and explain your plots.
(d) Briefly plot the magnitude and phase responses of 𝐻𝐻𝑎𝑎𝑎𝑎(𝑠𝑠).
5. (12%) We want to design a second-order IIR filter using pole-zero placement that satisfies the
following requirements: (1) the magnitude response is 0 at ω1 = 0 and ω3 = π (2) The
maximum magnitude is 1 at ω2,4 = ± π
4 and (3) the magnitude response is approximately 1
√2
at
frequencies ω2,4 ± 0.05
(a) Determine locations of two poles and two zeros of the required filter and then compute its
system function H(z).
(b) Briefly graph the magnitude response of the filter.
(c) Briefly graph phase and group-delay responses.
6. (8%) The following signals 𝑥𝑥𝑐𝑐(𝑡𝑡) is sampled periodically to obtained the discrete-time signal
x[𝑛𝑛]. For each of the given sampling rates in 𝐹𝐹𝑠𝑠 Hz or in T period, (i) determine the spectrum
X(eiω) of x[𝑛𝑛]; (ii) plot its magnitude and phase as a function of ω in 𝑟𝑟𝑟𝑟𝑟𝑟
𝑠𝑠𝑠𝑠𝑠𝑠 and as a function of
F in Hz; and (iii) explain whether 𝑥𝑥𝑐𝑐(𝑡𝑡) can be recovered from x[𝑛𝑛].
(a) 𝑥𝑥𝑐𝑐(𝑡𝑡) = 5ei40t + 3e−i70t , with sampling period T = 0.01, 0.04, 0.1
(b) 𝑥𝑥𝑐𝑐(𝑡𝑡) = 3 + 2 sin(16𝜋𝜋𝜋𝜋) + 10 cos(24𝜋𝜋𝜋𝜋) , with sampling rate 𝐹𝐹𝑠𝑠 = 30, 20, 15 Hz.
7. (12%) An 8-bit ADC has an input analog range of ±5 volts. The analog input signal is
𝑥𝑥𝑐𝑐(𝑡𝑡) = 2 cos(200𝜋𝜋𝜋𝜋) + 3 sin(500𝜋𝜋𝜋𝜋)
The converter supplies data to a computer at a rate of 2048 bits/s. The computer, without
processing, supplies these data to an ideal DAC to form the reconstructed signal 𝑦𝑦𝑐𝑐(𝑡𝑡). Determine:
(a) the quantizer resolution (or step),
(b) the SQNR in dB,
(c) the folding frequency and the Nyquist rate.
II Program Assignment (26%)
8. (4%) Compute and plot the phase response using the functions freqz, angle, phasez, unwrap, and
phasedelay for the following systems:
(a) y[n] = 𝑥𝑥[𝑛𝑛 − 15]
(b) 𝐻𝐻(𝑧𝑧) = 1+1.655𝑧𝑧−1+1.655𝑧𝑧−2+𝑧𝑧−3
1−1.57𝑧𝑧−1+1.264𝑧𝑧−2−0.4𝑧𝑧−3
9. (6%) According to problem 2 in paper assignment, plot magnitude response, phase response and
group-delay response for each of the systems.
10. (6%) MATLAB provides a function called polystab that stabilizes the given polynomial with
respect to the unit circle, that is, it reflects those roots which are outside the unit-circle into those
that are inside the unit circle but with the same angle. Using this function, convert the following
systems into minimum-phase and maximum-phase systems. Verify your answers using a polezero plot for each system(plot minimum-phase and maximum-phase systems for each question).
(a) H(z) = 𝑧𝑧2+2𝑧𝑧+0.75
𝑧𝑧2−0.5𝑧𝑧
(b) H(z) = 1−2.4142𝑧𝑧−1+2.4142𝑧𝑧−2−𝑧𝑧−3
1−1.8𝑧𝑧−1+1.62𝑧𝑧−2+0.729𝑧𝑧−3
11. (10%) Signal xc(t) = 5 cos(200πt + π6 ) + 4 sin(300πt) is sampled at a rate of Fs = 1 kHz to obtain
the discrete-time signal x[n].
(a) Determine the spectrum X(ejω) of x[n] and plot its magnitude as a function of ω in 𝑟𝑟𝑟𝑟𝑟𝑟
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
and as a function of F in Hz. Explain whether the original signal xc(t) can be recovered from
x[n].
(b) Repeat part (a) for Fs = 500 Hz.
(c) Repeat part (a) for Fs = 100
(d) Comment on your results.