EE3660 Homework #3 Solved

Description

1.  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:
   • y[n] = (𝑥𝑥[𝑛𝑛] + 𝑥𝑥[𝑛𝑛− 1]) − (𝑥𝑥[𝑛𝑛− 2] + 𝑥𝑥[𝑛𝑛− 3])
   • y[n] = 𝑥𝑥[𝑛𝑛] −𝑥𝑥[𝑛𝑛− 4] + 0.6561𝑦𝑦[𝑛𝑛− 4]

2.  Consider a periodic signal

x[n] = sin(0.1𝜋𝜋𝑛𝑛) +  sin(0.3𝜋𝜋𝑛𝑛) +  sin(0.5𝜋𝜋𝑛𝑛)

For each of the following systems, determine if the system imparts (i) no distortion, (ii) magnitude distortion, and/or (iii) phase (or delay) distortion.

• h[n] = {1_𝑛𝑛=0, −2,3, −4,0,4, −3,2, −1}
• y[n] = 10𝑥𝑥[𝑛𝑛− 10]

3.  An economical way to compensate for the droop distortion in S/H DAC is to use an appropriate digital compensation filter prior to DAC.
   • Determine the frequency response of such an ideal digital filter 𝐻𝐻_𝑟𝑟(𝑒𝑒^{𝑗𝑗𝑗𝑗}) that will perform an equivalent filtering given by following 𝐻𝐻_𝑟𝑟(𝑗𝑗𝑗𝑗)
   • One low-order FIR filter suggested in Jackson (1996) is

1      9             1

𝐻𝐻_{𝐹𝐹𝐹𝐹𝐹𝐹}(𝑧𝑧) = −  + 𝑧𝑧⁻¹ −𝑧𝑧⁻²

16     8           16

Compare the magnitude response of 𝐻𝐻_{𝐹𝐹𝐹𝐹𝐹𝐹}(𝑒𝑒^{𝑗𝑗𝑗𝑗}) with that of 𝐻𝐻_𝑟𝑟(𝑒𝑒^{𝑗𝑗𝑗𝑗}) above.

• Another low-order IIR filter suggested in Jackson (1996) is

𝐻𝐻𝐹𝐹𝐹𝐹𝐹𝐹(𝑧𝑧) =

Compare the magnitude response of 𝐻𝐻_{𝐹𝐹𝐹𝐹𝐹𝐹}(𝑒𝑒^{𝑗𝑗𝑗𝑗}) with that of 𝐻𝐻_𝑟𝑟(𝑒𝑒^{𝑗𝑗𝑗𝑗}) above.

4.  Consider the following continuous-time system

𝑠𝑠⁴ − 6𝑠𝑠³ + 10𝑠𝑠² + 2𝑠𝑠− 15

^{𝐻𝐻(s) =} 𝑠𝑠⁵ + 15𝑠𝑠⁴ + 100𝑠𝑠³ + 370𝑠𝑠² + 744𝑠𝑠 + 720 (a) Show that the system H(s) is a nonminimum phase system.

• Decompose H(s) into the product of minimum phase component 𝐻𝐻_{𝑚𝑚𝑚𝑚𝑛𝑛}(𝑠𝑠) and an all pass

component 𝐻𝐻_{𝑎𝑎𝑎𝑎}(𝑠𝑠).

• Briefly plot the magnitude and phase responses of H(s) and 𝐻𝐻_{𝑚𝑚𝑚𝑚𝑛𝑛}(𝑠𝑠) and explain your plots. (d) Briefly plot the magnitude and phase responses of 𝐻𝐻_{𝑎𝑎𝑎𝑎}(𝑠𝑠).

5. 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 ω_2,4 = ±  and (3) the magnitude response is approximately  at

frequencies ω_2,4 ± 0.05

• Determine locations of two poles and two zeros of the required filter and then compute its system function H(z).
• Briefly graph the magnitude response of the filter.
• Briefly graph phase and group-delay responses.

6.  The following signals 𝑥𝑥_𝑐𝑐(𝑡𝑡) is sampled periodically to obtained the discrete-time signal x[𝑛𝑛]. For each of the given sampling rates in 𝐹𝐹_𝑠𝑠 Hz or in T period, (i) determine the spectrum X(e^iω) of x[𝑛𝑛]; (ii) plot its magnitude and phase as a function of ω in _{𝑠𝑠𝑎𝑎𝑚𝑚}^{𝑟𝑟𝑎𝑎𝑟𝑟} and as a function of

F in Hz; and (iii) explain whether 𝑥𝑥_𝑐𝑐(𝑡𝑡) can be recovered from x[𝑛𝑛].

• 𝑥𝑥_𝑐𝑐(𝑡𝑡) = 5e^i40t + 3e^−i70t , with sampling period T = 0.01, 0.04, 0.1
• 𝑥𝑥_𝑐𝑐(𝑡𝑡) = 3 + 2 sin(16𝜋𝜋𝑡𝑡) + 10 cos(24𝜋𝜋𝑡𝑡) , with sampling rate 𝐹𝐹_𝑠𝑠 = 30, 20, 15 Hz.

7.  An 8-bit ADC has an input analog range of ±5 volts. The analog input signal is

𝑥𝑥_𝑐𝑐(𝑡𝑡) = 2 cos(200𝜋𝜋𝑡𝑡) + 3 sin(500𝜋𝜋𝑡𝑡)

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 𝑦𝑦_𝑐𝑐(𝑡𝑡). Determine: (a) the quantizer resolution (or step),

• the SQNR in dB,
• the folding frequency and the Nyquist rate.

II Program Assignment

 

9. Compute and plot the phase response using the functions freqz, angle, phasez, unwrap, and phasedelay for the following systems:
   • y[n] = 𝑥𝑥[𝑛𝑛− 15]
   • _𝐻𝐻(𝑧𝑧) = ₁₋₁1+1.₅₇.655𝑧𝑧⁻¹𝑧𝑧−1+1+1.₂₆₄.655𝑧𝑧⁻²𝑧𝑧−2−0+.₄𝑧𝑧𝑧𝑧−3⁻³
10.  According to problem 2 in paper assignment, plot magnitude response, phase response and group-delay response for each of the systems.
11.  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).

• H(z) = 𝑧𝑧²+22 𝑧𝑧+0.75

𝑧𝑧 −0.5𝑧𝑧

(b) H(z) = 1−21−1.4142.8𝑧𝑧−1𝑧𝑧+1−1.62+2𝑧𝑧.4142−2+0𝑧𝑧.729−2−𝑧𝑧𝑧𝑧−3−3

 

11. (10%) Signal xc(t) = 5 cos(200_πt + _{π6 ) + 4} sin(300_πt) is sampled at a rate of Fs = 1 kHz to obtain the discrete-time signal x[n].

• Determine the spectrum X(e^jω) of x[n] and plot its magnitude as a function of ω in _{𝑠𝑠𝑎𝑎𝑚𝑚𝑎𝑎𝑠𝑠𝑠𝑠}^{𝑟𝑟𝑎𝑎𝑟𝑟} and as a function of F in Hz. Explain whether the original signal xc(t) can be recovered from x[n].
• Repeat part (a) for Fs = 500 Hz. (c) Repeat part (a) for Fs = 100

(d) Comment on your results.