Digital Signal Processing, ELEC472-672

Final Exam Due May 3, 2000 by 8AM

1.

The system function H(z) of a causal linear time-invariant system has the pole-zero configuration shown in Figure 1. It is also known that H(z)=6 when z=1.

(a)

Determine the impulse response h[n] of the system. Your answer should be a mathematical expression in terms of n.

(b)

Determine the finite coefficient difference equation for this system. The equation should have y[n] as the output and x[n] as the input.

(c)

Determine the response, y[n], of the system to the input, x[n] which is obtained by sampling the continuous time signal

$$x(t)=50+10 \cos{20\pi t}+30 \sin{40\pi t}$$

with a sampling rate of 40Hz.

2.

When the input to a linear time-invariant system is x[n]=5u[n] the output is $y[n]=[2(\frac{1}{2})^n+3(-\frac{3}{4})^n]u[n]$.

(a)

Find the impulse response of the system for all values of n.

(b)

Write the difference equation that characterizes this system.

3.

A digital notch filter is characterized by the following transfer function:

$$H(z)=\frac{1+z^{-2}}{1+rz^{-2}}$$

(a)

What is the difference equation that describes this filter.

(b)

Determine the poles and zeros of this filter if r=0.9. Plot the poles and zeros in the zplane.

(c)

If this filter operates on a signal that was sampled at 1kHz, what frequencies in that signal will be zeroed out.

4.

An analog "mystery" signal has the form

$$y_a(t)=sin(2\pi f_1 t)+cos(2\pi f_2 t) +n(t)$$

in which n(t) is random noise. This signal is sampled 4000 times a second and the results are stored in the file myst.dat. Use the FFT to determine the frequencies f₁ and f₂ in Hertz. Is it possible to distinguish between f₁ and f₂. If so say what each is equal to. If not explain why not. Access this file in my public space in the directory ELEC472$\backslash$final.

5.

A signal consisting of two discrete time bandpass pulses, added together, and corrupted by noise is stored in the file, bpmyst.dat.

(a)

Use the FFT command to determine the center frequency and bandwidth of each pulse. Identify the lower frequency pulse as bp1 and the higher frequency pulse bp2.

(b)

Design 2 FIR filters. The first one, h₁[n], should be used to isolate bp1 and the second one, h₂[n], to isolate bp2. Plot the magnitude spectrum and phase for each filter.

(c)

Process the signal in bpmyst.m with each of the filters. Call the output of the first filter y₁[n] and the output of the second filter y₂[n]. Plot the time domain function y₁[n] and the magnitude of the frequency response $Y_1(\omega)$. Do the same for y₂[n].

(d)

(ELEC 672 or Xtra credit) How many memory locations (i.e. filter coefficients are required for each filter.

6.

An IIR filter can often accomplish the same function as an FIR filter with fewer coefficients. A typical 2 pole IIR filter has a transfer function of the form:

$$H(z)=\frac{1-z^{-2}}{1-2a z^{-1}\cos\theta +a^2z^{-2}}$$

This system has zeros at $\omega=0$ and $\omega=\pi$. The poles create a peak at $\omega=\theta$ with a 3 dB bandwidth equal to $2(1-a)/\sqrt{a}$. Use this form to design 2 IIR filters to isolate the the bandpass pulses in bpmyst.m. Again the first one, h₁[n], should be used to isolate bp1 and the second one, h₂[n], should be used to isolate bp2.

(a)

Plot the magnitude and phase of $H_1(\omega)$ and $H_2(\omega)$.

(b)

Process the signal in bpmyst.m with each of the filters. Call the output of the first filter y₁[n] and the output of the second filter y₂[n]. Plot the time domain function y₁[n] and the magnitude of the frequency response $Y_1(\omega)$. Do the same for y₂[n].

(c)

(ELEC 672 or Xtra credit) How many memory locations (i.e. filter coefficients are required for each filter.