EECS 451______________________PROBLEM SET #4______________________Fall 2009
ASSIGNED: Oct. 1, 2009. 1996 Text: Section 4.2 (skip 4.2.7-8).
DUE DATE: Oct. 8, 2009. 2007 Text: 4.2 (skip 4.2.7-8) and 4.3.
Applying z-transforms to difference equations; transfer functions.
    Application of 2-sided z-transform to link H(z) with h(n) and ARMA difference equations:
  1. [10] The Fibonacci numbers are {1,1,2,3,5,8,13,21,34...}; each is the sum of the previous 2.
    [1996: #3.33]. The ratio between successive numbers approaches 1.618=(1+√(5)/2.
    [2007: #3.31]. This is called the "golden ratio." It appears everywhere from
    the Mona Lisa to the Parthenon (in various aspect ratios) and flower patterns.
    Prove the Fibonacci numbers are the impulse response of y[n]=y[n-1]+y[n-2]+x[n].
    Compute impulse response, which is a closed-form expression for the Fibonacci numbers.
    Can also derive this formula using 1-sided z-transforms and initial conditions y(0)=y(1)=1.
    The point is that initial conditions can be regarded as coming from an impulsive input.
    Use the 2-sided z-transform here, and compute the zero-state response.
  2. [20] (2@10) Use 1-sided z-transforms to solve the following difference equations for y(n):
    y(n)-1.5y(n-1)+0.5y(n-2)=0; y(-1)=1; y(-2)=0 and y(n)=(½)y(n-1)+(1/3)n; y(-1)=1.
    [1996: #3.34bc. 2007: #3.49bc]. Drill on solving difference equations with initial conditions.
  3. [10] (2@5) Show the following 2 systems yield same outputs when given same inputs:
    [1996: #3.35]. #1: y(n)=0.2y(n-1)+x(n)-0.3x(n-1)+0.02x(n-2). #2: y(n)=x(n)-0.1x(n-1).
    [2007: #3.32]. Pole-zero cancellation. Why doesn't this happen in the real world?
  4. [10] [1996: #3.37]. Compute step response [to u(n)] if h(n)=(0.4)nu(n)+(3)nu(-n-1).
    [2007: #3.34]. Step response of noncausal system. Requires use of 2-sided z-transform.
  5. [15] (3@5) GIVEN: Input (½)nu(n)-¼(½)n-1u(n-1) produces output y(n)=(1/3)nu(n).
    (a) Compute h(n) and H(z) (b) Find a difference equation. (c) Is it stable?
    [1996 Text #3.43abd]. It is very important that you be comfortable with this.
    [2007 Text #3.40abd]. Most important problem in this set--heart of the z-transform.
  6. [15] (3@5) An LTI system has a zero {1} and poles {2,-½,-3}.
    What is the ROC of the transfer function if the system is known to be:
    (a) Stable (b) Stable and Causal (c) How many systems have these poles and zeros?
    [1996 Text: #3.51]. EECS 216 students would be scratching their heads at (c)!
    [2007 Text: #3.51]. ROCs. Don't find all the systems; just determine how many there are.
  7. [20] A simple deconvolution or inverse filtering dereverberation problem:
    1. [10] Run the Matlab program, which adds 31 echoes (reverberations) to the gong.
      Each reverberation is attenuated by a factor of 0.8 over the previous reverberation.
    2. [10] Fill in missing vectors B & A in filter so the 1st 32768 values of Z and X agree.
      This recovers original gong signal from the signal+reverberations. Explain why it works.
      You need a total of 3 nonzero coefficients. You can use z-transforms, but thinking is easier. 
      clear; load gong; N=[1:32768]; X=(y(N))'; Y=X; Y(65536)=0;
for I=1:31; Y=Y+[zeros(1,1024*I) (0.8^I)*X zeros(1,1024*(32-I))]; end 
subplot(2,2,1),plot(N,X); subplot(2,2,2),plot(N,Y(N)); Z=filter(B,A,Y);
subplot(2,2,3),plot(N,Z(N)); subplot(2,2,4),plot(N,Z(N)-X)