EECS 451______________________PROBLEM SET #2______________________Fall 2009
ASSIGNED: Sep. 17, 2009. 1996 Text: Sections 2.5 (SKIP 2.4); 3.1-3.3 on z-transforms.
DUE DATE: Sep. 24, 2009. 2007 Text: Sections 2.5 (SKIP 2.4); 3.1-3.3 on z-transforms.
THIS WEEK: Basic discrete-time signals and systems and convolution.
    Basic properties of discrete-time linear systems:
  1. [35] (14X5)@½ Determine whether each of the following systems
    has or does not have each of the following properties:
    (1) Static (2) Linear (3) Time-Invariant (4) Causal (5) Stable
    (a) y(n)=|cos(x(n)| (b) y(n)=x(n+1)+x(n)+x(n-1)+... (c) y(n)=x(n)cos(ωn) (d) y(n)=x(2-n)
    (e) y(n)=int[x(n)] (f) y(n)=round[x(n)] (g) y(n)=|x(n)| (h) y(n)=x(n) if n≥0
    (i) y(n)=x(n)+nx(n+1) (j) y(n)=x(2n) (k) y(n)=x(n) if x(n)≥0 (l) y(n)=x(-n)
    (m) y(n)=sign[x(n)] (n) y(n)=x(t=nT) (ideal sampler every T seconds).
    [1996 & 2007 Text #2.7]. Drill on basic system properties. Covers almost everything.
    No doubt you had a similar problem in EECS 216; it should be much easier this time.
    Put your answers in the form of a table--"Yes" or "No" for each problem is sufficient.
  2. [5] (8@½+1 for (i)) If each of 2 systems has the given property,
    does their cascade (series) connection also have it:
    (a) Linear (b) Time-Invariant (c) Causal (d) LTI (g) Nonlinear (h) Stable
    (e) If both systems LTI, interchanging their order doesn't change the overall system
    (f) If both time-varying, interchanging their order doesn't change the overall system
    (i) Give counterexamples to show the inverses of (c) and (h) are false.
    [1996 & 2007 Text #2.8]. Drill on properties of interconnected systems.
    Put your answers in the form of a table--"True" or "False" for each is sufficient.
  3. [10] The following three input-output pairs are observed for a system
    known to be time-invariant. What is its impulse response?
    (i) {1,0,2}→{0,1,2}; (ii) {0,0,3}→{0,1,0,2}; (iii) {0,0,0,1}→{1,2,1}.
    The first number of all sequences shown is at n=0. Is the system linear?
    The following input-output pairs are observed for a different, linear system:
    (i) {-1,2,1}→{1,2,-1,0,1}; (ii) {1,-1,-1}→{-1,1,0,2}; (iii) {0,1,1}→{0,1,2,1}
    The first number of all sequences shown is at n=0. Is the system time-invariant?
    [1996 & 2007 Text #2.10 (known time-invariant) and 2.11 (known to be linear)].
    Drill on discrete-time convolution:
  4. (a) [4] If y(n)=h(n)*x(n), prove ∑y(n)=∑h(n)∑x(n). Good check for the following:
    Just sum over n and note that Σx[n-i]=Σx[n] where both sums are over n. Pull out a Σ.
    (b) [16] {1,2,4}*{1,1,1,1,1}; {1,2,-1}*{1,2,-1}; {1,-2,3}*{0,0,1,1,1,1}; {0,1,4,-3}*{1,0,-1,-1}
    [1996 & 2007 Text #2.16ab(1,2,5,7)]. 4 short convolutions. Easy once you get the hang of it.
    For (1) only: Use conv and plot output using stem (this is preferred for discrete-time signals).
    You may check your results using Matlab's conv (not required), but DO them by hand first.
  5. [15] [1996 Text #2.47ab]. Write the ARMA equation and then plug in. Check results with
    [2007 Text #2.50ab]. Matlab: A=[1 -0.9]; B=[1 2 3]; X=[1 0 0 0 0 0]; Y=filter(B,A,X)
    (Note that I am implicitly giving you the answer--am I a nice guy or what? I know: "what.")
  6. Antialiasing filtering and downsampling:
    1. [15] (3@5) Run the bare-bones Matlab program below (label the plots better).
    2. Confirm the upsampled signal Z is identical to the original filtered signal X,
      even though it was reconstructed from the downsampled W, not the filtered X.
    3. Explain why low-pass filtering in the first step eliminates aliasing.
      clear; load handel; Y=y(1:65536);
FX=fft(Y);FX(16384:49154)=0;X=ifft(FX); 
subplot(3,2,1),plot(real(X)),axis tight
subplot(3,2,2),plot(abs(FX)),axis tight
W=X(1:2:65536); FW=fft(W);
subplot(3,2,3),plot(real(W)),axis tight
subplot(3,2,4),plot(abs(FW)),axis tight
Z=interpft(W,65536); FZ=fft(Z);
subplot(3,2,5),plot(real(Z)),axis tight
subplot(3,2,6),plot(abs(FZ)),axis tight