Exam 2, EE5350 Fall 2012 Find z-transforms of the following sequences in closed

Question

Exam 2, EE5350 Fall 2012

Find z-transforms of the following sequences in closed form, and their regions of convergence. Assume the R.O.C. of X(z) is |z| > .3

(a) u(n+9)    (b) -nx(n) (c) cnu(-n) (d) (n+8)+(n-9)   (e) n (cn/n!)×u(n)

An IIR digital filter has the impulse response

h(n)= (3 )nu(n)+ ( 5 )nu(n)

Find H(z) in closed form, and its region of convergence.

Give the poles of H(z).

Change h(n) to make it stable, without changing H(z). Give the new R.O.C. for H(z)

Give a stable set of recursive difference equations for the filter. Use the parallel form.

In the pseudocode below, which is based upon part (d), give correct expressions for A, B, C, D and E. Assume that allowable arguments for x(), y(),y1(), and y2() include only the integers from 0 to N.

y1(N) = A y2(N) = A

y(N) = y1(N) + y2(N)

For n = B to C y1(n) = D y2(n) = E

y(n) = y1(n) + y2(n) End

Let x(n) = 2n[u(n) – u(n-N)] , which is nonzero between n=0 and n=N-1.

Find X(z) and its R.O.C. Remember that x(n) is a causal, finite length sequence.

Find X(k), the DFT of x(n).

Multiply X(z) by (-.5z)/(-.5z) and give the new X(z) and R.O.C. Is the R.O.C. the same as in part (a) ?

If we find x(n) from the X(z) of part (c ), is it the same as in the problem statement ? Why ?

Find y(n) in terms of x(n) and h(n) (which can be complex) if

N -1

å          N                               N

(a) Y(k)= H*(k)× X *(k)   (b) Y(k)=      H(m - k)

m=0

× X(m - 2k )

A filter is needed to recover x2(n) from the signal, x(n) = x1(n) + x2(n) + x3(n), where x1(n) = sin(.5n), x2(n) = sin(1.4n), and x3(n) = cos(2.7n).

What kind of filter is required, LP,BP,HP, or BR ?

Specify a filter cut-off frequency or frequencies in radians, and find the corresponding cut-off sample or samples, k1 etc, for H(k).

Assume that H(k) is ideal in the sense that each of its samples is either 1 or 0. Find a closed form for the impulse response h(n) using H(k), the inverse DFT operation, and the cut-off sample symbol(s) ki.

Find a real expression for h(n) from part (c).

Explanation / Answer

a) x(n)=u(n+9)   

X(z)= z^9/(1-z^-1), ROC |z|>3

(b) x(n)=-nx(n)

X(z)= -z^-1/(1-z^-1)^2, ROC. |z|>3

(c) x(n)= c^nu(-n)

X(z)= 1/(1-cz^-1), ROC |z|<c, c>3

(d) x(n)=(n+8)+(n-9)  

X(z)= z^8+z^-9 , ROC | z|>3

h(n) = 3^n u(n)+5^nu(n)

H(z)= 1/(1-3z^-1)+1/(1-5z^-1) , ROC | z|>5

Let change h(n) to h'(n) to make it stable,

h'(n)= -3^nu(-n-1)-5^nu(-n-1)

H'(z) =  1/(1-3z^-1)+1/(1-5z^-1) , ROC | z|<3

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