Engineering

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1. Open book and notes.

2. Do not open this exam until 6:00 pm.

3. Put all youranswersin the appropriate space. If necessary, you can use extra work-sheets, but please try to put solutions in the correct spot on this exam.

4. Turn in your work and put your name at the top of all loose work-sheets. This work will be looked at for possible partial credit.

5.Justify all of your answers. A “correct” answer with no justification will not receive fullcredit. An incorrect answer with even partially correct partial derivation and/or justification will get partial credit.

6. The exam can be turned in at the front of the room before 8:50 pm. You must stop your work at 8:50 pm. Anyone working past this time will be penalized.

7. You can take bathroom breaks any time you wish and without special permission, during the exam, but cannot converse with anyone during these breaks.

8. This exam has total of 7 pages (including this page).

9. The total weight for the exam is 100 points.

Problem 1

Suppose we have a cascade systems shown in the following figure where1

1

( )

1

1

2

j

j

H e

e

w

-w

=

and2

1

( )

1

1

3

j

j

H e

e

w

-w

=

.

a) Find []hnsuch that the input []xnand output []ynsatisfy the relationship [ ] [ ] [ ]y nx nhn=*.(10 points)

b) Findg[n] such thatx[n] = g[n]*y[n].(10 points)

Problem 2

Consider the system shown in the following figure,

C/D L M D/C

The linear time-invariant filter ()jHewhas DTFT as shown in the following figure where we know .cLwp<Assume that the Fourier transform of the input,{( )}( )c cF x t=X jW, is band-limited, i.e.

( ) 0cX jW =for0|W|³ W.

a) We would like to choose a sampling periodTsuch that there is no aliasing by the C/D converter or, if there is any aliasing, the frequency component contaminated by aliasing is rejected by the filter ()jHew. Given0W,cwandL, determine the most general condition on sampling periodT.(10 points)

b) Let ( )jLX ew, ()jXew, ()jYew, ()j MY ewbe the discrete-time Fourier transforms of []Lx n, []x n, []y n, and []Myn, respectively. Express ( )jX ew, ( )jY ew, and ( )j MY ewrespectively in terms of ( )j

LX ewand, if needed, also ( )jH ew.(10 points)

d) Again, ifLM=, what is the overall equivalent continuous-time frequency responseeff(

(( ) ) )c cY j X H j j= W W W?(5 points)

Problem 3

A causal LTI system has system functionH(z) with the pole-zero plot shown in the figure below. You are also told thatH(1) 2.

a) Is ()Hzstable? Justify your answer. (Hint: Consider that01( ) 2.jz eHz) (5 points)

b) What is the region of convergence for ()Hz? Motivate your answer. (10 points)

c) Ish[n] real? Justify your answer. (Hint: you don’t need to solve forh[n] to answer this question.) (10 points)

d) What is the pole-zero plot and region of convergence for thez-transform of []hn? (5 points)

e) What is the region of convergence for the z-transform of2[ ]n jh n? (5 points)

Problem 4

Letx[n] be a causal stable sequence withz-transformX(z) . The complex cesptrum , which we might see used later for an estimator for undoing the effect of convolution (“deconvolution,”) is defined as the inverse transform of the logarithm of ()Xz; i.e.{ }1ˆ ( ) ln ( ) ˆ[ ]ZX z X z x n = «

where ln{ }log{ }e× = ×is the natural logarithm. The region of convergence of thisXˆ (z) includes the unit circle. (Strictly speaking, taking the logarithm of a complex number often requires some carefulconsiderations. Moreover, the logarithm of a validz-transform may not be a validz-transform. For now, just assume that this logarithm of a complex number operation is valid.) Given the sequencex[n]=d[n]+ad[nN], whereNis an integer anda<1.

a) Find ()Xz, thez-transform of this []xn. (Easy) (10 points)

b)Nis an arbitrary positive integer. Determine the complex cesptrum ˆ[]xnof the above sequence []xn . Your answer need not be in closed form. (It can be an infinite sum.) (Hint: feel free to use the infinite Mercator series definition:( ) ( )1 11 ln 1k k kx x k¥ + =- + =å) (5 points)

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