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EECS 401

Mid Term Exam I

1.   Consider the discrete-time signal {x [n]}which has a Fourier Transform (DTFT) over the interval −π≤ ω ≤ π given by:

X(ejω) = ⎨⎧⎪ 1 for ≤ ω ≤

⎪⎩ 0 otherwise

a.  Without computing {x[n]} , what is the energy in {x[n]}?

b.  Without computing {x[n]} , what is the power in {x[n]}?

2.   Consider a linear, time-invariant system that has an impulse response of: {h[n]} = {δ[n − 1]}+ {δ[n + 1]}

a.   Is the system causal?

b.   Is the system Bounded-input/Bounded-output stable?

c.   If the input signal is given by

{f[n]} = 2{δ[n]}+{δ[n − 1]}−{δ[n − 2]}

calculate and sketch the resulting output signal.

d.   What are the numerical values of the frequency response of the system at ω = 0  and

ω = π?

3.   Consider the system, S, consisting of the following consecutive (cascaded) operations:

i.      Filtering of the input signal by a real LTI system having a frequency response of H(ejω)

ii.      Time reversal of the output of the LTI system given in step i.

iii.      Filtering of the output of step ii by the real LTI system having a frequency response of H(ejω)

iv.      Time reversal of the output of the LTI system given in iii.

Show that the overall system, S, has a zero-phase frequency response.

4.  Consider the following bilateral z-transform of a discrete-time signal x[n] : X(z) =

a.   What are all of the possible (non-null) regions of convergence for X(z)?

b.   Determine the causal signal x[n] corresponding to X(z) .

5.  Consider a causal system described by the following difference equation where the input signal is {x[n]} and the output signal is {y[n]} :

y[n + 1]+y[n] = x[n + 1]

a.   What is the transfer function of the system

b.   What is the impulse response of this system

c.   What is the solution of the difference equation for y[0] = 1 and x [n] = (−1)n u[n]?