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Problem 1: Topics: Pole-Zero Plot, Stability, Causality, FIR vs. IIR

Given the Pole-Zero plot of a LTI discrete time filter:

a) Is this filter FIR or IIR?

b) Is this filter stable BIBO?

c) Is this filter Causal or Anticausal?

Problem 2: Topics: Stability, FIR vs. IIR, Difference Equation

Given the unit sample response of a LTI discrete time filter:

h[n] = (1/3)δ[n]+(1/3)δ[n-1]+(1/3)δ[n-2]

a) Is this a FIR or IIR filter?

b) Stable BIBO?

c) Write the filter difference equation.

Problem 3: Topics: Inverse Z transform

Given the Z transform of a signal X(z)=   z^-7/(1+0.9z^-1) with R.O.C  |z|< 0.9

a) Find x[n]=Z^-1{X(z)}

Problem 4: Topics: Fourier Series Coefficients

Given the following periodic signal: x(t)

a) Give an expression for the signal in the interval 0 <= t < 10

b) Find the Trigonometric Fourier Series coefficient a_n using the appropriate formula.

c) Find the Trigonometric Fourier Series coefficient a_n using any method.

d) Find the average power in x(t).

e) Write the integral expression that leads to the calculation of the Exponential Fourier Series coefficient c_n.  Include the actual values for the period T_o and ω_o (Do not perform the actual calculation).

Problem 5: Topics: Linear Convolution

Assume first entry in all depicted sequences correspond to discrete time zero: n=0. Given the discrete time linear time (shift) invariant system with input x[n] = [9  4  10  -1  6  -2]

a) Using any method find the filter output y[n].

Problem 6: Topics: Discrete Time Systems

Now consider a causal filter with difference equation y[n] =  x[n] + 0.8 y[n-1]

a) Is this filter IIR or FIR?

b) Find the System Function H(z) including R.O.C.   

c) Find the system unit sample response (impulse response) h[n] =

d) Sketch the Pole_Zero plot including R.O.C.

e) Find the filter unit-step response (output y[n] when the input is a unit-step function u[n]). 

f) Direct Form 1 Block Diagram realization of the filer.