EEL5701: Exam 2

1. (25 points)

   A stable digital filter has the following transfer function

   

   1. (5 points) Derive the poles and zeros of H(z) and sketch them on a pole/zero plot. State the ROC.

   2. (20 points) Derive a stable h[n] from the given H(z)

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2. (25 points) You are given the following difference equation:

   

   

   1. (5 points) Derive

   2. (10 points) Derive a causal impulse response h[n].

   3. (10 points) Provide a new set of difference equations (to replace (1) and (2) on the last page) such that a stable and causal system exists that has the same magnitude response as H(z). (Note: you can do this part even if you are not able to complete part b).

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3. (25 points) A generalized linear, FIR, causal filter has zeros at the following locations: 1, -1, and 1/2. (hint: some of these can be multiple zeros and there can be zeros at other locations).

   Answer the following questions (hint: you can answer them in any order)

   1. (5 points) What other zero must be present if this filter is one of the four filter types (I-IV)?

   2. (5 points) Can this filter be type I? Either give an example impulse response h[n] or explain why it can't be created with this type.

   3. (5 points) Can this filter be type II? Either give an example impulse response h[n] or explain why it can't be created with this type.

   4. (5 points) Can this filter be type III? Either give an example impulse response h[n] or explain why it can't be created with this type.

   5. (5 points) Can this filter be type IV? Either give an example impulse response h[n] or explain why it can't be created with this type.

4. (25 points) Short Answer
   1. (5 points) Can an all-pass filter be minimum phase? Either give an example or explain why it is not possible.

   2. (5 points) Is it possible for a minimum phase FIR filter to have linear phase? Either give an example or explain why it is not possible.

   3. (5 points) You are given a stable pole-zero system with all of its poles inside the unit circle. Does this system have to be causal? Either explain why the system has to be causal or provide a counter example.

   4. (5 points)

      Given the above pole/zero plot, sketch the magnitude of .

   5. (5 points) Given the following filter:

      

   Derive Y(z) as a function of X(z).