4.2.6. Exercises

1. Calculate the Z-transform of the step function \(u[n]\).

2. Show that for any function \(x[n]\), the bilateral and unilateral Z-transforms of the function \(x[n]u[n]\) coincide.

3. The standard form for a difference equation defining a LTI system is:

   \[y[n] = \sum_{m=0}^{M} b_m x[n-m] - \sum_{k=1}^{N} a_k y[n-k]\]

   Assuming that \(a_0=1\) this can be rewritten as:

   \[\sum_{k=0}^{N} a_k y[n-k] = \sum_{m=0}^{M} b_m x[n-m]\]

   Starting with the above equation calculate the transfer function \(H(z)\) for this LTI system.

4. Calculate the transfer functions for causal (!) running average filter over 5, 9 and 13 samples. Causality implies that a time delay is introduced compared with the running average over samples both in the past and future. Plot the Argand diagram with poles and zeros and the frequency response. This time you are NOT allowed to use the audiolazy package.

5. In the frequency response of the 5 point running average filter you see that there exists frequencies \(\omega\) for which the transfer function equals zero. What do the sinusoids for which this is true look like? What is the relation with the length (5 in this case) of the running average filter?

6. Find the transfer function for a band pass filter on the web. Tune this filter such that the band pass region is from around 1000 Hz to around 10000 Hz. Assume the signal is sampled at 44.100 samples per second.