3.7. Exercises

Exercise 3.1

For each of the following systems, what is its impulse response?

1. $y[n]=x[n]-x[n-1]$

2. $y[n]=2\cdot x[n]+x[n-2]$

3. $y[n]=\frac{1}{3}\cdot (x[n-2]+x[n-3]+x[n-4])$

Exercise 3.2

If $x$ is a signal of $N=100$ samples, an $h$ is an impulse response of $K=11$ samples, how many samples does $y=h\ast x$ have in each of the following modes?

1. Full

2. Valid

3. Same (centered)

Exercise 3.3

For each of the following systems, determine whether it is linear, shift-invariant, both, or neither.

1. $y[n]=-x[n]$

2. $y[n]=x[0]$

3. $y[n]=\frac{1}{2}\cdot |x[n]+x[n-1]|$

4. $y[n]=20$

5. $y[n]=n^2$

Exercise 3.4

Let $h=[1/K,1/K,\dots ,1/K]$ ($K$ times) denote a moving average filter so that if $y=h\ast x$, then $y[n]$ is the average of the previous $K$ samples in $x$.

Using an input recording $x$ of your choice (not more than 10 seconds), compute $y$ for different values of $K=1,4,16,64,256,1024$ and listen to each one.

How does each $y$ sound compared to the original signal? Do you notice any artifacts?

# Starter code for making a moving average filter
K = 8
h_K = 1/K * np.ones(K)