2.5. Exercises

Exercise 2.1

Imagine you are given a sampled signal $x[n]$ of $N$ samples, taken at $f_s=8000$ Hz. If you derive a new signal $y[n]$ by taking every other sample from $x[n]$:

$$y[n]=x[2\cdot n],$$

• What is the sampling rate $f_s$ of $y[n]$?

• What is the sampling period $t_s$ of $y[n]$?

• How many samples will $y[n]$ have?

Hint

Be sure to consider what happens when $N$ is even or odd.

Exercise 2.2

If the sampling rate is $f_s=100$ Hz, use the aliasing equation (2.3) to find two aliases for each of the following frequencies:

• $f=100$ Hz

• $f=-25$ Hz

• $f=210$ Hz

Exercise 2.3

Imagine that you want to sample a continuous wave $x(t)=\cos (2\pi \cdot f\cdot t)$ of some arbitrary (known) frequency $f>0$. Which sampling rates $f_s$ would you need to use to produce the following discrete observations by sampling $x(t)$? Express your answers as a fraction of $f$.

• $x[n]=1,1,1,1,\dots$,

• $x[n]=1,0,-1,0,1,0,-1,0,\dots$,

• $x[n]=1,\sqrt{1/2},0,-\sqrt{1/2},-1,\dots$.

Which (if any) of these sampling rates satisfy the conditions of the Nyquist-Shannon theorem?

Exercise 2.4

Using the quantize function from this chapter, apply different levels of quantization (varying n_bits from 1 to 16) to a recording of your choice. For each quantized signal, subtract it from the original signal and listen to the difference:

# We need to normalize the signal to cover the same range
# before and after quantization if we're going to compare them.
x_span = x.max() - x.min()
x_diff = x / x_span - quantize(x, n_bits) / 2**n_bits

What does it sound like for each quantization level?