Signals and Systems, Lecture 7

Sampling, and when it goes wrong

The Nyquist rate, aliasing, and how to choose a sampling frequency.

Department of Electrical EngineeringSample lectureWeek 7

Today

1. Why sample at all 5 min
2. The sampling theorem 15 min
3. Aliasing, demonstrated 10 min
4. Quiz and worked problem 20 min

Theorem

The Nyquist condition

f_s

Sampling frequency, samples per second

f_max

Highest frequency present in the signal

2f_max

The Nyquist rate, the minimum to avoid aliasing

Example

Audio to 20 kHz needs more than 40 kHz; CDs use 44.1 kHz

Demo

See aliasing in five lines

    import numpy as np
    t = np.arange(0, 1, 1/100)          # fs = 100 Hz
    x = np.sin(2*np.pi*90*t)            # a 90 Hz tone
    # 90 Hz is above fs/2, so it appears at 10 Hz
    plt.plot(t, x)
  

Check

A 6 kHz tone is sampled at 8 kHz. What frequency appears?

1. 6 kHz
2. 4 kHz
3. 2 kHz
4. 14 kHz

Take away

• Sample above twice the highest frequency.
• Filter first. An anti-aliasing filter removes what you cannot capture.
• Aliases cannot be undone after sampling.

Before next week

Reading

Chapter 4, sections 4.1 to 4.3Office hours Tuesday