Back in 2006 I taught an introductory Signals and Systems lecture in which, instead of only drawing plots on the board, I let the class listen to the concepts. Almost two decades later, here are those very sounds — all generated or recorded by me — so you can grasp three core ideas of the course with your own ears.
A signal is a function that maps a domain (often time) to a co-domain (a physical quantity, such as air pressure). And a system is a function that transforms signals into other signals. Sound : Time → Pressure.
1. Pure tones
The simplest signal is a pure sinusoid — a single frequency. The frequency sets the pitch of the sound.
PureTone(t) = P · sin(2π · f · t)
440 Hz (the A orchestras tune to) — [play/download]
1 000 Hz (a clear mid-range tone) — [play/download]
10 000 Hz (high-pitched — some ears no longer catch it) — [play/download]
2. Sum of sinusoids
Adding sinusoids of different frequencies produces richer signals — the seed of Fourier analysis: any sound can be seen as a sum of pure tones.
1 000 Hz + 10 000 Hz — [play/download]
3. Sampling and quantization
A computer does not store continuous signals: it stores samples taken at regular intervals, each with a finite number of bits. An audio CD uses 44 100 samples per second and 16 bits per sample. The clip below is just a phone-microphone recording of my voice — nowhere near studio quality — but at 44 100 Hz it is our full-rate reference. What happens if we drop to only 4 410 samples per second?
44 100 Hz (full rate — reference) — [play/download]
4 410 Hz, filtered (correct sampling: muffled but clean) — [play/download]
4 410 Hz, no filter (no anti-aliasing filter: the buzz of aliasing) — [play/download]
Compare the last two. Lowering the sampling rate loses the high frequencies (the sibilants: s, f, x). Done correctly (low-pass filtering first), the sound merely becomes muffled. Done naively (no filter), the high frequencies fold down as a metallic distortion — the famous aliasing, and exactly why the sampling theorem (Nyquist–Shannon) matters.
Where this meets the syllabus
- Signals & systems — a signal is a function (time → quantity); a system transforms signals.
- Sinusoids & frequency — a pure tone is sin(2πft); frequency sets pitch.
- Superposition & Fourier — any sound is a sum of sinusoids.
- Sampling — a continuous signal becomes a sequence of samples at rate fs.
- Quantization — each sample has finite resolution (16 bits on a CD).
- Sampling theorem (Nyquist–Shannon) — to reconstruct a signal you need fs > 2·fmax.
- Aliasing & anti-aliasing filter — without low-pass filtering before sampling, high frequencies fold down as audible distortion.
Exam-style question
A signal contains frequencies up to 20 kHz and is sampled at fs = 44.1 kHz.
(a) Does it satisfy the sampling theorem? What is the Nyquist frequency?
(b) Instead we now sample at fs = 4.41 kHz with no anti-aliasing filter. To what frequency does a 4 kHz component fold?
(c) How should we sample correctly at 4.41 kHz, and what is the audible effect?
Answers. (a) The Nyquist frequency is fs/2 = 22.05 kHz. Since 20 kHz < 22.05 kHz, the theorem is satisfied. (b) Now Nyquist = 2.205 kHz, and 4 kHz exceeds it, so it aliases to |f − fs| = |4 − 4.41| kHz = 0.41 kHz — the 4 kHz tone is heard as a spurious 410 Hz. (c) Low-pass filter the signal below 2.205 kHz before downsampling (anti-aliasing). The sound loses its highs (muffled) but contains no aliasing distortion.
References & credits
- H. Nyquist, “Certain Topics in Telegraph Transmission Theory,” Trans. AIEE, vol. 47, pp. 617–644, 1928.
- C. E. Shannon, “Communication in the Presence of Noise,” Proc. IRE, vol. 37, no. 1, pp. 10–21, 1949. (the sampling theorem)
- A. V. Oppenheim and A. S. Willsky, Signals and Systems, 2nd ed., Prentice Hall, 1997.
- E. A. Lee and P. Varaiya, Structure and Interpretation of Signals and Systems, 2nd ed., 2011.
- Course text: D. Valério, Sinais e Sistemas (SSM, Instituto Superior Técnico).
All sounds generated or recorded by Carlos Cardeira. Tones and sampling example from a 2006/07 Signals and Systems lecture; voice re-recorded in 2026. Approach inspired by Lee & Varaiya, “Structure and Interpretation of Signals and Systems”. Part of the Signals and Systems Curiosities series.