4.2 Signal Processing & Sampling

Frequency-domain thinking

The Signal Processing area is 5-8 of 110 questions and overlaps heavily with Linear Systems and Communications. The unifying idea is that any signal can be described as a combination of sinusoids. A Fourier series decomposes a periodic signal into a sum of harmonically related sinusoids (a fundamental plus integer-multiple harmonics). The Fourier transform generalizes this to aperiodic signals, mapping a time-domain waveform x(t) to a continuous spectrum X(ω):

X(ω) = ∫ x(t) e^(−jωt) dt

The payoff matches Linear Systems: a system's output spectrum is Y(ω) = X(ω)·H(jω), so filtering and modulation are easiest to reason about in frequency.

The sampling theorem and Nyquist rate

To process an analog signal digitally you must sample it — measure its value at uniform intervals Ts, giving the sampling rate fs = 1/Ts. The Nyquist–Shannon sampling theorem states that a signal band-limited to a maximum frequency fmax can be perfectly reconstructed only if:

fs > 2 · fmax

The quantity 2·fmax is the Nyquist rate — the minimum sampling rate. The frequency fs/2 is the Nyquist frequency — the highest frequency that a given fs can represent. Compact disc audio samples at 44.1 kHz precisely because human hearing tops out near 20 kHz, leaving headroom above the 40 kHz Nyquist rate.

Aliasing

If fs is too low — fs < 2·fmax — the spectral copies created by sampling overlap, and high-frequency content folds down to appear as a lower frequency. This is aliasing, and it is irreversible: once two frequencies map to the same sampled value, no later processing can separate them. A sinusoid at frequency f sampled at fs aliases to the apparent frequency |f − k·fs| for the integer k that lands the result in the band 0 to fs/2.

The fix is an anti-aliasing filter: an analog low-pass filter placed before the sampler that removes energy above fs/2. Because real filters are not brick-walls, designers either oversample or set the cutoff with margin below the Nyquist frequency.

Filter types

FE filter questions ask you to match a behavior to a filter class. The four basic magnitude responses are:

• Low-pass: passes frequencies below a cutoff, attenuates above (used for anti-aliasing and smoothing).
• High-pass: passes above a cutoff, blocks DC and low frequencies.
• Band-pass: passes a band between two cutoffs.
• Band-stop (notch): rejects a narrow band — e.g., removing 60 Hz line hum.

Digital filters split into FIR (finite impulse response — always stable, can be linear-phase, no feedback) and IIR (infinite impulse response — uses feedback, more efficient, but stability must be checked).

The z-transform

Discrete-time systems use the z-transform, the discrete analog of the Laplace transform:

X(z) = Σ_{n} x[n] · z^(−n)

A discrete LTI system has a transfer function H(z) = N(z)/D(z) with poles and zeros in the z-plane. The stability rule changes: instead of the left half-plane, a causal discrete system is stable when all poles lie strictly inside the unit circle, |z| < 1. The left-half-plane of the s-domain maps to the inside of the unit circle in the z-domain — a frequent FE distractor.