4.2 Signal Processing & Sampling

Frequency-domain thinking

The Signal Processing area is about 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 at f0 plus integer-multiple harmonics (2f0, 3f0, …). 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 the frequency domain. Time and frequency are two complementary views of the same signal — a narrow pulse in time is broad in frequency, and a pure sinusoid (infinitely long in time) is a single spike in frequency. This time–frequency reciprocity underlies why fast signals (sharp edges, short pulses) inherently occupy wide bandwidth and why band-limiting a signal necessarily smears it in time.

For a periodic signal, the Fourier-series coefficients give the amplitude and phase of each harmonic; the magnitude-versus-harmonic plot is the signal's line spectrum. A square wave, for example, contains only odd harmonics with amplitudes falling off as 1/n, which is why reconstructing sharp edges requires many high-frequency terms — a direct illustration of the time–frequency trade.

On the exam you are far more likely to be asked to interpret a spectrum (which harmonics are present, where the fundamental sits, what bandwidth a waveform occupies) than to compute Fourier coefficients by hand, so practice reading spectra and matching them to waveforms.

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 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 0 to fs/2.

Worked: A 30 kHz tone is sampled at fs = 50 kHz. Nyquist frequency is fs/2 = 25 kHz, and 30 kHz exceeds it, so it aliases. The apparent frequency is |30 − 1·50| = |−20| = 20 kHz — the 30 kHz tone masquerades as 20 kHz. The fix is an anti-aliasing filter: an analog low-pass filter placed before the sampler that removes energy above fs/2.

Filter types and the dB / cutoff math

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

• Low-pass: passes below a cutoff, attenuates above (anti-aliasing, 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.

The decibel is a logarithmic ratio, and the multiplier depends on the quantity: 20·log10 for amplitude/voltage ratios and 10·log10 for power ratios (because power ∝ voltage²). The simple first-order RC low-pass filter has cutoff (corner) frequency in hertz:

f_c = 1/(2πRC) (equivalently ωc = 1/(RC) in rad/s)

Worked dB/cutoff: For R = 1 kΩ and C = 0.1 µF, f_c = 1/(2π·1000·0.1×10⁻⁶) = 1/(6.28×10⁻⁴) ≈ 1.59 kHz. At f_c the output amplitude is 1/√2 = 0.707 of the input, which is 20·log10(0.707) = −3 dB — the half-power point. One decade above f_c the response is down −20 dB (a factor of 10 in amplitude).

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 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.

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 for a given selectivity, but stability must be checked against the unit circle). When a problem mixes domains, remember the mapping rule: stability in the s-domain (left half-plane) corresponds to the inside of the unit circle in the z-domain, and the jω-axis maps to the unit circle itself.