signal-processing-dsp-basics

# Signal Processing: DSP Basics

Digital signal processing is what happens after an ADC and before a DAC — or inside the world of purely discrete-time signals like audio files, sensor logs, and computed waveforms. The core ideas of DSP are deceptively simple (sample, add, multiply, feed back) but the traps are numerous (aliasing, leakage, numerical underflow, phase distortion, quantization noise). This skill covers the fundamentals every embedded or systems engineer needs to avoid making expensive analog problems digital.

**Agent affinity:** shima (DSP architecture and fixed-point implementation), horowitz (intuition and practical filter examples)

**Concept IDs:** elec-data-conversion-dsp, elec-signal-ac-analysis

## Sampling and the Nyquist Criterion

A continuous-time signal x(t) is sampled by taking its value at evenly spaced instants T_s apart, producing the discrete-time sequence x[n] = x(n * T_s). The sample rate f_s = 1 / T_s.

**The Nyquist-Shannon sampling theorem.** A band-limited signal whose frequency content is strictly below f_s / 2 can be perfectly reconstructed from its samples. Any frequency content above f_s / 2 gets aliased — folded back into the band from 0 to f_s / 2 — and becomes indistinguishable from legitimate signal content at the aliased frequency.

**The critical corollary.** The sample rate must be at least twice the highest frequency of interest. For audio at up to 20 kHz, the minimum is 40 kHz (compact disc uses 44.1 kHz, professional audio 48 or 96 kHz). For a temperature sensor whose highest frequency of interest is 1 Hz, 10 Hz is plenty.

**Anti-aliasing filters.** Before the ADC, an analog low-pass filter must attenuate everything above f_s / 2 below the noise floor of the ADC. No amount of post-processing can undo aliasing; once two frequencies fold onto each other, they are indistinguishable.

## Technique 1 — Quantization

An ADC represents each sample as one of 2^N levels, where N is the bit depth. The rounding error at each sample is a **quantization error**, uniformly distributed in the interval (-q/2, q/2) where q is the step size. Its RMS value is q / sqrt(12), and the resulting signal-to-quantization-noise ratio (SQNR) is approximately 6.02 * N + 1.76 dB.

**Practical numbers:**

| Bits | SQNR (dB) |
|---|---|
| 8 | 50 |
| 10 | 62 |
| 12 | 74 |
| 16 | 98 |
| 24 | 146 |

16-bit audio provides about 98 dB of dynamic range, which is near the limit of what consumer headphones can resolve. 24-bit audio is used in professional recording for headroom, not because the last eight bits are audible.

**Dithering.** Adding a small amount of random noise before quantization decorrelates the quantization error from the signal, trading distortion for noise. This is why 16-bit masters sound better than naive 16-bit truncation of a 24-bit original.

## Technique 2 — The Discrete Fourier Transform

The DFT of a length-N sequence x[n] is:

    X[k] = sum over n=0..N-1 of x[n] * exp(-j * 2π * k * n / N)

It represents the sequence as a sum of N complex exponentials at frequencies k * f_s / N. The magnitude |X[k]| shows how much energy is at each frequency; the phase shows the relative timing.

**The FFT.** The fast Fourier transform is an O(N log N) algorithm for computing the DFT; the naive computation is O(N^2). For N = 1024 the FFT is 100x faster, and the ratio grows with N.

**Worked example.** A 1024-sample buffer sampled at 48 kHz has frequency resolution 48000 / 1024 ≈ 46.9 Hz per bin. Bin k = 100 corresponds to 4.69 kHz. To resolve adjacent musical notes near 440 Hz (about 26 Hz apart between half-steps), a longer window is needed — e.g., 4096 samples gives 11.7 Hz resolution.

**Real vs complex.** A real-valued signal has conjugate-symmetric FFT: X[N-k] = conj(X[k]). Only the first N/2 + 1 bins carry independent information. DSP libraries provide real-FFT variants that exploit this for 2x speedup.

## Technique 3 — Windowing and Spectral Leakage

The DFT implicitly assumes the sequence is periodic with period N. If the actual signal is not an integer number of cycles in the window, the discontinuity at the boundary spreads energy across all frequency bins — **spectral leakage**.

**Windowing** multiplies the sequence by a tapered function that smoothly decays to zero at both ends of the buffer, suppressing the boundary discontinuity. Common windows:

| Window | Main lobe width | Side lobe level | Use |
|---|---|---|---|
| Rectangular | narrow | -13 dB | Baseline; worst leakage |
| Hann | medium | -31 dB | General-purpose |
| Hamming | medium | -43 dB | Speech analysis |
| Blackman-Harris | wide | -92 dB | Strong tones near weak tones |
| Kaiser (tunable) | variable | variable | When trade-off must be exact |

**Trade-off.** Narrower main lobe means better frequency resolution but worse side-lobe rejection. Tapered windows sacrifice some resolution for enormously better leakage suppression.

## Technique 4 — Convolution and FIR Filters

A finite impulse response filter has output:

    y[n] = sum over k=0..M-1 of h[k] * x[n - k]

where h[k] are the filter coefficients (the impulse response). This is a discrete convolution. FIR filters are always stable (no feedback, no poles) and can have exactly linear phase if the coefficients are symmetric — meaning all frequencies are delayed by the same amount, preserving waveform shape.

**Design methods:**

- **Window method.** Compute the ideal impulse response (sinc for a low-pass), truncate to M taps, multiply by a window.
- **Parks-McClellan (equiripple).** Optimize coefficients to minimize maximum error in the pass and stop bands.
- **Least-squares.** Minimize the sum of squared errors instead of the maximum error.

**Practical note.** FIR filter computation is dominated by multiply-accumulate (MAC) operations. DSP chips like TI's C6000 and ARM Cortex-M with DSP extensions can do one MAC per cycle, which is why they are called "DSP."

## Technique 5 — IIR Filters

Infinite impulse response filters use feedbac