Basic Signal Operations

Five fundamental time-domain operations that complement the existing signal measurements (energy, power, entropy) and FFT-based spectrum analysis. Together they give you a toolkit for analysing and manipulating signals before or instead of going to the frequency domain.

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RMS (Root Mean Square)

RMS is the standard measure of signal "loudness" — the square root of the mean squared amplitude:

\[\mathrm{RMS} = \sqrt{\frac{1}{N}\sum_{n=0}^{N-1} x[n]^2} \]

Reading the formula in C:

// x[n]^2  ->  a[i] * a[i]          square each sample
// sum     ->  accumulate in a loop  sum of squares
// 1/N     ->  / (double)N           divide by number of samples
// sqrt()  ->  sqrt()                take the square root
double sum = 0.0;
for (unsigned i = 0; i < N; i++)
    sum += a[i] * a[i];
double rms = sqrt(sum / (double)N);

A unit-amplitude sine wave has RMS = \(1/\sqrt{2} \approx 0.707\). A DC signal of value \(c\) has RMS = \(|c|\).

API:

double MD_rms(const double *a, unsigned N);

Quick example — measure the RMS of three signals:

    double rms_sine  = MD_rms(sine, N);
    double rms_noise = MD_rms(noise, N);
    double rms_mixed = MD_rms(mixed, N);

Verification tip: MD_rms(a, N) should always equal sqrt(MD_power(a, N)) to within floating-point precision.

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Zero-Crossing Rate

The zero-crossing rate (ZCR) counts how often the signal changes sign, normalised by the number of adjacent pairs:

\[\mathrm{ZCR} = \frac{1}{N-1}\sum_{n=1}^{N-1} \mathbf{1}\!\bigl[\mathrm{sgn}(x[n]) \ne \mathrm{sgn}(x[n-1])\bigr] \]

Reading the formula in C:

// Walk through the signal and count sign changes.
// Zero is treated as non-negative (standard convention).
unsigned crossings = 0;
for (unsigned i = 1; i < N; i++)
    if ((a[i] < 0.0) != (a[i - 1] < 0.0))
        crossings++;
double zcr = (double)crossings / (double)(N - 1);

A pure sine at frequency \(f\) with sample rate \(f_s\) has ZCR \(\approx 2f/f_s\). White noise has a higher ZCR than a low-frequency tone — this makes ZCR a simple proxy for distinguishing noise from tonal content.

API:

double MD_zero_crossing_rate(const double *a, unsigned N);

Quick example — compare ZCR of sine, noise, and their mix:

    double zcr_sine  = MD_zero_crossing_rate(sine, N);
    double zcr_noise = MD_zero_crossing_rate(noise, N);
    double zcr_mixed = MD_zero_crossing_rate(mixed, N);

Verification tip: for a sine wave with an integer number of cycles in the buffer, the ZCR should be very close to \(2f/f_s\).

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Autocorrelation

The autocorrelation measures the similarity between a signal and a delayed copy of itself. The normalised form divides by \(R[0]\) (the signal energy) so the output is bounded:

\[R[\tau] = \frac{\displaystyle\sum_{n=0}^{N-1-\tau} x[n]\,x[n+\tau]} {\displaystyle\sum_{n=0}^{N-1} x[n]^2} \]

Reading the formula in C:

// Denominator: sum of x[n]^2  (the signal energy, R[0])
double r0 = 0.0;
for (unsigned n = 0; n < N; n++)
    r0 += a[n] * a[n];
 
// Numerator for each lag tau: sum of x[n] * x[n + tau]
for (unsigned tau = 0; tau < max_lag; tau++) {
    double sum = 0.0;
    for (unsigned n = 0; n < N - tau; n++)
        sum += a[n] * a[n + tau];
    out[tau] = sum / r0;   // normalise so out[0] = 1.0
}

\(R[0] = 1.0\) always, and \(|R[\tau]| \le 1.0\). For a periodic signal, the autocorrelation peaks at the fundamental period (the inverse of the fundamental frequency) — this is the basis of time-domain pitch detection.

API:

void MD_autocorrelation(const double *a, unsigned N,
                        double *out, unsigned max_lag);

Quick example — autocorrelation of a noisy sine:

    MD_autocorrelation(mixed, N, acf, max_lag);

Verification tip: for a 200 Hz sine at 8000 Hz sample rate, the autocorrelation should peak at lag 40 (= 8000 / 200).

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Peak Detection

Peak detection finds local maxima in a signal. A sample \(a[i]\) is a peak if it is strictly greater than both immediate neighbours and above a given threshold. The min_distance parameter suppresses nearby secondary peaks.

Reading the algorithm:

// Left-to-right scan:
// 1. a[i] > a[i-1] AND a[i] > a[i+1]  (strict local maximum)
// 2. a[i] >= threshold                  (above noise floor)
// 3. distance from last peak >= min_distance  (suppress echoes)

Endpoints (i=0 and i=N-1) are never peaks because they lack two neighbours. A flat signal (all values equal) produces zero peaks.

API:

void MD_peak_detect(const double *a, unsigned N, double threshold,
                    unsigned min_distance, unsigned *peaks_out,
                    unsigned *num_peaks_out);

Quick example — find peaks in the autocorrelation to estimate pitch:

    MD_peak_detect(acf, max_lag, 0.3, 10, peaks, &num_peaks);

Verification tip: pass a hand-crafted signal like {0, 1, 3, 1, 0, 2, 5, 2, 0} with threshold 0 and min_distance 1. You should get peaks at indices 2 and 6 (values 3 and 5).

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Signal Mixing

Mixing computes the element-wise weighted sum of two signals:

\[\mathrm{out}[n] = w_a \cdot a[n] + w_b \cdot b[n] \]

Reading the formula in C:

for (unsigned i = 0; i < N; i++)
    out[i] = w_a * a[i] + w_b * b[i];

Equal weights of 0.5 produce the average. A weight of 1.0/0.0 passes one signal through unchanged. The output buffer may alias either input (in-place safe).

API:

void MD_mix(const double *a, const double *b, double *out,
            unsigned N, double w_a, double w_b);

Quick example — mix 80% sine with 20% noise:

    MD_mix(sine, noise, mixed, N, 0.8, 0.2);

Verification tip: for uncorrelated signals (e.g. a sine and white noise), the energy of the mix is approximately the sum of energies: \(E(\mathrm{mix}) \approx w_a^2 E(a) + w_b^2 E(b)\).