3. Signal Processing (Continuous to Discrete) · Concept 10 of 11
Wiener–Khinchin Theorem
It is a rule linking how a signal repeats itself over time to which frequencies it contains.
Autocorrelation (left) and power spectrum (right) are the same steady signal seen two ways, linked by the Fourier transform.
What it is
A rule: a steady signal's autocorrelation (self-similarity over time delay) and its power spectrum are a Fourier transform pair.
Key facts
Statement: Power Spectral Density (PSD) = Fourier Transform of the autocorrelation function. Same information, two views.
Autocorrelation R(tau): multiply the signal by a copy of itself shifted by delay tau, then average. tau = time shift in seconds.
PSD S(f): how much signal power sits at each frequency f, in power per hertz (e.g. V^2/Hz).
Pair: S(f) = integral of R(tau) * e^(-j*2*pi*f*tau) d(tau). f = frequency in Hz, j = imaginary unit, e = 2.71828.
R(0) = total average power of the signal = area under the whole PSD curve. Zero delay means perfect self-match.
Only valid for WSS signals (Wide-Sense Stationary): steady stats that do not drift. Hum, hiss, rumble qualify.
White noise: R(tau) is one spike at tau=0, zero elsewhere; its PSD is perfectly flat (equal power per Hz). Pure tone at f0: PSD is a single line at f0.
Named after Norbert Wiener (1930) and Aleksandr Khinchin (1934). Powers the Welch/periodogram method in every RTA.
Reading the result in dB: doubling power = +3 dB; doubling voltage/SPL = +6 dB; half-power point = -3 dB.
Audio band 20 Hz-20 kHz. Mains hum = 50 Hz in Australia plus harmonics 100, 150, 200 Hz. Speed of sound approx 343 m/s at 20 C.
How it works
Record a chunk of steady signal (e.g. the system hiss with the mic open).
Slide a copy of it against itself; at each delay tau measure how well it matches = autocorrelation R(tau).
Fourier transform R(tau). Out pops the power spectrum S(f).
Read the peaks: tall narrow peaks = tonal energy (hum, ring); flat carpet = broadband noise (hiss).
In practice analysers shortcut this: square the FFT magnitude and average many frames (Welch) = same PSD.
Real examples
Ringing out a wedge: the RTA carpet shows a sharp spike at one frequency = the feedback ring to notch.
A 50 Hz spike plus 100/150 Hz harmonics on the analyser = mains hum / earth loop, not the band.
Flat low rumble below 40 Hz = aircon or stage traffic; high-pass it.
Steady broadband hiss = preamp gain too high or noisy gear; its PSD is a flat shelf.
White-noise and pink-noise test signals: their known flat/sloped PSD is exactly what Wiener-Khinchin predicts.
How it helps in live sound
Use a long RTA average (slow/infinite) on steady noise so the PSD estimate settles and the floor smooths.
Hunt hum at 50 Hz and harmonics (100/150/200 Hz) first; fix earth loops before EQ.
Notch feedback by its spectral peak: narrow Q (1/10 octave) cut at the spiking frequency.
Pink noise + RTA to tune a room: a flat-ish PSD target on the analyser = balanced response.
High-pass vocals around 80-100 Hz to kill the low rumble seen as PSD energy under 50 Hz.
Average over seconds, not one snapshot: a single FFT frame is too noisy to trust a peak.
Everyday analogy
Like tapping a rhythm: noticing it repeats every second (self-similarity in time) instantly tells you there's a steady 1 Hz beat (a frequency).
Watch out
Myth: the spectrum tells you WHEN events happen. Wrong. The PSD throws away all phase/timing; two very different-sounding signals can share an identical power spectrum.
Fun fact
A signal and its time-reversed twin have the EXACT same power spectrum, because autocorrelation is symmetric, R(tau) = R(-tau). The analyser cannot tell them apart.
Key takeaways
Autocorrelation (time) and power spectrum (frequency) are two views of the same steady signal.
Self-similarity in time = structure in frequency.
R(0) = total power = area under the PSD.
Only holds for stationary (steady-stats) signals like hum, hiss, rumble.
It is the engine behind every analyser's noise-spectrum estimate.
Spectrum keeps power-per-Hz but discards all timing/phase info.