S4-5.5 Linear Systems with Random Inputs

What this is and why it exists

Everything in the course assembles here into one question: random signal in, linear system — what comes out? The answer is compact enough to feel like a reward: means pass through the DC gain, autocorrelations pass through a double convolution, and on the frequency side the output PSD is the input PSD times the squared magnitude of the frequency response. Hold that last relationship and you hold the analytical core of receiver design — and the same machinery reappears in DSP and machine learning wherever filtered noise is involved.

The vocabulary

• Convolution response — the output process: input convolved with the impulse response, sample function by sample function.
• Mean of the response — input mean times the sum (integral) of the impulse response: the DC gain.
• Mean-squared value — the output's average power; obtainable from the output autocorrelation at τ zero or by integrating the output PSD.
• Output autocorrelation — input autocorrelation convolved with the impulse response twice (once forward, once reversed).
• Input-output cross-correlation — for white input, proportional to the impulse response itself; the identity behind system identification.
• Output PSD — input PSD times the squared magnitude of H(f): the unit's central formula.
• Cross-power spectral density — the frequency-domain form of input-output cross-correlation: input PSD times H(f), once, unsquared.

The mental model

A filter is a doorman for frequencies, and a random signal arrives as a crowd. The PSD describes the crowd — how much power at each frequency — and the doorman's rulebook is the squared magnitude of H(f): at each frequency, power in times |H(f)| squared is power out. Squared, because PSD is power and H scales amplitude. Phase is absent from the output PSD — a doorman who delays guests changes no headcount — which is a genuine loss of information about waveform shape, and exactly why power spectra alone cannot distinguish systems differing only in phase.

The time-domain half tells the matching story. The output mean is the input mean scaled by the total weight of the impulse response — noise with zero mean stays zero-mean through any linear filter. The output autocorrelation is the input's, smeared twice by the impulse response: filters lengthen correlation. Pass white noise (zero-width correlation) through a narrow filter and the output wanders slowly, correlated over the filter's memory — colored, as the noise unit named it. The cross-relations are the practical gold: drive an UNKNOWN system with white noise and the input-output cross-correlation traces out the impulse response itself — system identification in one measurement, a trick used from channel sounding to room acoustics. On the frequency side the cross-power spectrum carries H(f) once and unsquared, phase intact, which is why identification methods prefer it.

Run the canonical example once with numbers: white noise of density N0 over 2 into an ideal low-pass filter of bandwidth B — output power N0 B, output autocorrelation a sinc function whose first zero falls at one over 2B. Every piece of the course participates: distributions (Gaussian in gives Gaussian out, by the closure from the operations unit), stationarity (WSS in, WSS out), Wiener-Khinchin bridging the two answers. Receiver noise-bandwidth calculations, matched-filter reasoning and estimator design all begin from exactly this computation.

What you should now be able to explain or do

Compute output mean, autocorrelation, PSD and total power for a given input process and system. Explain why phase vanishes from the output PSD but survives in the cross-spectrum. Derive the white-noise-through-low-pass results and interpret each. Describe white-noise system identification in two sentences.

Check yourself