S4-5.5 Linear Systems with Random Inputs

Work through these

• System response to random signals via convolution

  Everything in this subject assembles into one question here: put a random signal into a linear system, and describe what comes out. Convolution is the time-domain answer, and it is the starting point for the rest of the unit.

• Mean and mean-squared value of the response

  The mean of the output follows from the input mean and the system's response to a constant, and the mean-squared value gives the output power. These are the two quantities a design specification usually names.

• Autocorrelation of the response; input-output cross-correlations

  The output autocorrelation follows from the input autocorrelation and the system, and the input-output cross-correlations turn out to identify the system itself. That last fact is the basis of measuring an unknown system with noise.

• Power density spectrum of the response

  The frequency-domain answer is compact enough to memorise: output power spectral density equals input density times the squared magnitude of the transfer function. It is the analytical core of receiver and filter design.

• Cross-power density spectra of input and output

  The cross-spectra between input and output complete the picture and connect directly back to the correlation results. The same machinery reappears wherever filtered noise is analysed, including in signal processing and machine learning.