33 Linear Filtering of Stationary Stochastic Processes

A filter \(L\) can take a time varying signal and smooth it out over time, or limit the signal to a certain range of frequencies.
- Input signal: \(X(t)\), usually assumed to be a WSS stochastic process
- Output signal: \(Y(t) = L[X(t)]\)
A linear time-invariant (LTI) filter \(L\) satisfies
- (Linearity.) \(L[c_1 X_1 (t) + c_2 X_2(t)] = c_1 L[X_1(t)] + c_2 L[X_2(t)]\)
- (Time-invariance.) If \(Y(t) = L[X(t)]\) then \(Y(t-d) = L[X(t-d)]\).
A LTI filter is characterized in the time domain by its impulse response function, \(h(t)\), which represents the filter’s response at time \(t\) to a unit impulse at time \(t=0\): \[ h(t) = L[\delta(t)] \]
- In the time domain, the output signal is the convolution of the input signal with the impulse response \[ Y(t) = X(t)\star h(t) = \int_{-\infty}^{\infty} X(s)h(t-s)\, ds \]
A LTI filter is characterized in the frequency domain by its transfer function (a.k.a. frequency response), \(H(f)\): \[ H(f) = \mathcal{F}\{h(t)\} \]
If the input signal \(X(t)\) to a LTI filter \(L\) is WSS, then the output signal \(Y(t)\) is also WSS. (Furthermore, \(X(t)\) and \(Y(t)\) are jointly WSS.) \[\begin{align*} & & &\text{Time domain} & & \text{Frequency domain} \\ \text{mean} & & \mu_Y & = \mu_X \int_{-\infty}^{\infty} h(t)\, dt & \mu_Y & = \mu_X H(0) \\ \text{autocorrelation:} & & R_{Y}(\tau) & = R_{X}(\tau)\star h(\tau)\star h(-\tau) &\qquad \text{psd:} \quad S_{Y}(f) & = S_{X}(f)\,|H(f)|^2 \\ %\text{crosscorrelation} & & R_{XY}(\tau) & = R_{X}(\tau)\star h(\tau) & S_{XY}(f) & = S_{X}(f)\, H(f) & & \text{cross-psd} \end{align*}\]
\(|H(f)|^2\) is the power transfer function of \(L\)
Power spectral density of the filtered signal is the product of the power spectral density of the input signal and the power transfer function of the filter.
Recall: for complex number \(a+b j\), \(|a+b j|^2 = (a+bj)(a-bj) = a^2 + b^2\).

Example 33.1 (RC circuit.) A pure white noise signal is passed through a LTI filter with impulse response \[ h(t) = \frac{1}{RC}e^{-t/RC} u(t) \] where \(u(t)\) is the unit step function (\(u(t)=1\) for \(t\ge0\), and \(u(t)=0\) otherwise).

Find the transfer function.
Find the power spectral density of the output signal.
Find the expected power in the output signal.

Signal plus noise models are commonly used in communications.
\(X(t)\) is intended signal, assumed to be WSS
\(N(t)\) is noise, assumed to be WSS with mean 0 and independent of \(X(t)\)
\(X(t)+N(t)\) is received transmission, which is input to filter
Autocorrelation function of input signal \(X+N\): \[ R_{X+N}(\tau) = R_{X}(\tau) + R_{N}(\tau) \]
Power spectral density: \[ S_{X+N}(f) = S_{X}(f) + S_{N}(f) \]
Pass input \(X(t) + N(t)\) through LTI filter \(L\), output is \(L[X+N]\) \[ L[X(t) + N(t)] = L[X(t)] + L[N(t)] \]
One measure of the quality of the filter involves comparing the power signal-to-noise ratio of the input and the output \[ \text{input:}\quad \frac{P_X}{P_N} \qquad \text{versus} \qquad \text{output:}\quad \frac{P_{L[X]}}{P_{L[N]}} \]

Example 33.2 A signal can be modeled by \(X(t) + N(t)\) where:

The intended signal \(X(t)\) is WSS and has autocorrelation function \[ R_X(\tau) = 5000 + 30000\cos(2000\pi \tau) + 45000\, \text{sinc}^2(1800\tau) \]
The noise \(N(t)\) is WSS with mean zero and autocorrelation function \[ R_N(\tau) = 2000\exp(-10000|\tau|) \]
To filter out the noise, we pass the waveform through an ideal lowpass filter with transfer function \[ H(f) = 1, \quad |f| < 1800\, \textrm{Hz} \]

Find the power signal-to-noise ratio of the input signal.
Find and graph the power spectral density of \(X\).
Find and graph the power spectral density of \(N\).
Find and graph the power spectral density of \(L[X]\), the signal part of the output.
Find and graph the power spectral density of \(L[N]\), the noise part of the output.
Find the power signal-to-noise ratio of the output.