• 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).

1. Find the transfer function.

2. Find the power spectral density of the output signal.

3. 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 $H(f) = 1, \quad |f| < 1800\, \textrm{Hz}$
1. Find the power signal-to-noise ratio of the input signal.

2. Find and graph the power spectral density of $$X$$.

3. Find and graph the power spectral density of $$N$$.

4. Find and graph the power spectral density of $$L[X]$$, the signal part of the output.

5. Find and graph the power spectral density of $$L[N]$$, the noise part of the output.

6. Find the power signal-to-noise ratio of the output.