33 Linear Filtering of Stationary Stochastic Processes
- A filter
can take a time varying signal and smooth it out over time, or limit the signal to a certain range of frequencies.- Input signal:
, usually assumed to be a WSS stochastic process - Output signal:
- Input signal:
- A linear time-invariant (LTI) filter
satisfies- (Linearity.)
- (Time-invariance.) If
then .
- (Linearity.)
- A LTI filter is characterized in the time domain by its impulse response function,
, which represents the filter’s response at time to a unit impulse at time :- In the time domain, the output signal is the convolution of the input signal with the impulse response
- In the time domain, the output signal is the convolution of the input signal with the impulse response
- A LTI filter is characterized in the frequency domain by its transfer function (a.k.a. frequency response),
: - If the input signal
to a LTI filter is WSS, then the output signal is also WSS. (Furthermore, and are jointly WSS.) is the power transfer function of- 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
, .
Example 33.1 (RC circuit.) A pure white noise signal is passed through a LTI filter with impulse response
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.
is intended signal, assumed to be WSS is noise, assumed to be WSS with mean 0 and independent of is received transmission, which is input to filter- Autocorrelation function of input signal
: - Power spectral density:
- Pass input
through LTI filter , output is - One measure of the quality of the filter involves comparing the power signal-to-noise ratio of the input and the output
Example 33.2 A signal can be modeled by
- The intended signal
is WSS and has autocorrelation function - The noise
is WSS with mean zero and autocorrelation function - To filter out the noise, we pass the waveform through an ideal lowpass filter with transfer function
Find the power signal-to-noise ratio of the input signal.
Find and graph the power spectral density of
.
Find and graph the power spectral density of
.
Find and graph the power spectral density of
, the signal part of the output.
Find and graph the power spectral density of
, the noise part of the output.
Find the power signal-to-noise ratio of the output.