This module describes the random process statistics of a typical sonar receiver, where the sonar data is frequency shifted and then sampled to a digital time series.
Active sonar receiver model
Consider the sonar receiver processing chain shown below:
The sonar array input is heterodyned, low-pass filtered, sampled and scaled to generate a discrete time set of samples of the noise plus signal.
The input noise
is a real-valued, wide sense stationary random process with power spectral density
. Because
is wide sense stationary, the power spectral density and the autocorrelation function are related by
We assume that
is zero mean. A complex carrier
is applied to the random process
resulting in a complex random process
The mean value of
is given by:
The covariance of
is given by
,
which shows that
is wide sense stationary as well.
is passed through a band-pass filter to produce
. The frequency response of the band-pass filter is assumed to be low-pass with a bandwidth of
Hertz. That is we will assume that the filter transfer function
is given by:
The resulting
is a wide sense stationary random process with zero-mean and power spectral density:
, (1)
Where we have assumed that the bandwidth of the receiver is small relative to the center frequency of the signal we are trying to detect,
. The power spectral density of
can then be approximated by the power spectral density of the noise near
:
If
has a power spectral density given by Eq-1, then the autocorrelation function of
becomes:
Note that
.
Now if we choose a sampling interval
; then the samples at
have an autocorrelation given by
,
Hence
is a discrete time, wide sense stationary, white noise with intensity
.
For matched filtering applications, we scale the output of the Analog to Digital conversion process by
to conserve the signal energy over a time interval
This creates the discrete time process
.
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