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Properties of Active Sonar Matched Filtering
Matched filters are used extensively in coherent active sonar. The output of a matched filter is used for detection, classification and localization. This document develops some properties of matched filters, including the SNR response in ambient noise and the response to reverberation.
In a matched filter for active sonar, we are integrating the echo plus interference times the echo’s replica. When an echo passes through the matched filter, we are cross-correlating the echo with a scaled version of the echo, so that the output is a scaled version of the auto-correlation of the echo corrupted by noise. The autocorrelation of the echo has a peak in time whose duration is approximately the inverse of the echo’s bandwidth.
For some waveforms (such as the Sinusoidal Frequency Modulation pulse) the autocorrelation function will have multiple peaks, termed‘fingers’, due to the periodic structure of the pulse. Each autocorrelation finger has a time width approximately equal to the signal’s bandwidth.
The echo is written as
$e(t)=\sqrt{{E}_{R}}r(t)$ , where $\underset{0}{\overset{T}{\int}}{r}^{2}(t)\text{dt}=1$
This implies that the echo energy $\underset{0}{\overset{T}{\int}}{e}^{2}(t)\text{dt}$ is ${E}_{R}$ , measured in Pascal^2-seconds.
We can write the matched filter operation in continuous time as
$m(t)=\underset{t}{\overset{t+T}{\int}}y(\sigma )r(\sigma -t)\mathrm{d\sigma}$
$y(\sigma )$ is the receiver time series. In response to a target echo that arrives at ${T}_{D}$ seconds and without noise or reverberation, the receiver output is $y(t)=e(t-{T}_{D})$ . The output of the matched filter becomes:
$m(t)=\underset{t}{\overset{t+T}{\int}}e(\sigma -{T}_{D})r(\sigma -t)\mathrm{d\sigma}=\sqrt{{E}_{R}}\underset{t}{\overset{t+T}{\int}}r(\sigma -{T}_{D})r(\sigma -t)\mathrm{d\sigma}$
Hence $m({T}_{D})=\sqrt{{E}_{R}}$ . The peak power output of the matched filter, ${m}^{2}(t)$ , in response to a echo is ${E}_{R}$ .
We determine the matched filter response to noise next. Assume the input noise is white with variance ${\text{AN}}_{0}$ :
$E\left\{n(t)n(s)\right\}={\text{AN}}_{0}\delta (t-s)$
Note that the delta function has units of inverse seconds, and therefore ${\text{AN}}_{0}$ has units of Pascals^2/Hz, equivalent to a spectral density. From the definition of stationary random process autocorrelations and power spectral density, we know that the Fourier transform of the autocorrelation is the spectral density function for the random process. The Fourier transform of covariance becomes $\int {e}^{\mathrm{j2\pi f\tau}}{\text{AN}}_{0}\delta (\tau )\mathrm{d\tau}={\text{AN}}_{0}$ , which is the spectral density of the noise.
$\begin{array}{}E\left\{m(t)m(s)\right\}=E\left\{\underset{t}{\overset{t+T}{\int}}n(\sigma )r(\sigma -t)\mathrm{d\sigma}\underset{t}{\overset{t+T}{\int}}n(\beta )r(\beta -t)\mathrm{d\beta}\right\}=\\ {\text{AN}}_{0}\underset{t}{\overset{t+T}{\int}}\underset{t}{\overset{t+T}{\int}}\delta (\sigma -\beta )r(\sigma -t)r(\beta -t)\mathrm{d\sigma d\beta}={\text{AN}}_{0}\underset{t}{\overset{t+T}{\int}}{r}^{2}(\sigma -t)\mathrm{d\sigma}={\text{AN}}_{0}\end{array}$
Thus, the noise power response of a matched filter is the input spectral density, ${\text{AN}}_{0}$ .
We conclude that the signal to noise ratio (SNR) at the output of a matched filter is the ratio of the echo energy to the noise spectral density, ${E}_{R}/{\text{AN}}_{0}$ . This assumes that the noise is white, e.g. a flat spectral density at the input to the matched filter. This is a general result, independent of the signal waveform details, except for its energy ${E}_{R}$ .
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