New answers tagged filtering
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There's a very simple intuitive explanation that applies not just to matched filtering, but to any filter.
Assume we do the "convolution" without time reversal: you hold the impulse response "fixed", and slide (shift) the input signal on top of it.
So, imagine sliding the input signal back to $-\infty$ and then shifting it forward in time....
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If your "why the matched filter is a time reversed copy of the signal we hope to detect" hints at a minus sign of the index of summation in impulse response for the correlation function $y[k] = \sum_{-\infty}^\infty {(x[i]·h[k-i])}$, this form of linear coefficients $h[k,i]$ is not specific to formulas for matching filters. Considering, for example,...
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Does the "time-reversing" of one of the signals that occurs in convolution also bother you? I put that in quotes because the result of convolution does a great job of hiding that the output is simply the sum of scaled and time-delayed versions of the impulse response. You can think about it the other way around as well, where the signal is what is ...
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Picture the transmitted signal as a «signature». You want to find some process that maximize the probability of detecting that signature even when there is noise.
What do you do to find some signature buried in noise? You make a template that exactly match the known signature, and you slide it back or forth in time, noting how much the actual signal deviates ...
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To see why it makes sense, first recall the purpose of a matched filter: it implements cross-correlation between the input signal and the template that you're looking for. On the AWGN channel, correlation is the optimum method for detection of the presence of a particular waveform (represented by the matched filter).
Next, recall the definition of cross ...
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It's not a loophole. A causal linear phase filter is identical to a non-causal zero phase filter in series with a delay of half the filter length. If you check the results of your filter with an impulse input, you will probably see such a delay.
So it's not really real-time if you count the delay, but is often close enough to seem real-time if the filter's ...
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Overlap-add or overlap-save/scrap with zero-padded data are the common methods of using block based convolution on streaming data. Pad the convolution (FFT/IFFT fast, or linear) by at least the length of the impulse response above your desired noise floor, minus 1. The basic idea is that these methods save the remainder of the impulse response that doesn't ...
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