50

Since this question has multiple sub-questions in edits, comments on answers, etc., and these have not been addressed, here goes. Matched filters Consider a finite-energy signal $s(t)$ that is the input to a (linear time-invariant BIBO-stable) filter with impulse response $h(t)$, transfer function $H(f)$, and produces the output signal $$y(\tau) = \int_{-...


8

The time span of the zero crossings increases after the final RRC filtering (and the symbol sampling locations converge which is the goal for the benefit of zero ISI but the zero crossing increase in the process is to the detriment of timing recovery!). So if you are using a Gardner TED which is sensitive to this, it is better to have TED prior to RRC ...


7

There is no single "gain" value for any filter. The value that you're referencing as the sum of the taps is its DC gain: $$ H(0) = \sum_{n=0}^{N-1} h[n] e^0 = \sum_{n=0}^{N-1} h[n] $$ where $H(\omega)$ is the frequency response of the filter at angular frequency $\omega \in [-\pi, \pi)$. What you are concerned about is how large the peak can possibly be ...


6

There must be something wrong in your code because it's quite straightforward to show that the squared frequency response of a raised cosine pulse does not satisfy the Nyquist criterion. A raised cosine pulse satisfies the Nyquist criterion simply because a raised (co)sine function and an inverted raised (co)sine function add up to a constant: $$(1+\sin(x))...


5

You are right: A matched filter will maximize SNR at the instant of decision. Note that your proposal of a tall spike "filter" is not a filter, but actually a sampler (the sampler used at the decision point). The matched filter is a filter (i.e. linear time-invariant system) applied to the continuous input signal. The "spike at the point of decision" is ...


5

Let's consider a received signal $$Y(t)=Ap(t)+N(t)\tag{1}$$ where $A$ is the information symbol (modeled as a random variable), $p(t)$ is the transmit pulse, and $N(t)$ is additive white Gaussian noise (AWGN, modeled as a random process). For the sake of simplicity let's assume that all signals are real-valued (baseband case). Filtering with an LTI system ...


4

I think the problem is not as bad as you suspect it is. I wasn't around at the time, but from what I've read, early radar systems essentially connected the matched filter's output to an oscilloscope, and a trained operator would look at the phosphor and decide, from experience and intuition, when the signal raised above the noise ("the grass") indicating a ...


4

Short answer; I'll try to come back and improve this later. Recall that a matched filter is equivalent to a sliding correlator. At each time instant, the sliding correlator multiplies the last $T$ (where $T$ is the pulse duration) of the input signal by the conjugate of the pulse shape and coherently integrates the result over the pulse duration. If there ...


4

I'll illustrate with an example: detecting a rectangular pulse in noise, in Matlab. Let's start with defining the signal that contains the pulse. fs = 1/1000; t = 0:fs:1; s = [zeros(1,400), ones(1,100), zeros(1,501)]; First let's do detection in noise with small power. We calculate the received (noisy) signal: np = 0.1; % noise power (variance) n = sqrt(...


4

Two codewords $c_1$ and $c_2$ of length $n$, with elements in $\lbrace +1, -1 \rbrace$, and Hamming distance $d$, have a cross-correlation given by $$(n-d) -d = n-2d.$$ The reason is that there are $n-d$ bits that are equal and their product is $1$, and $d$ bits that are different and their product is $-1$. Note that: The larger the distance $d$, the ...


3

Yes, it is possible (at least on paper or code, since complex signal don't exist physically) to apply a matched filter to complex signals. This is one way to look at it that I think is illustrative. Assume that a pulse $p(t)$ is real and has energy 1, that is, $$\int_{-\infty}^\infty p^2(t) \, dt=1.$$ The filter matched to $p(t)$ has impulse response $p(-t)$...


3

Because a shift in time manifests itself as a phase angle shift in the frequency domain, perhaps comparing the spectral phase of the DFT of x(n) and the spectral phase of the DFT of y(n) would be useful. Just a thought.


3

It is usually useful to use normalized cross-correlation for finding position of small template on other longer signal. The value of normalized cross-correlation coefficient is invariant to change of amplitude and bias of signals. xcorr function does not calculate normalized cross-correlation coefficient for vectors with unequal length. I prepare code ...


3

Since the Matched Filter is used the Cross Correlation becomes "Auto Correlation" which is assured to be "Symmetric" relative to its maximum. Hence a good approximation would be to approximate the area around the sampled peak by a Symmetric Function as a parametric model of the function. The most trivial and easiest model for such function would be the ...


3

The matched filter is the maximum-likelihood receiver in the presence of additive white Gaussian noise. Thus, for equal prior symbol probabilities, it will yield optimum bit-error performance. This is equivalent on the AWGN channel to maximizing the signal-to-noise ratio, as you pointed out. You also correctly pointed out that this maximum-SNR condition is ...


3

Though the Matched Filter is the best tool detection of a known signals under AWGN it should work well here as well. To say something about the probabilities the question is, do you know something about the energy of the received signals? If you do, you should easily say something about the probabilities. Pay attention that if the assumption is a signal ...


3

Think of the time - frequency ambiguity of your matched filter like so: Frequency ambiguity means it will respond to a range of frequencies Time ambiguity means the response will be 'smeared' around the spatial location. If you you have 0% frequency ambiguity, the matched filter must look like a sine wave and go on for ever, which in the frequency spectrum ...


3

You need to define what you mean by SNR and processing gain, because Marcus Muller and Qasim Chaudhari can both be considered correct depending on your definitions of SNR and processing gain. A processing gain is usually taken to be the SNR at the output of a system divided by the SNR at the input of a system. If you consider the SNR to be the signal power ...


3

One first quick and dirty method could be to first store a sort of scale-space library of your original waveform $w(t)$, computed at different scales (discretized), and then perform a "filter bank" of matched filters in parallel on the same observed signal. This proposition is a kind of dual of the non-feasible solution you evoked. This way, you can save ...


3

First, please read this answer of mine for a detailed description of matched filters for real-valued signals. In particular, note that what I called the matched filter for a signal $x(t)$ is a(n LTI) filter with impulse response $h(t) = x(-t)$ which is better described as the time-reversed signal rather than the "inverse" of the impulse response as you call ...


3

It's often said that pulse compression gives you a gain proportional to the time-bandwidth product (otherwise known as the pulse compression ratio, or $PCR$). This is a really misleading statement, and it had me confused enough to sit down and think about it for awhile. I thought I'd share some of my findings that I pieced together from both reading the ...


3

In the ideal AWGN channel we have the received signal is $r(t)=s(t)+n(t)$, where $s(t)$ is the transmitted signal and $n(t)$ is white Gaussian noise. In this case, the transmitted symbols can be estimated using a matched filter whose output is sampled at the symbol rate. Note that in general the noise at the output of the matched filter is correlated, and no ...


3

If the input signal is real-valued, then the DFT returns a sequence of length $N$ that satisfies $$X[N-n]=X^*[n]$$ As one usually plots the magnitude of the spectrum, then the relationship of interest is: $$|X[N-n]|=|X^*[n]|\implies|X[N-n]|=|X[n]|$$ When you work with real-valued signals, one tends to "ignore" the second half of the FFT spectrum due to ...


3

Using the matched filter is right approach and typically a full length filter should be used. One of the advantages is that filter gain is approximately proportional to $10\log(T)$. If you don’t need full gain you can use a chunk of your matched filter and still have acceptable detection performance. One technique in active sonar is to use a broken up ...


3

If you have a reference signal you want to find in a different signal then your model matches almost perfectly (Up to the environment the signal to be found is in) to Matched Filter. So basically you need to do cross correlation between the Test Signal and the Reference Signal. Find the point of maximum correlation and create a cropping zone around it ...


3

The simple answer is the matched filter increases the signal to noise ratio of the reflected return signal. As the signal propagates from transmitter, the power that impinges on the target is proportional to $1/r_{target}^2$. The reflected energy back to the transmitter is also proportional to $1/r^2$, so the received energy is proportional to $1/r^4$. ...


3

A matched filter is matched to the received pulse shape. If the matched filter is implemented in the baseband, i.e., after down-conversion, the received pulse shape can be complex-valued, e.g., if the channel frequency response is not symmetric with respect to the carrier frequency. In that case the matched filter is a complex-valued low-pass filter matched ...


3

"Matched filter" or "matched filtering" is more of a concept than a specific object. It answers the question: given a template $t$, and a signal $x$, is there a filter $ \hat{h}$ that minimizes some nice function $f$ (usually a norm or a distance): $$ \hat{h} = \arg \min_h f(s-h\ast t)\,.$$ In the classical case, $f$ is the squared $L_2$ norm, and the ...


3

Convolution commutes $$ y[n]=\sum_k h[n-k]x[k]=\sum_k h[k] x[n-k] $$ for $n=0$ $$ y[n=0]=\sum_k h[-k] x[k] $$ If $H(e^{-\jmath \omega})$ is the Fourier Transform of $h[n]$, $H(e^{\jmath \omega})$ is the Fourier Transform of $h[-k]$, The maximization for the matched filter is at the point $n=0$. In communications $n=0$ corresponds to synchronization. ...


3

The calculation shown is the correlator equivalent of the LTI filter operation. The signal/sequence $x[\cdot]$ is being correlated (with zero time offset) with the signal/sequence $h[\cdot]$, and the value of the crosscorrelation function (at zero time offset) is defined to be $$C_{h,k}[0] = \sum_{k=-\infty}^\infty (h[k])^*x[k]$$ though there are people ...


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