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57

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 $$y(\tau) = \int_{-... 9 The time span of the zero crossings increases after the final RRC filtering (and the symbol sampling locations converge which is the goal for the benefit of zero ISI but the zero crossing increase in the process is to the detriment of timing recovery!). So if you are using a Gardner TED which is sensitive to this, it is better to have TED prior to RRC ... 6 There must be something wrong in your code because it's quite straightforward to show that the squared frequency response of a raised cosine pulse does not satisfy the Nyquist criterion. A raised cosine pulse satisfies the Nyquist criterion simply because a raised (co)sine function and an inverted raised (co)sine function add up to a constant:$$(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

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 ...

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 ...

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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 ...

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

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 ...

4

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 ...

4

Applying the discrete Fourier transform (DFT) twice to a time-domain signal, should produce the original time-domain signal in a circularly time-reversed, and linearly scaled (by N) form; as the inverse DFT is very similar to the forward DFT, except the sign of the complex exponential,a linear scale by 1/N. The inverse DFT of $X[k]$ is: $$x[n] = \frac {1}{N}... 4 It's often said that pulse compression gives you a gain proportional to the time-bandwidth product (otherwise known as the pulse compression ratio, or PCR). This is a really misleading statement, and it had me confused enough to sit down and think about it for awhile. I thought I'd share some of my findings that I pieced together from both reading the ... 4 Short answer; I'll try to come back and improve this later. Recall that a matched filter is equivalent to a sliding correlator. At each time instant, the sliding correlator multiplies the last T (where T is the pulse duration) of the input signal by the conjugate of the pulse shape and coherently integrates the result over the pulse duration. If there ... 4 The correlator is just a filtering operation. So in the case of passband pulse-amplitude modulation (PAM), such as QPSK or QAM, the (noiseless) received analytic signal before demodulation and sampling can be written as$$r(t)=e^{j((\omega_c+\Delta\omega) t+\phi))}\sum_{k}A_kp(t-kT)\tag{1}$$where \Delta\omega is a frequency offset, \phi is a phase ... 4 I'll illustrate with an example: detecting a rectangular pulse in noise, in Matlab. Let's start with defining the signal that contains the pulse. fs = 1/1000; t = 0:fs:1; s = [zeros(1,400), ones(1,100), zeros(1,501)]; First let's do detection in noise with small power. We calculate the received (noisy) signal: np = 0.1; % noise power (variance) n = sqrt(... 4 Two codewords c_1 and c_2 of length n, with elements in \lbrace +1, -1 \rbrace, and Hamming distance d, have a cross-correlation given by$$(n-d) -d = n-2d.$$The reason is that there are n-d bits that are equal and their product is 1, and d bits that are different and their product is -1. Note that: The larger the distance d, the ... 4 That's not true, it's not better. The thing is: the matched filter just implements the projection in the signal vector space, onto the signal vector itself (or a multiple thereof). (You'll find correlation is just an inner product in that space.) The line through that vector is the signal subspace, the plane to which that vector is normal is the noise space. ... 3 Yes, it is possible (at least on paper or code, since complex signal don't exist physically) to apply a matched filter to complex signals. This is one way to look at it that I think is illustrative. Assume that a pulse p(t) is real and has energy 1, that is,$$\int_{-\infty}^\infty p^2(t) \, dt=1.$$The filter matched to p(t) has impulse response p(-t)... 3 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 The function xcorr calculates the correlation of 2 signals. The correlation is known to be a good (The MLE) for delay estimation under Gaussian Noise. Yet, as can be seen in your data you're not using it in the cases it meant to be used. If we assume you have a model of a known signal with Additive White Gaussian Noise (AWGN or any other Additive White ... 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 If the auto correlation of the signal is sharp enough, you can do Matched Filter and search for local extreme points. Yet it seems the figure you'e displaying is in the frequency domain. But we can treat the frequency domain as it was time domain and work with the same "Trick". Namely searching for the frequency in the frequency is like searching for 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 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 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 ...

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In the frequency domain a signal with a carrier offset looks like the following- This is usually modeled as the desired baseband signal convolved (in the frequency domain) with a complex tone with frequency equal to the carrier offset. As you are probably already aware, convolution in the frequency domain is equivalent to multiplication in the time domain. ...

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