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In signal processing, two problems are common:
What is the output of this filter when its input is $x(t)$? The answer is given by $x(t)\ast h(t)$, where $h(t)$ is a signal called the "impulse response" of the filter, and $\ast$ is the convolution operation.
Given a noisy signal $y(t)$, is the signal $x(t)$ somehow present in $y(t)$? In other words, is $y(t)$...
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The two terms convolution and cross-correlation are implemented in a very similar way in DSP.
Which one you use depends on the application.
If you are performing a linear, time-invariant filtering operation, you convolve the signal with the system's impulse response.
If you are "measuring the similarity" between two signals, then you cross-correlate them.
...
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Personally I find Python one of the best choices out there and did myself some work in area of audio identification. You are welcomed to check for instance my software for automatic identification of birds from noisy audio recordings: Ornithokrites. The program is used by Department of Conservation of New Zealand and they are happy about it. Based on this ...
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"Is there a way to measure frequency (detect pitch) better than FFT, that is, with better resolution in less acquisition time?"
yes there is. or are. there are multiple better ways to do musical pitch detection in real time that are far, far better than running an FFT.
consider :
Average Magnitude Difference Function (AMDF)
$$ Q_x[k] = \sum_n |x[n] - ...
answered Nov 13 '15 at 9:08
robert bristow-johnson
16.3k3 gold badges29 silver badges68 bronze badges
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When you say that the "information content may remain the same," do you mean the information in the total signal, or the information of the desired signal? Hopefully this will answer both cases. I know Shannon entropy much better than Kolmogorov so I'll use that, but hopefully the logic will translate.
Let's say $X = S + N$ is your total signal ($X$), ...
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Though the Matched Filter is the best tool detection of a known signals under AWGN it should work well here as well.
To say something about the probabilities the question is, do you know something about the energy of the received signals?
If you do, you should easily say something about the probabilities.
Pay attention that if the assumption is a signal ...
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Note: I originally posted this answer for the Stack Overflow copy of this question, before realizing that it had also been asked here. It somewhat duplicates pichenettes' answer, but I felt it still worth (re)posting here, since it includes some extra details. (Whether those details are useful or not, I'll leave for you and the OP to judge.)
If you know ...
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@MathBgu I have read all above given answers, all are very informative one thing I want to add for your better understanding, by considering the formula of convolution as follows
$$f(x)*g(x)=\int\limits_{-\infty}^{\infty}f(\tau)g(x-\tau)\,d\tau$$
and for the cross correlation
$$(f\star g)(t)\stackrel{\text{def}}{=}\int\limits_{-\infty}^{\infty}f^*(\tau)g(...
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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 ...
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Maximum likelihood (ML) estimator
Here will be derived a maximum-likelihood estimator of the power of the clean signal, but it doesn't seem to be improving things in terms of root mean square error, for any SNR, compared to spectral power subtraction.
Introduction
Let's introduce the normalized clean amplitude $a$ and normalized noisy magnitude $m$ ...
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Maximium A Posteriori (MAP) and Maximum Likelihood (ML) are both approaches for making decisions from some observation or evidence.
MAP takes into account the prior probability of the considered hypotheses. ML does not. This set of probabilities, known as "a priori" probabilities or simply "priors", is often known imperfectly, but even rough approximations ...
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At the end what has proven to be the best solution was onset detection based on either high frequency or energy content. Before it could work I had to use high-pass filter to cut out first 1 kHz, since it contained too much noise.
Once I had noise-only area I could use its profile to reduce noise from rest of the sample.
One library I found particularly ...
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For peak detection a nice method is the following: apply a maximal filter to the data and find the places where the filtered data equals to the original one.
A maximal filter is simply sliding through the data and selecting the maximal element from the sliding window. Formally:
$$g_w[x] = \max\left(f[x-w], f[x-w+1], \dots , f[x+w-1], f[x+w]\right)$$
where ...
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You have a set of message set $m_i$, $0 \le i \le N-1$. (For example, QPSK will be $N=4$). For the transmitted message $m_i$, the corresponding symbol vector is $\textbf{x}_i$, and the received symbol vector is $\textbf{y} = \textbf{x} + \textbf{w}$, where $\textbf{w}$ is the AWGN at the receiver. The above is a simplified baseband model assuming a simple ...
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After signal detection, how to estimate the clean signal $s(t)$?
Matched filtering is used to detect the presence of a known signal in noise. There is no estimation part when you are talking about a matched filter. The estimate part comes after you have done the matched filter and need to estimate the symbols.
It looks like you are talking about a ...
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To add to the excellent information given by Cassman in his response, here is a block diagram of a carrier recover loop for QPSK and QAM modems using a decision directed approach. I have detailed the decision directed phase detector in this post Phase synchronization in BPSK and this one How to correct the phase offset for QPSK I-Q data, while the block ...
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There are many factors involved in understanding the theoretical limits to communication. What follows is just a brief introduction that only scratches the surface.
First, let's consider a simple scenario: there is no noise and no distortion of the signal being transmitted. We do allow for attenuation. Under these circumstances, you can transmit up to $R_p=...
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Actually all those "Angle of Arrival" algorithms can be used for that.
You can try:
MUSIC.
Pisarenko Harmonic Decomposition.
Sensor Array (MVDR).
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There is a book by Basseville and Nikiforov called "Detection of Abrupt Changes : Theory and Application" that they released to the public as a PDF several years ago (it's out of print, now, I believe).
That book looks at the basic CUSUM (cumulative sum) algorithm and how to choose appropriate thresholds for it.
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There is a lot of subtlety between the meanings of convolution and correlation. Both belong to the broader idea of inner products and projections in linear algebra, i.e. projecting one vector onto another to determine how "strong" it is in the direction of the latter.
This idea extends into the field of neural networks, where we project a data sample onto ...
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Communications systems are always designed under the assumption that both emitter and receiver know what "language" they will be speaking to each other.
AM modulation comms are standardized in frequency, channel width for example. So each receiver is materially designed to demodulate AM signals with the appropriate hardware.
When people start trying to ...
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If $x(t)$ is a deterministic signal, then you have $E[x(t)]=x(t)$. Furthermore, if $Y(t)$ is a random process you have $E[x(t)Y(t)]=x(t)E[Y(t)]$.
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In general LTI System is invertible if it has neither zeros nor poles in the Fourier Domain (Its spectrum).
The way to prove it is to calculate the Fourier Transform of its Impulse Response.
The intuition is simple, if it has no zeros in the frequency domain one could calculate its inverse (Element wise inverse) in the frequency domain.
Few remarks for the ...
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To be honest, I don't think CNNs, RNNs and LSTM are useful for this kind of problem – a bandpass filter followed by a threshold would be.
Now, that would have three parameters:
Lower cutoff frequency
Upper cutoff frequency
threshold value
and what is usually called "Machine Learning" is nothing but finding local minima over some (loss) function with real ...
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That's a very broad question, but I'll give it a shot.
1 . ... how much could signal processing actually do to help us here ? ...
That's highly dependent on the specific problem. Sometimes a lot, sometimes not all.
One example for "a lot" is the the acoustic echo canceller in your garden variety Smart Speaker (Amazon Echo, Sonos One, Google Home, ...
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Assuming you're looking for symmetry around $x=0$, you can decompose any function into a symmetric and an antisymmetric part:
$$ f(x) = \frac{1}{2}\left(f(x)+f(-x)\right) + \frac{1}{2}\left( f(x) - f(-x) \right)$$
Calling $f_+(x):=\frac{1}{2}\left( f(x) + f(-x) \right)$ and $f_-(x):=\frac{1}{2}\left( f(x) - f(-x) \right)$ you can easily verify that $f_+$ ...
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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 ...
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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 ...
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The Goertzel algorithm allows you to sample the DTFT of a signal with slightly lower complexity than direct computation (still $\mathcal{O}(N^2)$ to produce the DFT but with a smaller coefficient). I think of it as being advantageous in two ways:
If you know the frequency you are looking for (like in telephone dialing systems)
If you only want a small ...
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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 ...
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