### Fred Marshall (@Fred Marshall)

I would prefer to say it this way:There is temporal extent of (N+1)*T where T is the sample interval in time and N are the number of samples in time. The "+1"...

I'm not sure that I have all the points clear but here goes:1) The steady state output in time domain can be determined by substituting 'w' the digital frequency...

Just a few observations:1) "many books and references" are likely all different and some the same. Who is to know without the references?2) Real or Complex?3)...

"output amplitude" resulting from what INPUT??How can the "input amplitude" be off?? It is what it is, no?"Same output amplitude" as what?Sorry to be so picky...

I agree with everyone and, particularly, with Tim. The "edge" effects may be a real issue (or not).Here is a classical approach to time-domain interpolation using...

First, I suppose that you understand complex numbers and it appears that you do.Your question is a good one that comes up fairly often. The simple answer is that...

vrms or vpeak are just observation/measurement choices and have nothing to do with sensitivity, amplification, etc. as long as all those parts are *linear*, e.g....

Assuming that the transducer is specified at its center frequency then that's where the -196 dB ref vrms/upa applies. Other frequency outputs would have to be...

"Windowing" can mean a couple of things:1) A limitation in temporal or spatial or spectral extent (etc.) - whether intentional or simply a function of the limitations...

An FFT is a transformation that results in the same (or likely 2x) the data points. So, by itself, it won't help. "2X" because if the input is Real of "n" points...

Exhaustive search in a quadrant?Otherwise, I suppose that the details of the mathematical expression could be used to obtain the result analytically. But, to do...

From the spectral character it looks like you could reduce the sample rate to get better frequency resolution (i.e. a longer time window). Right now you're somewhat...

From the description, it seems evident that you're using an arbitrary time record.Here's a good test:Don't FFT the record (that you would have FFTd) at all. Just...

Well, physics may be working against you. You say STFT but without any parameters. What is the temporal length of the record or window?

OK. If the observation time is limited to 1ms then the frequency resolution will be limited to 1/1ms=1kHz. That rather leaves out "from a few Hz".There are a...

This is very much the same issue as when one is trying to build an "endless loop" of a sound record out of a limited amount of data. The issue there is "how long...

A few fundamentals:"phase" is a relative term that is tied to "time=0". Sorry if that's an obvious assertion but it may come up next...Any time delay causes a...

I didn't mean to ask if the target was present. I asked that you might consider if the target [when present] was within the reverberation spectrum [at the same...

Perhaps the signal isn't present within the reverberation spectrum in any significant way? In that case any reverberation rejection would be rather useless - which...

I'm not talking about a transient caused by connecting or disconnecting. I meant that the microphone should be disabled or not there at all. Normally, the input...

Somehow you are going to sample the room noise. What if you simply disconnect (or short out) the microphone and amplify the noise of your amplifiers, etc. until...

I'm a little confused. You seem to NOT want to use a random number generator and then refer to using a seed for a random number generator. So, which is it?

OK. Well, the suggestions still apply - only you can decide where to start and when. Good luck with your pursuit!

You have received some good suggestions but other than your current work in firmware, I have little idea where you might be starting. So, I'd start with first...

It's probably unlikely but just to be sure and in case you want to have better measurements:You didn't mention band-limiting before sampling. This means that any...

It seems to me that autocorrelation, or a variant thereof, would be a way to deal with this but best if the waveform of each period of the signal were identical. ...

Your last question implies that you want lossless compression. Yes, of course, these methods aren't strictly lossless... That is, if one assumes that the waveform...

It seems to me that tying the idea (or the objective) to a single sinusoid rather misses the point of the approach being more general. "Time compression" might...

What you want to do is reduce the sample rate. This means that the FT domain has to repeat more frequently (view the output as a single period of a continuous,...

Well, you can multiply the number of samples that I used by 2X as I was referring to complex or I/Q frequency samples. I think it's fair to say this:I and Q samples...

It's a bit easier to answer in concept than to get into all the intricacies of the steps to take. So here is a quick answer:You currently have FFT samples that...

A FIR filter, with symmetric coefficient structure (thus linear phase) will have some ripple in frequency. And, a FIR filter, with flat frequency response (thus...

I tried to acknowledge that in my last sentence with suggestion for language: "I'd rather say that they are correlated at a time compression factor of ...."

Well, OK, you have come up with an expression that relates two functions in an equality. Rarely will that apply because there is noise.A simple expansion using...

The sampling frequency has to be > 2X the highest frequency.And, getting back to your example of 4 samples, I'm afraid that I overlooked something that I only...

1) The N points of X is the observation window in frequency domain. Is this correct?No. As others have mentioned, I'd not call this an "observation window" but,...

The time frame is equal to the *number* of samples "N" times the sample rate.So, sample rate = 1kHz X number of samples of 1,000 would be 1 second time frame....

Detection is generally about:1) Matching, more or less, what's known about the signal to be detected to the input signal.That's where filtering and/or FFT processing...

No matter what, I'd test it with a single sample. And those single samples could be of various amplitudes. It won't tell you about summations and rounding but...

There's a whole range of capabilities:- PC-based software that shows a "cartoon" of the WiFi spectrum. It's not really a spectrum analyzer but shows a plot "as...

Starting with the definition which I grabbed off Wikipedia because it's handy:'The peak-to-average power ratio (PAPR) is the peak amplitude squared (giving the peak...

If one were to define the dynamic range as "from zero to the maximum input level without x% distortion as a function of frequency" then you could apply sinusoids...

This is the first time I've noticed "audio data". I thought that S1 was something you could manipulate...??I see temporal plots that look like radar or sonar.I...

Ah! OK. I'm a bit more used to physical systems for R1 or simulators of the same. If you have the ability to change S1 AND R1 then I should think that you...

OK. Well, I understand it better now. Thanks.Let me play this back:Dry Signal: S1Impulse response: R1Convolve S1*R1 = S2Wet Signal: Deconvolve S2 using R1 =...

I'm not sure what you mean by doing something to the "dry signal". If you did that then what happens to the reference? It seems like there's something missing...

Startibartfast gave you a good answer. I would only add:Envision that the coherence bandwidth is a *measure* and not a *band*. Then, if you shift the center...

Well, I worked on it a bit and end up wondering about a few things:As I read the Matlab "Hilbert" function, https://www.mathworks.com/help/signal/ref/hilbert.html,if...

.. figure out the SNR after performing low pass filtering and decimation?Start with the original signal, bandwidth (fs/2), and noise.When you lowpass filter with...

The issue is a specific example which has not been revealed. If there were an expression of the actual example (as with Matlab code) then maybe folks could comment. ...

Even when you say "the best frequency resolution", the answer isn't much different than the time span or the window length. I'd say, for practical purposes, not...

The biggest thing and the easiest to remember (for me at least) is the time span. The resolution is the reciprocal of the time span. So, that means: "the window...

It means that the signal is small enough that the nonlinearity is negligible. It's a measure of the dynamic characteristics of the amplifier (e.g. the linear differential...

You ask, how did the author reach this conclusion:First, if the mean is much larger than the standard deviation, Eq. 2-2 involves subtracting two numbers that are...

Both the signals are same except for the amplitude/power at the receiver. In this case, what difference does it make to the receiver and at what stage?You have only...

Does the power at the receiver matter at the RF or the baseband stage?Yes, it does. At the most fundamental level, if there's no signal then there's nothing to...

If I might suggest, perhaps you'd be well advised to back away from the immediate objective focus: "fit in small embedded systems" and get to the functional objective....

Well, when one asks a question about mathematics and definitions then I try to stick with the definitions as well as I might. I'm not trying to be critical of...

I don't understand this statement:"The one-sided Fourier transform has only positive frequency components and its amplitude is twice the amplitude of the double-sided...

dgshaw6 gave a very good reply to this.My own experience is in underwater acoustics. You might find "A Conceptual Model of Reverberation in the Ocean" to be of...

kaz, Thanks. I then still have to worry about what does RMS mean after all? Or, why do we measure it? One definition (which I don't endorse or deny) is "the...

Perhaps I'm missing something here...The *usual* RMS measure of a sinusoid starts with the sinusoidal amplitude as in:A*sin(wt) and the only thing of concern is...

From: Rick Lyons at:https://www.dsprelated.com/showthread/comp.dsp/101...It's a fred harris formula Atten/(22*(Fstop-Fpass)

For everyone who might want to know, the Matlab functions are easily reachable on the web. So Google fir1 and the top hit is:https://www.mathworks.com/help/signal/ref/fir1.htm...I...

Yes. But if you don't do the reversal, there is an opportunity to see the result. That can give insight.

In normal terminology, that signal is not a "sweep", it's a "pulse train" with varying pulse frequencies. It's loaded with transients and I'd not recommend it....

Others have given useful comments. Wanting something "more complex" seems to go against Einstein's "everything should be made as simple as possible, but no simpler"...

As others have already pointed out, a dynamic signal may well not exactly match your expectations when using steady-state analysis. You can't have both. But,...

I'll go a bit out onto a limb and suggest perhaps a different framework for this. So, I'll depart from the"why?" and come at it differently. Perhaps more like...

Since the component frequencies are different but harmonically related, the phase components will be related but can't be the "same". Consider that a phase shift...

The original math was a little terse for me. But I read it anyway...It occurs to me that the definition of the question leaves a bit to be desired. I would say...

Since there's a difference between frequency multiplication and frequency shifting, it's likely we need to ask which you want to do. If there's any modulation...

From a philosophical point of view, you would need to resolve the two sinusoids in some context. I don't think there's a way around that. But I could be wrong....

Is there a handy to start from a unit sample response?Is there a way to cascade filters or convolve the unit sample responses?

OK - thanks. I decided to dig into your ETS implementation to see if I could put it in more conventional terms so I could understand better. It seems that you're...

So if finding the base period remains the challenge then sometimes an iterative method will work. For example, you might match the slope and value of the end points...

Thank you, that clarifies things a lot for me.Is this a reasonable re-statement:"We want to observe a waveform with temporal resolution that is much shorter (i.e....

"due to the nature of the target for which i'm trying to apply this soln to, i can't afford to have very long FIR filter."That's interesting but we still don't know...

I'm sorry. I'm really trying to help but it seems you are mixing objective with method. You asked a question and I believe I gave a good answer. Now we seem...

I can see the first picture but not the 2nd one. Either way, it appears that you are computing the waveform using the expression in Matlab and then doing some...

Like Kaz, I believe that the mathematics should follow physics / understanding. It's a bit like sunglasses: it improves the image only if the image is there to...

There are lots of places where you can find a Windows executable of the Parks-McClellan program for FIR filter design. Such as:http://www.ece.umd.edu/~tretter/enee425/That's...

One rather straightforward approach (whether the first one here is effective enough is another matter) would be to bandpass filter the composite signal so that...

If you start with the FFT and zero out all the samples above 192kHz then that's similar to using a half-band filter. At higher downsample rates this can actually...

First, I will paraphrase what you've written:"The input signal is the sum of two sinusoids of different frequencies and of the same amplitude""This composite signal...

Tim Wescott and I gave similar answers earlier. Have you been able to come up with something yet? As before, using prediction methods with no model is likely...

Consider these things:1) Do you have a model of what's being measured? For example, if it's a tank of 500,000 gallons of water then there is at least some thermal...

"Phase mismatch" may not be what's happening. If you consider that one beamformer output is #1 and the other is #2 then they are simply NOT the same. So, any...

As I tried to point out, there should be no difference when the discrete and periodic nature is accounted for. So maybe saying a "weighted sum of sinusoids" is...

I have a little trouble following the question so I will offer some things that *may* help:- A "discrete periodic signal" by virtue of being periodic will have a...

I once worked on a system that did something like correlation - the details don't matter. What was interesting to me was that the temporal output could be noisy...

I would say YES also. That said, it's hard to know how one's mind works. I used to read my engineering texts and skip all the equations the first time through....

In a broad sense, it doesn't matter if you filter in time or in frequency. Only the details matter in implementation.If all you did was zero out the zero-frequency...

Michael,If it helps and to simply amplify what others have said: The maximum possible slope of any band edge of a filter response is inversely related to the length...

A few preliminaries:1) How accurate does this calculation have to be to meet your needs? Less accurate if the noise field is directional/unknown. Maybe your...

If there is no interference to be rejected then I have a hard time imagining that a classical beamformer isn't what you'd end up with. It's already been mentioned...

"David" gave you a good answer on StackExchange. I couldn't find where you get the assertion re: improved bearing resolution. There is bearing resolution and...

It's active sonar at relatively short range (?) so it may be that detection is never noise-limited anyway. Is it? I recall that the intention of signal processing...

In the article at the link you provided, there is an expression: "To achieve gain in computational efficiency, the following must hold:B<Fs/4"I'm sure you know...

Well, as before, using an FFT IFFT implementation of circular convolution (the proper term for it) can be way faster if the arrays are large enough. So maybe...

What I was trying to describe is that the "square in the frequency domain" is made up of a bunch of narrow frequency-shifted sincs that relate to the entire length...

Much of understanding this, I believe, boils down to the fact that unweighted / unwindowed finite-length sequences have an underlying sinc or (perhaps more accurately) circular...

The technique for saving operations is well known using an FFT IFFT process. And, the use of the largest possible arrays of samples helps. I found a reference:Efficient...

Any suggestions how to interpret the Sammon mapping output? All I understand is that there are 2 dimensions (or 3) with numbers in each. What do these dimensions mean?...

Well, I got the Sammon working in Matlab somewhat. I had to remove some of the less-than important data to get it to run. In the end I got some numbers which...

JOS: Yes I have plenty of historical inputs and outputs to work with and due to the short-memory that I anticipate, the results should be fairly well independent...

Y(J)S, I think I understand what you're suggesting. Let me paraphrase it back: Given a series of states of the inputs, can those inputs be classified into...

I will try to answer the questions but I'm not sure I'm doing us any favors....The inputs are decimal to 2 or 3 decimal places after the decimal point. I don't think...

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