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17 votes
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Why is the convolution of two sine waves a sinc function?

You are not convolving two sine waves, you are convolving two short snippets of sine waves. A snippet of a sine wave is the same as a real (infinite) sine wave multiplied with a rectangular window. ...
Hilmar's user avatar
  • 45.4k
9 votes
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Difference between RMS and a moving RMS

The total (standard) RMS, continuous time: $$ x_{RMS}=\sqrt{\frac 1{T_0} \int_0^{T_0} x^2(\tau)d\tau} $$ And the moving RMS, continuous time: $$ x_{RMS}(t)=\sqrt{\frac 1{T} \int_{t-T}^t x^2(\tau)d\...
Brethlosze's user avatar
  • 1,430
6 votes

Can every type of linear filter be modelled by a convolution?

No. It's only LTI (Linear and Time-Invariant) systems that can be modeled with convolution through a unique single impulse response. For example the systems $$ y(t) = g(t) x(t) $$ or $$ y[n] = \sum_{...
Fat32's user avatar
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5 votes
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How to filter out everything but a single frequency in the time domain?

Depending on the exact characteristics of the signal and implementation requirements (if any) I can think of a number of approaches to extract content at a "single" frequency. The most common: Apply ...
user883521's user avatar
4 votes
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Effect of low pass filter on the time domain

You have a finite length sequence $x[n]$, $n=0,1,\ldots, N-1$, with the discrete Fourier transform (DFT) given by $$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}\tag{1}$$ and its inverse DFT (IDFT) given ...
Matt L.'s user avatar
  • 90.4k
4 votes

Effect of low pass filter on the time domain

If you sample a pure sinewave for an integer number of periods, the samples will sum to zero (it will have equal "ups" and "downs"). If you zero an FFT bin, that's the same as subtracting a sinewave ...
hotpaw2's user avatar
  • 35.4k
4 votes
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Confused on the difference between the frequency spectrum of an entire song, and the frequency spectrum of a point in time

As far as my logic goes, I wont ge able to tell any of musical features just of an amplitude value in the time domain. That's wrong. The amplitude-over-time actually is the way the song is played by ...
Marcus Müller's user avatar
4 votes
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Index of stationarity of a time domain signal

This a very complicated question, and I would say a still open topic. The concept of stationarity is manifold, from pure statistics to applied DSP (strict, strong, wide-sense, quasi-stationarity, ...
Laurent Duval's user avatar
4 votes

Finite and Infinite support in time and frequency domain

The sinc is just fine an example as a signal with infinite support: Support being defined as the smallest interval in which the function has non-zero values, it's trivial to see that $\text{sinc}(x)=...
Marcus Müller's user avatar
4 votes

Optimal $ n $ -th Order IIR /AR Approximation of a Moving Average Filter

(Update: I just realized that the first part of this covering CIC structures is basically what Hilmar has already answered-- I'll leave this up since it offers more graphics and details in case that ...
Dan Boschen's user avatar
  • 52.1k
4 votes

Optimal $ n $ -th Order IIR /AR Approximation of a Moving Average Filter

This is not a full answer. It explores some basic approximations. For a moving average as long as 600 samples it is informative to look at impulse responses of continuous-time filters as ...
Olli Niemitalo's user avatar
4 votes
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Optimal $ n $ -th Order IIR /AR Approximation of a Moving Average Filter

I can show you some low order IIR approximations to an FIR moving average filter. In the figure below you see $3$ (infinite) impulse responses that approximate a moving average of length $N=600$. The ...
Matt L.'s user avatar
  • 90.4k
4 votes
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What is wrong with my residue partial expansion method? (Transfer Function into State-Space Modal/Diagonal Form)

The discrepancy between your derivation and matlab's computation results because of a convention mismatch you used during the partial fraction expansion: Given that the function to be expanded is $H(s)...
Fat32's user avatar
  • 28.3k
4 votes
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Can a 1-dimensional signal be in a domain other than the time or frequency domain?

If you have a one dimensional signal $s(\cdot)$, it somehow belongs to a combination of two different domains: the sampling domain: where samples are considered or taken, or how the signal is ...
Laurent Duval's user avatar
4 votes
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Where to start with DSP?

so I am wondering where can I go to form an understanding of DSP from the ground up. There is a great list of resources in this answer What Resources Are Recommended for an Introduction to Signal ...
Hilmar's user avatar
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4 votes
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Why aren't the integrator and the differentiator inverse systems?

The flaw is in the assumption that convolution is associative. This is only true if it is allowed to interchange the order of the corresponding integrals, which is not the case here because the ideal ...
Matt L.'s user avatar
  • 90.4k
4 votes

Measuring rise and fall times, of square wave

If it was necessary to get a low noise high quality estimate of the rise and fall time, one idea is to generate an eye diagram and from that create an averaged transition from which we could then ...
Dan Boschen's user avatar
  • 52.1k
4 votes

Applying Lowpass filter on a signal in time domain gives ringing artifacts - how to get rid of them

Why do I see the ringing artifacts after the decay of the impulse response and why are they pronounced at the end of the signal? DFT based frequency domain multiplication corresponds to circular (not ...
Hilmar's user avatar
  • 45.4k
3 votes

Difference between RMS and a moving RMS

A complement to the previous answer, which is really good. With a moving-average, or moving-RMS, the output rate of the RMS calculation is the same as the input rate. With a block average or block ...
Ben's user avatar
  • 3,777
3 votes

Time domain maximum from frequency domain data?

It's generally not possible to compute the exact maximum value, but you can compute a bound on the maximum value. Assuming your data are discrete-time, and you're using the discrete Fourier transform (...
Matt L.'s user avatar
  • 90.4k
3 votes

Stationary signal: time-domain vs frequency domain

Your definitions are not correct. For a Strict Sense Stationary process (signal) the joint distribution of your process' value for all instants of time must be independent of time, in other words if ...
Mohammad M's user avatar
  • 1,327
3 votes

Effect of low pass filter on the time domain

Your proposed condition is NOT true as written. What would be true is $$\sum_{t=-\infty}^{\infty}x_t = \sum_{t=-\infty}^{\infty}y_t$$ or $$\sum_{t=1}^{N}x_t = \sum_{t=-\infty}^{\infty}y_t$$ The ...
Hilmar's user avatar
  • 45.4k
3 votes
Accepted

Time-Frequency Analysis of Big Data - Data Size Reduction: averaging the most appropriate method?

So, first of all, CSV seems to me the least suitable format imaginable for this amount of data. It needs to be parsed, is memory hungry, and wastes precision, and isn't linearly addressable (ie. to ...
Marcus Müller's user avatar
3 votes
Accepted

Sampling Theorem: T-Sampled Signal

I will solve it for T=1. You can solve the others on your own. First, note that my definition of sinc is: $$sinc(t)=\frac{\sin(\pi t)}{\pi t}$$ Definitions of sinc tend do differ, so check what your ...
Maximilian Matthé's user avatar
3 votes
Accepted

Impulse response (general form for linear systems as two-variables function $h(t,\tau)$) applied to Time-invariant systems

If the response to $x(t)$ is given by $$y(t)=\int_{-\infty}^{\infty}h(t,\tau)x(\tau)d\tau\tag{1}$$ then the response to $x(t-T)$ is $$\tilde{y}_T(t)=\int_{-\infty}^{\infty}h(t,\tau)x(\tau-T)d\tau=\...
Matt L.'s user avatar
  • 90.4k
3 votes

Linear Constant Coefficient Differential Equations: Zero-Input and Zero-State responses

The two approaches should return the same solution. They are just two different ways to get to the same place. In the ZSR/ZIR method, you are solving two different IVPs - they have the same ...
Tendero's user avatar
  • 5,020
3 votes
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Finite and Infinite support in time and frequency domain

The coreof the question relies on how to define for the support of a function with $x$ in a domain $\mathcal{D}$. The most common notion in DSP is the closure of the set where $f$ does not vanish, i.e....
Laurent Duval's user avatar
3 votes

Optimal $ n $ -th Order IIR /AR Approximation of a Moving Average Filter

Depends a bit on your application. A moving average filter is a low pass filter and one with many lobes and pretty poor stop band rejection at that. Depending on what specific requirements you have, ...
Hilmar's user avatar
  • 45.4k

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