Questions tagged [fourier-transform]
The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum.
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Identifying sounds patterns from wav file
I have an audio recording stored as a .wav file. My goal is to identify exact sounds patters from it using Python.
The problem I'm facing is there's a clear "ting" sound in the audio file, ...
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Image Restoration by Solving Constrained Least squares in Frequency Domain (Frequency Domain Filtering)
I am trying to implement the constrained least squares filtering as described in Rafael C. Gonzalez, Richard E. Woods - Digital Image Processing 3rd Edition Section 5.9. The equation (...
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27 views
One integral inverse CWT
MATLAB's icwt docs state inversion to be done by a single integral:
$$
f(t) = 2 \Re e\left\{
\frac{1}{C_{\psi, \delta}} \int_0^\infty \left< f(t), \psi(t) \right> \frac{da}{a} \tag{1}
\...
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Fourier transform diagonalizes time-invariant convolution operators
I got the following paragraph from the book "A wavelet tour of signal processing" chapter one, page 2.
The Fourier transform is everywhere in physics and mathematics because
it diagonalizes ...
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25 views
PSD of linearly modulated signal using autocorrelation?
Consider a signal $v(t)$ given by
$$v(t)=\sum_{n=-\infty}^{\infty} b[n]p(t-nT).$$
Assume that $b[n]$ is uncorrelated with zero mean, i.e. $\mathbb{E}[b[n]b^*[m]]=\mathbb{E}[|b[n]|^2]\delta[n-m]$ and $\...
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Scipy fourier transform zero frequency spike (from DC offset) - de-meaning and hanning window have no effect
I am trying to plot the FFT of essentially a random signal that has a non-zero mean shown below.
The FFT of the signal is peaked over the zero frequency which usually indicates a DC offset. Although ...
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matlab code question with fractional fourier tranform
I want to estimate the parameters of a chirp signal using the fractional fourier transform. Following the paper estimating chirp parameters, the chirp rate $\mu$ can be obtained with (minus) the ...
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Taking FFT of 10 concatenated time traces with a time difference between each trace?
My goal - I'm interested in the white noise spectral region at higher frequencies, especially the phase information.
In this data acquisition instrument, digitised time data transfer is much faster ...
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78 views
Fourier transform of a signal and its autocovariance function
While I do know the difference between the two, in theory, I am not very sure about why we look at the Fourier transform of the auto-covariance function. What extra information does it give us over ...
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87 views
Optimally approximating the sign function by functions with compactly supported Fourier transform
I'm re-posting a question of mine from math.stackexchange in hopes that folks here might have the right kind of expertise.
I'm looking for a systematic way to approximate the sign function
$$\...
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1answer
87 views
Finding the equivalent filter H(u,v) in the frequency domain of a 3x3 spatial mask
I'm trying to find the equivalent frequency domain filter, $H(u,v)$, of a 3x3 spatial mask that averages all neighbours of a point $(x,y)$ in said 3x3 neighbourhood excluding the point itself. So far, ...
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53 views
Inverse Continuous Wavelet Transform derivation?
Wiki writes iCWT as
$$
f(t) = C_{\psi}^{-1} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} W_f(a,b) \frac{1}{|a|^{1/2}} \tilde\psi \left(\frac{t - b}{a}\right) db \frac{da}{a^2}, \tag{1}
$$
where $\...
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What is unit of time average of product of two signals?
Consider the signals $e(t)$ and $h(t)$ with units $\left[\frac{V}{m}\right]$ and $\left[\frac{A}{m}\right]$ and their Fourier transforms $E(\omega)$ and $H(\omega)$ with units $\left[\frac{Vs}{m}\...
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113 views
Why DFT is used for approximating CTFT when you can approximate CTFT-integral itself?
I was using MATLAB for approximating FTs. Why DFT is used if we can approximate the transform-integration using summation.
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Impulse invariance vs. DT representation of a CT system: Where is the inconsistency?
Suppose you have a continuous-time (CT) system $h_c(t)$, bandlimited to $B$. Your goal is to represent the system as a discrete-time (DT) system $h[n]$, sampled at $f_s \leq 2 B$. Clearly $h[n]$ won't ...
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Log derivative interpretation
In the origin paper on Synchrosqueezing Wavelet Transform, the phase transform, used to extract the instantaneous frequency of a signal $f(t)$, is defined as
$$
\omega (a, b) = -j[W_\psi f(a, b)]^{-1} ...
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Calculation of eigendecomposition of a signal in its Fourier domain?
I want to find the eigendecomposition of a 3-dimensional discretely sampled signal $X$, where each sample $X_{i,j,k}$ is treated as a vector $\langle i, j, k\rangle$ (with the origin at the middle of ...
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22 views
Right discrete cepstrum implementation in python
I know that the cepstrum is mainly computed as follow: $ C_{r}={\mathcal {F}}^{-1}\left\{\log({\mathcal {|{\mathcal {F}}\{f(t)\}|}})\right\} $
What I am wondering is if I should take the whole fourier ...
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2answers
86 views
Unclear time-to-frequency integration step
From here; $\hat f=\mathcal{F}(f)$, bar = complex conjugate:
Time-shift property: $x(t-b) \Leftrightarrow e^{-j\omega b}{\bf X} (\omega)$, so why is it $+$ (red)?
What at all is happening? Looks like ...
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Output of a stable LTI system
Let $\mathcal{L}$ be a stable LTI system. Is it true that if input has finite energy then output also has finite energy? I'm not sure about that. We know that $$\int_{-\infty}^{+\infty}|h(t)|dt\lt\...
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160 views
Autocorrelation for periodic signals
Autocorrelation for power signals is defined by $$R_x(\tau)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^Tx(t)x^*(t-\tau)dt\tag{1}$$ Is it true that for periodic signals $(1)$ can be computed by $$R_x(\tau)=...
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FFT followed by SVD leads to topography?
I am implementing the localizer method from this paper. One of the steps is not hard to understand and implement, but I don't understand why it is applied or what is the rationale behind it.
Each data ...
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Fast Fourier Inversion: Functions of a Complex Argument $f:\mathbb{C} \rightarrow \mathbb{R}$
I originally posted this question on math stack exchange, but I think it may be better suited for this community.
I'm interested in functions $f: \mathbb{C} \rightarrow \mathbb{R}$ with associated ...
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32 views
Discrete representation of a signal that has unevenly spaced samples in frequency
I'd like to describe a problem that I've been struggling with for a while. I want to apologize in advance due to the long text. I just want to be as clear as possible in my first post.
Consider the ...
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59 views
Windowing function for Inverse Fourier Transform
It is a common practice to apply windowing function, such as Hann or Hamming, to a time domain signal before FFT, in order to reduce spectral leakage. Often, we do 1) Windowing, 2) FFT, 3) frequency ...
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123 views
Alternative convolution theorem?
Instead of padding $x_1[n]$ and $x_2[n]$ then taking
$$
\text{iDFT}(\text{DFT}(x_1[n])\cdot\text{DFT}(x_2[n])), \tag{1}
$$
assuming we know $x_1(t)$ and $x_2(t)$, and their FT's, what if we do
$$
\...
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2answers
211 views
DFT of pure sinusoidal wave
I'm writing a program in which you can synthesize waves by adding to a sound's Fourier transform, and then inverse the transform to get the modified sound. In order to do this, I need to know what to ...
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How do I calculate the correct amplitudes of a discrete Fourier transform (DFT)?
I have a set of samples values in time domain. I know they are uncorrelated and I have to extract the correct amplitudes. However, the values are only ~88% of what they should be.
As a test see the ...
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107 views
How is signal to noise ratio actually measured by receiver equipment?
This sounds like quite a basic question but it surprised me, how is SNR actually measured?
You have the incoming signal:
It seems like the SNR is just the visual comparison of the peak signal '...
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Why is Fourier space not adequate for (theoretical or digital) filters?
As far as I have seen, almost all theoretical filter design occurs in Laplace or Z-space. Also, there is a pervasive connection to real life analog filters in the design. If one is just thinking in a ...
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DFT algorithm designed for a “sample-by-sample” senario
Suppose I have a system that wants to take an input signal (audio in this case) and wants to output a Discrete Fourier Transform of it in real time (ie every sample).
My initial thought is that if you ...
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Find CT Fourier transform of $ \left[ \frac{ \sin(\pi ~ t) }{\pi ~t} \right] \left[ \frac{ \sin(2\pi ~ (t-1)) }{\pi ~(t-1)} \right] $using properties
Use properties of Fourier Transform to solve the question.
The question is in the imgur link below.
I got $f_t$ of $\frac {sin(pi \cdot t)} {pi \cdot t}$ as rectangular pulse with value $1$ from -pi ...
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What is the DFT of $[x_1 -x_2 x_3 -x_4…x_n]$
If DFT of $[x_1 x_2 x_3... x_N]$ is $Y(k)$, what is the DFT of $[x_1 -x_2 x_3 -x_4,....x_N]$ in terms of $Y(k)$?
I have tried to formulate it but I cannot get a simplified expression for DFT of ...
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39 views
finding power spectral density from a vector
I have been given a vector:
\begin{equation}
v= \:\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{pmatrix}
\end{equation}
my job is to find the power spectral density from this vector
\...
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1answer
42 views
Bandwidth of a bandpass signal
If the Fourier transform of an aperiodic continuous time signal has signal components between the minimum frequency w1 and the maximum frequency w2, but not all the frequencies between w1 and w2, is ...
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1answer
49 views
Why is the continuous time Fourier series of DC signal an impulse?
In case of continuous time Fourier transform(CTFT), I can easily calculate the Fourier transform of DC signal by using Fourier duality or inverse CTFT. But I don't know how to calculate the continuous ...
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3answers
156 views
Does zero-padding distort the spectrum?
It's said to "sample the DTFT", revealing what "DFT fails to see". And I fail to see how this sampling isn't distortion.
The "spectrum" aims to provide the sinusoidal ...
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1answer
69 views
Find the length of the impulse response for the given output and input
Homework Question:
Consider a signal $x[n]=\alpha e^{j \omega_{0} n}+\beta e^{j \omega_{1} n}+\gamma e^{j \omega_{2} n} .$ What is the length of impulse
response $h[n]$ of a system (non-trivial) such ...
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Why does upsampling in the frequency domain produces replicas of the signal in spatial/time domain [duplicate]
The experiment is the following:
Given a 1d signal, e.g., a vector of values $f$. Let $F$ be its DFT, i.e., $F=fftshift(fft(f))$, shift is just to have DC centered. Then we upsample $F$ as $uF=...
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Can I sample at Nyquist rate if I know that my samples are taken only at the signal's maxima or minima?
I know that in general the sampling rate, $f_s$, must be greater than twice the highest frequency of the signal, $f$.
If I sample at the Nyquist rate, it can lead to the following:
However, if the ...
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4answers
139 views
what is the relationship between a spectrogram and the uncertainty principle heisenberg? [closed]
what is the relationship between these two things
Perhaps more resolution in a spectrogram is equivalent to knowing more the position of the electron and less resolution is knowing the velocity of the ...
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In computed tomography (CT), why is 'Inverse Problem of Radon transform' studied?
As I know, the Radon transform is a very important tool in CT by Beer's law. Thus, finding $f(x)$ of Radon transform $Rf(L):=\int_{L} f(x)dl(x)$ is helpful in CT. Nowaday, the Filtered back-projection ...
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33 views
expression for the FT of the frequency response of a system
I am trying to find an expression for the Fourier Transform of the frequency response of the cascade system seen here:
Here is my approach:
$(-1)^n = (-1)^{-n}$
$v[n] = x[n]e^{j\pi n}$
$V(e^{jw}) = X(...
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22 views
Is there anyway to find the frequency of DFT eigenvectors (basis) from its eigenvalues?
I've read this document which talks about the DFT. It describes that DFT bases are the eigenvectors of a circulant matrix. I know that every basis has a frequency in it, but I don't know what is it? ...
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42 views
Zero Padding in image reconstruction
I need to zero-pad the image for a better reconstruction but according to my project details, I will be given a Fourier-transformed image so can someone tell me how to pad the image in such a ...
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1answer
52 views
Bluestein's algorithm to evaluate the DFT from $f_o$ to $f_o + k\Delta_F$
Briefly, the convolution between $x(nT) e^{-j2\pi f_o nT} e^{-j \pi \Delta_F Tn^2}$ and $c(nT) = e^{j \pi \Delta_F T n^2}$ multiplied $e^{j \pi \Delta_F T k^2}$ allows me to find the DFT $X(f_k = f_o +...
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3answers
373 views
DFT coefficients meaning?
What "are" they? What's a sensible way to interpret the coefficients (and what isn't)? To pose specifics:
DFT coefficients describe the frequencies present in a signal
They describe the ...
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1answer
53 views
What do different Cepstral or MFCC coefficients represent intuitively?
I understand the explanation for separating slow and rapid changing log spectral components but i need to understand:
Why lower coefficients have higher and mostly positive magnitudes?
Why Higher ...
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34 views
Power spectrum of modified process
Suppose there is a process $x(t)$ with power spectrum $$S_x(\omega)=\lim_{T\to\infty}\frac{1}{T}\left|\int_{-T/2}^{T/2}\mathrm{d}t\,e^{j\omega t}x(t)\right|^2,$$ ignoring the expectation value for ...
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100 views
Why does time-domain convolution correspond to frequency-domain multiplication? (visual)
I seek a visual explanation of this. I've already seen the maths, and can derive the proofs - they amount to nill for an intuitive understanding. Any amount of math is welcome, as long as serving to ...
