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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Problems Using FFT to Compute Impedance in a Model Neuron

I'm a neuroscientist currently investigating the resonance properties of a single neuron model that a colleague and I have constructed. The language we code in is Julia, which I hope is similar enough ...
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Fourier transform of modulus of sum of weighted sines

$$ x(t) = |a \cos(\omega_0 t) + b \cos(\omega_1 t)| $$ with $a, b \geq 0$, $\omega_0, \omega_1 > 0$, but $a, b > 0$ or all $a, b$ (negatives included) is also acceptable, or replacing $\cos$ ...
3 votes
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Improving the intuition for the 2d fourier transform

As far as I understand, the 2d fourier transform is calculated as following: ...
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Derivation of Strehl ratio

I am in trouble proving the following fact: The Strehl ratio is the ratio of the volume of the aberrated OTF (optical transfer function) and the ideal OTF, i.e. \begin{align} \mathcal{S}=\frac{\int\...
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3 votes
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real time - active noise control

I am trying to implement an adaptive filter for system identification and active noise control for realtime signal processing on an FPGA using Labview. For system identification, I implemented the ...
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3 votes
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Convolution of signals sampled on a logarithmic grid

Is there a practical accelerated algorithm or a theoretical discrete (Fourier) transform based method to convolve discrete-time signals sampled on a logarithmic grid? What I mean is representing a ...
3 votes
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6k views

What's spectral "tilt"?

I am looking at article Speech-in-noise intelligibility improvement based on spectral shaping and dynamic range compression. In paragraph 2.2 the article mentions "tilt" of the spectral envelope. The ...
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3 votes
0 answers
73 views

Can you use Fourier transformations (or other) to read multiple superimposed barcodes?

If you printed bar codes on tracing paper/acetate etc. and then positioned several in front of one another, could you extract the individual codes from the aggregate overlaid image? I feel intuitively ...
3 votes
0 answers
364 views

Aliasing in the Short time Fourier Transform of a pulse

When attempting to take the Short Time Fourier Transform of a pulse, at the end of the pulse I'm running into problems. The signal looks like this at the end (it is a simple $sin^2$ pulse envelope, ...
3 votes
0 answers
181 views

Understanding Walsh coefficients

I am working with Walsh coefficients. I know the intuitive understanding is almost that that they are the degree of connectivity, but it is there a better way of thinking about it? What is the ...
3 votes
0 answers
454 views

Discrete Fourier transform in a multidimensional space

I want to measure the frequencies at which a point oscillates in a multidimensional space, let's take the example of a point on a 2d-surface. For now, I naïvely split the signal in two, along the ...
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3 votes
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Global Transforms besides the DFT?

This is a simple question. Fourier analysis gives us the DFT, which is known as a global transform of a signal. In contrast, the Discrete Wavelet Transform (DWT) has a plethora of wavelets, all of ...
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3 votes
3 answers
159 views

Using spectogram to speed up a signal - Time Scaling/Phase Vocoder

Background About half a year ago, while learning about spectograms as part of an Image Processing course I took, I was told you can speed up audio using spectograms as follows: Calculate the ...
2 votes
1 answer
45 views

Need help with DTFT problem

Prepping for exam and this is one of the practice problems: I just want some clarification on some of the steps my professor took. This is the answer in the answer sheet Only thing I dont understand ...
2 votes
0 answers
32 views

Removing once per revolution variation from data

I’m looking for help to find a robust technique to remove a once per revolution variation in some vehicle test data. The data is collected by driving a vehicle around a circular path at increasing ...
2 votes
0 answers
51 views

Answered-Question About Radar Pulse Modulation

I am trying to simulate a radar-transmitted signal with a 4.5 Hz clock frequency and 1.8 GHz carrier frequency. I generated the carrier signal and a rectangle shape pulse signal, then multiplied in ...
2 votes
0 answers
61 views

2D Cooley-Tukey FFT in Python

I've been trying to confirm the process for the Cooley-Tukey approach for FFTs. Currently I have a function that generates random input data for a matrix with $n_1$ rows and $n_2$ columns. The result ...
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2 votes
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pocketfft delivers wrong values

does anyone understand how to use the pocketfft by martin reinecke? Link: https://gitlab.mpcdf.mpg.de/mtr/pocketfft Basically it's just this snipped of code: ...
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2 votes
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Convolve sinc trains

$$ \begin{align} & \mathrm{sinc}(As + .5)\sum_{n=-\infty}^{\infty} \delta (s - n/A)\ \star \\ & \mathrm{sinc}(Bs + .5)\sum_{n=-\infty}^{\infty} \delta (s - n/B) \end{align} $$ How to compute? ...
2 votes
1 answer
49 views

making a laplace s-domain plot from numerical data to decompose signal into decaying sinusoids

I would like to make a 3D laplace s-domain plot from experimental data I have. The examples I have seen for this are when the function is already known and an analytical solution can be obtained (...
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2 votes
0 answers
29 views

Additional artefacts in limited angle Radon transform reconstruction using the Fourier Slice Theorem

I want to simulate the limited angle Radon transform reconstruction problem by employing the Fourier-Slice Theorem which states that $$ \mathbf{F}\left(\mathbf{R} f\right) (\theta, \sigma) = \mathbf{F}...
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2 votes
1 answer
180 views

FMCW radar: understanding of doppler fft

I am using fmcw radar to find out distance and speed of moving objects using stm32l476 microcontroller. I transmit the modulation signal as sawtooth waveform and I read the recieved signal in the ...
2 votes
0 answers
78 views

Discrete Fourier Transform of 2-D Images

I'm a high school student doing an essay on the applications of the Fourier transform on signal processing, but I haven't been able to find much information when applying the discrete fourier ...
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Multiple 1D gaussian filter

Given $$ \begin{cases} &f_0(x)=1 \\ &f_{n+1}(x)= (\varphi*(f\mathbb{I}_{[a_n,b_n]}))(x)=\int_{-\infty}^{+\infty}\varphi(x-t)f_n(t)\mathbb{I}_{[a_n,b_n]}(t)dt = \int_{a_n}^{b_n}\varphi(x-t)f_n(...
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calculate or decompose a Fourier transform signal amplitudes with unknown weights on sources

migrated from math-se... I am trying to calculate , or approximate the solution of following Fourier-sine transform problem that can be expressed as a contributions of periodic sources $f_i(x)$ and ...
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Welch Method FFT Python - Scaling factor?

I've been implementing a Welch method FFT and I am trying to work out the correct scaling factor that should be applied to the output of the function so the PSD is accurate because at the moment it's ...
2 votes
0 answers
223 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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Fourier Transform of an Exponential Sine Sweep

The Exponential Sine Sweep (ESS), according to Farina [1], can be described by the following formula: $$x(t)=\sin\left(\frac{2\pi f_1 T}{R}\left(e^{\frac{t R}{T}} -1\right) \right)$$ where, $t$ - ...
2 votes
1 answer
120 views

How to downsample a fourier transformed signal?

I have a signal of length 100000 timestamps sampled at a frequency of 25kHz. First I apply a high pass filtering at (300Hz) and then do the Fast Fourier Transformation. Then the absolute values are ...
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Find a LTI system such that $\mathcal{T}\{\frac{\sin t}{t}\} = \frac{\sin 2t}{t}$

Let $$x(t) = \frac{\sin t}{t} \qquad\text{and}\qquad y(t) = \frac{\sin 2t}{t}$$ Is it possible to find a LTI system such that $\mathcal{T}\{x(t)\} = y(t)$? If not, what's the reason for ...
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832 views

Is rotation of a Fourier transform the same thing as Fourier transform of a rotation?

I'm working on an image processing problem and wondering if DFT(rotation(image)) == rotation(DFT(image)) (1). My final goal is to apply rotations in the Fourier domain then do an inverse Fourier ...
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Power spectral density vs. Fourier Transform

I am trying to understand the difference between the Power Spectral Density and the Fourier transform. Specifically, I am trying to understand why the power spectral density is useful and in what ...
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Getting the right frequency (using FFT)

I am implementing the method from this paper: https://dspace.mit.edu/bitstream/handle/1721.1/66243/Picard_Noncontact%20Automated.pdf?sequence=1&isAllowed=y The main idea is cardiac pulse ...
2 votes
0 answers
131 views

Two versions of Constant Q Transform (CQT) doesn't match each other?

To my knowledge, there's two major CQT papers, the one by Brown in 1991, and the one by Schorkhuber in 2010. The 2010 paper claims to be a more computationally efficient implementation of the 1992 ...
2 votes
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31 views

What is a correct way to find or "guess" a kernel which transforms an image into another image using Fourier Transformations?

Assuming I have two images, apple and orange; also assuming a filter kernel that transforms an apple image into an orange image possibly exists, how would some series of Fourier Transformations (and ...
2 votes
0 answers
169 views

How to calculate the Fourier Transform of a solvable chaos waveform?

Recently I am stucking in frequency estimation of a solvable chaos waveform. Its local analytic expression in time domain is $$ z(t)=s_m(u_m-s_m)e^{\beta(t-mT)}\cos(\omega_0 t+\varphi),mT\leq t<(m+...
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299 views

How to derivate in the frequency domain

I have two Time Domain functions, $f_1(t)$ and $f_2(t)$. I have both Fourier Transforms, $F_1(\omega)$ and $F_2(\omega)$. Functions $f_1$ and $f_2$ are not independent and, in fact, $f_1$ is also a ...
2 votes
0 answers
52 views

Vector parameters in uncountably infinite-dimensional spaces

My question was, in an uncountably infinite-dimensional vector spaces, how to represent a vector by a list of parameters, as we do in finite-dimensional spaces? I was assuming that if we can not ...
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What is the correct way to do Short term Fourier transform and extract the phase information from local sections of a signal?

I took a triangular voltage signal and taking it's time derivative I obtained the current through a pure capacitor. Then I took different portion of voltage and current signals, took their Fourier ...
2 votes
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1k views

Relationship between Wavelet transform and Fourier Power Spectral Density

Is there anyway to obtain the Fourier Power Spectral Density from a wavelet transform of a time series? I am particularly interested in this problem because I was wondering if there is any ...
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Constructing $X(\omega/2)$ from the FFT of $x[n]$

Let $X(\omega)$ be the Discrete Time Fourier Transform (DTFT) of $x[n]$, I want to construct $X(\omega/2)$. Precisely, I use FFT function to compute the samples of $X(\omega)$ in one period, say $[0,...
2 votes
0 answers
208 views

$\tt ifft()$ function - absolute vs real form

I have multiple files from an experiment in frequency-domain that I would like to use ifft() function to convert to the time domain in R to apply signal processing ...
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Elementary proof of Fourier domain formula of multidimensional downsampling by $M$

I am trying to prove a well-know formula for the multidimensional downsampling by arbitrary downsampling integer matrix $M$ in $d$-dimensional case. The formula is $$ \hat{y}(\omega)=\frac{1}{\...
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Signal rescaling

Please help me out with this one. I think I've been given an impossible task. I'm working on a system that transmits an active signal and looks for strong reflections. The transmitter is actually ...
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Filtering and Differentiating phase-modulated signals

For a project work, I need to demodulate data from a Laser Doppler Vibrometer. The distance information is phase-modulated, therefore the velocity information is frequency-modulated: $$x \propto \...
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Optimal method to calculate Fractional Fourier for Chirp signals

There are several method exist in the literature to calculate fractional Fourier transform. My interest is in chirp signals and want to find time delay estimation using fractional Fourier transform (...
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Why does filtering the same discrete dataset while choosing different sampling rates yield different results?

If I have a two-dimensional discrete dataset (one space, one time), and I create subsets of this dataset by "sampling" (although the original dataset isn't continuous) it at different rates, should I ...
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Why doesn't the mel-scale use the twelth-root of 2 instead of a biased sample listener?

For the mel frequency spectrum, which is used extensively in audio-processing, the technique uses subjects to identify pitches of uniform distance from one another. Isn't this distance just the 12-th ...
2 votes
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Stationary processes and their power

The top answer to this question: Power spectral density vs Energy spectral density states: However if random process is stationary, then it is for sure that it has some finite power over some ...
2 votes
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A simpler method or more descriptive answer to the Fourier Transform

I'm trying to do a CT Fourier Transform of these two signals $$x_1(t)=e^{−a(t−1)} \cdot u(t−1)$$ and $$x_2(t)=e^{−a(t−1)} \cdot u(t)$$ Where $a$ is any real number, and $u(t)$ is the unit step ...

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