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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Effect of Passband Slope on QPSK Constellation
What is the effect of a passband slope on a QPSK (or any QAM) constellation? Specifically given a linear slope (in magnitude, not dB), and no phase distortion (linear phase or even zero-phase) how ...
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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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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 $\cos$ ...
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Bandwidth of Information Signal
I have trouble finding the bandwidth of a signal. Say I have an info bearing signal m(t)=sinc(2t/pi). I found the fourier transform of the sinc function and found that the angular frequency was 1/pi. ...
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Nuclear norm minimization of convolution matrix (circular matrix) with fast Fourier transform
I am reading a paper Recovery of Future Data via Convolution Nuclear Norm Minimization. Here, I know there is a definition for convolution matrix.
Given any vector $\boldsymbol{x}=(x_1,x_2,\ldots,x_n)^...
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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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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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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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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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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 ...
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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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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 ...
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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, ...
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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 ...
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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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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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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 ...
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Finding Discrete Fourier Transform (DFT) for different DFT size
$N$ is an even integer, $x[n]$ is a finite length signal over the interval $n \in [0,N-1]$, and $X[k]$ is the $N$-point DFT of $x[n]$. Analytically find the DFT of sequence below in terms of $X[k]$. ...
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In IDFT, can x[n] be reproduced if DFT is ranging from -1pi to 1pi?
In short, I can reproduce the original x[n] from DFT via IDFT. However this happens only when I take samples DFT from 0 to 2pi. When DFT is ranging from -1pi to 1pi, the reproduced x[n] is not correct....
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Extended CORDIC for general Lie groups and algebras via representations
CORDIC is a well-known method for quickly computing exponentials and logs, including trig functions and their inverses by decomposing the angle into conveniently computable increments and then ...
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Proof for the energy correction factor of DFT
I am looking for a mathematical proof for the energy correction factor in conteext of windowed discrete fourier transform.
In Spectrum and spectral density estimation by the Discrete Fourier transform ...
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For a real-world system of oscillating mechanical components, what kinds of frequencies should I seek in DFT?
I have a real world system I am analyzing consisting of actual mechanical components that oscillate by rotating back and forth in a fixed axle (kind of like those finger fidget spinners but my system ...
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How to obtain filtered impulse response from frequency response?
I am trying to find the reverberation time of a room using the Schroeder method (i.e., Reverse-time integration method). Therefore, impulse responses should be measured first.
There are many ways to ...
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Effect of Passband Slope on a BPSK Constellation
What is the effect of a passband slope on a BPSK constellation? (This is a companion question to the related "DSP Puzzle" for QPSK)
Specifically given a linear slope (in magnitude, not dB), ...
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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 ...
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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 ...
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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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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?
...
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In what cases can you get aliasing below the Nyquist frequency?
I took the one-sided FFT of a signal and plotted up until the Nyquist frequency. Then, I took the real part of this FFT multiplied by $i\omega$ following a calculation that I'm trying to do of a ...
user62718
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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
$$ \mathbf{F}\left(\mathbf{R} f\right) (\theta, \sigma) = \mathbf{F}...
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Error in the computation of PSF?
I am in trouble computing a PSF of an optical system in optical turbulence.
Background
The optical transfer function (OTF) for an imaging system in optical turbulence can be modeled as the product of ...
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Discrete Fourier Transform of 2-D Images
I'm a high school student doing an essay on the applications of the Fourier transform on signal processing, but I haven't been able to find much information when applying the discrete fourier ...
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Multiple 1D gaussian filter
Given
$$
\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 ...
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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,
$t$ - ...
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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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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 ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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,...
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$\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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