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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What is obtained from the cross correlation plot?

Let’s assume that we have two audio signals, x(t) and y(t) affected by the noise as shown below. And we would like to cross-correlate these two signals and the cross-correlation plot is shown as ...
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Fourier Transform Identities

We know the below, $$ \mathscr{F}\big\{x(t)\big\}=X(f) \tag{1} $$ $$ \mathscr{F}\big\{x(-t)\big\}=X(-f) \tag{2} $$ $$ \mathscr{F}\big\{x^*(t)\big\}=X^*(-f) \tag{3} $$ Now, if for some signal $$ x(-...
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Fastest implementation of fft in C++?

I have a MATLAB program that uses fft and ifft a lot. Now I want to translate it to C++ for production. I used OpenCV but I ...
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Why Do I Get This Crackling Noise on Zeroing out the High Frequencies?

I recently started playing with the Fourier transform (after spending a few weeks learning about the mathematics behind it). I decided to try hacking together a low-pass filter on the following sound ...
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Conceptually, how does a Fourier transform differ from an autocorrelation?

I realize the two are derived using different algorithms, and the units are different, but from a conceptual standpoint of the information they provide how do they differ? I'm thinking here about the ...
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Understanding the Gabor filter function

I need to implement a script for generating features from an input image by using the Gabor filter. I have no past experience of wavelets and I'm just learning Fourier analysis (I understand the basic ...
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Phase Correlation - Poor Performance on Noisy/Blurred Images?

I have successfully tested 1D phase correlation algorithm to determine vertical shift between two synthetic images. When I moved to real images, however, it is not able to detect translation at all (...
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Fourier transform artifacts

My starting point in what follows is a radially symmetric random field. Taking the Fourier transform of this (and plotting it in logarithm to highlight the patterns), I obtain the following image in ...
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How do I create a frequency vs time plot?

I'm a chemical engineer, not an EE, so this is a bit difficult. I'm trying to figure out how to take amplitude vs time data and transform it into frequency vs time. My first instinct is to slice my ...
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Spherical equivalent of Nyquist frequency

Let $\phi$ be a scalar function defined on the surface of a sphere. I have samples of $\phi$ at various locations on the sphere. I want to apply a spherical harmonic transform. I know that $\phi$ is '...
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Calculating smoothed derivative of a signal by using difference with larger step=convolving with rectangular window

I have a signal sampled at $\Delta t: fi(ti=i\Delta t)$ where i = 0..n-1. I want to find the first derivative of the signal: f'(t). My first thought was to estimate this by a central difference: $f&#...
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What is the $\mathcal Z$-transform of Bessel function $J_0(\alpha n)$ sequence

What is the $\mathcal Z$-transform of the sequence $J_0(\alpha n)$ for $n \in \mathbb{Z}$? The Fourier transform of zero$^{\rm th}$ order Bessel function $J_0(\alpha x)$ is known to be $\frac{2}{\...
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Mathematically Inclined Signal and Systems / Signal Processing Book Recommendations

I'm an electronics engineering student with high inclination to analysis and pure mathematics. I was just wondering if there was any book ( or any resource ) that treats signal and systems and signal ...
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Converting raw I/Q to dB

I am getting I/Q data from a software-defined radio. I want to do some stuff on signals in the data, but only if it exceeds a certain range. What is the general procedure to get dB (dBm, or anything)...
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How to generalise the Fourier transform?

The Fourier transform takes a signal and splits it into a series of sine and cosine waves. I am told that it's supposed to be possible to split a signal into some other set of functions. My question ...
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what is the difference between $X(j\omega)$ and $X(\omega)$ notation

I am trying to understand Fourier Transform and Laplace Transform. What is the difference between $X(j\omega)$ and $X(\omega)$ notation? what is the meaning of $j\omega$ ? Is it represent frequency? ...
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Inconsistency between the units of power spectral density and the definition that people often give

Perhaps someone can help me resolve something - this is my understanding: In deterministic signal analysis, for a signal $x(t)$ the signal energy is defined by $$E_{\textrm{s}} = \int^{+\infty}_{-\...
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Effect of windowing on noise

I understand that truncating a signal in time 'smears' the frequency response depending on the window chosen. In general, the shorter the signal duration, the more 'flattened' the frequency response, ...
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Using the Inverse Filter to Correct a Spatially Convolved Image (Deconvolution)

As part of a homework assignment, we are implementing the Inverse Filter. Degrade an image then recover with an Inverse Filter. I convolve the image in the spatial domain with a 5x5 box filter. I FFT ...
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Signal processing techniques for an accelerometer signal?

I am running some tests where I am recording accelerometer measurements. I am looking to use elements of signal processing on this signal, but I am unsure about where to begin, or what my approach ...
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Why are there so many windowing functions?

Many windowing functions are listed here in the Mathematica documentation. I tried using a few to reduce leakage when computing a Discrete Fourier Transform. From what I could tell it made little ...
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Phase shift and phase spectrum terms in multidimensional signal

I know about phase of a 1D signal. But when I go into higher dimensions like 2D,3D etc, it becomes headache to grasp the concept. What are the terms phase shift and phase spectrum mean in case of ...
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Example of Fourier Transform not existing for real-life signals?

I got curious based on this question here, but basically, is there ever a real-life signal that exists where its Fourier transform does not exist? If a signal is not finite energy, then its Fourier ...
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Complex conjugate and IFFT

I asked a question over on stack overflow. I'm having a slight problem however. As suggested by Paul R I am mirroring my lower $n/2$ bins into the upper $n/2$ bins. I have a few questions however. ...
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Difference between DFT and Z-Transform

I have searched this question but couldn't find the answer in this network. I know this is very confusing question for DSP beginners. Both DFT and Z-transform work for Discrete signal. I have read ...
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How one apply correctly FFT in image denoising

I'm writing program (Qt widgets/c++) for removing noise from images. As denoising method, i selected non local means method. This method has incredible quality of restored images (that's why it's the ...
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FFT of random binary data

I am trying to make sense of FFTs and binary data. Say I have a series of random binary data, which is measured with a repetition rate of 400Hz (interval time of 0.0025s). I have a total of 12489 ...
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Choice of Gaussian kernel parameters when lowpass filtering before image resampling?

I need to decimate a signal by a factor of q. More specifically my signal is a 3D "image": $\ I(x_i,y_j,z_k)$, which I need to downsample by a factor of two in the z direction. I want to do lowpass ...
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Phase correlation vs. normalized cross-correlation

I asked this over at Mathematics Stack Exchange, but since this sort of lies on the border of the questions normally asked over there and the questions you see over here I'll ask it here as well. (As ...
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Is a Fourier transform a sound way to analyse a transient signal?

I am currently working on a project that involves analysing transient signals from sensors. While not actually part of the analysis itself, I discussed it with the team, and they are using an fft to ...
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Spectral Leakage in layman's terms

I'm trying to understand the concept of spectral leakage for the DFT, without going deep to the mathematical intricacies (it's for practical purposes). I've read from the book "Introduction to Digital ...
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Coherent Sampling And The Distribution Of Quantization Noise

I have a question concerning the distribution of noise in the frequency domain in case of Coherent Sampling. I read up about Coherent Sampling and understood that in order for a frequency $f_{in}$ to ...
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Deriving 2-D discrete Fourier transforms

I have a problem in DFT. It was one of my past-year exam papers questions. Question: Let $F(u,v)$ be the 2-D Fourier transform of a 2-D continuous function $f(x,y)$. Derive in terms of $F(:,:)$ ...
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DFT and multiplication/convolution equivalence

Is there a simple or potentially intuitive explanation for, with the DFT, vector multiplication in one domain being equivalent to circular convolution of the transforms of the vectors in the other ...
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Intuition behind the scaling property of Fourier Transforms

The Fourier transform of $f(ax)$ is $\frac{1}{|a|}F(\frac{u}{|a|})$. So the frequencies are scaled horizontally but the magnitudes are also scaled when the graph of $f$ is scaled horizontally. On the ...
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Do $|s(t)|$ and $|S(f)|$ uniquely determine $s(t)$?

Consider a signal $s(t)$. My question is if you know $|s(t)|$ and $|\mathcal{FT}[s(t)](f)| = |S(f)|$ or equivalently $|s(t)|^2$ and $|S(f)|^2$ is it possible to determine $s(t)$? That is, is $s(t)$ ...
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Sense of zeropadding in a time domain

I have the task related to Radon transform which contains a subtask which uses resampling by means of DFT. Let's consider the non-periodical discretized signal (Fig.1) (for example the string of ...
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Removing a sinusoidal artifact from a set of movie frames

I am doing some post-hoc analysis of a dataset consisting of a series of movie frames that are contaminated by a strongly periodic artifact. I would like to remove this artifact from my frames. For ...
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Fourier transform of cosine to the power of 3

How can I find the Fourier transform of $$ f(x) = ( \cos(x) )^3$$ I know that for $ g(x) = \cos(x) $ $$\mathcal F \Big\{ g(x) \Big\} = \mathcal F \Big\{ \cos(x) \Big\} = \pi \Big [ \delta(w-\pi / 2)...
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Difference between DC component and zero frequency component of signal

We know that Fourier Transform of a signal exists if it is absolutely integrable and it exists for periodic signals if impulse functions are allowed. If we consider the fourier transform of $\text{...
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Why idft(dft(a) * dft(b)) not equal to convolve(a, b)?

I'm a little confused... I always thought the DFT of a convolution was equal to a product of DFTs, but when I tried this in Python: ...
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Why is not Fourier Transform Good for Non-linear Processes

I was reading through slides about Hilbert Huang Transform. In slide 14, which talks about the motivations of a new method instead of Fourier Transform (FT), the author provides those two reasons in ...
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Conjugation in Fourier Transform

I have a very simple question. In Oppenheim book, it says that: If CT Fourier transform of $x(t)$ is $X(j\omega)$ then, CT Fourier transform of $x^*(t)$ is $X^*(-j\omega)$. What I can't ...
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Why integrate over $2\pi$ in inverse DTFT?

In DTFT of a signal, the spectrum of a sequence is periodic with period $2\pi$ and all the information needed for derivation of the original signal from its spectrum is contained in $\pi <\omega &...
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What is $F(0)$ is “dc” component in the context of image processing?

It has always been said that $F(0)$ is the "DC component" in fourier transform. However, I don't get what it means to say that $F(0)$ is "DC" in the context of image processing. The zero in this ...
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What happens if we change the limits of integral in Fourier transform?

By definition of Fourier transform $$X(\omega)=\int_{-\infty}^\infty x(t) e^{-j\omega t} dt $$ Now what will happen to the answer of transform for example in case of $x(t)= \cos(\omega_0 t)$ if ...
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Scaling property of Fourier Transform

Problem 4.6(b) from Oppenheim, Wilsky & Nawab (2nd ed) reads: Given that $x(t)$ has the Fourier transform $X(j\omega)$, express the Fourier transform of $x(3t - 6)$ in terms of $X(j\omega)$. The ...
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What is the Fourier Transform of a constant signal?

I am trying to figure out what the fourier transform of a constant signal is and for some reason i am coming to the conclusion that the answer is 1. Or better yet a step function.
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Fourier Transform with both Time Delay and Frequency Shift

I know that the Fourier transform of a function with time delay can be written as: $$\mathscr{F}\big\{x(t-t_0)\big\}=X(f)e^{-j2\pi f t_0}$$ The Fourier transform of a function with frequency shift ...
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Zero Padding of FFT

There are many question related to the zero padding a time domain signal to get more frequency bins after performing Fourier transform. As I understand this process is equivalent to trigonometric ...

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