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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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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How to combine bins of my DFT

I have a time series and apply the FFT to get a spectrum. Let's assume that my sampling frequency and the length of the time sample are chosen such that I end up with a $\Delta f = 0.1$ Hz. As this ...
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How does Adobe After Effects generate its “audio spectrum” effect?

I'm trying to replicate the "audio spectrum" effect from Adobe After Effects. An example can be seen in this video: Obviously, it has to be some variant of a fourier transform, but I've tried taking ...
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Removing values from FFT result same as filtering?

I don't quite understand why the textbooks say it is impossible to implement an ideal low pass filter. If I was to take the FFT of a discrete signal x[n], with Matlab's fft function I'd be returned ...
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What's the difference between using DFT, IDFT or DCT to calculate cepstrum of a power spectrum?

I've seen different equations that calculate cepstrum from power spectrum, but the equations are not consistent. Some people use Fourier transform, some use the inverse Fourier transform, and some use ...
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FIR filter design by the Fourier transform method

I am having some problems understanding how the Fourier transform method is used to determine the FIR filter. As far as i have understood, you start by using the ideal impulse reponse for the ...
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Convolution in frequency domain

Simple math question. The convolution theorem states that multiplication in time domain is equal to convolution in frequency domain and vice versa. There is a condition that the signal has to be ...
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What happens with signal in frequency spectrum when it is time shifted in time spectrum?

I have some trouble to understand what is going on with signal in frequency spectrum when it is time shifted in time spectrum. I am hoping that somebody will help me to understand that. Thanks you ...
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How moving part pixel intensity values of video frames becomes dominant compared to stationary part intensities in reconstructed frames?

Hello everyone i want to do dynamic texture video sementation using the Fourier transform in MATLAB. I am applying 3-D fft on dynamic texture video frames (using matlab function 'fftn') and ...
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Covariance between real and imaginary parts of Fourier transform of a stationary time series

Since Fourier transform of a random stationary process in time (in the case of existence) is not necessarily real, my question is what is the relation between the covariance of real and imaginary ...
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Repeated Fourier transform - what happens? [duplicate]

I have a Fourier transformable complex function that is a function of independent real variable a. Now I take the Fourier transform of it, giving me a complex function of real variable b. Now I ...
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length of window and overlap rate in STFT

I want to use STFT to analyze my signal and am wondering what are differences between two solutions: Use short windows (for ex. 256 samples window) Use longer windows (to get higher resolution in ...
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Approximating Lorentzian Fourier transform with FFT

I have a Lorentzian frequency distribution $F(w) = \frac{1+iz}{1+z^2}$ Where $z = \frac{w-\Omega}{R}$ With $\Omega$ being the peak frequency and R the decay constant. I know that analytically ...
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Formulas of the Fourier transform family

It has annoyed me that there doesn't seem to be a source online where the complete complex Fourier transform family is presented with every variable defined. The lack of definitions can be a nuisance ...
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Can realizations of a filtered Gaussian white noise process be represented as a Fourier transform?

Suppose we have a noise process $V(t)$ which is the result of passing Gaussian white noise through a filter with frequency response function $H(\omega)$. Can we represent realizations of this process ...
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RMS calculation in frequency domain after windowing

I can calculate RMS in frequency domain as derived from Parseval's Theorem. But what if I have applied a windowing function before doing the FFT (in my case a Hann window)?. Now the RMS values are ...
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redundancy of sin and cos waves with real data

I have the following question. Isn't it true that when applying a fourier transform to a real function (i.e. computing a characteristic function for a density), we only ever need one of the two waves: ...
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Compressive Sensing Incoherence Principle

As people acquainted with Compressive Sensing would know, incoherence and sparsity are two main principles. I've been reading about compressive sampling and developed an interest into the topic. What ...
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How Much Zero Padding Do We Need to Perform Filtering in the Fourier Domain?

Consider an $M\times N$ image $f$ and an $G \times K$ filter $h$. Given that convolution in the spatial domain corresponds to multiplication in the Fourier domain, then we can perform a convolution of ...
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Difference Between Convolution and Multiplication [duplicate]

I read that multiplication is convolution in frequency domain. I also understand that convolution is just polynomial multiplication. Can somebody explain what are the advantages of doing convolution ...
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Relationship Between Sampled Continuous and Discrete Time Signals

Consider the sketched system below. $x_c(t)$ is an arbitrary, continuous-time signal at the input and $s(t)$ is an impulse train, defined as $s(t)=\sum_{n=-\infty}^{\infty} \delta(t-nT)$, where T is ...
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Multiplication property DTFT

I was truing to solve an example of DTFT which is following multiplication property. The problem is $$ a^n \sin(\omega_0 n) u[n]$$ we know that the definition of DTFT is $$ X(j \omega) = \sum _ {n=-\...
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Working around FFT windowing?

I have the following problem, that I ran into recently, when calculating the spectra of data that I obtain from a measurement technique, we are using in our group. In short what we do in the ...
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Can you decimate / downsample a signal in frequency domain just like you can interpolate / upsample it?

To interpolate a signal I can just zero pad it in the frequency domain. If I want to decimate the signal, can I just discard some part of the frequency domain? So in MATLAB this works: ...
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Difference between Fourier-Transform and FFT of rectangular pulse

I'm trying to find a link between the Fourier-Transformation of aperiodic Signals and the FFT of them. So to start with a basic example, let's take a rectangular pulse with width 0.1s and amplitude of ...
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Chop out frequencies outside human hearing range

I have a bunch of audio files all sampled at 44100 Hz sample frequency. I am trying to remove all the frequencies which are outside the human hearing range (I use the following as reference: Frequency ...
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Determining the period of a discontinuous function

I'm new to the field of DSP. I'm trying to determine the period and shift of the function. I've tried using FFT, but haven't had much luck. Seems like it should be simple. Signal (pastebin of ...
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DFT as convolution question

I have tried to make this question as readable and consistent as possible. The short of it, is that I am trying to ascertain how one gets from the math equation shown, (which I understand), to the ...
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783 views

What features describe audio signals? (Besides frequency and amplitude)

I recorded sounds with a microphone and I try to distinguish them in my Java program. The frequency works quite good, but if I look at the fourier transforms it seems like there should be more ...
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Fourier Decomposition

Hello everyone have a look at this video of Fourier Decomposition of an image(otherwise you can also refer the image which shows few plots of different extracted waves from an image) . We also know ...
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What is the interpretation of the discrete-time spectrum?

The CTFT of an analog signal is a representation of that analog signal in terms of the frequency parameter of sinusoidal (cosine specifically) functions whose weighted sum make up that signal. The ...
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Autocorrelation of power spectrum

Anyone have an idea of how I can implement autocorrelation of power spectrum of one image? I tried using: autocorrel = ifft( | fft(power spectrum) | ^ 2 ); but ...
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Phase spectrum of a Fourier transform

I am trying to compute the phase spectrum of the signal $$ s(t)=\frac{A}{\pi}\left[H\left(t+\frac{\tau}{2}\right)-H\left(t-\frac{\tau}{2}\right)\right] $$ The Fourier transform is $$ S(j\omega)=\...
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Unit of Energy Spectral Density

The continuous-time Fourier Transform (CTFT) of a signal $x(t)$ (with unit $unit$) is: $$X(\omega)=\int_{-\infty}^{\infty} x(t)e^{-i\omega t}dt$$ which should be in $unit\cdot sec$ or $\frac{unit}{...
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Infinite extent of spectrum, but also in time in Oppenheim's Discrete Time Signal Processing?

In Oppenheim's Discrete Time Signal Processing there's on p. 323 no limited band in both time and frequency - wouldn't that violate the Heisenberg Principle?
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Testing discrete data for periodicity

I have some data which looks roughly periodic - is there a nice way to measure this? This is an example I'm working on and I'd like a metric that I will be able to just threshold to give a decision ...
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Fourier descriptors: trying to classify objects

Describing my background: I have around 33 items labeled. For example, 3 pictures of the contour of a basil plant, 4 pictures of the contour of earphones, 7 of a mug, etcetera. I'm trying to ...
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How to recover $f(t)$ from Fourier Transform of its absolute value $\mathcal{F}|f(t)|$?

Let the Fourier Transform of a real signal, $f(t)$, be $\mathcal{F}(\omega)$. And the FT of the absolute value of the same signal, $|f(t)|$, be $\mathcal{F}(u)$. Can $\mathcal{F}(w)$ be recovered ...
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DFT symmetry vs DFT duality in Richard Lyons' “Understanding DSP”

I am reading Richard lyons, understanding dsp, chap 3. Article 3.2 is about property of dft symmetry but any where in this chapter, i am unable to find discussion about dft duality property I want ...
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Why do we need the power spectral density?

Since the power spectral density is just the squared of the fourier transform, why is it useful ? Can't I just replace every solution that requires the psd with the fourier transform ?
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Complexity of FFT derivation

I am confused regarding the complexity of the Fast Fourier Transform (FFT). The Discrete Fourier Transform is: $$\qquad X\left [ k \right ]=\sum_{n=0}^{N-1}x[n]W_{N}^{kn}\quad \text{where}\quad W_{N}...
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Implementation of the constant Q transform + property questions

I'm reading up on fourier theory, especially the transforms. I implement the math as spectrograms in C++ to get a better understanding of what is going on. I've made an implementation of the short ...
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Hartley Transform vs Fourier Transform

Can you explain to me in what way Hartley Transform differs from Fourier Transform? Is it even used today or is it some mostly forgotten, obsolete archaic thing? Please don't use equations - I dont ...
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A query on Power spectral density (PSD)

Say a narrow band signal $n(t)$ has Power Spectral Density (PSD) $S(f)$. If the signal $n(t)$ got multiplied by $\cos(2\pi Ft)$, then what will the PSD of resulting signal in terms of $S(f)$ be? Will ...
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How do you find the frequency and amplitude from a DFT?

I generated a time history of 200 points using the equation $$x(t) = A\cos(2\pi f t)+ B\sin(2\pi f t) $$ with $A = 1$, $B = 0.1$ and $f = 17.2 \mathrm{Hz}$ and a sample rate of 400 samples per ...
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Spectral structure of sinusoidal model

Let us consider the following code: ...
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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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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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Discrete Time Fourier Transform (DTFT) for an unstable system (Ideal Low Pass Filter)

The Dirchlet conditions state that if the signal is absolutely summable then it the DTFT of the signal definitely exists. This is a sufficient condition but not necessary condition. There are ...
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Amplitude after Fourier transform

How to obtain the correct amplitude after the numerical Fourier transform of a signal? Example: consider an exponential decaying wave $y(x)=e^{-x}\sin(100\pi x)$ with Fourier transform $y_f(x_f)$ ...