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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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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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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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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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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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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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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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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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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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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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 ...
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Why is the Periodogram Used as an Estimate of PSD?

Why is the periodogram method as given by Schuster (1898) - which is not a consistent estimate of the PSD - still used, as opposed to consistent non-parametric methods for estimating the PSD like ...
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1answer
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Fourier Transforms, Convolution, Cross-correlation: what is their physical unit exactly?

Let us assume we are talking about real, deterministic, electrical signals $x(t)$ and $y(t)$ (magnitude in Volts). There are different kind of Fourier Transforms. I made a table to summarize: NB: By ...
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Deriving Fourier Transform of Time-Windowed Discrete Signal

I'm trying to derive the Fourier Transform of a finite-length discrete signal to show the effect of windowing,e.g. spectral leakage and resolution, but I can't seem to arrive at the same answer. Just ...
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Correction of amplitude after zero padding for upsampling purposes

I have time sequence in which the data is sampled at 0.8 Hz. The data is related to chromatography (chemical analysis), that is why the sampling frequency is relatively low. The instrument cannot ...
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Morlet wavelet: DFT vs Fourier Transform

I find what seem to be contradictions between the two. "Morlet" defined here, and its Fourier Transform (FT) below it. DFT's imaginary component zeroes with large ...
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DFT derivative property?

Does it have one? The continuous variant does, $f'(t) \rightarrow j \omega F(\omega)$ - but $jkX[k]$ definitely isn't it for DFT. To find it there must be a useful simplification of $\text{DFT}(x[n] - ...
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Heterodyne modeling confusion in SDR

I've been reading this lab sheet which explains the signal processing math of the RTL-SDR radio dongle. http://www.eas.uccs.edu/~mwickert/ece4670/lecture_notes/Lab6.pdf In pages 5 and 6, the local ...
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How to determine the sine Fourier coefficients of discrete data?

The following relation gives me the measurements of interest $w$ at equally distanced locations $x_j$ in space: $$w_j=\sum_{m=1}^{11}A_m\sin\left(\frac{mπx_j}{L}\right)$$ where $A_m$ are the Fourier ...
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How to plot thermal noise in the “time domain”?

$P = 4kT$ (where $k$ = Boltzmann’s constant, and $T$ = temperature of the instrument ($K$)) And the mean voltage is thus, of course, $V^2/R = 4kT$ And the voltage is distributed as a Gaussian around ...
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How much zero padding I should do to an audio signal before fft?

I was working on project and I need to do fft to my audio signal. I was going through a code and found following line. Can anyone explain me the line of code. ...
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how to plot fft with imaginary and real part in 3d or calculate the degree of rotation

I would like to make a plot like 1 and see the real and imaginary part in a 3d space. I dont want to make exactly the same plot. for me it is okay if i see the peak for both signals shifted. it is ...
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Applying fourier transform twice (DSP course)

I stumbled upon a question in a DSP course (coursera) which I don't understand, shown below is a screenshot of the question and answer. The part which I don't understand is circled. Why is it equal ...
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The impact on the frequency of adding zero samples and non zero samples

Can someone give me a brief explanation of what is going on in here? I struggle to understand the full differences between the impact on the frequency of added zero samples and adding samples which ...
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1answer
44 views

Cosine of amplitude 1 but GNU Radio shows FFT has amplitude > 30,000

I am doing an FFT on a cosine wave which has an amplitude of 1. The FFT is amplitude is not 1 but over 30,000. What am I missing here? It's from the GNU Radio FFT wiki page:
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Calculate displacement from acceleration signal data using Fourier transformation

I recorded acceleration data from an accelerometer attached to a vehicle, and I want to calculate displacement using the Fourier Transformation Integration method. I used software called vibration ...
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Why does MATLAB's dualtree3 function not return the lowpass subband of the imaginary tree?

MATLAB has a function named dualtree3 which computes 3-D dual-tree complex wavelet transform. This function only retains the last low-pass coefficients/subband of ...
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Adding zeroes in between the samples of a discrete frequency domain leads to both zero padding and scaling

I won’t show ALL the calculations, but I’ll describe the problem fairly enough to understand. We’ve been taught that upsampling on one domain leads to padding zeroes or added time period which is zero ...
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Why we add imaginary part in inverse fourier transform since the time domain signal has only real value?

The equation of inverse Fourier transform is the following: $$ f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega $$ $$ f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\...
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Fourier transform of time division

I know that Fourier transform of $t^n f(t)= i^n \frac{d}{d\omega^n} F(\omega)$. But does this work when $n<0$? Is there any direct relation to compute the Fourier transform of $\frac{f(t)}{t}$?
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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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Discrete Fourier Transform on images using OpenCV

I applied DFT on an edge image (attached both). I expected the output image to not have very low frequencies (~0).[Refer this link for more info] But the output I got consisted of low and high ...
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What is the effect of wiping the right half of Fourier Transform bins?

I'm trying to change the pitch of a signal using a Fourier Transform (FFT) followed by an Inverse Fourier Transform (IFFT). I've found many examples, some of which zero out the right half of the real ...
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1answer
60 views

Prove a property using shift theorem and duality

I'm reading Lectures on the Fourier Transform and Its Applications and I'm going to prove shift theorem for the inverse Fourier transform using duality. According to the mentioned source, the duality ...
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59 views

Can you use Bartlett method in reverse? [closed]

I'm wanting to do an inverse Fourier transform. Can I use the Welch method to generate this inverse, by replacing the FT used within with an IFT?
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How to do Frequency Scaling on an Audio File

Please excuse me if my terminology is wrong. I'm from a music production background and have no experience in signal processing. I was wondering if it was possible to stretch out the overtones (...
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Does it make sense to express a given FT from 0 to 1 as a convolution?

I'm learning about Fourier transform, and was asked whether the following FT can be expressed as a convolution: $$X[k] = \sum\limits_{n=0}^{N=1} x[n]e^{-i\frac{2\pi}{N}kn}$$ There are two things I don'...
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How would I spectrally rotate/invert an audio signal (Matlab or Python)?

So I have a 4.5 minute wav audio clip. I want to spectrally rotate the frequencies such that the spatio-temporal characteristics of natural speech are retained, while having the new audio be ...
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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 $y_f(x_f)$ ...
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Inverse discrete time Fourier transform with differentiation

Consider a signal x[n] and its DTFT $X(e^{jω})$ . Assume $X(e^{jω})$ is differentiable. Compute the inverse DTFT of $j\frac{dX(e^{jω})}{d\omega}$ You should write your answer in terms of $x[n]$ and ...
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Is resizing of frequency spectrum a valid method of resampling?

One method of resampling is to perform a Fourier Transform on a signal, resize the resulting frequency spectrum, and then return to the resampled version of the signal with an inverse fourier ...
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I need help in understanding “Nyquist Criterion” definition

I am researching the split-step parabolic equation and its split step solution as in: Ozgun, Ozlem & Apaydin, Gokhan & Kuzuoglu, Mustafa & Sevgi, Levent. (2011). PETOOL: MATLAB-based one-...
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Time scale and Fourier transform

Consider the Fourier transform $F(\omega)$ of the function $f(t)$. The magnitude of $F(\omega)$ depends on $\omega$ and thus also depends on the scale of the $t$-axis. For example, when $f_1(t)$ is a ...
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What could cause fast Fourier transform to give complex conjugate of the intended result?

I have 2 real time series $x(t)$ and $y(t)$, after fft it should become $\tilde{X}(f)$ and $\tilde{Y}(f)$. Then I need to normalize $\tilde{X}(f)$ with $\tilde{Y}(f)$ : $\tilde{X}(f)/\tilde{Y}(f)=\...
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If a square wave is a sum of odd harmonic impulses, why is it continuous in the frequency domain?

A square wave is a sum of sinusoids so surely it should be represented as individual discrete impulses in the frequency domain, where all other frequencies are 0. Why instead are those intermediate ...
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Why do people use STFT as a preprocessing step to using CNN?

Just briefly looking up some research papers on audio data and I have come across some papers that use STFT as a preprocessing step to using CNN. Why is this the case? What are the advantages and ...
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Determine constant $A$ such that $x[n] = x[n] \star x[n]$

Let $x[n] = A\delta[n] - \frac{\sin(\frac{3n}{2})}{\pi n}$. Determine constant $A$ such that for all $n$ $$x[n] = x[n] \star x[n] \tag{1}$$ I think it's not possible since $(1)$ leads to $$X(e^{j\...
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FFT shows systematic deviations from analytical result

I am calculating the numerical Fourier transform of an exponential decay exp(-|t|) and compare it to the analytically calculated result, a Lorentzian. I find that the numerically calculated spectrum ...
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What's the relation between frequency band of $X(j\omega)$ and $\Phi_{xx}(j\omega)$?

in which: $x_{c}(t)$ is a continuous-time signal $X(j\Omega)$ is the Fourier Transform of $x_{c}(t)$ $\Phi_{xx}(j\Omega)$ is the Power Spectrum Density of $x_{c}(t)$ which defined as Fourier ...

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