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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Does Short-time Fourier transform impact quality of signal?

I am playing with spleeter to separate voice from an audio signal. They go to the frequency domain by computing the Short-time Fourier transform to create a spectrogram. As there are some ...
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Fourier Transform of a PSD and response of a PSD input

Does a Fourier Transform of a white noise exist? If so, what is its general form? It is possible to compute the fourier transform of a Power Spectral Density? My problem in detail. I have to compute ...
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Frequncy Domain Distribution

I have a complex signal in the time domain normally distributed. What will be its distribution in the frequency domain? I assumed since the frequency domain is a linear transformation the distribution ...
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Meaning of negative frequencies in Baseband non sinusoidal, non periodic signal

I can understand the meaning of negative frequencies in a sine or cosine signal, since by using Euler's identities, you have two complex phasors moving to different directions, which when added, give ...
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Fourier transform of the magnitude of the fourier transformed signal

I've come to realize that the Fourier transform of an already Fourier transformed signal gives the time-reversal signal. $$\mathcal F(\mathcal F(x(t)))=x(−t)$$ ref 1,ref 2 However, my question is, if ...
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215 views

How to perform spectral inversion in the frequency domain to convert a low-pass filter into a high-pass filter?

To convert a linear-phase FIR low-pass filter into a high-pass filter with the same cut-off frequency, we can invert the sign of the low-pass filter's impulse response $h(n)$ and then add one to the ...
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1answer
37 views

Behaviour of integrator at steady state

I wanted to calculate response of integrator of sinusoidal input at steady state via these two methods as mention in image but these two methods give two different answers at steady state, so where ...
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RMS of signal vs average amplitude

I am trying to estimate the average amplitude of some signal with frequency 6 Hz, sampled at ~300 Hz. See figures for a part of the signal and its dft calculated using matlab. I estimate the average ...
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What is Frequency Resolution?

Im trying to tackle the following problem while still not having a firm idea on what "frequency resolution" means : Suppose we sample a continuous time signal with sampling period Ts = 1/2000, and ...
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Frequencies of complex exponentials in Discrete Fourier Transform

What I understand is that using DFT, we are representing a given discrete signal using a basis of complex exponentials of different harmonic frequencies. If I am taking 16 point DFT of a signal ...
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MRI K-space to image: why track frequencies in two dimensions?

While trying to better understand how an MRI goes from k-space to an image, I came across this wonderful website that explains how you would represent an image as a collection of rows of pixels where ...
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Understanding the FFT phase spectrum with a simple example

I'm trying to compute the DFT using scipy's functions. I don't understand why the phase spectrum of a simple sine wave with 2 Hz frequency doesn't show $\pm\pi/2$ at the $\pm 2Hz$ frequencies. Instead,...
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Names of system functions in frequency domain

I was just trying to refresh my systems theory known from long ago, and I realized that I had forgetten the name of the basic functions. Specifically, what are the names of these functions ...
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What is the most lucid, intuitive explanation for the various FTs - CFT, DFT, DTFT and the Fourier Series?

Even after having studied these for quite sometime, I tend to forget (if I'm out of touch for a while) how they are related to each other and what each stands for (since they have such similar ...
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Tips for improving pitch detection

I'm working on a simple web app that allows the user to tune his/her guitar. I'm a real beginner in signal processing, so please don't judge me too harshly if my question is inappropriate. So, I ...
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2answers
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Showing error energy goes to zero

Let $$\hat{x}[k] = \frac{1}{2\pi}\int_{-W}^{W}X(e^{j\omega})e^{j\omega k}d\omega,\label{ift}\tag1$$ where $$X(e^{j\omega}) = \sum_{n=-\infty}^{+\infty} x[n]e^{-j\omega n}\label{dft}\tag2$$ Also, $$d[k]...
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How does one interpret an element of the “transfer matrix” used to calculate frequency domain granger causality (via VAR models)?

I am attempting to gain a better mathematical understanding for how autoregressive models can be used to infer frequency-domain granger causality. All freq. domain measures of causality that utilize ...
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GSP as an extenstion of DSP

I am a PhD. in pure mathematics. Could you please illustrate the following statement: the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development of graph-...
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1answer
45 views

Steady state variance of a stochastic differential equation - relation between the frequency and time domains

Consider a stochastic differential equation: $$ dx(t) = a x(t)dt + b y(t)dt \quad (1) $$ where $y(t)$ is a stochastic process satisfying $\langle y(t)y(t')\rangle = \delta(t-t')$. We will assume that ...
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Conjugate symmetry of the DFT of real-valued sequences

I have read about Fourier transformation that real signals are "mirrored" in the real and negative halves of the Fourier transform because of the nature of the Fourier transform. For ...
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Understanding index transformation in derivation of Fourier transform for sampling rate reduction

Was going over some notes regarding deriving fourier transform equation for Sampling Rate Reduction. Reference to Notes from below link https://ocw.mit.edu/courses/electrical-engineering-and-computer-...
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1answer
1k views

How to evaluate expressions without explicitly computing a DFT?

I am trying to learn this part but I dont seem to understand. if I have a finite length sequence, and a n-point DFT of it (in an interval)... is it possible to evaluate expressions of the n-point DFT ...
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1answer
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How do I make sense of the cosine wave having Fourier Transform coefficients which have infinite magnitude?

To illustrate my question better, consider the Fourier Transform of an aperiodic (as a periodic cosine wave has a Fourier Transform not Fourier Series) cosine wave $$f(x) = \begin{cases} \cos(2\pi ...
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“The Fourier transform cannot measure two phases at the same frequency.” Why not?

I have read that the Fourier transform cannot distinguish components with the same frequency but different phase. For example, in Mathoverflow, or xrayphysics, where I got the title of my question ...
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1answer
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Auto Tune Shifting

So i have an algorithm that gets the fundamental frequency of windows, and can shift the pitch of each window as well (by a factor of x). For example. if x is 2 and the f0 is 440, the resulting f0 ...
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Spectral analysis of a cross correlation function [duplicate]

So it's widely known that the Cross Correlation of 2 signals helps us in figuring out the time delay in those signals by analyzing the peak of the correlation coefficient in the time domain. For ...
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Fourier transform of an integrator filter

I have to find the Fourier transform , and $y(t)$ of an $ x(t) = e^{- \frac {t}{T} } u(t) $ that passes into a integrator filter. I know that $ Y(f) = X(f) H(f) $ so I first calculate the Fourier ...
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Duality Property for DFT

I was watching a youtube video for the duality property for continuous time Fourier transforms, which shows that if Fourier transform of $x(t)$ is $X(\omega)$ then Fourier transform of $X(t)$ is $2\pi ...
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Does it make sense to apply convolution in the frequency dimension of a fourier transformed signal?

What I have is more likely a theoretical question. Since I am not from a signal processing background it is hard for me to grasp the issue in using convolution in the frequency dimension of a Fourier ...
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How to interpret the window size and FFT size while performing FFT? [closed]

I have a discrete signal of length 98,000 samples and I am supposed to a FFT of the discrete signal with certain window size and FFT size. Can someone help me explain what is meant by the window and ...
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What is the gradient of fft?

I have a time-series of length N generated by the following equation: ...
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1answer
84 views

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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1answer
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Anyone explain to me this video?

I was watching a video in time 24:48 I would like to know where you got the value .9 (1.14z + .941) and 1.0232 + .757 Does anyone explain how he got those numbers?
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Proof of fourier transformation of multiplication of two signals

I've been trying to find a proof of the following, but still I m unable to proof it, can someone help me? $$ ℱ[x(t)g(t)] = \frac{1}{2\pi} [X(\omega)*G(\omega)] $$
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Understanding zero-phase filters

I'm reading the book "Digital Image Processing" by Gonzalez and Woods and I'm wondering how their definition of zero-phase-shift filter is equivalent to the one given here. "Digital Image Processing" ...
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Impulse response of a time scaling system

Assume a bandlimited signal $X(t)$. Given that the output for this signal is $X(t/2)$, what will be the impulse response $h(t)$ of such a system? \begin{array}{l} X( \omega ) \ =\ \int ^{\infty }_{-\...
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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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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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Zero-padding or Interpolation in 3D FFT

I'm trying to perform a FFT of a 3D regular grid and then compute the bin average (in spherical shell bins) of the Fourier transformed grid. The problem is that the resulted vector is very noisy as I'...
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Calculating Transfer function numerator and denominator from the rationalfit model

I have a frequency response data called 'AC_data' which is a vector of complex numbers (real and imaginary part) at different frequency points. I have calculated a rationalfit model for the AC_data ...
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Time domain to frequency domain conversion of audio signals to extract 1/3 octave frequencies

I am writing first time in this forum and I am not expert in programming and FFT. We have developed an android app (Noise Tracker) for noise measurement using smartphones. It displays noise levels in ...
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1answer
61 views

Relationship between real and imaginary part of a real-valued and causal system

I have one question about the real part of a real-valued and causal system with the imaginary part of its Fourier transform given by $$\textrm{Im}\big\{X(e^{j\omega})\big\}=3\sin(2\omega)-2\sin(3\...
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What is the DFT of a binomial filter?

I hope this question is not too simple, I just started learning digital image processing. The 1D binomial filter of size 2 is defined by $B_2 = \frac{1}{4}\begin{bmatrix}1 & 2 & 1\end{bmatrix}$...
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Why can we have non-integer frequency bins in FFT?

I am studying DFT/FFT and I'm very confused about one thing. I read online that the frequencies we can sample with DFT must be integer (Why does the frequency in the DFT have to be an integer?). Later ...
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1answer
56 views

Pitch shifting with bins creating large feedback

I am attempting to create a pitch correction algorithm. I started by performing a test. The test goes as such: Get WAV file Split it into bins of size n (512 in my case) Shift each bin by 2 semitones ...
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Wavelets vs Fourier Transforms

It's supposed that computing a spectrogram for a signal is faster using wavelets. However, wavelets need to be applied to the signal for every time step and frequency, giving the implementation a time ...
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Calculating SPL from pressure signal - Amplitude vs Power method

I have a pressure signal from a Fluent FFowcs-Williams Hawkings acoustics analysis. I converted this pressure signal into the frequency domain in order to get SPL values, using Matlab. I used the ...
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1answer
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Fourier transform of unit step

I was reading pdf by caltech and in one of its section, Fourier transform of Unit step signal is calculated but I am confused, how this can be possible if region of convergence for Laplace transform ($...
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Fourier Transform of the full Morlet wavelet

In 2014 someone asked here the Fourier transform of the Morlet wavelet; link below: Fourier Transform of Morlet wavelet Function? However, it was the approximated Morlet wavelet not written with the ...
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Discrete-time Fourier transform of $\frac{\cos(\frac{n\pi} 6)}{(n+3)\pi}$

Find the Discrete-time Fourier transform of $\frac{\cos(\frac{n\pi} 6)}{(n+3)\pi}$ I thought of making it to be a sinc, but at the bottom there is $n+3$ and if I replace $n+3$ then I don’t know how ...

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