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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Finding the impulse response of a system

I have the following transfer function. $$ H(j\omega) = \frac{1+0.5 e^{-j\omega}}{1-1.8 \cos(\frac{\pi}{16}) e^{-j\omega}+0.81 e^{-j2\omega}}$$ I'm trying to find the impulse response of the system. ...
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Discrete Time Fourier Transform (DTFT) cross correlation property

I came across this property of the Discrete Time Fourier Transform (DTFT) and I am having a tough time proving it. In general, consider two real signals $x[n] \: \& \: y[n]$. If $$ x[n] \...
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How To Find The Lowest Note of a Chord Being Played

The goal of my project is to find the lowest note in a signal coming from a guitar being played. I would like to do this programmatically and have sourced the equipment (raspberry pi, USB interface, ...
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Calculating 1/3 Octave Spectrum from FFT / DFT

I am not often on this forum and I am not an expert on the subject. I struggle with the theory of FFT / DFT and the 1/3 octave spectrum. Assume I have a DFT analysis of a given signal. It (the DFT ...
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Is it possible to recover the time-domain signal after manipulating it in the frequency-domain? [duplicate]

I've worked a fair amount with EEG signals, though I've never had formal training in signal processing, so please excuse my ignorance. The problem is this: my signal has noise at many, many ...
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$x[n]$ after sampling of $cos(16\pi t+\phi)$ at 12kHz

I'm not sure what the question really means, so this is just guesswork. I think options 1 and 4 can be ruled out as $w_0<\pi$. The CTFT of $cos(16\pi t+\phi)$ has two spikes at $16\pi$ and $-16\...
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Find integral of DTFT after sampling (Graph of CTFT given)

So for the first question: If this is sampled at 10kHz, then the amplitude is scaled by 10000. In the DTFT, the frequency 3.5kHz gets mapped to 3.5/10* 2pi=0.7pi. So this point lies outside the range ...
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Which frequency bins give the best interpolation for the derivative of a function?

A function $u:[0,2\pi]\to\mathbb R$ sampled over $N$ equidistant points $\theta_j=(2\pi/N)j,\, j = 0, \dots, N-1,$ can be interpolated by a set of functions $\{u_{k_0}\}$ enumerated by integers $k_0\...
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Why is a circular mask appropriate for Fourier filtering rectangular images?

Suppose I apply 2D DFT to an image with dimensions $H{\times}W$ where $H \neq W$, then shift the DC component to the center. Why does a circular mask capture the lowest frequency components, i.e. why ...
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Derivation of Nyquist Frequency and Sampling Theorem

I have been looking through different sites and questions over the internet about Sampling theory, but couldn’t find the clear definition of how nyquist frequency condition is derived? It would be ...
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Fourier Transform of finite time series

I have some signal 𝑠(𝑡) which is real data i.e. finite. The time runs from −𝑇 to +𝑇. The signal amplitude is large at 𝑡=0 and small (→0) at the ±𝑇 limits. I can do a finite (discrete) Fourier ...
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What is the form of the spectral derivative in the all-positive-frequency notation in DFT?

The Discrete Fourier Transform (DFT) of a function $u:[0,2\pi] \to \mathbb R$ sampled over $N$ equidistant points $\theta_j = 2\pi j/N,\, j = 0, \dots, N-1,$ is defined by $$ \tilde U_k = \frac1N \...
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34 views

Why zero padding the 2-d DFT interpolates images in spatial domain?

I was applying different image interpolation techniques and I came know to about interpolation in frequency domain. In this technique we first take 2d DFT of an image, padd it with zeros and take the ...
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Solving equation with convolution

I have the measured signal $y(t)$ that can be modeled in the frequency domain as $Y(f)$: $$Y(f) = X(f)\cdot A(f) - [X(f)\cdot B(f)] \ast C(f)$$ where $\ast$ is the convolution. I know $A(f)$, $B(f)$,...
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Inverse Fourier Transform Dirac impulse with scaled argument

Currently, I am dealing with the sampling problems and I don't understand how to calculate inverse Fourier transform of a scaling impulse function $\textrm{IFT}\{\delta(\Omega T)\} = ?$, $T$ is ...
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Difference between 2D DFT's and 1D DFT's of Linearized Matrices

I have recently left the safe and easy MATLAB environment and begun to use CUDA-C/C++ for image processing. Since CUDA doesn't allow 2D arrays to be passed into kernels I am now used to linearizing ...
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Sampling of frequency response

Let's consider any physical quantity depending on the frequency. For example, the impedance of a certain electrical component: $Z(f)$. Now, imagine to measure it in a continuous interval of ...
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Relationship between DFT and DTFT

Let $x[n]$ be an $N$-point sequence, i.e. $x[n] = 0$ for $n < 0$ and $n \ge N$. Let $X[k]$ be the $N$-point DFT of $x[n]$. Let $$y[n]= \left \{ \begin{array}{ll} \displaystyle\sum_{l=-\infty}^{+\...
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Fourier transform of a periodic/aperiodic signal

Generally speaking, I know that periodic signals (continuous-time domain signals) with period 2pi/wo have a spectrum with equidistance Delta-impulses of distance w0. My question is that, if we have ...
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Palmprint Identification - Why do we align the images before we use the Fourier Transform?

I have been reading the paper PALMPRINT IDENTIFICATION BY FOURIER TRANSFORM by WENXIN LI, DAVID ZHANG and ZHUOQUN XU about identifying persons based on an image of their palm, one version of the paper ...
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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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FFT freqency bin center in R

I'm trying to do a spectral analysis in R. I learned it in Python from Allen Downey's ThinkDSP book. What is the R equivalent of the Python numpy function, numpy.fft.fftfreq? If you provide a window ...
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Optical realization of the Radon Transform

I am trying to reproduce this paper concerning the Optical realization of the Radon Transform, especially the simulation (section 3). Shortly, the experiment is just a "Fourier filtering" with a 4F ...
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Multiply signal $x[k]$ with $\cos(2\pi\nu_0k)$, then given $X(\nu)$ draw resulting function in frequency domain?

Let $$y[k]=x[k]\cdot \cos(2\pi\nu_0k) .\tag{1}$$ Then, given a signal $x[k]$ with the DTFT $X(\nu)$ according to the following figure what will the frequency domain for $Y(\nu)$ look like for a ...
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Fourier transform 4 times = original function (from Bracewell book)

I was glancing through "The Fourier Transform & Its Applications" by Ronald Bracewell, which is a good intro book on Fourier Transforms. In it, he says that if you take the FT of a function 4 ...
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Two Consecutive Inverse Fourier Transforms [duplicate]

What happens to a function F(w) if you take two consecutive inverse Fourier transform of it?
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264 views

Which Approach Is Better for Decomposing an Image into High Frequency and Low Frequency Components?

Which approach is better or there is mathematical justification for using Bilater filter and Fourier Transform to decompose a image into High Frequency and Low Frequency Component. Both Bilateral ...
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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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Frequency Domain Signal to Noise Ratio

I am doing some research on low-cost air pollution sensors. I'm measuring the "ground truth" with a single low-noise sensor, and I'm trying to use it to calibrate a low-cost sensor that has high noise....
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Distortion in sound after multiplying frequency spectrum by constant

I make a simple sound equalizer that operates in frequency domain and lets user to adjust frequencies in sound by using 4 sliders. The first one responsible for 0 - 5kHz, the fourth one for 15-20kHz. ...
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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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If the cosine function is periodic, why does it have a Fourier Transform? [duplicate]

As far as I understand Fourier Transforms are for non-periodic signals and Fourier Series for periodic signals. So why is it we can take the Fourier Transform of a cosine when it is a periodic ...
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How does minimum-latency partitioned convolution reverb work when you receive input samples in chunks, rather than one at a time?

I'm writing a reverb system where I receive an input block of samples 480 elements long, do some operation on them, and pass the block on to the next effect. I've been reading up on partitioned ...
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DTFT of even and odd samples

Here to find DTFT of $h(2n)$ they have scaled omega, while in RHS to find DTFT $x(2n+1)$ they didn't, why is that?
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How to find a Matched Filter Transfer Function from large signal sample

Lets say I have a system where I have a small sample of a signal with no noise $\hat{x}(t)$ and a lot of a similar signal with noise $y(t) = \hat{x}(t) + n(t)$, and from $\hat{x}(t)$ I want to create ...
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Inverse Discrete Time Fourier Transform of $1$

$\textrm{DTFT}(\delta[n]) =1$, but $\textrm{IDTFT(1)} = \frac{\sin(\pi n)}{\pi n}$. Why it is not equal to the unit impulse $\delta[n]$?
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What effect does rotation in the spatial domain has on phase in Fourier transforms?

More precisely, let's say I apply a 45 degrees rotation to an image (in the spatial domain) say, in Matlab : Ir=imrotate(myImage,45,'crop'); FT_I=fft2(I); In the ...
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Matlab FFT not producing symmetric spectrum

I am plotting a FFT of a sampled RC pulse but my resulting spectrum isn't symmetric - it's offset. ...
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Why the spectral coherence is unity for all frequencies between single-frequency time series and itself

In the example below, I am plotting the coherence between time series and itself. The time series do has one frequency.The coherence magnitude was one for all frequencies. I wonder why it is not zero ...
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What determines peaks in FFT?

I ran FFT on three audio files and found that the results for some have more peaks than the other. Could anyone give me any conceptual explanation as to what determines these peaks? Below are plots of ...
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Signal processing using numpy python

To process a .wav audio file with numpy (using fast Fourier transform algorithm). I want to process an audio signal at a particular interval with a sampling frequency 44100hz and sampling rate of 20ms ...
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Fourier Spectra : Significance of the Negative Amplitude [duplicate]

For example, for an aperiodic gate pulse, the Fourier Transforms for the continuous time case is a sinc function, while the discrete time case gives a sine over sine periodic kind of a function. In ...
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Is there a version of Welch's method that doesn't look for power?

Welch's method splits a time signal, $x(n)$ into $M$ periodograms $P_m$, $P_{x_m,M }(k) = \frac{1}{M}|F_k(x_m)|^2$ and averages them to give the Power Spectral Density (PSD), $S_{x}(k) = \frac{1}{K}...
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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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Relation between two k-spaces phase-frequency and spatial frequencies in

When I see MRI explained, two types of 2D k-space images seem to be described as if they were the same. Axes are the two spatial frequencies. This images is directly fourier-transformed into the ...
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How do I decide which frequencies are signal and which are noise?

I have an arbitrary recorded digial signal, on which I have run a Fourier transform. I'm not sure what conventions are on a case like this, but I have 1024 frequency bins. Second bin is the highest ...
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2answers
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Problem identifying the analytic expression of such determined signal

I came across this problem I am supposed to find the Fourier transform of $g(t)$, but I am not able to find the analytical expression of such signal. The teacher suggests that I should consider ...
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Fourier Transform: interpretation of continuous spectrum at specific frequencies

B. P. Lathi in his book "Principles of Linear Systems and Signals" mentions in the Fourier Transform: When $x(t)$ is periodic, the spectrum is discrete, and $x(t)$ can be expressed as a sum of ...
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155 views

How to calculate the Fourier transform of a mean filter in Matlab?

In Matlab, how can I calculate the discrete-space Fourier transform of a mean which takes the average of 4 adjacent points, with this kernel $$\begin{pmatrix} 0 &1& 0\\ 1 &0&...
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1answer
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Applying duality property to fourier transform of unit step function

For Continuous time aperiodic signals, the duality property of Continuous Time Fourier Transform (CTFT) is following $$\mathscr{F}\Big\{x(t)\Big\} = X(f), \qquad\text{then} \quad \mathscr{F}\Big\{X(t)...