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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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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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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Are the FFT coefficients symmetric in image processing?

On page 11 of Fundamentals of Image Processing by Ian T. Young, Jan J. Gerbrands, Lucas J. van Vliet (pdf) the results of the Fourier transform are shown (Figs 4a and 4b) and it appears to me (please ...
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Centered Fourier transform

What is the difference between the non-centered and centered Fourier transforms? In other words, when should you use one instead of the other? Non-Centered: $\quad \displaystyle X_1(f)=\sum\limits_{n=...
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Link between DFS, DFT, DTFT

My understanding of DFT is as follows For a signal $x[n]$ of finite-length, the DFT is DFS of the periodic extension, $\tilde{x}[n]$, of that signal $x[n]$ and also another way to view DFT is that it’...
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Fast Hartley Transform Implementation in MATLAB

I want to implement Fast Hartley Transform (Specifically Discrete Hartley Transform) in a script file in MATLAB. Does anyone know have a reference implementation of this in MATLAB or another language ...
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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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Fourier series expansion of $\exp[-j2kA\sin(\omega t)]$

I've been going through this paper where we exploit the well known Fourier expansion of the model signal as shown in the image below. I've never come across this well known fourier expansion before. ...
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How to Flip Spectrum Around DC?

Is it possible to flip a signal's spectrum around DC? I have a simple spectrum that I made up (MATLAB code): ...
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Regarding Bode plots; $H(s)$ and $H(j\omega)$

In circuit analysis, I understand the use of Laplace Transforms to obtain the impedance of a linear RLC circuit, ie transforming from the time domain to the frequency domain. In most texts I have seen ...
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How can I detect clicks/ticks in a wav file?

I have a long recording of a train (here is a small sample) there are a number of clicks as the wheels go over joins in the tracks. I would like to detect the location of these. Looking at the ...
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Why do we have to rearrange a vector and shift the zero point to the first index, in preparation for an FFT?

I am trying to learn how to implement the FFT as a way to approximate the continuous-time Fourier transform, and as a "nice easy example" I have chosen to test it with a simple Gaussian pulse in the ...
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Why Does 2D FFT of Gaussian Looks More Sharper than Gaussian Itself?

I am trying to understand why 2D FFT is done on a Gaussian process in a particular code. From my understanding from these posts: https://www.researchgate.net/post/...
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Is there a name for the procedure of taking the FT over separate consecutive small time-blocks?

Suppose we have a continuous time-interval $I=[a,b]$, and a signal $x \colon I \to \mathbb{R}$. A procedure that is sometimes carried out (e.g. when doing bispectral analysis) is to partition $I$ ...
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what is nyquist rate of $h(t)\cdot h(t)$ and $h(t)\circledast h(t)$

Let's say we have $h_c(t)$ as a continuous-time signal with bandwidth $B$ and we would like to sample it. To be able to reconstruct it correctly, the sampling rate must be greater than $2B$. Now ...
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Derive DTFT of $x[2n]$

If the DTFT of discrete sequence $x[n]$ is $X(e^{j\omega})$, what is the DTFT of $g[n] = x[2n]$? I see the textbook answer is \begin{align*} G(e^{j\omega}) &= \frac{1}{2} \left( X(e^{j\omega/2}...
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why do we use $X(e^{j\omega})$ instead of $X(j\omega) $ in Discrete Time FT

I am studying DT-FT. But I cannot figure out why we use $X(e^{j\omega})$ instead of $ X(j\omega) $ in DT FT Thanks in advance..
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Conceptual question on FFT and chirp signal

If I take the FFT of a sinusoid I will get a plot whit all the energy of the signal concentrated at the sinusoid frequency. But what happens if I have a signal in which the frequency keeps changing?(...
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DSP interview question: use of the identity in development of a significant transform

I'm preparing interview and found this question. But I don't really understand what is the question. Does it ask about Fourier transform or Z transform? How the simple identity $$xy=\frac{1}{2}x^2 + ...
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What does an image of Fourier Transformation of an image tell us?

First time studying image processing... I just don't understand what does fourier transformed image of an image describe? For example consider given following pictures, The first one is the image, and ...
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Right algorithm for fourier transform on physical heights

I have data from a LIDAR unit that I would like to get the spectral density of. Unfortunately, the only thing I remember from my Fourier analysis class are the methods that I know will not work. The ...
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Interpreting N in DFT as the Number of Points vs. Number of Intervals

The "N" is DFT is understood to be the number of data points in a given sequence or in other words the length of the sequence. We recently have had discussions here Indexing in DFT (from an ...
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The Fourier transform of a damped cosine and the units of the result

If I take a simple transient voltage signal of the form $$f(t) = V_p e^{-t/\tau} \cos(\omega_0 t)$$ and take the Fourier transform in the normal way $$F(\omega) = \frac{V_p}{\sqrt{2 \pi}} \int^{+\...
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Basic difference between Fourier transform and laplace transform? [duplicate]

I have read few links about difference between Fourier transform and Laplace transform but still not satisfied Please correct me if i am wrong Simply put, the main difference between Fourier ...
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How to calculate the Fourier transform of absolute-valued sinusoid?

The signal $x(t) = |1+a\sin(\omega t)|,(a>0)$ is a continuous waveform. In order to extract the frequency parameter $\omega$, I conduct the FFT of it and obtain its spectrum showed as follows. The ...
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Fourier Transform Signals - Time Transformations

I was going over some review problems and came across an interesting one. Using the techniques of (linearity, time shifting, and time scaling) what are some approaches I could use to turn the ...
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2D Fourier transform of Sobel kernel

Can someone explain me the highlighted text parts regarding this image ? Here is a pseudo-code of how it was created: ...
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Zero-padding the middle of a signal

I sample a signal at a certain frequency for a finite amount of time to get a sequence $$(x_n)_{n=1}^N = (x_1, x_2, ... , x_N)$$ with the intention of analyzing its power spectral density by ...
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How do I convert a timeseries to a different frequency band?

I have a real time series sampled at 32 MHz. So, when I channelized it and plot it against time, I get a Frequency vs. Time image plot where the frequency domain spans 16 MHz. To elaborate, this ...
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Magnitude of the Gradient in Frequency Domain

I'm learning some basics of image processing. Recently I've read about image filtering and two-dimensional Fourier transform, because I'm preparing for exam. And I have one question I don't know ...
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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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Fourier Transform of infinite sum

I'm looking at last year's exams, and I found an exercise I can't solve: (Roughly translated) Consider $x(t)=\sum\limits_{k=1}^{+\infty}\left(\frac{1}{2}\right)^k \cos(k2\pi t)$ the input to a LTI ...
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Why Is Super Resolution (SR) Possible?

I've been reading about Super Resolution Image reconstruction (Reconstruction of high resolution image from multiple low resolution aliased images contain sub pixel shifts), and i want to know why SR ...
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How to find the envelope of a wide-band signal?

I would like some feedback on possible techniques that one may use to determine the envelope of a broad-band time domain signal. I have heard anecdotally, that it is not as straight-forward as it ...
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FFT of color images incorporated into an Object Recognition method?

Is there any instance where Fourier transforms of color images have been used in conjunction with other object recognition method? Any instance of usage of Fourier transforms in color images? I ...
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Response of an ideal integrator to a cosine wave

It sounds like a very elementary question on system theory but I got quite confused about it, so hopefully you guys can enlighten me. I'm considering an ideal analog integrator, i.e., a system with ...
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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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Orthonormal Dictionaries for Band Limited Signals

If $\mathbf{x} = [x_0, x_1, \ldots, x_{N-1}]^T$ is the time sampled input signal and $\mathbf{Y} = [Y_0, Y_1, \ldots, Y_{N-1}]^T$ is the Fourier transform of the input signal, then a linear ...
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How does sampling rate of $x[n]$ relate to sampling rate in frequency domain after DFT transformation?

I've got an analog signal $x(t)$ sampled at frequency $F_s$ to obtain samples: $$ x[n] = x(t) \bigg|_{t=n/F_s} $$ I transform this signal with DFT defined as: $$X[k] = \sum_{n=0}^{N-1}x[n]e^{-i2\pi ...
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Impulse response of ideal filters

I am aware that an ideal low-pass filter in both continuous time and discrete time has a $\mathrm{sinc}$ impulse response. What would the impulse response of an ideal high-pass or band-pass filter ...
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Time-Bandwidth Product

The following text is cited from a textbook, "Spotlight Mode Synthetic Aperture Radar: A Signal Processing Approach", I would like to ask if anyone knows the proof to the following statements, as the ...
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How does a Hermitian FFT work in Numpy?

Say, I create a Hermitian complex signal using, import numpy as np t = np.arange(-4, 4) z = np.exp(1j * t) Here z should be a ...
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Mathematical relationship between highpass and lowpass filtering

Let $g, h_{HP}, h_{LP}: \mathbb{R} \rightarrow \mathbb{R}$ and $G, H_{HP}, H_{LP}$ denote their continuous Fourier transforms under the Fourier operator $\mathcal{F}$. Let $*$ denote the continuous ...
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Discrepancy between Gaussian FFT and its Fourier Transform

I am trying to do the FFT of a Gaussian signal and comparing it to the theoretical Fourier transform. For infinitely small time step $dt$ and infinitely long signal length $T$, the 2 should become ...
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Analyzing a particular discrete-time LTI system for input signal $x[n]=(1/3)^n$ for *all* $n$

I'm considering the following problem from some course notes. Suppose the following is known about a discrete-time LTI system: Given the input $x[n]=(1/3)^n$ for all $n$, the system ...
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Fourier transform exercise

I have this signal: $$ X(f)= 2\delta(f)+ \delta\left(f-\frac 1{T_0}\right)+\delta\left(f+\frac 1{T_0}\right)+\textrm{rect}\left(\frac{f-\frac 4{T_0}}{\frac 2{T_0}}\right)+\textrm{rect}\left(\frac{f+\...
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Shifting in DFT (FFT)

Input image Listing-1 i = imread('Untitled.png'); i = rgb2gray(i); F = fft2(i); %%%F = fftshift(F); F = abs(F); F = log(F+1); F = mat2gray(F); imshow(F); ...
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Find state space model from transfer function

Let's suppose we have: G(s) = (s+1)/(s^2-2s+1) how can we find the state space representation of the transfer function: x_dot = x2 x2_dot = 2*x2-x1+u where u is an arbitrary input. I am very new ...
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Fourier transform of $\cos(n\omega t)$

My question is probably very stupid, but I've been strugling for a while on it now... In need to find the Fourier transform of $1+\cos^3(2\pi ft)$. I wrote that : $$\cos^3(2\pi ft)=\frac{\cos(6\pi ...
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Difference between Fourier Transform and DFT? - Example

I have read many excellent answers to similar questions, but never one this specific. Here is another way to ask it. Why is the modulation transfer function (MTF) of $\textrm{rect}(x/5) = \textrm{...

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