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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Applicability of Fourier transform to periodic signals? [duplicate]

Can we use Fourier Transform for periodic signals or only we can use fourier series with periodic signals? I am asking this question because I read on page 301 of alex palamides as shown highlighted ...
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Cepstrum Calculation of Rational Function H(z)

I am trying to solve my first problems at cepstrum calculation. I want to calculate the complex cepstrum $\hat{h}[n]$ of a signal $h[n]$ with Z-Transform: $$H(z)=\frac{(1-0.5z^{-1})(1+4z^{-2})}{(1-0....
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Calculating an image's fourier spectrum by hand?

Suppose I have a $4x4$ image with the following values as its grey-level intensity for each pixel like this: I want to get its Fourier spectrum. Usually, I would just punch into Matlab and run a fft ...
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Inverse Fourier of Two-Pole Transfer Function

I would appreciate if someone could walk me through this derivation. I have a transfer function in the frequency domain, which has two poles $$\tilde{H}(\omega) = \Big(\frac{1}{1 + i \omega \tau_1}\...
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Fourier transform of a tilted line function

Assume a line function (line segment to ensure integration): $$y = a\cdot x + b$$ What is the Fourier transform of the line segment? Intuitively, it may give a sinc function, but actually not.
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Inverse Sliding DFT

From paper: Bradford R., Dobson R., ffitch J. - Sliding is Smoother than jumping In chapter 6 - Signal Reconstruction, the inverse of the sliding DFT can be achieved by this formula: $$f_0=\...
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Where is the flaw in this derivation of the DTFT of the unit step sequence $u[n]$?

This question is related to this other question of mine where I ask for derivations of the discrete-time Fourier transform (DTFT) of the unit step sequence $u[n]$. During my search for derivations I ...
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Difference between discrete time fourier transform and discrete fourier transform

I have read many articles about DTFT and DFT but am not able to discern the difference between the two except for a few visible things like DTFT goes till infinity while DFT is only till N-1. Can ...
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What is the type number of a discrete time system given $H(z)$?

Given a continuous time impulse response $h(t)$, if I take the Laplace transform and count the no. of poles at origin, that gives the type number of the system. For e.g., $$H(s) = \frac{2}{s(s+2)}$$ ...
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Simple $\sin(2\pi 1000t)$ Fourier transform in PSpice not behaving as expected

So I have this very basic circuit show below which I am simulating with PSpice. Now, when doing the Fourier transform of $\cos(2\omega1000t)$, I expect to see two impulses(one at -1kHz and one at ...
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Output of an LTI system given its transfer function and input

Given the transfer function $$T(s) = \frac{100}{1 + \frac{s}{10^{6}}}$$ and the input $$v_i(t) = 0.1 \sin(100t)$$ find the output, $v_o(t)$. My approach was to use $v_o(t) = \mathcal{L^{-1}}\left\{T(...
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Intuitive understanding of Fourier transform of images [duplicate]

I am trying to have an intuition about the Fourier transform of images For example the image on the right is the Fourier transform of the image on the left, my question is : 1)Why are the ...
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What is the interpretation of Fourier Transform containing only imaginary part?

The FT of a unit step function is taken as: $$ X(\omega) = \int_0^\infty e^{-j\omega t}dt = \frac{-1}{jw}e^{-j\omega t} \Biggr |_{0}^{\infty} = \frac{j}{\omega} $$ The transform only has the ...
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Why is the ROC of Laplace transform independent of imaginary part of s?

An integral is defined as converging if it yields a finite value upon application of limits of integration. It is divergent otherwise. Now sticking to the mathematical notation of Laplace transform, ...
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Fourier Transform of the Hilbert Transform of cos(t) (using Fourier time-shifting property)

If $x(t)=cos(t)=\frac{1}{2}e^{jt}+\frac{1}{2}e^{-jt}$, then $X(\omega)=\pi \delta(\omega-1)+\pi \delta(\omega+1)$. If $y(t)=cos(t-\frac{\pi}{2})=\frac{1}{2}e^{j(t-\frac{\pi}{2})}+\frac{1}{2}e^{-j(t-\...
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What is the physical interpretation of the absolute value of a fourier transformed signal, $\left| F(t)\right|$?

If one has some oscillating voltage signal, for example: $$V(t) = V_{max}\cos(2 \pi \nu_{0}t) e^{-\gamma t}$$ and you take the Fourier transform of this in the usual way to get: $$\hat{V}(\nu) = V_{...
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Fastest implementation of fft in C++?

I have a MATLAB program that uses fft and ifft a lot. Now I want to translate it to C++ for production. I used OpenCV but I ...
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Why do the two methods give different answers for the Fourier transform of $Y = \cos(\omega_0 t + \phi)$?

Why do the following two methods give different answers (or are they the same) for the Fourier transform of $Y = \cos(\omega_0 t + \phi)$, with respect to $t \to \omega$ ?
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Fourier transform pair for $ln(ln(…))$ cascade?

I need to analyze real signals $y_i$ in the frequency domain. $y_i$ are defined like: $y_1 = ln(ln(x))$ $y_2 = ln(ln(ln(x)))$ $y_3 =~ ...$ $...$ Are there Fourier transformation pairs for this ...
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How is the decay of a signal exemplified in a Fourier Transform?

Is there any way to tell if a signal is decaying from its fourier transform?
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Using MATLAB to plot the input and the magnitude spectrum of the signal

I have an aperiodic signal $v_{out} = e^{-t} u(t)$ (real exponential signal) from discharging capacitor. I was trying to plot using MATLAB 15 seconds of this signal in time domain? I am thinking how ...
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Meaning of Real and Imaginary part of Fourier Transform of a signal

Say $f$ is a signal of time $t$, $F$ its Fourier transform of the variable $v$. It is known that in polar coordinate, $|F(v)|$ tells us how much the frequency $v$ is present over the signal, and $Arg(...
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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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What is a periodic signal in image processing?

In the context of image processing (and computer vision), the concept of convolution comes up a lot. Convolution is quite related to the concept of Fourier transform and DFT. In the context of image ...
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49 views

DTFT frequency range

$$X(e^{j\omega}) = \sum_{n=-\infty}^\infty x[n] e^{-j\omega n} $$ The frequency term $\omega$ in DTFT is normalized as $\omega = \frac{\Omega}{f_\mathrm{s}}$ $\Omega= 2 \pi f$ is the angular ...
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How were windows originally conceived?

I am aware of the common types of windows, (Hamming, Hanning, Kaiser, Tukey, etc etc). However while many books describe them - almost none tell me just how exactly they were derived. What is so ...
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“Dirac Comb” vs “Ones Comb”

While learning sampling theory - I noticed that examples of continuous signal sampling always achieved the goal via multiplying the signal with a "Dirac Comb". I was intrigued by the requirement to ...
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Ideal sampling using sinc funcion

Let $ x(t) $ be a bandwidth limited signal such as $ \forall |\omega|>\frac{\pi}{T} : X^F(\omega)=0 $ while $ X^F(\omega)$ is the signal's Fourier Transform. Let us denote $y[n]=\int_{-\infty}^{\...
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Reconstructing a signal from FFT by adding individual signal components

I'm attempting to reconstruct a signal from the DFT of the signal. I tried to do it by extracting the individual sinusoids and adding them up, but the answer I get is incorrect. ...
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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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Interpreting FFT Coefficients from System Matrix That Is Originally Toeplitz (Not Circulant)

If I have a measured signal $y$, true signal $x$, and a convolution matrix $A$ that is a Toeplitz but not circulant matrix, I can write the convolution as \begin{equation} y = A x \ . \end{equation} ...
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Count Matches to a Kernel

So, I have this problem where I want to apply a kernel to an image and count the number of matches that happened. So for example, if I have the kernel: $$\begin{bmatrix} 1 & 2 & 1\\ 1 & ...
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Fourier Transform of Alternating Periodic Rectangular Pulse

I'm having trouble determining Fourier transform of signal. I have 2 ideas on how to solve this problem. Given the signal is periodic I could use formula for Fourier transform of periodic signals: $$...
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How to get descrete fourier tarnsfom [closed]

Could anyone explain me please how to produce descrete fourier transform of such signal? There are no anymore information besides the picture in this task.
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59 views

Applying frequency-domain filters on a centered Fourier transform

I understand why we shift the Fourier transform such that the 0-frequency is centered for visualization. In the shifted DFT(u,v) of an M*N 2-dimensional image, the top-left corner of the 4th quadrant ...
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Significance of modular arithmetic in DFT?

In what ways does modular arithmetic plays a part in DFT? Why is it a so integral part of DFT?
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Finding the input from the impulse response and output

I have $y,h,x$ which are all vectors. From $y[n]=x[n]*h[n]$ which is basically how I got $y[n]$. I also know $h[n]$. I put this through a Fourier transform. Let's assume that the capitalized ...
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What should my reference value be when converting FFT bin amplitudes to dB?

I want to transform my FFT output values into a dB scale, but I'm struggling to determine the function I should run each bin amplitude through. My understanding of the decibel scale is that a value ...
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53 views

Discrete Time Fourier Analysis

Suppose we're given the following: $ x[n] = 2 + (-1)^n $, and are given the impulse response $ h[n] = u[n] a^n $, of an LTI system where $ |a| < 1$. We're asked to find the output $y[n]$, if $x[n]$...
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What is the correct solution for Fourier transform of unit step signal?

The unit step signal defined as $$ u[n]= \lbrace 1; n>=0; \\ \qquad0; n<0 \rbrace $$ has three possible solutions for its Fourier domain representation depending on the type of ...
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Plotting the Phase Response

I would appreciate it very much if someone would be able to provide some clarity on plotting phase responses. For instance, given that the frequency response of a filter can be written as H(exp(j*&...
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60 views

Fourier Transforms, symmetry, real/imaginary

I was hoping to clarify if the following was correct: -a real function (neither even nor odd) in time exhibits conjugate symmetry in frequency, so the real part of the frequency response is even, and ...
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1answer
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How would I find the function given the magnitude plot and the phase response?

I'm wondering how I'd find the Fourier Transform X(jw) given the following information: My understanding is that the expression for the continuous time fourier transform (CTFT) is magnitude(CTFT)exp(...
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Implementing Convolution in Frequency Domain?

Suppose, we have a bitmap image represented as a 2D integer array, int [,] image2D; whose FFT is Complex[,] fftImage2D; ...
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Basic Confusion About the DFT and Convolution

I am learning DSP (with Digital Images) and I have some elementary confusion about the convolution between two discrete periodic signals. Specifically, I have learnt that when filtering an image, we ...
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Kernel Convolution in Frequency Domain - Cyclic Padding

I don't know whether this is the right place to post this, but I suppose it is. I know that frequency multiplication = circular convolution in time space for discrete signals (vectors). I also know ...
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1answer
154 views

Applying Image Filtering (Circular Convolution) in Frequency Domain

To filter an image we can: Use a 3x3, 5x5, 7x7, etc. filter, that is convolve the image and the filter in the space domain. Use a FFT on both the image and the filter, multiply them together in the ...
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Inverse Fourier Transform From Plots (2019 edition) [closed]

Hello I borrowed the title for another post. I cannot figure out how to find the inverse fourier transform from this spectrum. I know what the transform is I'm sorry for the plot being hard to ...
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Low pass filter transfer function

I am calculating the transfer function of a low pass RC filter and I have gotten $\frac{1}{1+jωRC}$ which is correct. But somehow it seems $ωRC = \frac {ω}{ω_0}$ that refers to the cutoff freqency ...
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776 views

Convolution in spatial domain is multiplication in frequency domain

I have to prove convolution in spatial domain = multiplication in frequency domain using two matrices. $$ x(m, n) = \begin{bmatrix} 1 && 2 \\ 3 && 4 \end{bmatrix} $$ $$ h(m, n) = \...