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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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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Why am I not getting the intended coefficients in this 2D Fourier demonstration?

I am trying to demonstrate how the 2D Fourier decomposition of an image works with Matlab and a very, very simple example. I create a 4 x 4 pixel image with cosine basis vectors as such: ...
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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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Overlapping in real time fourier transform?

I have an algorithm and I need to record audio and perform short time Fourier transform to obtain which frequency is the most common. I am using a Hanning window to try and reduce spectral leakage as ...
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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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36 views

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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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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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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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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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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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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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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68 views

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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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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51 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 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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56 views

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)...
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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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174 views

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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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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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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113 views

From Fourier transform to Laplace Transform

It's well known that you can estimate the Fourier Transform $X(f)$ of a signal $x(t)$ via its Laplace Transform $X(s)$, just by setting $s = j2\pi f$ to the latter, as long as the region of ...
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138 views

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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60 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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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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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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61 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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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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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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55 views

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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Help me understand the stages involved in filtering a signal using Discrete Fourier Transform

I have a series of discrete values measured from a sensor. I want to filter the frequencies coming from this sequence of values. Then, if I understood the process correctly this is what I do: I ...
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180 views

Extrapolate a 2D array using Fourier Transform

I need to extrapolate a given 2D array to a larger domain, keeping the spatial frequency. This is the original field: (the data file in numpy npz format and a Jupyter notebook to plot it can be found ...
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The Fourier Transform of a periodic function and it's series

Let $X(f)$ be the Fourier transform of $x(t)$: $$ X(f) \triangleq \mathscr{F}\Big\{ x(t) \Big\} = \int\limits_{-\infty}^{\infty} x(t)\,e^{-j 2 \pi f t} \ \mathrm{d}t $$ $$ x(t) \triangleq \mathscr{F}...
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289 views

Normalized cross-correlation in frequency domain

I never worked with signal processing and never really used Fourier transforms before, still I am working on a project consisting on taking the output of an accelerometer to detect some movement ...
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How to find Fourier series coeffecients of convolution of two periodic continuous functions with different time periods?

Above given is my solution Plz check if it is correct and I got struck at last step .Plz help
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Why does shortening window widen its Fourier Transform?

On page 76 of the book Discrete Time Speech Signal Processing by Thomas F.Quatieri, the author states "shortening the window widens its Fourier Transform". Can anyone explain why?
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When a stochastic process would be a beneficial model in terms of noise

Let's say we have an image/signal with some noise in it. When would it be beneficial to model the signal as an outcome of a stochastic process? More specifically: How significant would noise have to ...
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Evaluate Fourier coefficients at arbitrary point using Python

Lets say I have a sinusoidal function $s$ that looks like ...
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63 views

On the spectral representation of deterministic and random signals

I went back to many references in order to fix some of the confusions that I have on many concepts in signal spectral representation. I concluded that: 1) Deterministic signals may be represented ...
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Seperating two (image) signals in Fourier Space / Denoising

i have a problem here which is not so easy to solve. I have a measurement system which provides me two signals: -The "contrast" channel" - this is the main channel on the object -A second channel ...
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Frequency analysis : relation between FFT size and sampling

I am currently trying to perform an experiment given an audio signal. I am therefore sampling the audio frequency range (0 to 22kHz) with a 48k sampling rate. When performing a FFT on my signal, I ...
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55 views

Finding n Amplitudes by DFT, what is correct normalization

Please forgive me if this has already been asked. Let us assume an example with $x(t) = \sum_{i=1}^N A_i \sin(2\pi f_i t) $ under a given sampling frequency $f_s$, frequencies $\omega_i$ and ...
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Checking Parseval's Theorem for Gaussian Signal by Using Scipy

I'm trying to check Parseval's theorm for Gaussian signal. It's well known that fourier transform of $\exp(-t^2)$ is $\sqrt{\pi}\exp(-\pi^2 k^2)$. So I implement it by using quad and simps. I think ...
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What is a correct way to find or “guess” a kernel which transforms an image into another image using Fourier Transformations?

Assuming I have two images, apple and orange; also assuming a filter kernel that transforms an apple image into an orange image possibly exists, how would some series of Fourier Transformations (and ...
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Is there a way to find a picture inside another picture, using only 2D Fourier Transforms of both images?

Can Fourier Transforms of both images tell anything about "likeliness" of two pictures? If yes, how precise? Can it still work if only several pixels are different or can it tell they are same if one ...