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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How to Find "pitch" from Fourier Series

The end goal of my project is to create an autotune program, But the problem I'm trying to solve right now concerns finding the pitch of someone singing a note. I have written some code that performs ...
2 votes
1 answer
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Need help with DTFT problem

Prepping for exam and this is one of the practice problems: I just want some clarification on some of the steps my professor took. This is the answer in the answer sheet Only thing I dont understand ...
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2 answers
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What are some approaches / algorithms for reducing size of numerical data of large size with redundancies?

I'm dealing with bunch of .asc(ascii) files that are the output of continous monitoring of various electronic equipments for certification purposes. We monitor various parameters of the equipments ...
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Removing once per revolution variation from data

I’m looking for help to find a robust technique to remove a once per revolution variation in some vehicle test data. The data is collected by driving a vehicle around a circular path at increasing ...
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1 answer
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Fourier transform why can I convert one of the axes into an imaginary number?

Contextualizing This question is inspired by the following video: https://www.youtube.com/watch?v=-qgreAUpPwM&t=60s&ab_channel=3Blue1Brown I own a sign, a drawing of a square with 200 points ...
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2 answers
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Fourier transform of periodic functions

The Fourier transform is derived from the Fourier series by considering a non-periodic signal, thinking of it as a infinitely long periodic signal, putting it into the Fourier series and making this ...
2 votes
1 answer
46 views

Rayleigh Bandwidth Calculation-Radar

I am trying to generate a simple Gaussian pulse that has a 1 ns pulse width. However, when I generate the pulse, I realized that I did not meet the condition that calculates Rayleigh Bandwidth (1/...
1 vote
1 answer
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How to find the inverse Fourier transform of $u(\omega) e^{-j \frac{\pi}{2}} + u(-\omega) e^{j \frac{\pi}{2}}$?

I have been trying to find the following inverse Fourier transform but without success: $$ H(\omega) = \begin{cases} e^{-j \frac{\pi}{2}} & \omega \gt 0 \\ e^{j \frac{\pi}{2}} & \omega \lt 0 \...
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What is the frequency response of binning 2x2 pixels of an image into 1 pixel in software?

What is the frequency response of binning 2x2 pixels into 1 pixel in software? Can the binning introduce aliasing? Since the Fourier of the 2D boxcar function is a 2D sinc I would intuitively think ...
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Is an interval for a function and its Fourier transform based on the time constants?

The Fourier transform of an exponential function is a Lorentzian. For the sum of multiple exponential functions with time constants $k$, is it only meaningful to define the function and its Fourier ...
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Amplitude ratio is larger than 0.5 on a RC-filter if RC = 1.7684e-03

I have two signals ...
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Answered-Question About Radar Pulse Modulation

I am trying to simulate a radar-transmitted signal with a 4.5 Hz clock frequency and 1.8 GHz carrier frequency. I generated the carrier signal and a rectangle shape pulse signal, then multiplied in ...
1 vote
2 answers
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Why use sinc function to downsample an image in fourier domain?

I'm very confused about downsampling in image processing and the use of sinc function to do it. I read this post [1]: 2D Fourier downsampling some time ago that talked about my own doubt, that is to ...
3 votes
1 answer
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Practical applications of wavelets

I know wavelets were all the rage a few years ago, but I missed that boat and am wondering if it is worth putting significant effort into learning about them. My impression is that they were a little ...
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5 votes
3 answers
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Period and wavelength of a noise signal?

How do I determine the wavelength of a noise signal like the one below? I find it easy to understand for sine waves, but it gets tricky for me when the signal is more complex like a noise signal. If I ...
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Applying 1D wiener filter radially to 2D image

I am new to image processing. Using python, I have constructed a 1D wiener filter from the power spectrum of a Noisy Image, with Noise as a function of k. They both have the same dimension which is ...
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2 votes
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2D Cooley-Tukey FFT in Python

I've been trying to confirm the process for the Cooley-Tukey approach for FFTs. Currently I have a function that generates random input data for a matrix with $n_1$ rows and $n_2$ columns. The result ...
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perform fourier-type integration using DFT misunderstanding

I need to perform the following integral ($i^2 = -1$) (with a fixed value of $m$): $ \int_{\phi=0}^{\phi=2\pi} \psi(\phi) e^{-im \phi} \hspace{0.7mm} d\phi $ for all values of $m$ in $\{-N_{\theta}, -...
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Scaling factor in DFT: pure math or bandwidth issues?

I'm trying to match the amplitudes of a signal before performing DFT and after. So, let's consider a 64-point sine signal with amplitude of $1$: The DFT of such a signal will give us the amplitude (...
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3 votes
3 answers
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Effect of overlapping percentage on STFT output

I know STFT is generally applied to non-stationary signals but I tried to apply it to a stationary signal to get a working knowledge. I created a stationary signal composed of three frequencies as ...
1 vote
1 answer
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Showing Fourier slice theorem and Radon transform relation in MATLAB

I wrote some code to demonstrate the Fourier slice theorem and it's relation to the Radon transform. However the sampled FFT from the 2D FFT and the 1D FFT of the projection at the same angle don't ...
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1 vote
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How do I generate the approximate Fourier Transform of the signal using Principle of Stationary Phase?

The following excerpt is taken from the textbook "Digital Processing of Synthetic Aperture Radar" by Ian G. Cumming The Principle of Stationary Phase (POSP) can be briefly explained as ...
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2 votes
0 answers
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pocketfft delivers wrong values

does anyone understand how to use the pocketfft by martin reinecke? Link: https://gitlab.mpcdf.mpg.de/mtr/pocketfft Basically it's just this snipped of code: ...
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3 answers
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Verify FFT results without equation of waveform

I used python to convert the time-domain signal below into a frequency spectrum so that I can analyze the harmonics. To do this I used Python libraries, specifically numpy and called fft to get the ...
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1 vote
1 answer
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How to explain the meaning of the intersection in a Fourier series representation of periodic signals?

I saw a piece of code on github which transforms the planetary movement into the Fourier wave function. These circles are given by the $x$ and $y$ ordinates: $x=\cos (\omega t)$, $y=\sin (\omega t)$, ...
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Multiply and divide by the same function in convolution

I am calculating the convolution of two functions $F(x), G(x)$ in $\mathbb{R}^{n}$, n-dimensional space. I have another function $h(x)$ that is a Gaussian. What effect does multiplying $F(x)$ by $h(x)$...
1 vote
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Bring two Fourier transforms to same range to add them

I have Fourier transforms of two images which I wish to add (Basically I have an input Fourier transform which I mask, reconstruct the underlying image using an algorithm, and then try to replace the ...
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3 votes
1 answer
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Finding a periodic signal knowing its period, mean value and power

I've found an interesting exercise which I have been trying to solve for a couple days, without success. Let $x(t) \in \mathbb{R}$ be a periodic signal with fundamental period $T_0 = \tfrac{1}{4}$, ...
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3 votes
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Improving the intuition for the 2d fourier transform

As far as I understand, the 2d fourier transform is calculated as following: ...
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1 vote
1 answer
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Can I reduce the complexity of multiplication with FFT if the input vector is repeating?

I have a Fourier matrix $F$ with size $N \times N$, such that $y = F \times x$, if I have the vector $x$ contains four identical parts, for example $x = [x_1, x_2,x_3,x_4]’$ and $x_1 = x_2 = x_3 = ...
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6 votes
3 answers
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What are the criteria for a change-of-basis transform to be doable in $O(n \log(n))$?

The Fourier basis is a common choice for transformations, but a lot of times, it's not the best for a specific application. For instance, wavelet bases give us better spatial / temporal locality than ...
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11 votes
5 answers
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What is the frequency representation of nonuniform sampling?

Uniform sampling can be thought of as multiplication of a function $x(t)$ with a Dirac comb function: $$\text{III}_T(t) = \sum_{k=-\infty}^{\infty}\delta(t-kT)$$ Multiplication of $x(t)$ with $\text{...
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2 votes
1 answer
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Problem with the existence of inverse DTFT

I am having trouble on the following exercise and I can't figure out where I am doing something wrong: Given an LTI system described by the following difference equation: $$y(n)=x(n)+2x(n-2)+y(n-1)$$ ...
1 vote
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Fourier transform of a top-hat function in the Faraday Measurement synthesis context

I'm currently trying to calculate the Fourier transform of a top-hat function in the context of Faraday Measurement Synthesis. This is pretty straightforward, however, I cannot understand why I cannot ...
3 votes
1 answer
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Validity of taking an inverse $\mathcal{Z}-$ transform instead of taking an inverse DTFT

I have the following problem: I am using the Convolution Theorem and have got an expression of $H(z)X(z)$ and now I need to take $\text{DTFT}^{-1}(H(z)X(z))$, namely I have to take the inverse DTFT ...
1 vote
1 answer
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Axes of Discrete Fourier Transform

Problem Given $X[k] = \sum_{n=0}^{N-1} x[n]e^{-j2\pi kn/N}, k = 0, ..., N-1$ What are the units on the x-axis and y-axis? Note that for the x-axis there are two answers. Attempted solution My first ...
1 vote
0 answers
62 views

Convolution of infinite sums

We have $$ \sum_{n=-\infty}^{\infty}f(n) \sum_{m=-\infty}^{\infty}g(m) = \sum_{n=-\infty}^{\infty} \sum_{m=-\infty}^{\infty}g(m) f(n) $$ and $$ \int_{-\infty}^{\infty}f(x) dx \int_{-\infty}^{\infty}g(...
-1 votes
2 answers
92 views

Convolution of squares / boxcars

$$ \Pi(t/A) \star \Pi(t/B) $$ where $$ \Pi(t) = \begin{cases} 1,\ -1/2 \leq t \leq 1/2 \\ 0,\ \text{otherwise} \end{cases} $$ How to compute? Derivation/steps optional but welcome. Note: I'm aware of ...
0 votes
0 answers
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creating a power vs frequency graph on for different harmonics Matlab

I am trying to create a Power vs Frequency graph for different harmonics(x_5(t), x_20(t), and x_100(t)) of the half triangular wave. I want to see how the difference in power between the original and ...
2 votes
0 answers
53 views

Convolve sinc trains

$$ \begin{align} & \mathrm{sinc}(As + .5)\sum_{n=-\infty}^{\infty} \delta (s - n/A)\ \star \\ & \mathrm{sinc}(Bs + .5)\sum_{n=-\infty}^{\infty} \delta (s - n/B) \end{align} $$ How to compute? ...
4 votes
2 answers
522 views

Fourier transform of modulus of sum of sines

$$ x(t) = |\cos(\omega_0 t) + \cos(\omega_1 t)| $$ with $\omega_0, \omega_1 > 0$. Is there a known result for $\mathcal{F}\{x(t)\}$? Derivation not needed but is welcome. Of main interest is the ...
4 votes
0 answers
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Fourier transform of modulus of sum of weighted sines

$$ x(t) = |a \cos(\omega_0 t) + b \cos(\omega_1 t)| $$ with $a, b \geq 0$, $\omega_0, \omega_1 > 0$, but $a, b > 0$ or all $a, b$ (negatives included) is also acceptable, or replacing $\cos$ ...
0 votes
1 answer
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How to maintain DFT symmetry for zero padding in frequency domain? [duplicate]

I am looking for a mathematically correct way of zero-padding in the frequency domain when we have an even number of points. (i) If we zero pad in the center of the DFT, after the Nyquist value, the ...
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Fractional Fourier Transform of a Scaled and Shifted Function

I am trying to figure out what the Fractional Fourier Transform of the signal $\sqrt{c} x(c(t-\tau))$ would be with respect to that of $x(t)$. According to the paper "The Fractional Fourier ...
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DCT vs Inverse Fourier Transform in the Final Step for Computing MFCCs [duplicate]

I've been a little confused lately about the use of these two functions in the final step of computing the MFCCs. I often see them used interchangeably, but they do the opposite thing. And my ...
2 votes
1 answer
49 views

making a laplace s-domain plot from numerical data to decompose signal into decaying sinusoids

I would like to make a 3D laplace s-domain plot from experimental data I have. The examples I have seen for this are when the function is already known and an analytical solution can be obtained (...
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2 votes
1 answer
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Interpretation of complex time-domain signal resulting from time-shift property of Fourier transform

I am currently working on simulating RF transmissions for beamforming and other applications in Matlab. One of the fundamental properties that I need to simulate is signal propagation delay due to ...
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0 votes
2 answers
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Sampling of the DTFT causes the inverse transform to become periodic?

As you can see the above equation, DTFT is calculated from sample x[n] which is discrete sample of x(t). But calculated X(w) is continuous, even though it is calculated from discrete value of x[n] as ...
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How to calculate DFT of signal which occurs from 3 to 8?

How can I calculate discrete fourier transform of this signal? I'm confused as in the normal cases signal always start from 0 to N-1 but in this case it starts from 3 to 8 instead. So what would be ...
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4 votes
2 answers
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Why aren't negative frequencies folded in reconstruction of the aliased signal?

I'm working on the problem 1.9 from the book Introduction to Signal Processing by Sophocles J. Orfanidis. The pdf version and solution is freely available here. This is the solution for part a of the ...
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