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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Envelope of a signal regarding shifts

Assume I have a Ricker wavelet. I can compute the envelope of this wavelet as shown below: This is the normal condition we usually see. However, if I shift the Ricker wavelet to be wholly negative, ...
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Can the magnitude of a discrete-time Fourier transform be negative?

Consider the discrete-time system $$ H(z) = 1 + z^{-1} + z^{-2} + z^{-3} $$ To obtain the magnitude of the discrete-time Fourier transform, I substitute $z = e^{j\omega}$ to get \begin{align} H(\omega)...
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Fourier transform of x[n] = e^(jn/8)*u[n]? [closed]

Fourier transform of x[n] = e^(jn/8) is non-zero for only w=w0, w0=1/8 here. It is an impulse at w=w0. But what is the Fourier transform of x[n] = e^(jn/8)*u[n] ? I couldn't find out.
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How to calculate magnitude of frequency response?

I'm trying to write a function in Python that calculates the magnitude of an FIR filters frequency response. I tried doing it by first calculating the Fourier transform with np.fft.fft and then ...
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zero-centered and causal zero padding

I have followed the link below to simulate two different zero-padding methods (zero-centered and causal) https://ccrma.stanford.edu/~jos/mdft/Zero_Padding.html Sample code ...
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I'm having problems simplifying this discrete-time fourier tranform

I have this problem, and I can't get to the solution. $$X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} {(0.6)^{|n|}[u(n + 10) − u(n − 11)]}e^{-j\omega n}$$ The solution is $$X(e^{j\omega}) = \frac{0.64 − ...
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misunderstanding of spectral leakage

I want to understand spectral leakage. I understand that whenever we feed $N$ time-samples of a periodic, continuous, signal into a FFT algorithm we are multiplying in time-domain the true periodic, ...
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How to find the difference between maximum and minimum of a time signal, having only Fourier series cofficients

I currently know the Fourier coefficients of a signal $c_n$, from the exponential Fourier series of this form: $$f(t) = \sum^{+\infty}_{n=-\infty} c_ne^{in\omega t}$$ Using these Fourier coefficients, ...
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1answer
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inverse fourier transform coefficients

Context I want to implement (real) cepstrum on stock data (for example MSFT stock) and achieve cepstral coefficients of this time series. as noted in "Cepstral-based clustering of financial time ...
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Fourier transform of $\textrm{sinc}^2(100\pi t)$

I'm confused about a tutorial problem concerning the Fourier transform of the $\textrm{sinc}^2$ function. Specifically, the question involves the Fourier transform of $\textrm{sinc}^2(100\pi t)$, ...
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Fourier Transform Definition

In Wikipedia the definition of the Fourier transform is: "The Fourier transform of a function of time is a complex-valued function of frequency, whose magnitude (absolute value) represents the ...
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Computing Phase Difference Between Two Channels from SDR

I'm trying to compute the phase difference between two channels of a software-defined radio. I'm providing code for two different methods that I'm trying below. In both cases, I'm transmitting a pure ...
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How to interpret fft2() plots for images in MATLAB? What are represented by axes?

I got the frequency spectrum of a grayscale image using ...
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1answer
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Is there an efficient algorithm to precisely find the spectrum of a finite discrete signal consisting of incommensurable frequencies?

If we periodically sample a continuous band-limited signal at a sufficient sampling rate, we can estimate its spectrum by using Discrete Fourier transform (DFT). DFT of a finite discrete signal sample ...
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what are the application of fourier transform in finding frequency response? [duplicate]

i need the application of Fourier transform in finding frequency response in mathematical terms
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Connection from Fourier to Laplace Transform

I have a basic understanding of Laplace and Fourier but having trouble making a connection. Every time I attempt to look at reasons these are connected I'm told about the s-plane and regions of ...
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Implementation of Inverse Discrete Fourier Transform for a 3-dimensional signal with plain python [migrated]

I am trying to calculate inverse Discrete Fourier Transform for a 3D Numpy array. I have already implemented the same for a 1D signal. Please can somebody assist me with converting this code for a 3D ...
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Which conditions must fulfill a time-limited signal (so of unlimited bandwidth) $f(t)$ to have a bounded maximum slew rate?

Which conditions must fulfill a time-limited signal (so of unlimited bandwidth) $f(t)$ to have a bounded maximum slew rate? I want to know about which conditions must fulfill a real-valued time-...
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Find Fourier Transform of Unit Step using the $z$-Transform [duplicate]

Since the unit step $u[t]$ is not absolutely summable, it has no Fourier Transform. In the DSP book (Proakis), the Fourier Transform of the unit step is formed by evaluating its $z$-Transform on the ...
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CTFT of $ X(j\omega) $ vs $ X(jf) $

The CTFT of a signal as a function of $ \omega $ and $ f $ is identical: $$ X(j\omega) = \int_{-\infty}^{\infty} x(t) \, e^{-j \omega t} \operatorname{dt} \;\;\;\;\bigg|\;\;\;\; X(j f) = \int_{-\infty}...
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Is there an order to apply time shifting and frequency shifting to signals in DTFT?

For instance in one question applying frequency shifting first and then applying time shifting yields a different answer wrt applying time shifting first. Please elaborate i am clueless. if i use ...
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Finite Precision DFT looks broken

I calculate the FFT of a sine wave. In Matlab I get the result as I would expect with a nice real and imaginary part. (blue is real, red is imaginary) Now I also calculate a FFT on a FPGA with fixed ...
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How could I do a Discrete Fourier Transform in Python if my data is non uniform?

I have been trying to find a way to transform my time series data in an equivalent manner to the discrete Fourier transform. What I wish to find is something like: ...
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Inverse DCT4 by using Inverse DFT

Say I have an inverse DFT function ready for use. I am struggling to find online or mathematically deduce the additional steps needed to compute an inverse DCT4 by using said inverse DFT function. If ...
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FFT filtering by hand in python - iffy issue - cannot get filtering right

I am trying to filter an image by hand (with no using the image processing libraries) to understand how fft.shift works in python. No luck. I take a png an image from MNIST, e.g.. Here is the code: <...
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Difference between reference and measurement signal?

Which differences can we find on various signals? We assume that we have a reference signal (determined theoretically) and measured signal. Which methods exist for determining the similarities ...
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Derivation of Strehl ratio

I am in trouble proving the following fact: The Strehl ratio is the ratio of the volume of the aberrated OTF (optical transfer function) and the ideal OTF, i.e. \begin{align} \mathcal{S}=\frac{\int\...
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Identifying a signal by its power spectrum

Background: There is a method in optics for determining the electric field of light (both intensity and phase) via a three step process: Add a known phase shift to the light (called the "...
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1answer
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Fourier transform of an impulse-train sampled signal

I'm trying to calculate the Fourier transform of an impulse-train sampled signal in two differnt ways but I end up with different results. Impulse-train sampling of a continous signal $x(t)$ with ...
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1answer
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How to interpret Fourier transform?

I am very new to this topic. I ran a Fourier transform with the scipy fft function. I than plotted the return values: I am assuming the x-axis means how many cycles there are in all the data and y-...
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Impulse Response and Frequency Response of a FBLMS filter

Suppose I work with sampling frequency FS and block length of L. I implemented the Frequency-domain Block LMS (FBLMS) algorithm ...
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1answer
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How to specify parameters in scipy.stft to reproduce published analysis

I am starting a project on this Keggle dataset containing EEG registrations (sampled at 128 Hz) of several subjects. What I am really interested in is the final ...
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How to prove that the integral of Hilbert transform is not equal to the Hilbert transform of the integral?

To prove that $\int_{-\infty} ^\infty \mathcal{H}(g(t))(t)\text{d}t\neq\mathcal{H}(\int_{-\infty} ^\infty g(t) \text{d}t)$, where $\mathcal{H}(\cdot)$ is the Hilbert transform operator My approach to ...
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What does it mean to calculate the FFT of an integer sequence?

The Fourier transform is used to map functions to and from time/frequency domains. I can make sense of what it means to calculate the Fourier transform of, say: $$y(t) = e^{j\omega_0t}$$ which is $$Y(\...
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1answer
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Fourier transform and Zero Order Hold

I was asked to explain why a zero order hold (ZOH) filter is not a good choice for signal reconstruction. To answer this question, I was thinking about studying the Fourier transform of the ...
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1answer
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Fourier transform of shifted periodic function

Assuming $x(t)$ is a periodic function of period $T$ and having the Fourier transform $X(\omega)$, it is required to calculate the Fourier transform of the signal $x(t)+x(t-T)$. Since x(t-T) is equal ...
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1answer
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Expression of the convolved image

I have an image of a 2D-sinusoidal pattern $f(x, y)$ with wavelength $\lambda$ which I would like to convolve with a 2D circular pill-box function $h(x,y)$ of radius $r$. The image is given by $$ f(x,...
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Why do the lengths of the sampled signals $x_1, \: x_2$ have to be $\text{length}(x_1)+\text{length}(x_2)-1$?

We know that convolution in time is equivalent to multiplication in frequency (Fourier). $$x_1(t) \ast x_2(t) \leftrightarrow X_1(\omega)X_2(\omega) \tag1$$ However, for a sampled signal, this ...
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Frequency domain processing

I'm a little fuzzy on how the Fourier transform works. It's supposed to map time domain to frequency domain, and vice versa, so my intuition says that if I wanted to make an equalizer, I'd take the ...
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1answer
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$H(e^{j\omega})$ represented with a limited no of samples

I have a low pass filter defined as: $$ H_{lp}(e^{j\omega})=\left\{ \begin{aligned} &1 &|\omega|>\omega_c \\ &0 &\omega_c<|\omega|<\pi \end{aligned} \right. $$ and the ...
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1answer
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FFT for Arbitrary time history

I have two Time-Speed histories one is real ( real measurements) the other is synthetically generated using formula. I need to compare both in the frequency domain using FFT given that: the time step ...
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Units of a Fourier Transform (FFT) and Spectrogram [closed]

I am very interested in analyzing signals in the frequency domain. Units of a Fourier Transform (FFT) when doing Spectral Analysis of a Signal 1.In the previous question, answer the unit for the ...
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How to Use Convolution Theorem to Apply a 2D Convolution on an Image

How do I actually apply the convolution theorem? I have my fourier transformed image matrix, and a Fourier transformed kernel, but how do I actually multiply these together to achieve the intended ...
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Prove Convolution Property for DFT using duality

If $x_1[n]$ and $x_2[n]$ are finite length sequences of length $N$ $$\mathcal{DFT}(x_1[n] \circledast x_2[n]) = X_1[k]X_2[k]$$ where $X_1[k]$ and $X_2[k]$ are the DFTs of$x_1[n]$ and $x_2[n]$, ...
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Reconstruction of a Signal from Sub Sampled Spectrum by Compressed Sensing

Context Attempting to reproduce an illustrative example of compressive sampling from Candes-Wakin 2008. Specifically, the L1 recovery of a sparse signal shown on pg 5 in Fig. 2. Using my code (below), ...
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1answer
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How to interpret the Fourier Transform 𝑋(𝜔) (Foundational Questions)

The motivation behind the fourier transform is to somehow represent a non-periodic signal as a sum of sinusoids just as we do with the fourier series for periodic signals, correct? With the Fourier ...
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Discrete Fourier Transform of 2-D Images

I'm a high school student doing an essay on the applications of the Fourier transform on signal processing, but I haven't been able to find much information when applying the discrete fourier ...
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1answer
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Frequency mixer shows 2 FFT spikes

I'm attempting to use a frequency mixer to shift one frequency range to another. Right now, I have a complex sine wave at $43\ \rm kHz$. My imaginary value on the complex object is 0. If I output a ...

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