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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Is Fourier series a sampled version of Fourier transform?

I recently learned about dtft and how dft/dfs is the sampled version of dtft. I was wondering if Fourier series is also obtainable by sampling Fourier transform? I am a noob in the subject so sorry if ...
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FFT of input length 1536

Does anyone knows can i find a FFT of 1536 length input. Its a specification given in 3gpp Lte and we need a transform of 1536 input size which is neither a power of any number i would say. I just ...
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Proof of complex conjugate symmetry property of DFT

According to the Proof : \begin{align} X_n &= \sum_{k=0}^{N-1}x_ke^{-j\frac{2\pi k n}{N}}\\ X_{N-n} &= \sum_{k=0}^{N-1}x_ke^{-j\frac{2\pi k (N-n)}{N}}\\ &=\sum_{k=0}^{N-1}x_k e^{-j 2\pi ...
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What is phase congruency?

I am beginner in Image processing.I am studying the importance of phase in signal.Can anybody explain what is phase congruency ?
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Zero-pad before or after windowing for FFT

What's the correct way. Should I zero-pad a signal before or after applying a windowing function?
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Energy calculation in frequency domain

I was just wondering... The formula I learned to calculate the energy of the signal is expressed in the time domain: $$E_x^{\text{time}} = \sum_{n=-\infty}^{\infty} |x[n]|^2$$ Then, what does the ...
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Chop out frequencies outside human hearing range

I have a bunch of audio files all sampled at 44100 Hz sample frequency. I am trying to remove all the frequencies which are outside the human hearing range (I use the following as reference: Frequency ...
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Analysing 2500 frequencies using FFT with an input vector of 2048 samples?

I am currently reading the paper A Highly Robust Audio Fingerprinting System and on page 4 one can read about the technical parameters they use: Sampling rate of 5000 Hz, frames of 2048 samples as ...
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Books that explain DSP well to those not directly in engineering?

I do work with computer graphics and am dipping my toes into ray tracing. That field involves a good number of the subjects covered in DSP (Fourier transform, time vs frequency space, etc) but I was ...
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Why are Fourier analysis and transform only applicable for LTI systems?

Why are Fourier analysis and transform only applicable for LTI systems? What if the system is not LTI, won't Fourier analysis or transform be possible?
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Does sampling in the frequency domain cause time-domain aliasing?

Let's say I have an impulse response $h[n]$. I analyze the power spectrum of that impulse response similar to fourier transformed $h[n]$ corresponding to roughly $H[f]$. Now I compare $H[f]$ with ...
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Derive Frequency Representation of Impulse Train Function

I want to walk through the derivation of the frequency representation of an impulse train. The definition of the impulse train function with period $T$ and the frequency representation with sampling ...
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How condition for existence of Fourier transform is valid?

The condition for Discrete time Fourier transform to exist for function $f(n)$ is given as $$\sum_{-\infty}^\infty |f(n)| < \infty.$$ In case of continuous Fourier transform the difference is ...
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How to Zero Pad in Order to Perform Filtering in the Fourier (Frequency) 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's the impact of aliasing in the time domain?

I've been studying digital audio and come across something I can't understand. There appears to be something like a consensus (among those capable of understanding such things) that the impact of ...
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Why can't DFT be used when samples are not equally spaced in time?

I found the following comment here The DTFT can be used when the samples are not equally spaced in time, the DFT cannot My initial thought was that this had to do with periodicity of the basis ...
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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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Difference Between Convolution and Multiplication [duplicate]

I read that multiplication is convolution in frequency domain. I also understand that convolution is just polynomial multiplication. Can somebody explain what are the advantages of doing convolution ...
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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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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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Fourier transform of given signal

This is the signal whose FT i need to find, at first i thought that i could solve this as a convolution of two rectangular pulses, but i could not find pulses that fit into this (it turns out that ...
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Performing DFT twice on an image. Why am I getting an inverted image? [duplicate]

I was asked to perform DFT on an image twice as a part of my school assignment. Why am I getting a blurry inverted image when I perform DFT on an image twice? Sorry, I'm new to image processing and ...
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Why does the Fourier Transform of the impulse look so different from the Fourier Transform of the impulse train?

The fourier transform of the impulse functions is: $$ \delta(t) \longleftrightarrow 1$$ The shifted delta: $$ \delta(t-nT) \longleftrightarrow e^{-j \Omega nT}$$ But the fourier transform of the ...
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What will be the filtered output?

I tried to solve this question from basic Here is my work Image 1 Image 2 But the correct answer is Option $(B)$.What is the mistake i am doing?
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Which information do we get from magnitude and phase spectrum?

I am learning image processing. I want to ask very basic question related to FFT topic Which information do we actually get from "phase spectrum" and "magnitude spectrum" about an image?
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Finding out modulation index and DC offset

I have a question form my teachers, and I cannot understand why I can find out the modulation index form the figure. The question provide a Figure like this: And the information signal is a ...
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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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Can I assume system is LTI when given by DTFT of impulse response

I'm having hard time to grasp it probably because i don't fully understand it. I understand that when a system is given by $h(t)$ (in general $h(t-\tau)$) i can assume that it is a LTI system. So i ...
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The Number of Sine and Cosine Waves in an $ N $ Point DFT

This is bound to be an embarrassingly simple question, but here it goes... I was reading the chapter on discrete Fourier transforms (DFT) of this really didactic online book, The Scientist and ...
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Does “keying on” a sine wave at a zero-crossing reduce its bandwidth?

I understand that a pure sine wave of infinite duration occupies no bandwidth, i.e. it is only the modulation of a carrier that gives it sidebands. Does the exact timing of a sudden modulation make ...
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$2\pi$ periodicity of discrete-time Fourier transform

In my signals and systems course, we have learned that the discrete-time Fourier transform is $2\pi$ periodic, but the continuous-time Fourier transform is not periodic in general. For reference, we ...
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How does Adobe After Effects generate its “audio spectrum” effect?

I'm trying to replicate the "audio spectrum" effect from Adobe After Effects. An example can be seen in this video: Obviously, it has to be some variant of a fourier transform, but I've tried taking ...
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Interpreting the inverse fourier transform from a graph

I'm given a graph of the fourier transform of some function $x(t)$. The graph is labelled $F(X(\frac{\omega}{\pi}))$ on the y-axis and $\frac{\omega}{\pi}$ on the x-axis. The graph is plotted only ...
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Implementation of wideband beamformer for planar array

I would like to obtain a good reference (or references) on the implementation of a wideband beamformer for a small planar (rectangular) array comprised of 4 rows by 6 columns, for 24 elements in total....
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Whats the optimal window function to use for analyzing real-time data samples?

Say you wanted to run a X point FFT on the last X audio samples that were played. The problem being, using a normal hann window function would place emphasis on the "middle" of the audio sample. ...
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Why Is the Total Time Equal to $ N \cdot {T}_{s} $ and Not $ \left( N - 1 \right) \cdot {T}_{s} $ In the Context of DFT?

In the definitions of the DFT DFT $$ X(j)=\sum_{k=0}^{N-1} x(k) \exp \left(-i 2 \pi\left(\frac{j}{N}\right) k\right) $$ Let us say, if we have $10$ points, $N=10$, each sampled at $0.2$ seconds, why ...
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Image Restoration by Solving Constrained Least squares in Frequency Domain (Frequency Domain Filtering)

I am trying to implement the constrained least squares filtering as described in Rafael C. Gonzalez, Richard E. Woods - Digital Image Processing 3rd Edition Section 5.9. The equation (...
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Why DFT is used for approximating CTFT when you can approximate CTFT-integral itself?

I was using MATLAB for approximating FTs. Why DFT is used if we can approximate the transform-integration using summation.
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Intuition behind image derivative using Fourier Transform for edges detection

This equation can be shown mathematically: $\frac{\partial f}{\partial x}=\frac{2\pi i}{N} \mathcal F^{-1}\left(u\cdot \mathcal F(f(x,y)\right)$ I am struggling to understand the intuition behind it ...
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Calculate the Inverse DTFT of the DTFT Derivative in Terms of $ x \left[ n \right] $

Consider the signal $ x \left[ n \right] $ and its DTFT transform $ X \left( {e}^{j \omega} \right) $. Assume $ X \left( {e}^{j \omega} \right) $ is differentiable. What is the Inverse DTFT of: $$ j ...
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Why edge sharpening produces high frequency?

I have a low-resolution image in which the high frequencies are missing. When I apply an edge sharpening filter some of the missed high frequency is recovered. I am wondering why this edge sharpening ...
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A kind of Phase Retrieval problem

I know there are lots of papers proposing algorithms for the problem of reconstructing a signal from modulus of its Fourier Transform (so-called Phase Retrieval Problem). Also, recently it is studied ...
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Signal Processing using Fourier Transform

So I'm trying to understand how MRI machines work. I understand all the concepts of it, the parts, what they do, how the machine works, etc. The part I'm having trouble with is the fourier transform ...
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FFT, How to decide if there is a signal among noise?

I have sets of data of different deep sky objects. My job is to check for any periodicity. I use IDL to run an FFT and wavelet methods to check for a signal. To test my code I ran the IDL built in ...
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Recovering a signal after nonuniform sampling

Let $x(t)$ be a bandlimited signal such that $X(j\omega) =0 $ when $|\omega|>M$. Also $p(t) = p_1(t) - p_1(t-\Delta)$ is a nonuniformly spaced periodic pulse train where $$p_1(t) = \sum_{k = -\...
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Bridging CTFT and DTFT for a cosine

I'm trying to understand how I can start from the CTFT of a signal and end up with a DTFT. For example if I take a basic example: $$\begin{aligned} x(t) &= \cos(\omega_x \cdot t) = \frac{1}{2} \...
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How can understand periodicity of a Signal from frequency domain representation?

Is it possible to say a signal is periodic from its frequency domain representation? A periodic signal is sum of its sinus and cosinus. Frequency translation of sinus and cosinus functions are ...
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Response of a system to a step function (heaviside)

I'd like to compute the response to a step function of a electrical/thermal system. Generally I can "easily" compute the transfer function $H$: $$H(\omega) = \frac{V_{out}(\omega)}{V_{in}(\omega)}$$ ...
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Is it possible to do single vehicle tracking using Fourier transform?

I am working on a project in image processing which is based on importance of phase only reconstruction of a signal obtained using Fourier transform.For more information about phase only ...

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