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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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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743 views

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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Image Processing and applicability of 2D Fourier Transform

As a newbie in the world of signal processing, I am having a hard time in appreciating image 2-D fourier transforms. I am fully able to appreciate the concept of 1-D Fourier transform. Essentially, ...
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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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DFT of discrete signals, why do we only analyze frequency bins equal to number of input samples?

If we have a signal $x[n]$ such that we have $N$ samples i.e. $n=0, \ldots, N-1$, then when we analyze the DFT $X[k]$ we only analyze for $k=0,\dots,N-1$ as well. Why is the range of $k$ tied to the ...
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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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525 views

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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123 views

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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629 views

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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What is difference between terms $X(j \omega) ,X(\ e^{j \omega })$ and $ X(\omega)$?

While studying frequency transforms ,I get confused with the terms like $X(j \omega) ,X(\ e^{j \omega })$ and $ X(\omega)$ ,where $ \omega = 2 \pi f $. So what is the difference between them ?
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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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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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821 views

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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803 views

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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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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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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Is $O (N \log N)$ FFT speed the fastest we can ever attain?

I am wondering about whether or not there is a theoretical limit as to the speed at which we can compute a DFT. We all know that the FFT executes in $O (N \log N)$ time. However, is this a lower bound ...
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how to compute a discrete fourier transform on fragmented data

I have a list of collected data points that I need to take a DFT of, however with the problem that about one quarter of the data points in the middle are missing, and so even though the existing data ...
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What should the amplitude be when plotting 1-sided Amplitude Spectrum?

I have a continuous signal x(t) such that $x(t)=12cos(6\pi t)+6cos(24\pi t)+3cos(30 \pi t)$ and is asked to sketch a 1-sided Amplitude Spectrum of the signal x(t) if sampled above the minimum ...
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From MFCC to Machine learning. What are the steps?

I was able to get a dataset with MFCC coefficients. However, depending on the length of my sound file I get a different sized matrix. As in, 13 (13 MFCC coefficients) by XXX, where XXX will vary ...
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MP3 Filterbank + MDCT: Why?

What is the rationale for the two-step process the MP3 format performs, first decomposing the input into 32 subbands of 6/18 samples each then performing MDCT on each subband individually? Why not ...
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Signal Processing using Fourier Transform

How can I derive the fourier transform of ...
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508 views

Is it possible to decompose an image into basis images using MATLAB?

I have read that: Using Fourier decomposition any arbitrary image can be represented as summation of orthogonal basis images. I want to see the basis images for any image say Lena or Cameraman, ...
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172 views

Malaysia Flight 370 Image Processing

I wish that all those people on that flight were with us, and maybe they still are, as I like to think they are on an island somewhere waiting to us to find them ... however, it does not look good... ...
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Fourier Transform Computer Vision Textbook

I am required extremely fast to fill the gap and learn Fourier Transform with application in Computer Vision. It was easy to find mathematical aspects of Fourier Transform, but I am more interested ...
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How to get coefficients for sine/cosine function from complex FFT?

I'm working on a control system that measures the movement of a vibrating robot arm. Because there is some deadtime, I need to look into the future of the somewhat noisy signal. My idea was to use the ...
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847 views

Deriving the integration property of the Fourier Transform

I want to derive the property of the Fourier Transform that states that if $X(j\omega) = \mathcal{F} (x(t))$ then $$\mathcal{F} \left( \int_{-\infty}^{t} x(\tau) \mathrm{d} \tau \right) = \frac{1}{j\...
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832 views

2D Fourier Transform of Rotated Discrete Domain Signal

Assume we know that the Fourier transform of a signal $x(n_1,n_2)$ is $\mathcal{F}(x(n_1,n_2))=X(\omega_1,\omega_2)$. What is the Fourier transform of the signal after being transformed by a rotation ...
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How do optical anti-aliasing filters work from a frequency domain perspective?

To prevent aliasing caused by the finite number of pixels on a sensor, a blurring filter is commonly used. How does that work from a frequency domain perspective? What is the transfer function of such ...
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506 views

3D wiggle plot for an analytic signal: Heyser corkscrew/spiral

Just reading The Analytic Impulse, A. Duncan, 1988, I met the name "Heyser corkscrew" for the first time in my DSP life, for a 3D display of a cisoid or complex exponential $e^{i\omega }$ (often ...
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How does taking the absolute value of a complex signal reflect in the frequency domain?

I have a frequency-domain representation $X(e^{i\omega})$ of the complex discrete one-dimensional signal $x[n]$: $X(e^{i\omega})=\mathcal{F}\{x[n]\}$. Is there a frequency-domain transformation of $X(...
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Why is level of power spectrum dependent on FFT resolution?

I created a sinusoidal wave in some noise, and plotted the power spectrum of the signal using two periodogram estimates (welch procedure). One estimate is 'high resolution' - i.e. it uses a longer ...
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172 views

1-D Fourier Transform Of A 2-D Image But At An Arbitrary Orientation

If one has a 2-D array and would like to take the 1-D Fourier transform along a direction $\theta$ degrees off the horizontal, is there a better/faster way to do this rather than rotating the image by ...
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940 views

What's the difference between using DFT, IDFT or DCT to calculate cepstrum of a power spectrum?

I've seen different equations that calculate cepstrum from power spectrum, but the equations are not consistent. Some people use Fourier transform, some use the inverse Fourier transform, and some use ...
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How does causality (i.e. unit step) affect the DTFT of a sine or cosine wave?

Tables of common Discrete-Time Fourier Transform pairs list the transform of a sine wave: $ \sin(\omega_0\ n) $ and its transform: $ -j\pi\ [d( \omega\ - \omega_0\ ) - d( \omega\ + \omega_0\ )] $ ...
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Problems Using FFT to Compute Impedance in a Model Neuron

I'm a neuroscientist currently investigating the resonance properties of a single neuron model that a colleague and I have constructed. The language we code in is Julia, which I hope is similar enough ...
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Periodicity of a constant signal!

This can be a very silly question, but I'm quite confused: If we take the Fourier transform of any constant signal, we get an impulse at zero, which says that its frequency is zero and, hence, it is ...
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331 views

Motivation of time-frequency analysis

Can anyone give me an example of two signals with different temporal waveforms having the same Fourier transform (FT)? Would the inverse Fourier transform still be able to recover correctly each ...
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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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Does DFT produces the same output as FFT?

In my journey about learning what / why / how of DFT, I tried to implement a DFT on MATLAB and then I compared its output with fft output and then I noticed it was ...
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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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FFT for a single frequency

I was looking for a more efficient way of finding the magnitude and phase of a signal at a certain frequency without performing an FFT because it produces more information than I need and I came ...
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549 views

A system that perfoms Fourier Transform operation - is it an LTI system?

If a system takes input as the time domain signal and outputs the frequency domain signal, is such a system an LTI system? For if the input time domain signal can be represented as a linear ...
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Online DFT Algorithm

I have a discrete audio stream $x$ that needs to be processed in real-time. Specifically, as the each new sample is received, I would like to compute a Fourier transform of the last $n$ samples of the ...
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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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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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What happens with signal in frequency spectrum when it is time shifted in time spectrum?

I have some trouble to understand what is going on with signal in frequency spectrum when it is time shifted in time spectrum. I am hoping that somebody will help me to understand that. Thanks you ...
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Difference between CTFT and DTFT?

I have tried to read different articles but still confused in difference between continuous time Fourier transform and discrete time Fourier transform?
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FFTs of a complex signal - separating the real and imaginary parts

I have a complex time varying signal at a single frequency x = a + jb where a represents the contribution from the cosine basis ...

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