The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum.

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FFT in different coordinate systems

Is there a transformation that will enable one to calculate the FFT in an arbitrary coordinate system? What I am interested is the following two cases: The space is Euclidean and infinitely ...
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How do I interpret the result of a Fourier Transform?

For example, I entered the following "equation" into Wolfram|Alpha: FourierTransform[Piecewise[{{sin[t],t > 0 and t < 2*pi}}, 0], t, \[Omega]] So as to ...
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FFT over a fixed and equal numbers

I've got a vector of $100 000$ numbers. All numbers are equal ($7000$ for example). If I perform FFT over this vector, what will I get? From my understanding, I should receive a fixed DC line. Is ...
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Fourier transform of $\cos(n\omega t)$

My question is probably very stupid, but I've been strugling for a while on it now... In need to find the Fourier transform of $1+\cos^3(2\pi ft)$. I wrote that : $$\cos^3(2\pi ft)=\frac{\cos(6\pi ...
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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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160 views

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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102 views

Difference between Fourier Transform and DFT? - Example

I have read many excellent answers to similar questions, but never one this specific. Here is another way to ask it. Why is the modulation transfer function (MTF) of $\textrm{rect}(x/5) = ...
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30 views

Fourier transform of $ne^{-an}u[n]$

I need to find the Fourier transform of the following signal: $$ne^{-an}u[n]$$ The answers start by using the rule of the basic signal: $$a^nu[n] \rightarrow \frac{1}{1-ae^{-j\omega}} $$ and then ...
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amplitude at exact frequency in wide band signal

Could anyone suggest the most computationaly efficient method for finding amplitude of exact frequency having a noisy wide band signal. To be more specific about a task. I have some physical ...
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26 views

Transfer functions from wavelet transfrom

So I have this problem where I need to measure the phase of a signal and correct for a delay associated with the travel time of the signal while simultaneously determining the transfer function of my ...
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644 views

Why real part of FFT converts image into rotation + original?

I have read this image: taken its FFT (2D) and then Inverse FFT to get exactly the image back. Code is provided for reference: ...
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1answer
27 views

How would Fourier and Cosine Transforms responds to summation of cosines with same frequency but different phases?

For example, if I have two signals, $\cos(2\pi ft+\frac\pi4)+\cos(2\pi ft+\frac\pi3)$, what would be different in both transforms (Fourier and cosine) how would the spectrum changes? And What would ...
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How to convert a spatial frequency in a 2D-DFT into the units radians per pixel?

Let's say I have a 2D image, and I take the discrete Fourier Transform (via FFT) of that image. In the frequency domain, I get the following image: In this image, let's just assume all the spatial ...
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1answer
47 views

Computation of Only Even or Odd Frequency Bins of DFT

I have an algorithm where I am computing the FFT of a large signal. However, I desire only the even or odd terms of the DFT of the signal, but not both. Currently, I discard these undesired terms. Is ...
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38 views

Hilbert transform linearity

Explain why the Hilbert transform of $f(t)=\operatorname{sinc}(at) \cos(2 \pi \nu_c t)$ is $$\hat{f} (t) = \operatorname{sinc} (at) \sin(2 \pi \nu_c t),$$ where $0<a<\nu_c.$ Attempt: I have ...
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1answer
29 views

Hilbert transform from analytic signal

Show that the Hilbert transform of $h(t) = m(t) \cos(2 \pi \nu_c t)$ is $$\hat{h} (t) = m(t) \sin(2 \pi \nu_c t),$$ where $m(t)$ is a real valued, band-limited function (i.e. we have Fourier ...
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1answer
21 views

How is Linear Canonical Transform a generalization of Fractional Fourier Transform?

I have studied that Fourier transform changes the domain of a signal from time to frequency, and in that way it is a 90 degree shift. When it comes to Fractional Fourier Transform a generalization of ...
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66 views

Hilbert transform pair proof

I am looking for the proof that the Hilbert transform of $\displaystyle\frac{\sin(at)}{at}$ is given by $$\frac{\sin^2(at/2)}{at/2}.$$ How do we prove this? This is a $\operatorname{sinc}(at)$ ...
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33 views

What does it mean for the DFT phase to be relative to a cosine wave?

The following paragraph from Understanding Digital Signal Processing got me puzzled: The answer is: The DFT phase at the frequency $mf_s/N$ is relative to a cosine wave at that same frequency of ...
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28 views

Deriving the DFT magnitude of $A\cos(2\pi nk/N)$

Given that $$x(n) = A\cos(2\pi nk/N),$$ the $N$-point DFT of $x(n)$ can be expressed as follows—the derivation can be found in here: $$X(m) = \color{red}{\frac{A}{2}\sum_{n=1}^{N-1}e^{-j2\pi ...
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30 views

Where does $\frac{N}{2}$ came from in approximating an N-point DFT?

I've came across the author saying that ... for a real cosine input having k cycles in the N-point input time sequence, the amplitude response of an N-point DFT bin in terms of the bin index m is ...
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45 views

Why is N-point DFT approximated by the sinc function?

While looking into DFT leakage, I've came across the author saying that "..., the amplitude response of an N-point DFT bin in terms of the bin index m is approximated by the sinc function." ...
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18 views

Should S-Matrix components for scattering response in a Polarimetric Radar be Real or Complex?

I am trying to implement some polarimetric decomposition algorithms on my MIMO polarimetric radar. At first, I will try the Pauli Decomposition algo. But instead of doing signal processing on the ...
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42 views

Why does Fourier zero-fill interpolation pad the middle frequencies?

In this: http://dspguru.com/dsp/howtos/how-to-interpolate-in-time-domain-by-zero-padding-in-frequency-domain the author says that we can interpolate a signal by zero-padding the middle frequencies in ...
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what is nyquist rate of $h(t)\cdot h(t)$ and $h(t)\circledast h(t)$

Let's say we have $h_c(t)$ as a continuous-time signal with bandwidth $B$ and we would like to sample it. To be able to reconstruct it correctly, the sampling rate must be greater than $2B$. Now ...
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1answer
18 views

The signal in MRI

I am trying to understand the signal formation in MRI and have a confusion. I understand that in the presence of the external magnetic field $B0$, the protons are precessing at the frequency given by ...
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21 views

How to analyze baby speech time series

I've been collecting speech data for my baby brother (who is now 6 months old) with the intention of doing computational analysis of the development of his speech patterns. I haven't much deep ...
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42 views

How to interpret output of matched filter with complex input?

I have implemented a matched filter based on the Fourier Transform approach. In the real numbers domain that means that I use as the coefficients of my filter (B) the inverted time-samples of the ...
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1answer
29 views

Bandwidth range for Fast Fourier vs principal component analysis?

I've read somewhere that the Fast Fourier is only applicable to those processes exhibiting bandwidth. Where as principal component analysis can be applied to a process exhibiting any finite ...
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28 views

Calculate 2D Windowed Sinc Kernel

I am implementing a Windowed Sinc Filter for Scaling images by the factor of 2. Therefore I need to calculate a Filter Kernel. I already know how to calculate the kernel including a Blackman Window ...
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137 views

Fourier transform of a Fourier transform

I generate a Gaussian noise and then I filter it with a passband FIR Kaiser window filter. When I perform the Fourier transform of the output of the filter and plot its magnitude spectrum, it is ...
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24 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 cisoid. This representation is quite common for analytic signals, but ...
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33 views

Why Fourier transform and Stockwell-transform retain the absolute phase information of one signal?

Hello friends am studying the topic of signal processing and the Fourier transform and the s-transform and in most books as for example "Time-Frequency Signal Analysis and Processing. 2nd" of Boashash ...
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1answer
23 views

Autocorrelation of a noisy linear map

I am interested in calculating the autocorrelation function of a linear map with some noise (model given below) but am slightly confused in doing so. At first, I did not realize there were two ...
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2answers
35 views

In fourier space, how to apply transfer function with n frequencies to input data with m>>n frequencies

I have a transfer function in Fourier space with $N=2028$ frequencies $(\frac {0, 1}{(N\cdot dx)} \dots ) $ Where $dx = 0.1m$. I need to apply this transfer function to a signal with 20000 samples ...
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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 ...
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70 views

Why are the basis functions for DFT so?

When you get a DFT of a signal, you use the basis functions as: $e^{-j2\pi kn/N}$ Why is it so? Why don't we use the conjugate, $e^{j2\pi kn/N}$, or any other function?
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107 views

DFT of real sinusoids - why sum over -$N/2$ to $N/2-1$ as opposed to $0$ to $N-1$?

I'm going through a Coursera course on signal processing, and we're just introduced to DFTs. We are told that if you have a complex sinusoidal signal $x[n]$ where $n=0,1...\ N-1$, its DFT is given ...
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36 views

Convolution theorem for cross-correlation

Forgive me is this is an ill-posed question. Is there any such thing as a 'convolution theorem' for the cross-correlation. Namely, the convolution theorem states that: $$ x[n] * h[n] = ...
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28 views

How to find Galois Field $\textrm{GF}(p)$ from any $n$-point DFT

How to find a finite field $\textrm{GF}(p)$ (where $p$ is a prime number) (here I want to find the value of $p$ only)that is as small as possible and efficient computation can be performed for (let's ...
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2answers
41 views

Polar form of the Fourier transform of $\sin(t)$

I came across the Fourier transform of sin(t). It ends up being a purely imaginary (dirac delta) impulse pair. But when considering the frequency domain representation of a signal, we consider the ...
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8 views

A 2 input 1 output model with time delay

What assumptions should be taken into consideration if I need to build a 2 by 1 black box process model on Matlab where the inputs samples I have are sampled each minute to result eventually after 1 ...
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29 views

Approximation of Step Response from Data with delay

How can I approximate a Step Response curve from a measured data of a single input and a single output using Matlab while the delay between the input and the output is 30 minutes?
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1answer
58 views

How do I understand Fourier descriptors more visually and intuitively?

I read the book Image Processing, Vision and Machine Vision and find the concept Fourier descriptors hard to understand, although literally its derivation is somewhat reasonable. Can anyone give me a ...
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1answer
55 views

I have a pressure signal and want to do SPL analysis on it

Signal I have an acoustic signal from a Ffowcs Williams Hawkings CFD analysis and would like to convert it to the frequency domain and see the SPL and OASPL. I know I need to use fft() but I am ...
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117 views

Relation between the DTFT and the spectrum of a sampled signal

In the $\rm DTFT$ (Discrete Time Fourier Transform) the spectrum is periodic with period of $2\pi$ . A continuous signal when sampled has a spectrum which is a repeated version of its original ...
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60 views

$2\pi$ 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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45 views

What is the interpretation of the discrete-time spectrum?

The CTFT of an analog signal is a representation of that analog signal in terms of the frequency parameter of sinusoidal (cosine specifically) functions whose weighted sum make up that signal. The ...
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38 views

square wave frequency representation

New to this signals stuff and i'm confused about the frequency representation of the square wave. Correct me if i'm wrong, a periodic square wave is composed of odd harmonics sine waves which are ...
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61 views

Anomaly at bin centre frequency

I'm building an additive synth using a real IFFT. Spectrum size 512, Samplerate 44100. I've noticed that when I alter the fine tune of one my oscillators, I get a blip sonic artefact exactly at the ...