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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 utilize product graphs in graph signal processing?

I'm working with dynamic graph signal processing. How to plot time variation and spatial variation separately and take the cartesian product of those two graphs? Any relavant Matlab codes?
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1 answer
91 views

How can sound waves be modeled in a manner that distinguishes individual voices but also recognizes words?

More specifically, I'm curious how we can represent sound waves in a way that would both distinguish between individual voices and also recognize, e.g. that recordings of different people saying the ...
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4 answers
138 views

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 ...
2 votes
1 answer
210 views

In what cases can you get aliasing below the Nyquist frequency?

I took the one-sided FFT of a signal and plotted up until the Nyquist frequency. Then, I took the real part of this FFT multiplied by $i\omega$ following a calculation that I'm trying to do of a ...
1 vote
1 answer
55 views

Why Does This Sliding Window Goertzel Filter Trick Work?

I'm reading an analog signal at a constant interval on an embedded device, and I want to extract a specific frequency component. Since the signal is continuously being read I want to constantly scan ...
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Sinc droop in a simple sampling mixer

This question is about the conceptual interpretation of a track-and-hold mixing operation implemented conceptually below, where an output $y(t)$ tracks an input $x(t)$ for half of a period $T_{LO}$ ...
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1 answer
246 views

Can someone explain me how the phase spectrum of trigonometric fourier series is related to phase spectrum of exponential fourier series of a signal?

Suppose we take a periodic signal and perform fourier analysis over it . Now we have two ways of representing the fourier series of this particular signal , one is trigonometric fourier series and ...
1 vote
1 answer
146 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? ...
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1 answer
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How does MATLAB's edgetaper function work?

MATLAB has a function edgetaper that does something I haven't seen before: J = edgetaper(I,PSF) blurs the edges of the input ...
1 vote
1 answer
158 views

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
393 views

Time domain to frequency domain conversion of audio signals to extract 1/3 octave frequencies

We have developed an android app (Noise Tracker) for noise measurement using smartphones. It displays noise levels in Leq sound pressure levels. I want to implement A & C weighting. As per my ...
1 vote
1 answer
2k views

How to calculate the spatial frequency of an image in cycles per image

I have been attempting to calculate the spatial frequency of a colour image. I first converted it to grayscale and used a Fast Fourier transformation (fft2) that codes the quantity of each spatial ...
2 votes
4 answers
669 views

For the square wave signal, why does a narrower square wave correspond to more spread in the frequency domain?

Consider the square wave. As we decrease the pulse width $T_1$, it seems that the positive Fourier coefficients (located in the central area/lobe) are spreaded out across more frequencies. In fact, if ...
3 votes
1 answer
221 views

What is the relation between the domain of Fractional Fourier transform and time and frequency?

This refers to Wikipedia's page on Fractional Fourier Transform. In the definition section it clearly writes that: The FrFT argument $u$ is neither a spatial one $x$ nor a frequency $\xi$. We will ...
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1 answer
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How do I properly compute a convolution kernel given the convolved and initial signal?

I'm trying to for solve for $ b $ in the equation $C = a*b$ where $C,a,b$ are 1D arrays of size $k$. $F(b)$ is also $= \frac{1}{|F(a)|}$. I have tried doing the following procedure in order to isolate ...
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1 answer
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Is the analytical fourier transform of an exponential decay the same as the FFT of a time series of the same signal?

Analytical Fourier Transform of $e^{-t/2}$: FFT of a time series representation of $e^{-t/2}$: I'm trying to find the decay coefficiencts in tri-eponential decay signals, and as a logical starting ...
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1 answer
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Fourier Transform of product of sinc function and its raised powers

Is it possible to find the fourier transform of $$\left(\frac{\sin(x)}{x}\right)^3$$ and $$\frac{\sin(x)}{x} \frac{\sin(x^2)}{x^2} $$
2 votes
1 answer
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How should I go about completely decorrelating a digital signal?

So I'm working on real time signal compression, and I need to come up with the best convolution to minimize the entropy of incoming data (which I will then compress), which I understand is achieved by ...
3 votes
2 answers
63 views

Border effects / artifacts for spectral filtering of images

I am working on a spectral filtering problem as follows: I'm given an image with a periodic distortion pattern (see below), and I want to remove the distortion pattern while preserving the main ...
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1 answer
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Problem observing the correctness of a Fourier transform property

I'm studying "Signals & Systems" by Oppenheim et al. On page 391, for a real-valued $x[n]$ whose Fourier transform is $X(e^{j\omega})$: $$x[n] \leftrightarrow X(e^{j\omega})$$ $$\...
1 vote
1 answer
52 views

Signal response amplitude depends on the time interval in simulation

I have already tried to look for an answer, but I do not find existing answers satisfactory. I am interested in the absolute value of the response function of a damped oscillator (or any time series). ...
0 votes
2 answers
94 views

How do you properly cut out negative frequencies from FFT of a real signal if it reduces sample size?

If a real time signal has $N$ samples, then the magnitude of its FFT will also have $N$ samples--half of which will have positive frequencies and half of which will have negative frequencies. The ...
1 vote
1 answer
90 views

How To Calculate Length Of Sequences And A Suitable N?

I have been struggling to understand how to calculate the length of a sequence and also the minimal N to choose in order to avoid aliasing. Most sources tell me to take the (last non-zero value - ...
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2 answers
1k views

How to understand FFT results of scipy.fftpack?

I calculated FFT for a speech wav-file using scipy.fftpack. How do I read (understand) the return of FFT? I have read that it supposed to be like so: y[0] is 0Hz loudness, y[1] is 1Hz loundess, ... y[...
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3 answers
115 views

Rigorous derivation of autocorrelation of white noise

It is said that the autocorrelation of white noise is the dirac delta function $\delta(\tau)$, but I don't know how to derive that... Since white noise is a function with constant power spectral ...
1 vote
1 answer
85 views

Noise color in multidimensional frequency space - is there an angular effect or dependence?

I understand that if there is a non-constant level of noise in one-dimensional frequency space, e.g. noise variance increasing linearly with frequency gives "blue noise" in the iFFT space. ...
1 vote
2 answers
728 views

Discreteness and periodicity in Fourier transform

Why discreteness in time / frequency domain dictates periodicity in the other frequency / time domian? For example the DTFT is perodic in frequency? Why it doesn't contain all the frequencies? Why ...
1 vote
1 answer
828 views

Why does the bandwidth of a signal need to be half of the sampling rate? [duplicate]

Suppose I perform a DFT on some function with sampling rate of $\frac{1}{\Delta t}$. According to this page, the bandwidth, which is the maximum frequency that can be analyzed when performed the DFT, ...
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1 answer
51 views

Looking for an online course

I work for a company with expertise in electromagnetics physics, specifically dealing with time domain-based simulation software. We routinely need to transform our results into frequency domain, ...
1 vote
4 answers
2k views

Negative frequency in the Fourier Transform [duplicate]

Why use this instead of a positive frequency? I asked on the math forum but it just wasn't helpful. I understand the idea of a negative frequency is important in general since many real signals like ...
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1 answer
47 views

Comparing Spectral Leakage in Sine and Cosine Functions

I am not from a signal processing background, kindly excuse me for my vocabulary. When I compute the power spectrum of a periodic function, where the "real" space variables are generated by <...
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3 answers
392 views

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 ...
2 votes
4 answers
2k views

Sawtooth wave Fourier coefficients

I have to calculate the Fourier coefficients of this signal. I found that signal equation is $$ y = \frac {A(2t-T)}{T} $$ To find Fourier coefficients I wrote $$ x_k = \frac{2A}{T} \int_{0}^{T/2} \...
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2 answers
2k views

Show Equivalence Between Multiplication in Time Domain to Convolution in Frequency Domain

My goal is to compute the Fourier of the product between two discrete time signals, y1 and y2. This can be done by computing the ...
1 vote
2 answers
145 views

Why i am not getting same convolution plots?

I am trying to implement convolution using fourier transform property but i am not getting exactly same shape of output that i get by using conv command First portion of graph obtained using conv ...
1 vote
2 answers
120 views

Why is white noise uncorrelated for any two different samples?

Consider white noise with spectrum density $N_0/2$, it is known that the autocorrelation is given by: $$R(\tau) = \frac{N_0}{2}\delta(\tau)$$ Meanwhile the definition of delta function in the ...
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1 answer
94 views

Decomposing a signal so that the magnitude spectrum of the decomposed signals lie in some specific amplitude range

Suppose I have the following signal: I can compute its power magnitude spectrum: PSD = np.abs(np.fft.rfft(signal)) Now I would like to pick a specific amplitude ...
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Can I get a resonance frequency from FFT power spectrum result?

I'm going through vibration analysis of cantilever by using FFT device but impact hammer input data isn't showing for some reason. So instead, I'm trying to get the resonance frequency by acceleration ...
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1 answer
85 views

realistic sampling - where am I wrong?

I’m given a signal $x(t)$, it's convolved with $h(t)$ and sampled at rate T=1. The result is called $\tilde{x}[n]$. For $$h(t) = \begin{cases} 1 & -0.5<t\le 0.5 \\ 0 & \text{else} \end{...
1 vote
1 answer
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transform time series different length and sampling rate with frequency domain

i want to transform my time series (each same scale) to frequency domain. There are 2 things to conside: some time series are longer then the other i have different measuring intervals which means e....
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1 answer
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FFT comparisons between g.USBamp software vs BCI2000 software

Update: I followed the instructions in g.USBamp Demo how to download the codes into Matlab (btw the product page is at https://www.gtec.at/product/gusbamp-research/ ) Here is the exact Matlab code ...
1 vote
2 answers
91 views

Channel impulse response (CIR) and OFDM relationship

In Chapter 3 of the [Goldsmith book] (https://faee.sut.ac.ir/Downloads/AcademicStaff/1/Courses/7/Andrea%20Goldsmith-Wireless%20Communications-Cambridge%20University%20Press%20(2005).pdf) we can see ...
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3 answers
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very basic confusion about the bandwidth of constant signal function

I have a very basic question about bandwidth of constant function $f(t) = K$. As the Fourier transform of a constant pulse is a delta-function centered at zero frequency, the bandwidth (maximum value ...
1 vote
2 answers
190 views

Taking an FFT of a signal that wraps around

I have an embedded application that does some phase modulation and streams back the phase offsets it's applying over UDP. I now want to take the FFT to see the spectrum of the applied modulation. ...
0 votes
1 answer
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Understanding symmetry in DFT magnitude plot

I am trying to intuitively understand the mystery of why the DFT of a complex vector with real values produces apparent symmetry in the magnitude plot (see the second plot here for example). In DFT $X[...
1 vote
1 answer
83 views

Decay of the impulse response for poles contained in the unit circle

I've been struggling with the following exercise in Ljung's "System Identification: Theory for the User" (Problem 3G.1): Given a rational transfer function $G(z)$ such that its poles are all ...
6 votes
1 answer
375 views

Effect of Passband Slope on QPSK Constellation

What is the effect of a passband slope on a QPSK (or any QAM) constellation? Specifically given a linear slope (in magnitude, not dB), and no phase distortion (linear phase or even zero-phase) how ...
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Fourier Transform and Audio Processing

Most audio capturing software automatically distills the audio into distinct frequencies with Fourier transform. This makes audio editing much easier. I have a project where I want to show the ...
12 votes
5 answers
2k views

Inconsistency between the units of power spectral density and the definition that people often give

Perhaps someone can help me resolve something - this is my understanding: In deterministic signal analysis, for a continuous signal $x(t)$ the signal energy is defined by $$E_{\textrm{s}} = \int^{+\...
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2 answers
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Fourier series - finding its period from frequency representations

I’ve been given the following signal: $$X^F(\omega) = \sum_{n=-\infty}^{\infty} 2\pi a[n] \delta(\omega - \omega_0 n)$$ and I was asked to: find it’s period given $|X^F(\omega)| \ne 0$ only at $|\...

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