Questions tagged [fourier-series]
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Advantages of the Karplus algorithm for sound synthesis
For a personal project I am trying to make a software to synthesize plucked and hammered string instruments. This is a whole research topic and there are already some models and algorithms such as the ...
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Fourier transform of a time discrete signal
I would like some help to better understand the Fourier transform of a discrete time signal. My doubts are:
The sampling of a signal can be seen as $x_s(t)=x(t) \cdot
\sum_{k=-\infty}^{+\infty} \...
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positivity of the spectrum of quasi-stationary signals
I am working on the "System identification : theory for the user" by Lennart Ljung (freely available here) and it is one of these books which contains exercises but no answers... My exercise ...
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Creating infinite continuous time series out of a finite discrete spectrum
I have a power spectrum $S(\omega)$: a $1 \times N$ real vector that matches frequencies from 0 to 125Hz. I would like to create a time series $S(t)$ $[-\infty<t<\infty]$ that matches this ...
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Trying to understand how to get this basic Fourier Series
I'm sorry if this kind of question isn't allowed, but I'm starting to learn Fourier series and I'm still not entirely sure what's going on... in this specific case, I'm trying to find the Continuous ...
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Can we control the maximum norm of a continuous signal whose finitely many Fourier coefficients are fixed?
Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous signals $x:\mathbb{R}\to \mathbb{R}$.
Fix $n\in \mathbb{N}$ and put
$$\Lambda_n=\{y\in C_{2\pi}: \mathcal{F}(y)[k]=0 ~\...
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Discrete Fourier series of an odd signal
Assuming the signal shown below :
I have found an expression for fourier series coeffecients as the following:
$$a_{k} = \frac{1}{5}+\frac{j}{5}\sin{\frac{2\pi}{5}k}$$
Which matches with what the ...
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Is my solution correct?
$\textbf{Question:}$
$y_a(t)$ is a rectangular waveform defined as:
$$\
y_a(t) =
\begin{cases}
2 &t \in [0,1/25)s\...
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Finding $A_k$ coefficients
I was able to demonstrate that for a signal $x(t)$ real, we can write the truncated Fourier series as:
$x_N (t) = A_0 +\sum\limits_{k=1}^{N}A_k\cos(kω_0t + \varphi_k)$, but now I've been given the ...
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Fourier coefficients of two discrete-time signals of different periods
I'm trying to understand the Fourier series coefficients of the sum of two discrete-time periodic signals.
Consider two discrete-time periodic signals $x[n]$ and $y[n]$. $x[n]$ has period $N$, its ...
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Finding the Fourier Coefficients
Up until now, I have dealt with finding Fourier Coefficients for functions: $f(t) > 0$
Which made it convenient calculating the Fourier Analysis Integral. However, I am now presented with ...
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What happens to sidebands when they enter "negative" frequencies?
I am working with PWM signals. These signals are generated by comparing a modulating (at frequency $f_m$), and a carrier (at frequency $f_c$), as shown in the following image:
In the resulting ...
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What is $z[n]$ , when $Z[k]=X[2k]H[2k]$ with $y[n] = x[n]∗h[n]$ and $Z[k] = X(\omega)H(\omega)$ at $\omega = \frac{4πkn}{N}$
How do I express $z[n]$ in terms of $y[n]$, with:
$$y[n] = x[n]∗h[n]$$
$x[n]$ and $h[n]$ being 16 length sequences.
$X(\omega)$ and $H(\omega)$ are DTFT of $x[n]$ and $h[n]$
$Z[k]$ is defined as
$$Z[k]...
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Bandwidth of cosine of bandlimited signal
I have a signal $x(t)$ with bandwidth $B_x$, and I am taking its cosine to create $y(t) = cos(x(t))$. After checking the spectrum with FFT, it seems that $y(t)$ is also bandlimited. But, is there a ...
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How to generate a sound closer to a saxophone using sinusoids after Fourier Transform?
Generate a sound wave of saxophone frequency and compare it to the original sound clip and play both to listen to if there is a good match.
I am trying to generate a sound closer to a saxophone using ...
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How to reconstruct original signal using IFFT after cutting past Nyquist limit
I'm working on a pitch shifting program. Everything works up to the point where I try to do the IDFT. Because I cut the DFT array past the Nyquist limit, when I run the IDFT, I don't get the same ...
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Pitch successfully changes with Phase Vocoder, but there's an issue
I've been working on a phase vocoder program. The goal is to change the pitch of a recording of my voice. While doing research on how to change pitch, I came across this from a paper on phase vocoders ...
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How to change fundamental frequency with DFT?
I'm working on a voice changer. My plan is to make it so that it can change your voice in various different ways, but right now I'm just trying to make it change your voice to "chipmunk voice&...
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How to Find "pitch" from Fourier Series
The end goal of my project is to create an autotune program, But the problem I'm trying to solve right now concerns finding the pitch of someone singing a note. I have written some code that performs ...
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Fourier transform of periodic functions
The Fourier transform is derived from the Fourier series by considering a non-periodic signal, thinking of it as a infinitely long periodic signal, putting it into the Fourier series and making this ...
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Improving the intuition for the 2d fourier transform
As far as I understand, the 2d fourier transform is calculated as following:
...
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Can I reduce the complexity of multiplication with FFT if the input vector is repeating?
I have a Fourier matrix $F$ with size $N \times N$, such that $y = F \times x$, if I have the vector $x$ contains four identical parts, for example $x = [x_1, x_2,x_3,x_4]’$ and $x_1 = x_2 = x_3 = ...
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Sampling of the DTFT causes the inverse transform to become periodic?
As you can see the above equation, DTFT is calculated from sample x[n] which is discrete sample of x(t).
But calculated X(w) is continuous, even though it is calculated from discrete value of x[n] as ...
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How do I plot a phase spectrum of rectangular pulse with Matlab?
As you can see, I made a code about rectangular pulse like this. And, I plotted complex-exponential-coefficient and Magnitude-Spectrum-of-Complex-Exponential-Form, Phase-Spectrum of-Complex-...
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What's the difference between male and female voice? [duplicate]
If I record the voice of a man and a woman, what are the main differences I get in the various spectra and harmonics in Fourier analysis?
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Confusion understanding Fourier series line spectra?
I am reading book signals and systems Laboratory with MATLAB where I am studing chapter 5, Fourier series and i trying to understand magnitude spectrum and phase spectrum but i have certain confuisons ...
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What is the Fourier convolution theorem range of application (example of Dirac comb times rectangular window)?
$\DeclareMathOperator{\sinc}{sinc}$
I have questions regarding the Fourier transform of the product of functions or distributions and the range of application of the convolution theorem.
Context
When ...
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What is the reason of existence of Fourier transform? (Why we use Fourier transform?)
I'm currently trying to understand Fourier transform and I've got curious about why Fourier transform exists.
Let's suppose that we have a 10 seconds of non-periodic wave. For example:
As far as I ...
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Proving real and odd function has imaginary and odd Fourier Transform
Cheers, I am trying to prove that a real and odd function/signal has imaginary and odd Fourier Transform. Although it seems fairly easy, I can't find a way to achieve it, and searching online hasn't ...
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Are there Fourier methods for VERY low frequency repeating signals?
Suppose I have a very low frequency pattern of sound. For example a 10 second music file. Then 10 seconds of silence. Then the same 10 second music file repeated again. The whole sequence repeats ...
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How to apply Butterworth high pass filter in the frequency domain?
I have a time series of measurements which I want to high pass with Butterworth filter.
Python scipy package has a built in function for Butterworth filter (signal.butter) and I know how to apply it ...
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Fourier Series of a piecewise function
I've been given the task to find the Fourier Series Representation. All I'm given is this $$x(t)= \begin{cases}-t & \text { for } 0 \leq t<1 \\ 1 & \text { for } 1 \leq t<2 \\ 0 & \...
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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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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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Approximation of Periodic Parabolic Function by Fourier Series!
I've just tried to approximate the periodic-parabolic signal by Fourier Series. I know, this sounds a bit strange. I am just trying to figure out relationship between Fourier Series and Taylor ...
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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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Prove Discrete Time Fourier Series Multiplication property
Note: This is not a homework problem. I'm just stalled at a point because I think I might be interpreting the duality property incorrectly.
If $x_1[n]$ and $x_2[n]$ are periodic with period N, then if ...
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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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On discrete fourier coefficient convolution of indivdual periodic signals with different frequencies
It is a well-known result that when two signals with the same period $T$ get multiplied in the time domain, the resulting signal's Fourier coefficients are given by the discrete convolution of ...
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How is this given impulse response of infinite duration? Isn't it just from -π to +π?
How is hd(n) infinite duration when it is from -π to +π. Book says as it is infinite duration, we in the next step take-:
h[n]=hd[n] from n=-(N-1)/2 to (N-1)/2 and 0 otherwise.
I can't see how this ...
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Compare two Fourier series to depict the signal smoothness
I have several signals, that I am trying to find a metric to compare the signal smoothness.
By signal smoothness I mean, the signal that the distance between the peak to trough become smaller (getting ...
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Why do we use the DFT instead of the DTFS? Or, why was the FFT algorithm built for the DFT instead of the DTFS?
As we know, both the DTFS (discrete-time Fourier series) and the DFT (discrete Fourier transform) are used to represent discrete-time periodic signals for all time (or the periodic extension of ...
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Meaning of Rect and Train of Rect Spectra
The Fourier transform of $x(t)=\operatorname{rect}(t)$ is $X(f)=\operatorname{sinc}(f)$
The Fourier transform of a periodic train of rectangular pulses $x(t)=\sum\limits_{n=-\infty}^{\infty}\...
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FFT of square wave - what does output represent?
I am really new to FFT and signal processing. I am doing an analysis of square waves with FFT and I am trying to understand why the FFT output on the frequency domain has a downward slope for square ...
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transform signal
Hello everyone I need help solving a Fourier transform for the given signal, I know it will be a frequency convolution for the first function it will be a window function and for the second function I ...
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Finding original signal $x(t)$
For given 4 conditions, I have to find out what is $x(t)$ with period of 3, and I don't know if $x(t)$ is real or not.
For fourier coefficients $x_k$,
$$1.\ x_k=x_{k+2}$$
$$2.\ x_k=x_{-k}$$
$$3.\ \...
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Unable to understand the time-shifting property of CTFS
The CTFS of $x(t)$ is $c_{k}$ the Fourier series coefficients. Furthermore, $x(t-t_{0})$ is known to be $e^{-j\omega t_{0}}c_{k}$, the proof is given as follow :
$$
\begin{aligned}
\mathscr{F}\left(f\...
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How to visualize this statement regarding Conjugate Symmetry
A property of real signals states that if $x(t)$ is real then the Fourier series coefficient (frequency spectrum) is given by :
$$
c_{k}=c_{-k}^{*}
$$
In polar form, this can be expressed as :
$$
|c_{...
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Square pulse test of Upwind Finite Differences
I`m analyzing the numerical methods for the 1D convection equation for stability, consistency, and accuracy. I want to see How does this square pulse move in domain and time?
Here is my code
% ...
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Indetermination in a complex fourier series
I determined the complex Fourier series of a sinusoidal signal and arrived at the following expression:
$$\sum_{n=\infty}^{\infty} \left[\frac{4e^{-j \frac{\pi}{2}n}}{\pi(1-n^2)}(e^{-j\pi n}+1)\right]...
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