Questions tagged [fourier-series]

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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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1 answer
61 views

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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2 answers
31 views

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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1 answer
51 views

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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1 answer
69 views

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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2 answers
130 views

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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What is the trigonometric form of the discrete-time Fourier series or inverse discrete Fourier transform?

As we know, the continuous-time Fourier series (CTFS or just FS) has three forms: the trigonometric, the amplitude-phase or compact trigonometric, and the complex exponential. I've found formulas for ...
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2 answers
282 views

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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4 answers
67 views

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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1 answer
75 views

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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2 votes
1 answer
122 views

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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1 vote
1 answer
203 views

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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1 answer
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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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1 answer
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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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1 answer
114 views

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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1 answer
141 views

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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2 votes
2 answers
112 views

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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2 answers
893 views

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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-1 votes
1 answer
91 views

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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1 vote
0 answers
123 views

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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1 answer
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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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1 vote
2 answers
130 views

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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1 answer
23 views

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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1 vote
1 answer
151 views

Why Does ASK Modulation Create Fourier Sidebands?

I know why analog amplitude modulation has side bands, it is related to (fc+fd) and (fc-fd). But what about DAM? ASK(DAM) is a type of digital modulation, and there are only two state: carrier signal ...
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0 votes
1 answer
57 views

How to find inverse Fourier transform of summ of delta functions?

I am practicing for my exam that I have this semester and I stumbled upon this one. How can i find inverse Fourier transform given: $$ X(j\omega) = \sum_{k=-\infty}^{\infty}\delta(\omega-2k+1) $$
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1 vote
1 answer
36 views

Finding the discrete time Fourier series for signal

I think I did everything correctly here, but I must be missing something still. We have the following signal: My approach: We are told that the signal has period $N = 4$ We know $$Y[k] = \frac{1}{N}\...
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1 answer
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Solution verification for this Fourier series problem

We have a signal with period $T = 2$ We want to find the continuous time fourier series for this signal. Since $T = 2$, $\omega = \pi$. All we have to do know is find the frequency domain. $$x(t) = \...
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9 votes
2 answers
665 views

Fourier series of cycloid

What is the Fourier series representation of a cycloid? The parametric representation of the curve is as follows. $$ t=\dfrac{\theta-\sin\theta}{\pi}\\ x=\dfrac{1-\cos\theta}{\pi} $$ The period is $2$...
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1 answer
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Finding the discrete time fourier coefficients to this problem

I'm trying to find the fourier series to this discrete time signal. $$x_1[n] =\begin{cases} +\frac72&\text{if }0\le n \le 4\\ -\frac72&\text{if }5\le n \le 9 \end{cases}$$ My approach: We see ...
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1 answer
65 views

Represent DFT coefficients with respect to Continuous time-Fourier series coefficients

Does anyone know how to represent the Discrete Fourier transform (DFT) coefficient, $X[k]$, with respect to the Continuous time-Fourier series (CT-FT) coefficient, $X_k$? I come to the conclusion as $...
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3 votes
1 answer
80 views

The magnitude spectrum of a sharpening filter

I'm trying to derive an expression for the magnitude spectrum of the following sharpening filter. $$ g(m,n) = \delta(m,n)+\lambda (\delta(m,n) - h(m,n)) $$ where $\lambda$ is some positive constant ...
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1 answer
71 views

Why is the Fourier Series a special case of the Fourier Transform and not the other way around?

I was reading a text book on the frequency domains in signal processing and my understanding is that the Fourier Transform considers signals that are a-periodic in time while the Fourier Series ...
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1 vote
1 answer
76 views

MIT 6.003 HW#8 Problem 4 - Fourier Coefficients of Triangle Wave

In the mentioned homework, part of the solution involves finding the Fourier coefficients of the triangle wave. The solution mentions that we can express this function as follows: What does that ...
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1 vote
2 answers
406 views

Problems computing the DFT of finite length sequence

I am having trouble finding the same answer as the solution manual for this sequence. The problem asks to compute the DFT of $$ x[n] = \begin{cases} 1 & \text{for even } n \in \{0\...
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1 answer
53 views

multiplication of a function with a Fourier-transformed equals to Fourier-transformed with a function

I already showed b item using the fact that it is $h\left(0\right)=\int \:f\left(t\right)g\left(0-t\right)dt$ I struggle a lot of hours trying to find the trick in item C. Can anyone help please ?
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2 votes
1 answer
69 views

Fourier coefficients of discrete difference of a square wave

I have a discrete square wave $f(t)$ where $t \in \mathbb{N}$, of amplitude $A$, period $T$ and duty cycle $1/T$ $$ f(t) = \left\{\begin{matrix} A, & \mathrm{if}\;t=Tn\\ 0, & \mathrm{if}\;t\...
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1 vote
0 answers
80 views

Forecasting that FFT result

I have the following FFT result charts. Searching for a way to forecast the future AMPLITUDE steps from the Time Domain Plot if possible, or the next maximum/minimum deviation based on some previous ...
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0 votes
1 answer
226 views

If a time-series has odd number of samples does it have no energy at Nyquist frequency?

Suppose I have real time series A with n samples and time-spacing dt and I want to analyze ...
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-1 votes
1 answer
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Output of LTI (in time and frequency $\omega$ domain) : when input goes through LPF

I would like to raise a mathematical question : Let's say we are been given : $$x(t) = \begin{cases} \cos(\pi t) & |t| \leq 0.5 \\ 0 & \textrm{otherwise} ...
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0 answers
39 views

How to get coefficient of the discrete fourier Series from the fourier transform

Given $X(v)$ the Discrete Fourier transform of a discrete periodic signal $x(n)$, it's possible to arrive to the $c_k$ of the Fourier series $$x(n)=\sum_{k=0}^{n-1} c_k \exp(2\pi i k t) $$ directly?
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1 vote
3 answers
67 views

Sum of equidistant exponents

Consider the next sum \begin{equation} \sum_{k = 0}^{N - 1}e^{-j\frac{2\pi}{N}k} \end{equation} Its geometric meaning is the sum of uniformly distributed vectors on the unit circle. Thus, we can say ...
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1 vote
0 answers
22 views

Evaluate a complex exponential at negative infinity [duplicate]

I am learning about the properties of the Fourier Series (FS), which is defined by: $$x(t) = \sum_{k=-\infty}^{\infty}c_{x}[k]e^{j2\pi kt/T}\tag{1}$$ where $$c_{x}[k] = \frac{1}{T}\int_{T}x(t)e^{-j2\...
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