# Questions tagged [fourier-series]

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### What is the most lucid, intuitive explanation for the various FTs - CFT, DFT, DTFT and the Fourier Series?

Even after having studied these for quite sometime, I tend to forget (if I'm out of touch for a while) how they are related to each other and what each stands for (since they have such similar ...
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### A good mathematical explanation of Gibbs phenomenon

I was explaining to someone how Fourier series work in context of constructing signals that are not everywhere differentiable, e.g. square waves, sawtooth waves, etc. When I mentioned the Gibbs ...
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### Graphical fourier series of a square wave

This is probably off-topic since it isn't really a question, but I thought that this GIF of the fourier series of a square wave was too cool not to share.
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### Slow Down Music Playing While Maintaining Frequency

Playing a piece of music audio at a slower speed would lower its pitch (frequency). Is there a tool and theory to slow down the song playing while keep the frequency the same? I suppose one can do ...
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### Intuition behind the scaling property of Fourier Transforms

The Fourier transform of $f(ax)$ is $\frac{1}{|a|}F(\frac{u}{|a|})$. So the frequencies are scaled horizontally but the magnitudes are also scaled when the graph of $f$ is scaled horizontally. On the ...
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### Scaling property of Fourier Transform

Problem 4.6(b) from Oppenheim, Wilsky & Nawab (2nd ed) reads: Given that $x(t)$ has the Fourier transform $X(j\omega)$, express the Fourier transform of $x(3t - 6)$ in terms of $X(j\omega)$. The ...
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### How to get Fourier coefficients to draw any shape using DFT?

I'm teaching myself about Fourier Series and the DFT and trying to draw a stylised $\pi$ symbol by fourier epicycles as detailed by Mathologer on youtube (from 18:39 onwards), and the excellent ...
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### Is Fourier series a sampled version of Fourier transform?

I recently learned about dtft and how dft/dfs is the sampled version of dtft. I was wondering if Fourier series is also obtainable by sampling Fourier transform? I am a noob in the subject so sorry if ...
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### 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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### About Fourier transform of periodic signal

In Fourier transform for periodic signal, I checked different books and I found a different explanation in each book. Let's take the explanation in Signals and Systems by Rajeshwari & Rao: The ...
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### Derive Frequency Representation of Impulse Train Function

I want to walk through the derivation of the frequency representation of an impulse train. The definition of the impulse train function with period $T$ and the frequency representation with sampling ...
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### Fourier series - time shift and scaling

What will be the new Fourier series coefficients when we shift and scale a periodic signal? Scaling alone will only affect fundamental frequency. But how to calculate new coefficients of shifted and ...
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### Gibbs phenomenon in Hamming's digital filters

In 'Digital Filters' by Hamming there is a cryptic section where he describes how the Gibbs phenomenon can be viewed as the displacement between the centers of two functions as they are convolved ...
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### 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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### Estimate the Discrete Fourier Transform / Series of a Signal with Missing Samples

Assuming we have a discrete signal ${ \left\{ x \left[ n \right] \right\}}_{n = 1}^{N}$. Which has a Discrete Fourier Transform / Series. Now, assume I'd like to estimate its Discrete Fourier ...
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### How to Remove the Periodic Oscillations from a Signal

The task that I have is to remove the annual and semiannual oscillation from a set of temperature measurements, taken over several years, by means of least squares method. I found the method ...
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### Signal Processing using Fourier Transform

How can I derive the fourier transform of ...
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### What to do after this last step?

I am solving a question from book in which I have to use summation. It is as follows: $$\frac{1}{10}\sum_{n=0}^{9} e^{-jk\omega_0n}$$ The value of $\omega_0$ is $\frac{2\pi}{10}$. What I ...
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### Convolution in frequency domain

Simple math question. The convolution theorem states that multiplication in time domain is equal to convolution in frequency domain and vice versa. There is a condition that the signal has to be ...
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### Fourier series expansion of $\exp[-j2kA\sin(\omega t)]$

I've been going through this paper where we exploit the well known Fourier expansion of the model signal as shown in the image below. I've never come across this well known fourier expansion before. ...
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### Given the Graph of a Fourier Series $\sum c_k e^{2\pi ikx}$ Find the Graphs of $\sum c_{3k} e^{2\pi ikx}$ and $\sum (c_k)^2 e^{2\pi ikx}$

Define a 1-periodic function on $\mathbb{R}$ by: $f(x) :=$ $\left\{\begin{matrix} 1 & if & 0<x<\frac{1}{10}\\ 0 & if & \frac{1}{10}<x<1 \end{matrix}\right.$ with ...
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### The Fourier Series Of This Triangle Wave

I am using matlab to study digital signalling and have come across a problem which i was wondering if anyone with more experience could help me with. I need to work derive the Fourier series of a ...
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### Finding the fundamental frequency of a periodic signal

Suppose we have the signal $$x(t) = e^{j\omega_1 t} + e^{j\omega_2 t} + e^{j\omega_3 t},$$ where all the frequencies are rationally related (that is, the ratio of any pair of frequencies is a rational ...
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### When is the Fourier transform of a signal periodic?

I mean not the time-domain signal being periodic, but the Fourier transform being periodic.
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### What is the moment when all oscillators aligned to make a jump called?

Say we have a square wave and its Fourier series. When the wave doesn't jump, its oscillators aren't aligned: But if they are aligned, the wave will jump: What is this moment called? They might not ...
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### Benefit to know Fourier series for image processing? [closed]

I know there's a benefit of knowing the Fourier Transform for image processing, but is there a benefit to know Fourier series, or could you just skip them? Would you recommend skipping Fourier series ...
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### integration property of fourier series

Please help me sort this issue out. The integration property in Fourier series is as follows: So, for proving the above property, i took this approach: This is where my doubt is. Some books and ...
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### Why can't we just make all wireless networks use integer multiples of base frequency?

I always wondered why transmission capacity depends on bandwidth. For example, let us say that there is an isolated island. In this island, people decide that all wireless networks use frequencies ...
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### Gain function calculation (frequency response)

Define moving average process $y_t := 0.5 x_t + 0.5 x_{t-1}$ where $x_t := e^{i2 \pi t}$. Its frequency response is then: $$H(f) = 0.5 + 0.5 e^{-i2\pi f}$$ Recall that the frequency response in ...
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### Spectrum of Cosine in Complex Form

The complex exponential form of cosine $$\cos(k \omega t) = \tfrac{1}{2} e^{i k \omega t} + \tfrac{1}{2} e^{-i k \omega t}$$ The trigonometric spectrum of $\cos(k \omega t)$ is single amplitude of ...
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I have to find the expression of this graphic and after find the inverse Fourier transform of it. First of all I found that the expression of the graphic is $$X(f) = \frac{1}{2} tri (\frac{f+f_0}{B}) ... 1answer 407 views ### Fourier coefficients of sum of two functions with different fundamental periods? If we assume \quad x(t)\leftrightarrow a_k\: and it is periodic with fundamental period T. How can we determine the fourier coefficients of the sum x(t-7)+x(-2t+3) I know that x(t-7)\... 1answer 737 views ### Fourier coefficients of product of two periodic signals question: If x(t) and y(t) are two periodic signals(both with period T) with Fourier coefficients c_{n} and d_{n} respectively then, Fourier coefficient of z(t)=x(t)\cdot y(t) is: (a) \... 1answer 61 views ### Rationally related frequencies and the Fourier Series representation Suppose that we have the signal$$x(t) = e^{j\omega t} + e^{j\frac{3}{2} \omega t},$$and we want to find a Fourier Series representation for that signal. Is this possible? According to my ... 1answer 2k views ### How to do simple extrapolation with Fourier transformation? I have 1024 sample points, and I would like to do really simple extrapolation using Fourier transformation. First I apply Fast fourier transformation on the data. My first intuition was that I just ... 1answer 656 views ### What is the time-integration property in the Fourier series analysis? In the continuous Fourier series properties for a periodic continuous-time signal, we have time-integration property:$$ \int_{-\infty}^t x(\alpha)d\alpha \leftrightarrow \frac{a_k}{jk\omega_0} $$... 3answers 297 views ### Can I study continuous time Fourier Transform and treat the rest as special cases Say I learned the theoretical result of continuous time Fourier transform. And I want to extends some results(say "convolution rule") to Lapace transform, Z transform, DTFT, DFT, Fourier sequence ... 1answer 837 views ### Discrete Fourier Transform and Opposite Convolution Theorem I am reading the Wiki for DFT. There is a part for circular convolution theorem which sounds a bit odd saying:$$ \mathcal{F} \left \{ \mathbf{x\cdot y} \right \}_k \ \stackrel{\mathrm{def}}{=} \sum_{...
I am trying to compute the Fourier series representation of a full wave rectifier output. The equation of the signal is: $x_8(t) = | \cos (2 \pi f_o t)$ | I have tried to find the Fourier series ...