# 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 ...
8k views

### 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 ...
11k views

### The difference between DFT and DFS

In the literature, I've found that DFS and DFT are one and the same. If they are one and the same why to use two different names for them? If there is really a difference what is it and what is the ...
4k views

### 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 ...
3k views

### 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.
19k views

### 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 ...
830 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 $2$...
28k views

### 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 ...
806 views

### 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 ...
6k views

### 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 ...
2k views

### 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 ...
4k views

### 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 ...
3k 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? 11k views

### 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 ...
9k views

### 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 ...
178 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 ...
277 views

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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 ...
137 views

### 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 ...
6k views

### When is the Fourier transform of a signal periodic?

I mean not the time-domain signal being periodic, but the Fourier transform being periodic.
2k views

### 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 ...
2k views

### 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 ...
733 views

### 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. ...
378 views

### 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 ...
3k views

### Periodicity of Fourier series's coefficients

Why exactly continious Fourier-series's coefficients aren't periodic like coefficients of discrete Fourier series (DFS)? $e^{-j2\pi}$ is periodic in both sequences.
283 views

### Why do waveforms that are symmetrical above and below their horizontal centerlines contain no even-numbered harmonics?

All About Circuits site states that waveforms that are symmetrical above and below their horizontal centerlines contain no even-numbered harmonics. Can somebody explain this mathematically, or point ...
6k views

### 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 ...
348 views

### Fourier Series Coefficients

Question: The fourier series coefficients is given as: $$c_k= \begin{cases} 1 \qquad & k \ \text{ even} \\ 2 \qquad & k \ \text{ odd} \\ \end{cases}$$ the period of the signal is $T=4$, ...
1k views

### FFT of SIN waves with different phase delays

I have come across a peculiarity of FFTs which has got me somewhat baffled. I've simply summed up 101 sine waves and taken the FFT using this matlab script : ...
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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 ...
85 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 ...
111 views

### Fourier coefficients of 1/(1+it)

I have to find the Fourier coefficients of $$\frac{1}{1+ t^{2}}$$ I tried with $$\frac{1}{T}\int_{0}^{T} \frac{1}{1+it}e^{-i 2 \pi f_0 T }$$ but I should do at least two integrals by parts , so I ...
4k views

### 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 ... 2k views

### A clarinet has no even harmonics. What would produce no odd harmonics?

According to this link, the waveforms of clarinets do not have even-numbered components in their harmonic series: A closed cylindrical air column will produce resonant standing waves at a ...
2k views

### What is the physics behind the width of a main lobe?

We know that the square window gives the lowest main lobe width possible, and that other windows after that trade main lobe width for side lobe height. I also understand that the main lobe width is ...
67 views

### Number of zeros of a sum of Shah functions by applying Rice's formula?

There is a Dirac pulse train following the scheme of the Shah function (or $\delta$-cumb function) with its Fourier series of the form: \varsigma(t,T)=\sum_{n=-\infty}^{\infty}\delta (t-nT)=\frac{1}...
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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 ...
76 views

### 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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### Improving the intuition for the 2d fourier transform

As far as I understand, the 2d fourier transform is calculated as following: ...