Questions tagged [fourier-series]

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12
votes
2answers
6k 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 ...
12
votes
1answer
2k 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.
10
votes
3answers
2k 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 ...
6
votes
2answers
2k 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 ...
4
votes
4answers
716 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 ...
4
votes
2answers
2k 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 ...
4
votes
4answers
1k 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 ...
4
votes
1answer
4k 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 ...
4
votes
1answer
241 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 $T=4$, ...
4
votes
1answer
1k views

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 ...
4
votes
2answers
764 views

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 ...
4
votes
3answers
6k views

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 ...
4
votes
1answer
167 views

Signal Processing using Fourier Transform

How can I derive the fourier transform of ...
3
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2answers
558 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 ...
3
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2answers
6k 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 ...
3
votes
2answers
341 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 ...
3
votes
1answer
238 views

DSP interview question: use of the identity in development of a significant transform

I'm preparing interview and found this question. But I don't really understand what is the question. Does it ask about Fourier transform or Z transform? How the simple identity $$xy=\frac{1}{2}x^2 + ...
3
votes
2answers
98 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 ...
3
votes
2answers
702 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 : ...
3
votes
1answer
370 views

Estimate the Discrete Fourier 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 Series. Now, assume I'd like to estimate its Discrete Fourier Series ...
3
votes
1answer
828 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 ...
3
votes
1answer
53 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}...
3
votes
3answers
11k views

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 ...
3
votes
2answers
2k 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 ...
2
votes
2answers
118 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 ...
2
votes
6answers
3k 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.
2
votes
4answers
88 views

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 ...
2
votes
2answers
1k 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?
2
votes
1answer
320 views

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 ...
2
votes
2answers
362 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. ...
2
votes
1answer
118 views

Does the Fourier series coefficient of AC components remains same if DC component is subtracted form the given signal?

Suppose a signal is defined by $ x(t)= \begin{cases} t & 0\leq t \leq 1 \\ 2-t & 1\leq t\leq 2 \\ \end{cases} $ Since $x(t)$ has even symmetry, I can calculate fourier coefficient as $$ a_n = ...
2
votes
1answer
156 views

Mathematical model of a signal in Compressive Sampling

Currently I am reading a paper on Compressive Sampling, and trying to understand each and every parts of it. When I came across the mathematical model of the signal in paper, I got confused. I have ...
2
votes
2answers
212 views

Trignometric Fourier series representation of a continous time signal

While learning Fourier series I read the definitions of representation for a continuous time signal $x(t)$ as: $$x(t)=A_0 + 2 \sum_{k=1}^{\infty} A_k \cos(k \omega_0 t) - B_k \sin(k \omega_0 t) \tag{...
2
votes
2answers
116 views

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 ...
2
votes
1answer
567 views

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 ...
2
votes
1answer
642 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 ...
2
votes
1answer
504 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) $\...
2
votes
1answer
56 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 ...
2
votes
1answer
232 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)\...
2
votes
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 ...
2
votes
1answer
565 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: $$ \int_{-\infty}^t x(\alpha)d\alpha \leftrightarrow \frac{a_k}{jk\omega_0} $$ ...
2
votes
3answers
289 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 ...
2
votes
0answers
46 views

Fourier series representation of a full wave rectifier output

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 ...
2
votes
1answer
1k views

FFT has unexpected DC component

I have a mixture of Gaussians and I want to look at their power power spectrum. The spatial distribution looks like this: It's already been convolved with a Gaussian window function. I subtract the ...
1
vote
2answers
45 views

inner product zero?

I am studying about Fourier series from book"Signals and Systems Laboratory with MATLAB" I came across topic "Orthogonality of Complex Exponential Signals" I am confused in case when m=k, will the ...
1
vote
2answers
4k views

Fourier series - time and frequency domain confusion

I am computing the fourier series of the following function between $[-0.5, 0.5]$ $$\displaystyle f(t) = \frac{1}{2} - |t|$$ According to the definition of Fourier Series the coefficients are given by ...
1
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4answers
995 views

Formulas of the Fourier transform family

It has annoyed me that there doesn't seem to be a source online where the complete complex Fourier transform family is presented with every variable defined. The lack of definitions can be a nuisance ...
1
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2answers
116 views

why fourier series gives lower amplitude for max value of signal

I want to approximate below signal using fourier series on Matlab. My code is below ...
1
vote
3answers
338 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 ...
1
vote
1answer
82 views

fftshift in MATLAB with even number of data points in double sided spectrum

I have a question with reference to this Table. With even N, the frequency axis extremes should be $\pm$Fs/2, where Fs is the sampling frequency. However in the array we have only one value ...