Questions tagged [fourier-series]
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189
questions
13
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 ...
7
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 ...
5
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 ...
5
votes
4answers
3k 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
4answers
865 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
1answer
5k 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
2answers
7k 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 ...
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
837 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
1answer
433 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 ...
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
168 views
3
votes
2answers
679 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
votes
2answers
343 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
239 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
$$xy=\frac{1}{2}x^2 + ...
3
votes
2answers
113 views
Why do waveforms that are symmetrical above and below their horizontal centerlines contain no even-numbered harmonics?
All About Circuits site states that waveforms that are symmetrical above and below their horizontal centerlines contain no even-numbered harmonics. Can somebody explain this mathematically, or point ...
3
votes
1answer
246 views
Fourier Series Coefficients
Question:
The fourier series coefficients is given as:
$$c_k= \begin{cases}
1 \qquad & k \ \text{ even} \\
2 \qquad & k \ \text{ odd} \\
\end{cases}$$
the period of the signal is $T=4$, ...
3
votes
2answers
760 views
FFT of SIN waves with different phase delays
I have come across a peculiarity of FFTs which has got me somewhat baffled.
I've simply summed up 101 sine waves and taken the FFT using this matlab script :
...
3
votes
1answer
77 views
Fourier coefficients of 1/(1+it)
I have to find the Fourier coefficients of $$ \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 } $$ but I should do at least two integrals by parts , so I ...
3
votes
1answer
888 views
What is the physics behind the width of a main lobe?
We know that the square window gives the lowest main lobe width possible, and that other windows after that trade main lobe width for side lobe height. I also understand that the main lobe width is ...
3
votes
1answer
56 views
Number of zeros of a sum of Shah functions by applying Rice's formula?
There is a Dirac pulse train following the scheme of the Shah function (or $\delta$-cumb function) with its Fourier series of the form:
$$\varsigma(t,T)=\sum_{n=-\infty}^{\infty}\delta (t-nT)=\frac{1}...
3
votes
3answers
12k 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
3k 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
122 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
4k 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
90 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
2k 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
331 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
4answers
11k 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 ...
2
votes
2answers
406 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
156 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
$$
a_n = ...
2
votes
1answer
157 views
Mathematical model of a signal in Compressive Sampling
Currently I am reading a paper on Compressive Sampling, and trying to understand each and every parts of it. When I came across the mathematical model of the signal in paper, I got confused. I have ...
2
votes
2answers
219 views
Trignometric Fourier series representation of a continous time signal
While learning Fourier series I read the definitions of representation for a continuous time signal $x(t)$ as:
$$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
120 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
580 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
1k 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
2answers
70 views
Inverse Fourier transform Of a triangular impulse
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 $$ X(f) = \frac{1}{2} tri (\frac{f+f_0}{B}) ...
2
votes
1answer
310 views
Fourier coefficients of sum of two functions with different fundamental periods?
If we assume $\quad x(t)\leftrightarrow a_k\:$ and it is periodic with fundamental period T.
How can we determine the fourier coefficients of the sum
$x(t-7)+x(-2t+3)$
I know that $x(t-7)\...
2
votes
1answer
612 views
Fourier coefficients of product of two periodic signals
question:
If $x(t)$ and $y(t)$ are two periodic signals(both with period T) with Fourier coefficients $c_{n}$ and $d_{n}$ respectively then, Fourier coefficient of $z(t)=x(t)\cdot y(t)$ is:
(a) $\...
2
votes
1answer
60 views
Rationally related frequencies and the Fourier Series representation
Suppose that we have the signal $$x(t) = e^{j\omega t} + e^{j\frac{3}{2} \omega t},$$ and we want to find a Fourier Series representation for that signal. Is this possible? According to my ...
2
votes
1answer
2k views
How to do simple extrapolation with Fourier transformation?
I have 1024 sample points, and I would like to do really simple extrapolation using Fourier transformation. First I apply Fast fourier transformation on the data. My first intuition was that I just ...
2
votes
1answer
621 views
What is the time-integration property in the Fourier series analysis?
In the continuous Fourier series properties for a periodic continuous-time signal, we have time-integration property:
$$
\int_{-\infty}^t x(\alpha)d\alpha \leftrightarrow \frac{a_k}{jk\omega_0}
$$
...
2
votes
3answers
292 views
Can I study continuous time Fourier Transform and treat the rest as special cases
Say I learned the theoretical result of continuous time Fourier transform. And I want to extends some results(say "convolution rule") to
Lapace transform,
Z transform,
DTFT,
DFT,
Fourier sequence ...
2
votes
1answer
785 views
Discrete Fourier Transform and Opposite Convolution Theorem
I am reading the Wiki for DFT. There is a part for circular convolution theorem which sounds a bit odd saying:
$$ \mathcal{F} \left \{ \mathbf{x\cdot y} \right \}_k \ \stackrel{\mathrm{def}}{=} \sum_{...
2
votes
0answers
64 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
51 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
4answers
615 views
Why Fourier series if Fourier transform can be calculated for both periodic and aperiodic?
While learning about Fourier Transform after Fourier Series, That we can calculate Fourier transform of periodic signals too. If we can take the Fourier transform of periodic signal too then my ...
