Questions tagged [fourier-series]
The fourier-series tag has no usage guidance.
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Synthesis discrete time signal from fourier coefficients
Following information is given about a signal $x[n]$
$x[n]$ is real and even signal
$x[n]$ has a period $N=10$ and Fourier coefficients $a_k$
$a_{11}=5$
$\frac1 {10}\sum_{n=0}^9 |x[n]|^2=50$
How can ...
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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:
$$x(t)=A_0 + 2 \sum_{k=1}^{\infty} A_k \cos(k \omega_0 t) - B_k \sin(k \omega_0 t) \tag{...
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Why does the Fourier series not include phase information?
From my understanding, the Fourier series is a way to describe an arbitrary continuous signal in terms of sinusoids of varying frequencies and amplitudes, as shown here in Equation 1.
Why are the ...
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Discrete Fourier Transform and Opposite Convolution Theorem
I am reading the Wiki for DFT. There is a part for circular convolution theorem which sounds a bit odd saying:
$$ \mathcal{F} \left \{ \mathbf{x\cdot y} \right \}_k \ \stackrel{\mathrm{def}}{=} \sum_{...
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Why we need fourier transform of periodic signal although we have fourier series for periodic signal?
I was going through some of the basics of fourier series and fourier transform. And I came across one topic "Fourier transform of periodic signal". But I am not able to understnad why we need to go ...
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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 ...
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Fourier Transforms and Series for the NON mathematically inclined.
This would most likely be the opposite of this question ( Mathematically inclined Signal and Systems/Signal Processing book? ) I figured I'd ask here if there are any good books that while, ...
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Reason for bimodal behavior while low second fourier coefficient
If I have a time series (for eg. for 23 timestamps) and if I plot it and see that it is bimodal, that means it might be having high value of second fourier coefficient (with frequency = 2). But when I ...
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How to check if Fourier components are in phase of out of phase?
I have a time series (of 23 timestamps) of which I take the Fourier transform. Now the fourier transform has 23 imaginary values and each has an amplitude and a phase. When I get the phase angle, it ...
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Similarity theorem in Fourier analysis
I have a homework problem that I'm not quite sure how to complete. The problem is as follows:
PROBLEM
Write the definition of the Fourier coefficients, and show that
$$f(t + \frac{1}{2}T) = f(t) \...
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Calculation of cosine in frequency domain instead of calculatin in time-domain followed by a FFT
I got an $N$, in my case 512, point FFT of a real-valued signal. Based on some calculation in my application I determine the parameters $k \in [1, N-1]$, the number of oscillations per period, $\phi \...
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Disadvantages of wavelet transform
I have a question related to wavelet transform: we know that while the Fourier transform is good for a spectral analysis or which frequency components occurred in signal, it will not give information ...
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Book recommendations on Gabor filter and Fourier series
I'm starting to learn about Gabor filters and Fourier series. I need to make a presentation on Gabor filters in a few months, so I need quality references for the presentation.
Does anyone have any ...
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How to prove $cos(t) + cos(\pi t)$ is non periodic function? Also can I represent this signal using fourier series?
I would just want to prove $\cos(t) + \cos(\pi t)$ is non periodic.
I don't know where to start it. Also I know that individually these signals ie $\cos(t)$ and $\cos(\pi t)$ are periodic with ...
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Two basic questions related to complex Fourier series
Doing excersies from Richard Hammings book about digital filters I've got two questions about them:
1) Fourier expansion of $g(x) = \sin^5(x)$. Provided answer is: $\sin^5(x)=5\sin(x)-20 \sin(3 x)+16\...
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Multiply FS Coefficients by 0 or 1 to get Low Pass Filter?
I have written a bit of code to upsample and interpolate a sample waveform in MATLAB. It's at the point where I have taken a signal, upsampled it 2x and filled in the gaps with 0's, and then I found ...
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g(x)=x odd and even expansions
I'm studying book about digital filter by Richard Hamming. And there is exercise to get odd and even expansion of g(x)=x where x is from 0 to $\pi$. I understood even expansion, but can't get into odd ...
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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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Ambiguity in the term 'dimension'?
We used to classify signals as 1D and 2D etc ie one dimensional and two dimensional. For example a periodic square wave signal is 1D and an image is a 2D signal etc (reference - Signals and systems by ...
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Reason of Fast Fourier and Inverse Fast Fourier [closed]
I am not very good at mathematics
I was doing some image processing so I came to know about FFT and IFFT
I was learning about ...
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The Fourier Series, Amplitude and Phase Plot of a Saw Tooth Waveform
I am trying to find the amplitude and phase plots of the saw tooth waveform pictured.I have already computed the Fourier series of the waveform but I don't know how to derive the amplitude and phase ...
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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 ...
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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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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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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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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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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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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
...
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A Laymans Fourier transform and harmonics explanation?
Please bear with me, I know some of you will scoff but I have looked on Wiki and in various literature (see below) and can't quite get a handle on a few things.
I am a general business programmer, ...
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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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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 ...
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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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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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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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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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