Questions tagged [fourier-series]
The fourier-series tag has no usage guidance.
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Energy of a sinc signal
My book give me two signals to demonstrate that the temporal translation does not alter the energy and area. It gave me
$$ x(t)=\operatorname{sinc}(t) $$
and
$$ s(t)=x(t-T)$$
and I found that ...
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Fourier transform properties
I have to find the Fourier transform of $$ x(t)= \frac{1}{T}e^{-\frac{t-T}{T}}u(t-T) $$
First I applied traslation property , so $$ F[x(t-T)] = X(f) e^{-i 2 \pi f T} $$
after I applied time scaling ...
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Fourier coefficients of odd and even part of a signal
I have this signal and I have to find the Fourier coefficients of the odd and even part. First I found that $$ x_p(t) = \frac{1}{2} ( x(t) + x(-t) ) $$ and I made the graphic of this part and I ...
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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 ...
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E of a signal using Rayleigh
I have to find The energy of a signal using Rayleigh th. the signal is $$ x(t) = A e^{-At } u(t) $$ assuming A>0
Using the classic definition of E , I found that it should be $$ \frac{A}{2} $$
Using ...
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2
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Fourier coefficients of |A cos (x)|
I have to find Fourier coefficients of $x(t) = |A \cos (2 \pi f t )|$. The problem also give me $y(t) = |x(t)|$. To find Fourier coefficients I wrote
$$ \frac{1}{T_0} \int_{0}^{T_0} |A \cos (2 \pi f ...
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Discrete signal - fourier transform coefficient period [closed]
if I have an example signal in the picture, how can i decide its period?
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How to calculate fourier coefficients of sum of two discrete time with different fundamental periods
Assume that we have two discrete-time signals named x[n] with fundamental period 3 and fourier coefficients ak (k from 1 to 3), and y[n] with fundamental period 5 and fourier coefficients bk (k from 1 ...
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Complex signal sine reconstruction
Noob here, I read that any signal can be made by putting together sines and cosines, it always shows some kind of basic harmonic wave with constant amplitude such as square wave.
I understand that ...
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Understanding Fourier Transforms in abstract math terms
I am having a hard time implementing a method that computes Fourier transform coefficients for the complex form using the trapezoid rule.
I have floated questions in the math and stackoverflow ...
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Fourier Series representation of a signal
Use the defining equation for the Fourier Series coefficients to evaluate the Fourier Series representation of the following signal:
$$x(t)=\sum_{m=-\infty}^{+\infty}=(\delta(t-m/3)+\delta(t-2m/3))$$...
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Fourier series expansion of $ x_1(t) = \sum _{-\infty}^{\infty} \Delta (t-2n) $
I want to evaluate the Fourier series expansion of $ x_1(t) = \sum _{-\infty}^{\infty} \Delta (t-2n) $, where $ \Delta (t) $ is a triangular function defined as:
I have done the following ...
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Fourier series representation of a full wave rectifier output
I am trying to compute the Fourier series representation of a full wave rectifier output.
The equation of the signal is:
$ x_8(t) = | \cos (2 \pi f_o t) $ |
I have tried to find the Fourier series ...
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Fourier Transform: interpretation of continuous spectrum at specific frequencies
B. P. Lathi in his book "Principles of Linear Systems and Signals" mentions in the Fourier Transform:
When $x(t)$ is periodic, the spectrum is discrete, and $x(t)$ can be expressed as a sum of ...
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Aliasing when interpolating with DFT?
I'm coming from an understanding of the continuous-time Fourier Transform, and the effects of doing a DFT and the inverse DFT are mysterious to me.
I have created a noiseless signal as:
...
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Is Fourier series a sampled version of Fourier transform?
I recently learned about dtft and how dft/dfs is the sampled version of dtft. I was wondering if Fourier series is also obtainable by sampling Fourier transform? I am a noob in the subject so sorry if ...
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Proof of First Difference Property for Fourier Series
I am having trouble with deriving a proof for the first difference property for the Fourier Series.
Here is my attempt at the derivation:
$$
y[n] = x[n] - x[n-1]
$$
Fourier Series Representation:
...
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How do I obtain the fourier series coefficients for a signal obtained by multiplication of two signals of different frequency?
What i assume here is that LCM of time periods of the two taken signals exist that is signals periods are not like pi/2 and 1 but are rather like 1 and 2 (just an example)
I am given fourier series ...
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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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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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Phase diagram of a rectangular pulse with Fourier Series - help understanding
I understand perfectly fine how to plot the magnitude of a Fourier series, but I'm having serious trouble understanding how to plot the phase spectrum. Below is a picture of a rectangular pulse. The ...
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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 ...
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Is there a generalized method to get the input with the given output and the impulse response?
$y(t)=y(t+12), y(t) = x(t) \ast h(t)$
The continuous time signal output $y(t)$ is a periodic square wave, 50% duty cycle pulse.
The impulse response is a box function.($A = 1, T = 2$)
By using ...
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2
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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 ...
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Is there an analogy of the Fourier-decomposition in the Laplace space to decompose a signal to a few components?
I have a signal from which I know, that it is the sum of a few, exponentially decaying components. I want to find these components.
If it would be a sum of some sinusiod waves, it would be easy to ...
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Fourier Transforms, symmetry, real/imaginary
I was hoping to clarify if the following was correct:
A real function (neither even nor odd) in time exhibits conjugate symmetry in frequency, so the real part of the frequency response is even, and ...
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The Fourier Transform of a periodic function and it's series
Let $X(f)$ be the Fourier transform of $x(t)$:
$$ X(f) \triangleq \mathscr{F}\Big\{ x(t) \Big\} = \int\limits_{-\infty}^{\infty} x(t)\,e^{-j 2 \pi f t} \ \mathrm{d}t $$
$$ x(t) \triangleq \mathscr{F}...
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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 ...
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Sampling Frequency and Spectral Regrowth
Sampling a cosine wave of 10 Hz at Fs = 64 and number of samples Ns = 256
I setup my time vector for the cosine wave as n=(0:Ns-1)*(1/Fs)
If I change the sampling frequency from 64 to 64.0005 I get ...
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How can I improve my fit of cosines to periodic data using Python?
I have a space-separated csv file containing a measurement. First column is the time of measurement, second column is the corresponding measured value, third column ...
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A question on Fourier Series and the frequency of the sinusoids
On studying about Fourier series, I encountered 2 doubts:
How is it that a non-periodic function has a Fourier series?
When expressing a periodic function as summation of sinusoids, why is the ...
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What is the basic idea behind Fourier transform? [closed]
What is the basic idea behind (discrete and continuous) Fourier transform (FT)? In short, what is the difference between discrete and continuous FT?
I have read multiple answers on the web related to ...
user40095
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Correct form of discrete-time Fourier series representation
As I see in this slides, Fourier series representation for discrete-time signal $s[n]$ with period $N$ is $\sum_{k = 0}^{N-1} c_k e^{j2\pi k n / N}$
According to Wiki, Fourier series representation ...
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How to find period of signal, when spectrum is known?
Here is signal in frequency domain .Division of frequencies does not give an integer number or real number. May it can be here another method of finding period for ths signal.
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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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How to do convolution in Fourier Series?
Two signals are given to me :
$$x(t)=\cos(4\pi t)$$
$$y(t)=\sin(4\pi t)$$
I have founded their coefficients as follows:
$$a_k = a_1=a_{-1}=\frac{1}{2} $$
$$b_k = b_1=b^*_{-1}=\frac{1}{2j}...
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Does the Fourier series coefficient of AC components remains same if DC component is subtracted form the given signal?
Suppose a signal is defined by
$
x(t)= \begin{cases}
t & 0\leq t \leq 1 \\
2-t & 1\leq t\leq 2 \\
\end{cases}
$
Since $x(t)$ has even symmetry, I can calculate fourier coefficient as
$$
a_n = ...
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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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How was this result on discrete Fourier series achieved?
I was trying to do the question 10, part b of the following document (https://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/assignments/MITRES_6_007S11_hw10.pdf)
I was going through ...
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Find Fourier series $f(t) = e^{jx t}$ , $−\pi < t < \pi$ [closed]
I need to find the Fourier series of the $f(t) = e^{jxt}$ , $− \pi < t < \pi$
What will be the first step to solve it?
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What does it mean for a function to have frequencies?
In a lecture, my professor mentioned that $\cos$ has two frequencies. I see that using the inverse Euler's formula we can express $\cos$ as a some linear combination of complex exponentials, each with ...
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Why do frequencies of analog signals range from $-\infty$ to $\infty$ while frequencies of digital signals are restricted to $[0,2\pi]$?
In Fourier analysis while dealing with discrete-time signals, frequencies range from $0$ to $2\pi$ why? Intuitively how can i understand it?
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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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Fourier series of $cos(\omega_0 t)$ in continuous time
Can any one please help me with understanding how we can calculate the Fourier series of Cos(w0t) using the formula:
I saw that they did the following calculus, but I Don't really understand how we ...
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From Fourier (k space) to wavelet domain in MRI sensing
In compressed sensing MRI (cSENSE MRI) technology the idea seems to entail sampling from the Fourier domain (k space) in a way that, when transformed to the wavelet domain ("sparsification"), the ...
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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)\...
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Is there any special when all Fourier components have the same angle?
When a square wave doesn't jump, its oscillators aren't aligned:
But if they are in sync, the wave will jump to its extrema:
However this is just an example of square waves. The tool Understanding ...
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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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How to determine which Fouriers Series terms to use to approximate a signal?
I have a signal (a time-series of air temperature values) that I can approximate quite well with a Fourier series. However, the number of terms in the series grows rapidly, to the point that 30 - 40 ...
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characterization of DC component
Consider the following two statements:
In time axis:
A signal without a DC component is a signal which doesn't have the zero frequency (the DC frequency)
A signal without a DC component is averaged ...
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