Questions tagged [fourier-series]
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173
questions
0
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0answers
13 views
Which frequency bins give the best interpolation for the derivative of a function?
A function $u:[0,2\pi]\to\mathbb R$ sampled over $N$ equidistant points $\theta_j=(2\pi/N)j,\, j = 0, \dots, N-1,$ can be interpolated by a set of functions $\{u_{k_0}\}$ enumerated by integers $k_0\...
1
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0answers
25 views
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 ...
2
votes
0answers
32 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 ...
0
votes
2answers
27 views
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 ...
0
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2answers
148 views
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:
...
4
votes
4answers
658 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 ...
1
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1answer
62 views
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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0answers
47 views
How to find fundamental frequency of two signals?
I am facing difficulty with finding fundamental frequency of signals
I mean by fundamental frequency=(1/Time period)
Correct me if I am wrong
consider two continuous time signals with
Time period ...
-1
votes
1answer
36 views
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 ...
4
votes
4answers
797 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 ...
1
vote
3answers
229 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 ...
0
votes
3answers
91 views
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 ...
1
vote
1answer
54 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 ...
0
votes
1answer
35 views
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 ...
0
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0answers
9 views
2D Fourier Synthesis (IDFT) not yielding expected result
I am trying to recover a 2D signal using inverse DFT, to my understanding the IDFT outputs the coefficients of the fourier series of the original function up to the Nyquist frequency.
So for example ...
1
vote
2answers
37 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
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0answers
45 views
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 ...
0
votes
1answer
69 views
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 ...
1
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0answers
181 views
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}...
0
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0answers
49 views
How to find Fourier series coeffecients of convolution of two periodic continuous functions with different time periods?
Above given is my solution Plz check if it is correct and I got struck at last step .Plz help
3
votes
2answers
95 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 ...
1
vote
1answer
22 views
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 ...
1
vote
1answer
491 views
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 ...
1
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2answers
181 views
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 ...
0
votes
2answers
99 views
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 ...
0
votes
0answers
18 views
Intuition behind FT of Dirac Comb [duplicate]
What is the intuitive explanation behind a dirac comb having a dirac comb as Fourier transform? How to calculate this is clear, but I cannot picture why this intuitively makes sense.
0
votes
1answer
28 views
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 ...
-1
votes
1answer
44 views
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.
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} $$
The value of $\omega_0$ is $\frac{2\pi}{10}$.
What I ...
0
votes
1answer
50 views
How to do convolution in Fourier Series?
Two signals are given to me :
$$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}...
2
votes
1answer
105 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
$$
a_n = ...
2
votes
1answer
377 views
Spectrum of Cosine in Complex Form
The complex exponential form of cosine
$$\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 ...
0
votes
1answer
62 views
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 ...
0
votes
1answer
35 views
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?
0
votes
2answers
398 views
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?
4
votes
2answers
1k 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 ...
1
vote
2answers
58 views
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 ...
0
votes
0answers
43 views
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 ...
1
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1answer
179 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)\...
0
votes
1answer
57 views
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 ...
2
votes
4answers
87 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 ...
0
votes
1answer
59 views
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 ...
0
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1answer
293 views
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 ...
1
vote
1answer
2k views
Proof of the convolution property of Fourier Series in continuous time
I am facing problem in understanding the proof of Convolution property of Fourier Series (FS) in continuous time CT;
that is: $$\mathrm{FS} \big\{x_1(t)\star x_2(t)\big\}=T\sum_{n=-\infty}^{\infty}...
0
votes
1answer
116 views
Uniqueness of Fourier Series Representation and the Fourier Transform of Periodic Signals
If we are given a signal of the form $$x(t) = \sum_{k = -\infty}^{+\infty} a_k e^{j k \omega_0 t},$$ can we call it a Fourier Series representation of $x(t)$ right away?
Suppose we are given the ...
-2
votes
1answer
80 views
Fourier Coefficients
Consider the signal
$$x(t)=\cos(2\pi t)$$
Since $x(t)$ is periodic with a fundamental period of $1$, it is also periodic with a period of $N$, where $N$ is any positive integer.
What are the ...
0
votes
1answer
179 views
What is the exact meaning of the output of the Discrete Fourier Transform
I'm fairly new to the subject, but so far my understanding that this would be a transform you could use to go from a discrete set of data, say [1, 0, 1, 2] to a continuous sinusoidal function in the ...
0
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1answer
191 views
Inverse Discrete-Time Fourier Transform of $X(Ω)=jΩ$
I am trying to solve it by using the properties but I can’t seem to find the same solution as on my book.
0
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1answer
81 views
Fourier components of $\cos(2\pi f_1t)$
I have the signal $s(t) = \cos(2\pi f_1t)$ and I am looking for its components vs the Fourier basis, over the interval $[0, T]$. The formula for computing the coefficients is
$$
s_n = \int_{t_0}^{t_1} ...
2
votes
1answer
420 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) $\...
