Questions tagged [fourier-series]
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201
questions
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1answer
53 views
Fourier transform of the sampled signal
I want to calculate Fourier transform of the sampled signal in two ways. Let $$s(t) = \sum_{k = -\infty}^{\infty}\delta(t - kT)$$And $z(t) = x(t)s(t)$. So we have $$z(t) = \sum_{k = -\infty}^{\infty}x(...
1
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1answer
73 views
Duality in the discrete-time Fourier series
Suppose $g[n]$ is periodic with fundamental period $N$ and $f[k]$ being its Fourier coefficients i.e. $$ f[k] = \frac{1}{N}\sum_{n=<N>}g[n]e^{-j\frac{2\pi}{N}nk}$$ In more convenient notation $$...
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0answers
27 views
How to apply FT to real-life signal that labeled in seconds, not radians
In training examples we always do a transformation on signals which have t-scale in labeled in radians.
I understand that Pi is just a number, but I still have some troubles to understanding how to ...
0
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2answers
34 views
Where did the length of time’s period disappear in Periodic Fourier Series Discrete Time
In continuous time Periodic Fourier Series has smallest n as possible, since it is an integral and a length of the repeating time (period time) which is T0. In discrete time however we don’t have ...
0
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2answers
29 views
How can it be that there is a series of integrals in Fourier Series if it’s a projection on a continuous basis?
If the process of finding Fourier coefficient is finding the projection of a signal on a member from an orthonormal basis, basis which is continuous in frequency. How can it be that Fourier ...
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0answers
33 views
frequencies in frequency spectrum with no correlation together
I have a lack of understanding of the following questions:
If I have a signal from a motor that is recorded with an accelerometer. And the rotating speed of the motor is 150Hz(rpm 9000 ), I can see in ...
1
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2answers
34 views
Where did the k of $a_k$ disappear from Fourier Reverse Transform if $\omega=\omega_0k$?
Where did the $k$ of $a_k$ disappear from Fourier Reverse Transform if $\omega=\omega_0k$?
We turn $\omega0$ to be $d\omega$, but $\omega=\omega_0k$, so shouldn’t there be a $k$ in the reverse ...
0
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1answer
41 views
harmonic waves as integer multiple in spectrum
i have a motor that is rotating with a certain frequency. If i check the frequency spectrum it contains a peak on 150 hz. Also i have peaks at 300,450,600 ... i guess that those peaks are harmonics.
...
0
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1answer
66 views
Fourier transform of t*rect(t)
In my previous post I asked for help for a Fourier transform of $$ t \text{rect} ( t- \frac{1}{2} ) $$ and I think I’ve understand the process. Now I’ve 2 another similar Fourier transform to do , I ...
0
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3answers
95 views
Fourier transform of a rect*half triangle
I have to calculate the analytic expression of Fourier transform $$ x(t) = t{\rm rect} ( t- \frac{1}{2} ).$$ First I made the graph of these two signals and I obtained the graph I posted.
Now I ...
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1answer
54 views
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48 views
Filtering using fourier series
Suppose I have a measured signal $M$ that has frequency components from 0 to 50 Hz. I plot the specturm of this signal using FFT and I observe its frequency content (power vs frequency). Then, I ...
1
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1answer
45 views
Given a signal and its Fourier transform, find FS coefficient of the shifted sum of the signal
Given $x_1(t),X_1(j\omega), x_2(t)=\sum_{k=-\infty}^{\infty}x_1(t-6k)$, find Fourier series coefficient of $x_2(t)$.
Looking up the FT table, I got $X_2(j\omega)=\sum_{k=-\infty}^{\infty}e^{-j\omega ...
1
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3answers
91 views
Complex exp. Fourier series, finding $x(t)$ when $X(j\omega)$ is given as magnitude and phase plot
I'm watching Neso Academy series on Signals and Systems, and in one of the videos the problem is to find $x(t)$ when magnitude and phase plot are given. The plot looks like this:
When he finishes ...
0
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2answers
114 views
How to find fundamental frequency of Fourier Series
The original problem is from the Problem Set 7 of MIT OpenCourseware: Find the Fourier series coefficients for
$$
x(t)=sin(10\pi t+\frac{\pi}{6})
$$
What I did is to rewrite it in exponential form $\...
0
votes
1answer
59 views
What kind of periodic signals cannot be represented with the Fourier Series?
Oppenheim et al. state in Signals and Systems that there exist periodic signals which cannot be represented with Fourier series. What signals are these?
Although Euler and Lagrange would have been ...
0
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1answer
113 views
How to do the Fourier Transform of bounded function?
I was trying to solve a Fourier transform of a function using the properties of Fourier transforms. The function is given as:
$\frac{At}{2}$ for $-2<t<2$ and $0$ for all other $t$. Doing the ...
2
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2answers
86 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 $$ X(f) = \frac{1}{2} tri (\frac{f+f_0}{B}) ...
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0answers
47 views
express pass band filter as sum of low pass filter
I have to find impulsive response of an ideal pass band filter, but I have a problem to express $$ H_{BP} (f) $$ as a sum of $$ H_{LP} (f) $$. I mean that $$ H_{BP} (f) = rect ( \frac{f-f_0}{B} ) + ...
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67 views
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 ...
0
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2answers
69 views
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 ...
1
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1answer
80 views
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 ...
3
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1answer
86 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 ...
0
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1answer
23 views
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 ...
0
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2answers
128 views
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
$$ \frac{1}{T_0} \int_{0}^{T_0} |A \cos (2 \pi f ...
-1
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1answer
19 views
Discrete signal - fourier transform coefficient period [closed]
if I have an example signal in the picture, how can i decide its period?
1
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0answers
35 views
How to calculate fourier coefficients of sum of two discrete time with different fundamental periods
Assume that we have two discrete-time signals named x[n] with fundamental period 3 and fourier coefficients ak (k from 1 to 3), and y[n] with fundamental period 5 and fourier coefficients bk (k from 1 ...
0
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0answers
26 views
I want to invert my fourier transform components to waves again
Hi I am using R to analyze some data
I basically did fft(data) and got a vector of complex numbers but from that now I want to remove certain harmonics from my actual wave but how do I convert one of ...
0
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2answers
43 views
Complex signal sine reconstruction
Noob here, I read that any signal can be made by putting together sines and cosines, it always shows some kind of basic harmonic wave with constant amplitude such as square wave.
I understand that ...
1
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1answer
113 views
Understanding Fourier Transforms in abstract math terms
I am having a hard time implementing a method that computes Fourier transform coefficients for the complex form using the trapezoid rule.
I have floated questions in the math and stackoverflow ...
1
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1answer
45 views
Fourier Series representation of a signal
Use the defining equation for the Fourier Series coefficients to evaluate the Fourier Series representation of the following signal:
$$x(t)=\sum_{m=-\infty}^{+\infty}=(\delta(t-m/3)+\delta(t-2m/3))$$...
0
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0answers
20 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\...
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0answers
35 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
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0answers
86 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
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2answers
171 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
268 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
976 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
143 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:
...
-1
votes
1answer
46 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 ...
5
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4answers
5k 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 ...
2
votes
3answers
648 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
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3answers
328 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
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1answer
153 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
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1answer
75 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 ...
1
vote
2answers
56 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
61 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
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2answers
224 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 ...
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0answers
263 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}...
3
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2answers
133 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
24 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 ...
