Questions tagged [fourier-series]
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246
questions
0
votes
1answer
80 views
Why do we use the DFT instead of the DTFS? Or, why was the FFT algorithm built for the DFT instead of the DTFS?
As we know, both the DTFS (discrete-time Fourier series) and the DFT (discrete Fourier transform) are used to represent discrete-time periodic signals for all time (or the periodic extension of ...
2
votes
2answers
56 views
Meaning of Rect and Train of Rect Spectra
The Fourier transform of $x(t)=\operatorname{rect}(t)$ is $X(f)=\operatorname{sinc}(f)$
The Fourier transform of a periodic train of rectangular pulses $x(t)=\sum\limits_{n=-\infty}^{\infty}\...
0
votes
2answers
108 views
FFT of square wave - what does output represent?
I am really new to FFT and signal processing. I am doing an analysis of square waves with FFT and I am trying to understand why the FFT output on the frequency domain has a downward slope for square ...
0
votes
1answer
73 views
transform signal
Hello everyone I need help solving a Fourier transform for the given signal, I know it will be a frequency convolution for the first function it will be a window function and for the second function I ...
0
votes
0answers
121 views
Finding original signal $x(t)$
For given 4 conditions, I have to find out what is $x(t)$ with period of 3, and I don't know if $x(t)$ is real or not.
For fourier coefficients $x_k$,
$$1.\ x_k=x_{k+2}$$
$$2.\ x_k=x_{-k}$$
$$3.\ \...
0
votes
1answer
19 views
Unable to understand the time-shifting property of CTFS
The CTFS of $x(t)$ is $c_{k}$ the Fourier series coefficients. Furthermore, $x(t-t_{0})$ is known to be $e^{-j\omega t_{0}}c_{k}$, the proof is given as follow :
$$
\begin{aligned}
\mathscr{F}\left(f\...
1
vote
2answers
75 views
How to visualize this statement regarding Conjugate Symmetry
A property of real signals states that if $x(t)$ is real then the Fourier series coefficient (frequency spectrum) is given by :
$$
c_{k}=c_{-k}^{*}
$$
In polar form, this can be expressed as :
$$
|c_{...
0
votes
0answers
15 views
Square pulse test of Upwind Finite Differences
I`m analyzing the numerical methods for the 1D convection equation for stability, consistency, and accuracy. I want to see How does this square pulse move in domain and time?
Here is my code
% ...
0
votes
1answer
20 views
Indetermination in a complex fourier series
I determined the complex Fourier series of a sinusoidal signal and arrived at the following expression:
$$\sum_{n=\infty}^{\infty} \left[\frac{4e^{-j \frac{\pi}{2}n}}{\pi(1-n^2)}(e^{-j\pi n}+1)\right]...
1
vote
1answer
72 views
Why Does ASK Modulation Create Fourier Sidebands?
I know why analog amplitude modulation has side bands, it is related to (fc+fd) and (fc-fd). But what about DAM?
ASK(DAM) is a type of digital modulation, and there are only two state: carrier signal ...
0
votes
1answer
55 views
How to find inverse Fourier transform of summ of delta functions?
I am practicing for my exam that I have this semester and I stumbled upon this one.
How can i find inverse Fourier transform given:
$$
X(j\omega) = \sum_{k=-\infty}^{\infty}\delta(\omega-2k+1)
$$
1
vote
1answer
26 views
Finding the discrete time Fourier series for signal
I think I did everything correctly here, but I must be missing something still.
We have the following signal:
My approach:
We are told that the signal has period $N = 4$
We know $$Y[k] = \frac{1}{N}\...
0
votes
1answer
30 views
Solution verification for this Fourier series problem
We have a signal with period $T = 2$
We want to find the continuous time fourier series for this signal.
Since $T = 2$, $\omega = \pi$. All we have to do know is find the frequency domain.
$$x(t) = \...
9
votes
2answers
546 views
Fourier series of cycloid
What is the Fourier series representation of a cycloid?
The parametric representation of the curve is as follows.
$$
t=\dfrac{\theta-\sin\theta}{\pi}\\
x=\dfrac{1-\cos\theta}{\pi}
$$
The period is $2$...
0
votes
1answer
40 views
Finding the discrete time fourier coefficients to this problem
I'm trying to find the fourier series to this discrete time signal.
$$x_1[n] =\begin{cases}
+\frac72&\text{if }0\le n \le 4\\
-\frac72&\text{if }5\le n \le 9
\end{cases}$$
My approach:
We see ...
1
vote
1answer
33 views
Represent DFT coefficients with respect to Continuous time-Fourier series coefficients
Does anyone know how to represent the Discrete Fourier transform (DFT) coefficient, $X[k]$, with respect to the Continuous time-Fourier series (CT-FT) coefficient, $X_k$? I come to the conclusion as $...
2
votes
1answer
61 views
The magnitude spectrum of a sharpening filter
I'm trying to derive an expression for the magnitude spectrum of the following sharpening filter.
$$
g(m,n) = \delta(m,n)+\lambda (\delta(m,n) - h(m,n))
$$
where $\lambda$ is some positive constant ...
0
votes
1answer
44 views
Why is the Fourier Series a special case of the Fourier Transform and not the other way around?
I was reading a text book on the frequency domains in signal processing and my understanding is that the Fourier Transform considers signals that are a-periodic in time while the Fourier Series ...
1
vote
1answer
52 views
MIT 6.003 HW#8 Problem 4 - Fourier Coefficients of Triangle Wave
In the mentioned homework, part of the solution involves finding the Fourier coefficients of the triangle wave.
The solution mentions that we can express this function as follows:
What does that ...
1
vote
2answers
275 views
Problems computing the DFT of finite length sequence
I am having trouble finding the same answer as the solution manual for this sequence.
The problem asks to compute the DFT of
$$
x[n] = \begin{cases}
1 & \text{for even } n \in \{0\...
0
votes
1answer
49 views
multiplication of a function with a Fourier-transformed equals to Fourier-transformed with a function
I already showed b item using the fact that it is $h\left(0\right)=\int \:f\left(t\right)g\left(0-t\right)dt$
I struggle a lot of hours trying to find the trick in item C.
Can anyone help please ?
0
votes
0answers
37 views
How to compose a Discrete Prolate Spheroid (DPSS) dictionary?
I have a model of signal as
$$
Y=AX + N
$$
where $Y$ is received data in a linear array, $A$ is steering matrix, $X$ is data of sources and $N$ is noise. If $A$ has the form of $A=\exp(\alpha \sin(\...
2
votes
1answer
57 views
Fourier coefficients of discrete difference of a square wave
I have a discrete square wave $f(t)$ where $t \in \mathbb{N}$, of amplitude $A$, period $T$ and duty cycle $1/T$
$$
f(t) = \left\{\begin{matrix}
A, & \mathrm{if}\;t=Tn\\
0, & \mathrm{if}\;t\...
1
vote
0answers
57 views
Forecasting that FFT result
I have the following FFT result charts. Searching for a way to forecast the future AMPLITUDE steps from the Time Domain Plot if possible, or the next maximum/minimum deviation based on some previous ...
0
votes
1answer
69 views
If a time-series has odd number of samples does it have no energy at Nyquist frequency?
Suppose I have real time series A with n samples and time-spacing dt and I want to analyze ...
0
votes
1answer
59 views
Output of LTI (in time and frequency $\omega$ domain) : when input goes through LPF
I would like to raise a mathematical question :
Let's say we are been given :
$$x(t) = \begin{cases} \cos(\pi t) & |t| \leq 0.5 \\
0 & \textrm{otherwise} ...
0
votes
0answers
32 views
How to get coefficient of the discrete fourier Series from the fourier transform
Given $X(v)$ the Discrete Fourier transform of a discrete periodic signal $x(n)$, it's possible to arrive to the $c_k$ of the Fourier series $$x(n)=\sum_{k=0}^{n-1} c_k \exp(2\pi i k t) $$ directly?
1
vote
3answers
64 views
Sum of equidistant exponents
Consider the next sum
\begin{equation}
\sum_{k = 0}^{N - 1}e^{-j\frac{2\pi}{N}k}
\end{equation}
Its geometric meaning is the sum of uniformly distributed vectors on the unit circle.
Thus, we can say ...
1
vote
0answers
21 views
Evaluate a complex exponential at negative infinity [duplicate]
I am learning about the properties of the Fourier Series (FS), which is defined by:
$$x(t) = \sum_{k=-\infty}^{\infty}c_{x}[k]e^{j2\pi kt/T}\tag{1}$$
where
$$c_{x}[k] = \frac{1}{T}\int_{T}x(t)e^{-j2\...
2
votes
5answers
621 views
Finding Fourier series coefficients for discrete time signal
Let $x[n]$ be a periodic sequence with period $N$ and Fourier series representation $$x[n] = \sum _{k=<N>}a_ke^{jk\frac{2\pi}{N}n}$$ Determine the Fourier series coefficients for
$$y[n] =
\...
0
votes
0answers
73 views
Rectangular Pulse Train and Sinc Function
I wanted to ask that in frequency domain the rectangular pulse is a sinc function, so is this sinc function periodic or aperiodic?
Also if signals that are continuous in time domain then they are ...
-1
votes
1answer
33 views
Why do we scale bins in FFT in this code?
Hi I am learning FFT I am confused about this bit of code:
what is the reason for scaling the sampling frequency and what is bin scale and why and when do we use it? thank you
...
1
vote
2answers
445 views
Autocorrelation for periodic signals
Autocorrelation for power signals is defined by $$R_x(\tau)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^Tx(t)x^*(t-\tau)dt\tag{1}$$ Is it true that for periodic signals $(1)$ can be computed by $$R_x(\tau)=...
0
votes
0answers
111 views
Can different Discrete-Time-Fourier-Series(DTFS) coefficients have the same discrete sequence in the time domain?
Please, check the following discrete periodic sequence when the period $N=2$.
$x[k]=\exp(j\frac{2\pi}{N}k), N=\text{period}$
$..., x[0]= 1, x[1]= -1, x[2]= 1, x[3]= -1, ... , N=2$
According to my ...
1
vote
1answer
88 views
Why is the continuous time Fourier series of DC signal an impulse?
In case of continuous time Fourier transform(CTFT), I can easily calculate the Fourier transform of DC signal by using Fourier duality or inverse CTFT. But I don't know how to calculate the continuous ...
0
votes
2answers
111 views
Can I sample at Nyquist rate if I know that my samples are taken only at the signal's maxima or minima?
I know that in general the sampling rate, $f_s$, must be greater than twice the highest frequency of the signal, $f$.
If I sample at the Nyquist rate, it can lead to the following:
However, if the ...
0
votes
1answer
140 views
How to save Fourier series approximated signal to a WAV file
I changed this Matlab/Octave code to approximate square wave by using combination of Fourier series and Fejér taper:
...
0
votes
1answer
57 views
Mistake or not - Fourier Series of x(2t+3)
I have a couple of resources I have from my university I had being checking and I found this:
Find Fourier Series coefficients of x(2t+3). x(t) is continuous and periodic by T.
I see this solution:
...
1
vote
1answer
86 views
Fourier transform of time division
I know that Fourier transform of $t^n f(t)= i^n \frac{d}{d\omega^n} F(\omega)$.
But does this work when $n<0$?
Is there any direct relation to compute the Fourier transform of $\frac{f(t)}{t}$?
0
votes
0answers
25 views
Redundancy when using Nyquist rate for a time series
I sample temperature at one sample per second (the hardware I am using takes the temperature at this rate, so this is the max I can efficiently sample.) The Nyquist rate for this signal would be 2 ...
asked Aug 4 '20 at 16:26
HH- Apologize to Carole Baskin
4371 gold badge5 silver badges15 bronze badges
0
votes
1answer
76 views
why does the additive synthesis method for a triangle wave require amplitude scaling by 8/pi^2?
I had to make a bunch of band limited digital triangle waves recently, so I went to (where else) wikipedia for the equations.
I noticed that there is a constant amplitude scalar of ...
1
vote
1answer
256 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 $$z(t) = \sum_{k = -\infty}^{\infty}x(...
1
vote
1answer
380 views
Duality in the discrete-time Fourier series
Suppose $g[n]$ is periodic with fundamental period $N$ and $f[k]$ being its Fourier coefficients i.e. $$ f[k] = \frac{1}{N}\sum_{n=<N>}g[n]e^{-j\frac{2\pi}{N}nk}$$ In more convenient notation $$...
0
votes
0answers
32 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 ...
1
vote
2answers
48 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
votes
2answers
30 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 ...
0
votes
0answers
34 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
vote
2answers
36 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
votes
1answer
51 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
votes
1answer
167 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 ...
