Questions tagged [fourier-series]
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224
questions
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1answer
35 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
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1answer
38 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
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2answers
215 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
$$
x[n] = \begin{cases}
1 & \text{for even } n \in \{0\...
0
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1answer
19 views
multiplication of a function with a Fourier-transformed equals to Fourier-transformed with a function
I already showed b item using the fact that it is $h\left(0\right)=\int \:f\left(t\right)g\left(0-t\right)dt$
I struggle a lot of hours trying to find the trick in item C.
Can anyone help please ?
0
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0answers
35 views
How to compose a Discrete Prolate Spheroid (DPSS) dictionary?
I have a model of signal as
$$
Y=AX + N
$$
where $Y$ is received data in a linear array, $A$ is steering matrix, $X$ is data of sources and $N$ is noise. If $A$ has the form of $A=\exp(\alpha \sin(\...
2
votes
1answer
53 views
Fourier coefficients of discrete difference of a square wave
I have a discrete square wave $f(t)$ where $t \in \mathbb{N}$, of amplitude $A$, period $T$ and duty cycle $1/T$
$$
f(t) = \left\{\begin{matrix}
A, & \mathrm{if}\;t=Tn\\
0, & \mathrm{if}\;t\...
1
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0answers
43 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
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1answer
55 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
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1answer
47 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 :
$$x(t) = \begin{cases} \cos(\pi t) & |t| \leq 0.5 \\
0 & \textrm{otherwise} ...
0
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0answers
29 views
How to get coefficient of the discrete fourier Series from the fourier transform
Given $X(v)$ the Discrete Fourier transform of a discrete periodic signal $x(n)$, it's possible to arrive to the $c_k$ of the Fourier series $$x(n)=\sum_{k=0}^{n-1} c_k \exp(2\pi i k t) $$ directly?
1
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3answers
58 views
Sum of equidistant exponents
Consider the next sum
\begin{equation}
\sum_{k = 0}^{N - 1}e^{-j\frac{2\pi}{N}k}
\end{equation}
Its geometric meaning is the sum of uniformly distributed vectors on the unit circle.
Thus, we can say ...
1
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0answers
21 views
Evaluate a complex exponential at negative infinity [duplicate]
I am learning about the properties of the Fourier Series (FS), which is defined by:
$$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
184 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
$$y[n] =
\...
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0answers
50 views
Rectangular Pulse Train and Sinc Function
I wanted to ask that in frequency domain the rectangular pulse is a sinc function, so is this sinc function periodic or aperiodic?
Also if signals that are continuous in time domain then they are ...
-1
votes
1answer
30 views
Why do we scale bins in FFT in this code?
Hi I am learning FFT I am confused about this bit of code:
what is the reason for scaling the sampling frequency and what is bin scale and why and when do we use it? thank you
...
1
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2answers
161 views
Autocorrelation for periodic signals
Autocorrelation for power signals is defined by $$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)=...
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80 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
49 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
99 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
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1answer
127 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
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1answer
33 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
56 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
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0answers
23 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
4151 gold badge5 silver badges13 bronze badges
0
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1answer
54 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
78 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 $$z(t) = \sum_{k = -\infty}^{\infty}x(...
2
votes
1answer
136 views
Duality in the discrete-time Fourier series
Suppose $g[n]$ is periodic with fundamental period $N$ and $f[k]$ being its Fourier coefficients i.e. $$ f[k] = \frac{1}{N}\sum_{n=<N>}g[n]e^{-j\frac{2\pi}{N}nk}$$ In more convenient notation $$...
0
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0answers
28 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
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2answers
44 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 ...
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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
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
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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
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1answer
45 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
80 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
120 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
62 views
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53 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
47 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
vote
3answers
100 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
votes
2answers
693 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
81 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
252 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
128 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 $$ X(f) = \frac{1}{2} tri (\frac{f+f_0}{B}) ...
0
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0answers
52 views
express pass band filter as sum of low pass filter
I have to find impulsive response of an ideal pass band filter, but I have a problem to express $$ H_{BP} (f) $$ as a sum of $$ H_{LP} (f) $$. I mean that $$ H_{BP} (f) = rect ( \frac{f-f_0}{B} ) + ...
0
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0answers
105 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
votes
2answers
74 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
vote
1answer
131 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
votes
1answer
90 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
votes
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
votes
2answers
150 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
votes
1answer
20 views
Discrete signal - fourier transform coefficient period [closed]
if I have an example signal in the picture, how can i decide its period?
