Questions tagged [fourier-series]
The fourier-series tag has no usage guidance.
283
questions
0
votes
0
answers
34
views
Creating infinite continuous time series out of a finite discrete spectrum
I have a power spectrum $S(\omega)$: a $1 \times N$ real vector that matches frequencies from 0 to 125Hz. I would like to create a time series $S(t)$ $[-\infty<t<\infty]$ that matches this ...
3
votes
3
answers
339
views
Trying to understand how to get this basic Fourier Series
I'm sorry if this kind of question isn't allowed, but I'm starting to learn Fourier series and I'm still not entirely sure what's going on... in this specific case, I'm trying to find the Continuous ...
- 31
1
vote
1
answer
46
views
Can we control the maximum norm of a continuous signal whose finitely many Fourier coefficients are fixed?
Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous signals $x:\mathbb{R}\to \mathbb{R}$.
Fix $n\in \mathbb{N}$ and put
$$\Lambda_n=\{y\in C_{2\pi}: \mathcal{F}(y)[k]=0 ~\...
- 315
2
votes
1
answer
44
views
Discrete Fourier series of an odd signal
Assuming the signal shown below :
I have found an expression for fourier series coeffecients as the following:
$$a_{k} = \frac{1}{5}+\frac{j}{5}\sin{\frac{2\pi}{5}k}$$
Which matches with what the ...
2
votes
1
answer
77
views
Is my solution correct?
$\textbf{Question:}$
$y_a(t)$ is a rectangular waveform defined as:
$$\
y_a(t) =
\begin{cases}
2 &t \in [0,1/25)s\...
- 143
1
vote
0
answers
57
views
Finding $A_k$ coefficients
I was able to demonstrate that for a signal $x(t)$ real, we can write the truncated Fourier series as:
$x_N (t) = A_0 +\sum\limits_{k=1}^{N}A_k\cos(kω_0t + \varphi_k)$, but now I've been given the ...
- 53
3
votes
1
answer
93
views
Fourier coefficients of two discrete-time signals of different periods
I'm trying to understand the Fourier series coefficients of the sum of two discrete-time periodic signals.
Consider two discrete-time periodic signals $x[n]$ and $y[n]$. $x[n]$ has period $N$, its ...
- 75
0
votes
0
answers
41
views
Finding the Fourier Coefficients
Up until now, I have dealt with finding Fourier Coefficients for functions: $f(t) > 0$
Which made it convenient calculating the Fourier Analysis Integral. However, I am now presented with ...
3
votes
0
answers
63
views
What happens to sidebands when they enter "negative" frequencies?
I am working with PWM signals. These signals are generated by comparing a modulating (at frequency $f_m$), and a carrier (at frequency $f_c$), as shown in the following image:
In the resulting ...
- 125
0
votes
0
answers
46
views
What is $z[n]$ , when $Z[k]=X[2k]H[2k]$ with $y[n] = x[n]∗h[n]$ and $Z[k] = X(\omega)H(\omega)$ at $\omega = \frac{4πkn}{N}$
How do I express $z[n]$ in terms of $y[n]$, with:
$$y[n] = x[n]∗h[n]$$
$x[n]$ and $h[n]$ being 16 length sequences.
$X(\omega)$ and $H(\omega)$ are DTFT of $x[n]$ and $h[n]$
$Z[k]$ is defined as
$$Z[k]...
0
votes
1
answer
61
views
Bandwidth of cosine of bandlimited signal
I have a signal $x(t)$ with bandwidth $B_x$, and I am taking its cosine to create $y(t) = cos(x(t))$. After checking the spectrum with FFT, it seems that $y(t)$ is also bandlimited. But, is there a ...
- 125
0
votes
0
answers
42
views
How to generate a sound closer to a saxophone using sinusoids after Fourier Transform?
Generate a sound wave of saxophone frequency and compare it to the original sound clip and play both to listen to if there is a good match.
I am trying to generate a sound closer to a saxophone using ...
0
votes
1
answer
201
views
How to reconstruct original signal using IFFT after cutting past Nyquist limit
I'm working on a pitch shifting program. Everything works up to the point where I try to do the IDFT. Because I cut the DFT array past the Nyquist limit, when I run the IDFT, I don't get the same ...
1
vote
0
answers
47
views
Pitch successfully changes with Phase Vocoder, but there's an issue
I've been working on a phase vocoder program. The goal is to change the pitch of a recording of my voice. While doing research on how to change pitch, I came across this from a paper on phase vocoders ...
0
votes
1
answer
124
views
How to change fundamental frequency with DFT?
I'm working on a voice changer. My plan is to make it so that it can change your voice in various different ways, but right now I'm just trying to make it change your voice to "chipmunk voice&...
2
votes
1
answer
189
views
How to Find "pitch" from Fourier Series
The end goal of my project is to create an autotune program, But the problem I'm trying to solve right now concerns finding the pitch of someone singing a note. I have written some code that performs ...
0
votes
2
answers
241
views
Fourier transform of periodic functions
The Fourier transform is derived from the Fourier series by considering a non-periodic signal, thinking of it as a infinitely long periodic signal, putting it into the Fourier series and making this ...
3
votes
0
answers
78
views
Improving the intuition for the 2d fourier transform
As far as I understand, the 2d fourier transform is calculated as following:
...
- 131
1
vote
1
answer
72
views
Can I reduce the complexity of multiplication with FFT if the input vector is repeating?
I have a Fourier matrix $F$ with size $N \times N$, such that $y = F \times x$, if I have the vector $x$ contains four identical parts, for example $x = [x_1, x_2,x_3,x_4]’$ and $x_1 = x_2 = x_3 = ...
- 670
0
votes
2
answers
50
views
Sampling of the DTFT causes the inverse transform to become periodic?
As you can see the above equation, DTFT is calculated from sample x[n] which is discrete sample of x(t).
But calculated X(w) is continuous, even though it is calculated from discrete value of x[n] as ...
0
votes
0
answers
189
views
How do I plot a phase spectrum of rectangular pulse with Matlab?
As you can see, I made a code about rectangular pulse like this. And, I plotted complex-exponential-coefficient and Magnitude-Spectrum-of-Complex-Exponential-Form, Phase-Spectrum of-Complex-...
0
votes
1
answer
79
views
What's the difference between male and female voice? [duplicate]
If I record the voice of a man and a woman, what are the main differences I get in the various spectra and harmonics in Fourier analysis?
- 15
0
votes
1
answer
182
views
Confusion understanding Fourier series line spectra?
I am reading book signals and systems Laboratory with MATLAB where I am studing chapter 5, Fourier series and i trying to understand magnitude spectrum and phase spectrum but i have certain confuisons ...
- 1,041
1
vote
0
answers
59
views
What is the Fourier convolution theorem range of application (example of Dirac comb times rectangular window)?
$\DeclareMathOperator{\sinc}{sinc}$
I have questions regarding the Fourier transform of the product of functions or distributions and the range of application of the convolution theorem.
Context
When ...
- 11
0
votes
2
answers
193
views
What is the reason of existence of Fourier transform? (Why we use Fourier transform?)
I'm currently trying to understand Fourier transform and I've got curious about why Fourier transform exists.
Let's suppose that we have a 10 seconds of non-periodic wave. For example:
As far as I ...
- 57
1
vote
2
answers
943
views
Proving real and odd function has imaginary and odd Fourier Transform
Cheers, I am trying to prove that a real and odd function/signal has imaginary and odd Fourier Transform. Although it seems fairly easy, I can't find a way to achieve it, and searching online hasn't ...
0
votes
4
answers
118
views
Are there Fourier methods for VERY low frequency repeating signals?
Suppose I have a very low frequency pattern of sound. For example a 10 second music file. Then 10 seconds of silence. Then the same 10 second music file repeated again. The whole sequence repeats ...
- 101
0
votes
1
answer
831
views
How to apply Butterworth high pass filter in the frequency domain?
I have a time series of measurements which I want to high pass with Butterworth filter.
Python scipy package has a built in function for Butterworth filter (signal.butter) and I know how to apply it ...
- 3
0
votes
1
answer
89
views
Fourier Series of a piecewise function
I've been given the task to find the Fourier Series Representation. All I'm given is this $$x(t)= \begin{cases}-t & \text { for } 0 \leq t<1 \\ 1 & \text { for } 1 \leq t<2 \\ 0 & \...
- 1
1
vote
1
answer
111
views
How to interpret Fourier transform?
I am very new to this topic.
I ran a Fourier transform with the scipy fft function.
I than plotted the return values:
I am assuming the x-axis means how many cycles there are in all the data and y-...
- 115
2
votes
1
answer
202
views
Fourier transform of shifted periodic function
Assuming $x(t)$ is a periodic function of period $T$ and having the Fourier transform $X(\omega)$, it is required to calculate the Fourier transform of the signal $x(t)+x(t-T)$. Since x(t-T) is equal ...
0
votes
0
answers
138
views
Approximation of Periodic Parabolic Function by Fourier Series!
I've just tried to approximate the periodic-parabolic signal by Fourier Series. I know, this sounds a bit strange. I am just trying to figure out relationship between Fourier Series and Taylor ...
- 71
1
vote
1
answer
413
views
Prove Convolution Property for DFT using duality
If $x_1[n]$ and $x_2[n]$ are finite length sequences of length $N$
$$\mathcal{DFT}(x_1[n] \circledast x_2[n]) = X_1[k]X_2[k]$$
where $X_1[k]$ and $X_2[k]$ are the DFTs of$x_1[n]$ and $x_2[n]$, ...
- 191
0
votes
1
answer
443
views
Prove Discrete Time Fourier Series Multiplication property
Note: This is not a homework problem. I'm just stalled at a point because I think I might be interpreting the duality property incorrectly.
If $x_1[n]$ and $x_2[n]$ are periodic with period N, then if ...
- 191
1
vote
1
answer
108
views
How to interpret the Fourier Transform 𝑋(𝜔) (Foundational Questions)
The motivation behind the fourier transform is to somehow represent a non-periodic signal as a sum of sinusoids just as we do with the fourier series for periodic signals, correct?
With the Fourier ...
- 149
1
vote
1
answer
75
views
On discrete fourier coefficient convolution of indivdual periodic signals with different frequencies
It is a well-known result that when two signals with the same period $T$ get multiplied in the time domain, the resulting signal's Fourier coefficients are given by the discrete convolution of ...
- 11
0
votes
1
answer
87
views
How is this given impulse response of infinite duration? Isn't it just from -π to +π?
How is hd(n) infinite duration when it is from -π to +π. Book says as it is infinite duration, we in the next step take-:
h[n]=hd[n] from n=-(N-1)/2 to (N-1)/2 and 0 otherwise.
I can't see how this ...
- 21
5
votes
1
answer
163
views
Compare two Fourier series to depict the signal smoothness
I have several signals, that I am trying to find a metric to compare the signal smoothness.
By signal smoothness I mean, the signal that the distance between the peak to trough become smaller (getting ...
- 51
0
votes
1
answer
255
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 ...
- 103
2
votes
2
answers
291
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}\...
- 284
0
votes
2
answers
2k
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 ...
- 103
-1
votes
1
answer
96
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 ...
1
vote
0
answers
124
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 $x_k$,
$$1.\ x_k=x_{k+2}$$
$$2.\ x_k=x_{-k}$$
$$3.\ \...
- 103
0
votes
1
answer
44
views
Unable to understand the time-shifting property of CTFS
The CTFS of $x(t)$ is $c_{k}$ the Fourier series coefficients. Furthermore, $x(t-t_{0})$ is known to be $e^{-j\omega t_{0}}c_{k}$, the proof is given as follow :
$$
\begin{aligned}
\mathscr{F}\left(f\...
- 129
1
vote
2
answers
190
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 :
$$
|c_{...
- 129
0
votes
0
answers
21
views
Square pulse test of Upwind Finite Differences
I`m analyzing the numerical methods for the 1D convection equation for stability, consistency, and accuracy. I want to see How does this square pulse move in domain and time?
Here is my code
% ...
0
votes
1
answer
26
views
Indetermination in a complex fourier series
I determined the complex Fourier series of a sinusoidal signal and arrived at the following expression:
$$\sum_{n=\infty}^{\infty} \left[\frac{4e^{-j \frac{\pi}{2}n}}{\pi(1-n^2)}(e^{-j\pi n}+1)\right]...
- 13
1
vote
1
answer
200
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 ...
- 13
0
votes
1
answer
61
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:
$$
X(j\omega) = \sum_{k=-\infty}^{\infty}\delta(\omega-2k+1)
$$
- 11
1
vote
1
answer
42
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}\...
- 149