Questions tagged [fourier-series]

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14 votes
7 answers
11k views

The difference between DFT and DFS

In the literature, I've found that DFS and DFT are one and the same. If they are one and the same why to use two different names for them? If there is really a difference what is it and what is the ...
0 votes
0 answers
25 views

Advantages of the Karplus algorithm for sound synthesis

For a personal project I am trying to make a software to synthesize plucked and hammered string instruments. This is a whole research topic and there are already some models and algorithms such as the ...
2 votes
1 answer
146 views

Fourier transform of a time discrete signal

I would like some help to better understand the Fourier transform of a discrete time signal. My doubts are: The sampling of a signal can be seen as $x_s(t)=x(t) \cdot \sum_{k=-\infty}^{+\infty} \...
1 vote
1 answer
33 views

positivity of the spectrum of quasi-stationary signals

I am working on the "System identification : theory for the user" by Lennart Ljung (freely available here) and it is one of these books which contains exercises but no answers... My exercise ...
1 vote
1 answer
132 views

Exact formula for alias of Discrete Fourier transform for periodic sigals

Suppose that $f(t): \mathbb{R} \to \mathbb{C}$ is a $T$-periodic signal, with highest frequency $f_h$. Now suppose that our sampling rate frequency is lower than $f_h$, and is not any multiples of $1/...
0 votes
0 answers
43 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
366 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 ...
1 vote
1 answer
48 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 ~\...
2 votes
1 answer
50 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 ...
3 votes
1 answer
84 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\...
1 vote
0 answers
61 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 ...
3 votes
1 answer
98 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 ...
0 votes
0 answers
45 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
76 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 ...
0 votes
0 answers
47 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
75 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 ...
-1 votes
1 answer
100 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 ...
0 votes
0 answers
47 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
219 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
55 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
139 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
264 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
328 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 ...
12 votes
3 answers
4k views

Slow Down Music Playing While Maintaining Frequency

Playing a piece of music audio at a slower speed would lower its pitch (frequency). Is there a tool and theory to slow down the song playing while keep the frequency the same? I suppose one can do ...
2 votes
1 answer
410 views

Why does Hilbert filter distorts the shape of the signal?

If all the harmonics composing a the signal are shifted by the same amount, this would be the same as sampling later or earlier in time. I think. Take a simple pulse train as an example. If Fourier ...
3 votes
0 answers
94 views

Improving the intuition for the 2d fourier transform

As far as I understand, the 2d fourier transform is calculated as following: ...
1 vote
1 answer
77 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 = ...
0 votes
2 answers
55 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 ...
1 vote
1 answer
77 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 ...
-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 ...
0 votes
0 answers
215 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-...
-1 votes
1 answer
104 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
1 answer
95 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?
0 votes
1 answer
223 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 ...
2 votes
1 answer
398 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 ...
1 vote
0 answers
67 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 ...
8 votes
1 answer
806 views

Estimate the Discrete Fourier Transform / Series of a Signal with Missing Samples

Assuming we have a discrete signal $ { \left\{ x \left[ n \right] \right\}}_{n = 1}^{N} $. Which has a Discrete Fourier Transform / Series. Now, assume I'd like to estimate its Discrete Fourier ...
0 votes
2 answers
206 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 ...
1 vote
2 answers
1k 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 ...
1 vote
3 answers
434 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
4 answers
152 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 ...
0 votes
1 answer
921 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 ...
1 vote
0 answers
111 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
1 answer
92 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 vote
1 answer
147 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: ...
3 votes
1 answer
5k views

How to do simple extrapolation with Fourier transformation?

I have 1024 sample points, and I would like to do really simple extrapolation using Fourier transformation. First I apply Fast fourier transformation on the data. My first intuition was that I just ...
3 votes
1 answer
85 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 ...
1 vote
1 answer
263 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 ...
1 vote
1 answer
150 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 $\frac{f(t)}{t}$?
1 vote
1 answer
122 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-...

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