Questions tagged [fourier-series]

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Separating 'heart sound' from 'lung sound'

I have audio files recorded from electronic stethoscope and in those files I want to filter out heart sounds and retain just the breathing sounds. How can I do this using just the signal processing ...
hamza nawaz's user avatar
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1 answer
62 views

Relationship between fourier transform and fourier series

Let $$x(t) = A\sin(2 \pi f_0 t + \alpha)$$ its Fourier transform is given by $$ X(\omega) = \frac{A \pi}{i}(e^{ia}\delta(\omega-2\pi f_0) - e^{-ia}\delta(w+2\pi f_0)). $$ the Fourier series complex ...
MOHAMED SALHI's user avatar
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1 answer
120 views

Magnitude and phase spectrum of a periodic signal

Let the following T-periodic signal : I found that $$ x(t) = \frac{A \cdot t}{T} \qquad 0 \le t < T $$ and its Fourier series is : $$ x(t) = \frac{A}{2} - \frac{A}{\pi} \sum_{n=1}^\infty \frac{\...
MOHAMED SALHI's user avatar
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1 answer
80 views

Is there a Fourier Transform generalization that lets you analyze arbitrary complex frequencies?

Suppose you have a function that can be described as $$f(s) = \sum_{n=0}^{\infty} a_n e^{f_n s}$$ where each $f_n$ is a complex number. I am looking for a transform $T$ to act on $f$ which produces a ...
Sidharth Ghoshal's user avatar
1 vote
1 answer
86 views

Finding a discrete signal using some information about its Fourier coefficients

I'm struggling to solve the following question. I've solved it partially, but I can't get complete it. We have the given information about a signal of the form ...
Danialz's user avatar
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1 answer
138 views

Complex exponential Fourier series coefficient of periodic convolution

Let the complex exponential Fourier series coefficients of two periodic signals $x_1(t)$ and $x_2(t)$ be $C_{1n}$ and $C_{2n}$, respectively, with $T_0$ being the fundamental time period of both the ...
Kushagr Jaiswal's user avatar
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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 ...
Agustín's user avatar
3 votes
1 answer
193 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} \...
Maghreb_1911's user avatar
1 vote
1 answer
40 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 ...
NokiYola's user avatar
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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 ...
Daniel Polyakov's user avatar
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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 ...
Fern Mendiz's user avatar
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1 answer
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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 ~\...
ABB's user avatar
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2 votes
1 answer
62 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 ...
Ait-Gacem Nabil's user avatar
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1 answer
89 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\...
user avatar
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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 ...
Ema Martins's user avatar
3 votes
1 answer
113 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 ...
Miumiu's user avatar
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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 ...
JellyTree's user avatar
3 votes
0 answers
127 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 ...
Olayo's user avatar
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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 ...
Olayo's user avatar
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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 ...
Adrian Surani's user avatar
0 votes
1 answer
395 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 ...
BigChungus443's user avatar
1 vote
0 answers
72 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 ...
BigChungus443's user avatar
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1 answer
239 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&...
BigChungus443's user avatar
2 votes
1 answer
412 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 ...
BigChungus443's user avatar
0 votes
2 answers
658 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 ...
ozgun can's user avatar
3 votes
0 answers
183 views

Improving the intuition for the 2d fourier transform

As far as I understand, the 2d fourier transform is calculated as following: ...
dmmpie's user avatar
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1 vote
1 answer
95 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 = ...
Gze's user avatar
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2 answers
70 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 ...
Nervous Hero's user avatar
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295 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-...
user299980's user avatar
0 votes
1 answer
151 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?
Luca Leone's user avatar
0 votes
1 answer
334 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 ...
DSP_CS's user avatar
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1 vote
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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 ...
kapytaine's user avatar
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3 answers
312 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 ...
Doohyeon Won's user avatar
1 vote
2 answers
2k 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 ...
average_discrete_math_enjoyer's user avatar
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4 answers
277 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 ...
Daron's user avatar
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1 answer
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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 ...
Judita's user avatar
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0 votes
1 answer
111 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 & \...
ian's user avatar
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1 vote
1 answer
140 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-...
Borut Flis's user avatar
2 votes
1 answer
280 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 ...
Mathpdegeek497's user avatar
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0 answers
190 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 ...
LunaLOVEGOOD's user avatar
1 vote
1 answer
610 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 $x_2[n]$, ...
Orpheus's user avatar
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1 answer
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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 ...
Orpheus's user avatar
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1 vote
1 answer
120 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 ...
BigBear's user avatar
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1 vote
1 answer
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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 ...
user58865's user avatar
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1 answer
92 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 ...
achhainsan's user avatar
5 votes
1 answer
246 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 ...
Saeed's user avatar
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1 answer
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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 ...
alejnavab's user avatar
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2 votes
2 answers
538 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}\...
Kinka-Byo's user avatar
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0 votes
2 answers
4k 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 ...
Sam's user avatar
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-1 votes
1 answer
97 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 ...
Orel Lachiani's user avatar

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