# Questions tagged [fourier-series]

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### 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 ...
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### Why are Fourier analysis and transform only applicable for LTI systems?

Why are Fourier analysis and transform only applicable for LTI systems? What if the system is not LTI, won't Fourier analysis or transform be possible?
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### Where does $\frac{N}{2}$ came from in approximating an N-point DFT?

I've came across the author saying that ... for a real cosine input having k cycles in the N-point input time sequence, the amplitude response of an N-point DFT bin in terms of the bin index m is ...
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### Signal's Fundamental Frequency is different from Plotted Signal

I've been attempting to plot the following function using MATLAB: $$x(k)=\sum_{n=11}^{50} \sqrt{n} \sin (2n\pi k) +\sum_{n=1}^{40}\sqrt{n} \sin (3n\pi k)$$ Note that $k$ is a continuous variable....
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### square wave frequency representation

New to this signals stuff and i'm confused about the frequency representation of the square wave. Correct me if i'm wrong, a periodic square wave is composed of odd harmonics sine waves which are ...
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### Relation between Fourier Series & Fourier transform [duplicate]

So i was just revising some basic DSP concepts. Just wanted to verify this fact. Fourier series represents a periodic signal $\hat{x}(t)$ with period P as a countably infinite sum of sinusoids of ...
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### How do I calculate the eigenvalues of a covariance matrix which contains harmonic functions

I have read that $ce^{-\frac{\phi_i-\phi_j}{\rho}}$ is a harmonic function of the form $e^{-in\phi_i}$, and therefore it's eigenvalues are one of the Fourier components of $(|\phi_i-\phi_j|)$. My ...
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### Derive Frequency Representation of Impulse Train Function

I want to walk through the derivation of the frequency representation of an impulse train. The definition of the impulse train function with period $T$ and the frequency representation with sampling ...
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### Fourier Transform/Series DFT/DFS textbook problem (simple?)

Suppose $x_c(t)$ is a periodic continuous time signal with period 1 ms and for which the Fourier series is \begin{align*} x_c(t) &= \sum\limits_{k=-9}^9 a_k e^{j(2000 \pi k t)} \\ \end{align*} ...
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### Proof of properties of Fourier series in CT

I feel problem in understanding the proof of Fourier series properties Time scaling \begin{align} b_k &= \frac{1}{T}\int_{T}x(t)e^{jk\omega_0t}dt\\ & = \frac{a}{T}\int_{T/a}x(at)e^{jk(\...
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### magnitude and phase Fourier coefficients

While solving Fourier series coefficients in example, i found couple of things which confuse me. How the minus sign changes to plus sign $a_1= 1-\frac{1}{2j} = 1+\frac{1}{2}j$? After plotting the ...
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### Mathematical model of a signal in Compressive Sampling

Currently I am reading a paper on Compressive Sampling, and trying to understand each and every parts of it. When I came across the mathematical model of the signal in paper, I got confused. I have ...
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### Proof for determining Fourier coefficients

While determining Fourier coefficients we have this equation $$\int^{T}_{0} x(t) e^{-jn\omega_0t} dt = \sum^{+\infty}_{k\ =\ -\infty} a_k [\int^{T}_{0} e^{j(k-n)\omega_0t}dt]$$ I want to ask that how ...
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### how to create synthetic $1/f$ noise? [duplicate]

I am writing an app to work with synthetic time series data from a physics experiment. In our experiments we always have $1/f$ noise in our time series, but I haven't been able to find code/packages ...
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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 $1/... 3answers 291 views ### Can I study continuous time Fourier Transform and treat the rest as special cases Say I learned the theoretical result of continuous time Fourier transform. And I want to extends some results(say "convolution rule") to Lapace transform, Z transform, DTFT, DFT, Fourier sequence ... 1answer 167 views ### Signal Processing using Fourier Transform How can I derive the fourier transform of ... 2answers 1k views ### A clarinet has no even harmonics. What would produce no odd harmonics? According to this link, the waveforms of clarinets do not have even-numbered components in their harmonic series: A closed cylindrical air column will produce resonant standing waves at a ... 1answer 102 views ### Fourier series qn determine the fourier series coefficients Can someone please help me with this Fourier series question: Determine the Fourier series coefficients of$x(t)$given as$x(t) = > \cos4t +\sin8t+3$? 1answer 225 views ### Why does Fourier Series have$\sin$and$\cos$Components While we look at Fourier Series there are both$\sin$and$\cos$components.But I think$\sin$component is ony needed to describe wave.why there is also an$\cos$component in Fourier Series?$S_n(x)...
thanks for taking the time to help with this problem! I have to prove the time-scaling property: $$x_{(m)}[n] = \begin{cases} x[n/m], & n=0,\pm m, \pm 2m,...\\ 0, & otherwise \end{cases}$$ ...
I'm preparing interview and found this question. But I don't really understand what is the question. Does it ask about Fourier transform or Z transform? How the simple identity xy=\frac{1}{2}x^2 + ...