# 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 ...
1 vote
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### 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 ...
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### 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 ...
1 vote
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1 vote
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### 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-...
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 ...
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 ...
1 vote
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 $x_2[n]$, ...
848 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 ...
1 vote
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 ...
1 vote
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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 ...
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 ...
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 ...
430 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 ...
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}\...