Questions tagged [fourier-series]
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286
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Proof of the convolution property of Fourier Series in continuous time
I am facing problem in understanding the proof of Convolution property of Fourier Series (FS) in continuous time CT;
that is: $$\mathrm{FS} \big\{x_1(t)\star x_2(t)\big\}=T\sum_{n=-\infty}^{\infty}...
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Uniqueness of Fourier Series Representation and the Fourier Transform of Periodic Signals
If we are given a signal of the form $$x(t) = \sum_{k = -\infty}^{+\infty} a_k e^{j k \omega_0 t},$$ can we call it a Fourier Series representation of $x(t)$ right away?
Suppose we are given the ...
user33568
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Fourier Coefficients
Consider the signal
$$x(t)=\cos(2\pi t)$$
Since $x(t)$ is periodic with a fundamental period of $1$, it is also periodic with a period of $N$, where $N$ is any positive integer.
What are the ...
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What is the exact meaning of the output of the Discrete Fourier Transform
I'm fairly new to the subject, but so far my understanding that this would be a transform you could use to go from a discrete set of data, say [1, 0, 1, 2] to a continuous sinusoidal function in the ...
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Inverse Discrete-Time Fourier Transform of $X(Ω)=jΩ$
I am trying to solve it by using the properties but I can’t seem to find the same solution as on my book.
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Fourier components of $\cos(2\pi f_1t)$
I have the signal $s(t) = \cos(2\pi f_1t)$ and I am looking for its components vs the Fourier basis, over the interval $[0, T]$. The formula for computing the coefficients is
$$
s_n = \int_{t_0}^{t_1} ...
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Fourier coefficients of product of two periodic signals
question:
If $x(t)$ and $y(t)$ are two periodic signals(both with period T) with Fourier coefficients $c_{n}$ and $d_{n}$ respectively then, Fourier coefficient of $z(t)=x(t)\cdot y(t)$ is:
(a) $\...
user33321
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Finding the fundamental frequency of a periodic signal
Suppose we have the signal $$x(t) = e^{j\omega_1 t} + e^{j\omega_2 t} + e^{j\omega_3 t},$$ where all the frequencies are rationally related (that is, the ratio of any pair of frequencies is a rational ...
user33568
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Rationally related frequencies and the Fourier Series representation
Suppose that we have the signal $$x(t) = e^{j\omega t} + e^{j\frac{3}{2} \omega t},$$ and we want to find a Fourier Series representation for that signal. Is this possible? According to my ...
user33568
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Basis signals for the discrete-time Fourier Series
I am using Alan Oppenheim's Signals and Systems and I am a bit confused by the notion of discrete-time periodic exponentials as basis signals for the discrete-time Fourier Series (and later the DTFT). ...
user33568
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Pulse wave question
Wikipedia, fount of all knowledge (Ha! LOL), gives a formula for a pulse wave here:
The formula is:
$$f(t)=\frac{\tau}{T}+\sum_{n=1}^{\infty}\frac{2}{n\pi}\sin\left(\frac{\pi n \tau}{T}\right)\cos\...
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Fourier Series of Aperiodic convolution of periodic functions
we were given the following classic exercise:
Given two periodic signals $x(t), y(t)$ with fundamental period $T$
with Fourier series coefficients $c_m^x, c_m^y$ respectively, find the
Fourier ...
user33199
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Consider an ideal low pass filter $H(\omega)$, and the input to this filter is the periodic square wave $x(t)$. Find the output $y(t)$
The solution to the problem is $$y(t) = 5 + \frac{20}{\pi} \sin(\pi t) + \frac{20}{3\pi} \sin(3 \pi t) $$ and to get that the solution says to find the Fourier series expansion of $x(t)$ and I am ...
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Fourier series coefficient of signal when Time period is twice the fundamental period
My try:
First of all I tried observing the symmetry but I did'nt find any.So I tried to calculate the fourier series coefficient of the signal like this
First I differentiated the signal $x(t)$ so ...
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How to do simple extrapolation with Fourier transformation?
I have 1024 sample points, and I would like to do really simple extrapolation using Fourier transformation. First I apply Fast fourier transformation on the data. My first intuition was that I just ...
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Efficient format for 2D signal?
I'm trying to solve the following problem. I got a "low frequency" input 2D signal over a square region. I'll collect a few samples, somewhere around 10-30 maybe, the exact sample count will be ...
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Explanation of fundamental filtering's consequences on signal
Can anyone explain why exactly an "Overshooting" phenomena is observed when the fundamental harmonic is removed as seen on the figures? Is it technically right to call this "overshooting" at all ? If ...
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Fourier transform
Fourier transform of a DC signal is an impulse at the origin.
Now if the spectrum of a signal is the plot of the amplitudes of the respective frequencies of each harmonic, then the Fourier transform ...
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Fourier Series Coefficients
Question:
The fourier series coefficients is given as:
$$c_k= \begin{cases}
1 \qquad & k \ \text{ even} \\
2 \qquad & k \ \text{ odd} \\
\end{cases}$$
the period of the signal is $T=4$, ...
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Fourier series - time shift and scaling
What will be the new Fourier series coefficients when we shift and scale a periodic signal?
Scaling alone will only affect fundamental frequency. But how to calculate new coefficients of shifted and ...
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Dirichlet Conditions
What is meaning of a signal having a "finite number of maxima and minima during any single period of time"?
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Difference between frequency components and harmonic components - Fourier
What is the difference between frequency components and harmonic components? The first concern the continuous domain of frequency, while the second concern the discrete domain of frequency ($f_{k}=kf_{...
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Can I compute Fourier series without looping through all frequencies?
I need to compute Fourier series of an audio stream.
But DFT/FFT is slow.
Are there any ways to compute Fourier series of a signal without using the Fourier transform to check if whether a frequency ...
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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 ...
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How to get the Fourier series using Python's $\tt fft$
I Would like to be able to reconstruct every individual sinusoid that makes up a Discrete signal. I Have the following signal: (I am working in Python)
The signal is essentially an array with about ...
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Why doesn't a sudden loud noise sound high pitched?
The jumpiness or high change of a signal is due to the higher frequency components in the signal. So if I have a sound signal that increases suddenly in amplitude, why doesn't the signal sound very ...
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Fourier series approximation: DC component and fundamental frequency
In the linked image below, what is meant by plotting the DC component and fundamental frequency for a Fourier series approximation?
For dot point 1 does it want me to graph just the DC component ...
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How can $a_n$ for the trigonometric Fourier series be $0$ for a non-odd square signal?
Can $a_n$ for the trigonometric Fourier series be $0$ for a non-odd wave? Since at the moment the square wave is not even nor odd. Square wave is given by:
$$
s_2(t)=\begin{cases}
-4, & 0\leq t\...
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Why can Fourier series be used on a non-repetitive function?
I was just wondering why Fourier series can be used on the function in the linked image. This is since I thought the function had to repeat itself to use Fourier series on it. Or is it saying a period ...
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Basic Fourier Series Not Understanding how to interpret n values [closed]
I have a question regarding $n$ in the Fourier series. If a question states "find the Fourier series for (any function) and find the values of an $b_n$ etc", say I'm finding it from the trig way when $...
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How can understand periodicity of a Signal from frequency domain representation?
Is it possible to say a signal is periodic from its frequency domain representation? A periodic signal is sum of its sinus and cosinus. Frequency translation of sinus and cosinus functions are ...
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Effects of interchanging sine terms with cosine terms
Suppose we have a real signal $x(t)$. Now, we know that $x(t)$ can be represented as a sum of sines and cosines.
w be the angular frequency.
If $a(\omega)$ be the coefficients of the cosine terms, ...
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Why are wavelet transforms (Multi Resolution Analysis) used more in practice for compression rather than Fourier series?
I know that both Fourier and wavelet can be used for compression of signals.
The Fourier series guarantees that it gets the closest approximation of the original signal in the least squared sense. ...
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Unipolar Transmission Spectrum
I am currently trying to understand mathematically the amplitude shift keying modulation technique. My question is, say I wanted to compute the Fourier series of a basic pulse train, it is quite ...
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Range of function $ \frac{1}{n\pi}(1-\cos(n\pi))\sin(\frac{n\pi}{2})$ and determination of Fourier coefficients
I am computing the Fourier series expansion of the given signal $x(t)$. In that I am having difficulties in calculating range of function $ \frac{1}{n\pi}(1-\cos(n\pi))\sin(\frac{n\pi}{2})$ (or) I am ...
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fourier series fitting matlab
I am using cftool in Matlab to fit time series values to Fourier model with 8 terms:
...
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RMS of absolute-valued sinusoid
A signal $$x(t)=\left|\sin(2\pi(1000)t)\right|$$ ($\left|x\right|$ is the absolute value function) is fed to an ideal low-pass filter with cutoff frequency of $1500\,\mathrm{Hz}$ to produce an output $...
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Fourier-series odd vs even square wave?
I really need to know the difference when doing a fourier seires between even and odd square waves. I've been trying to understand but I just get the same results and the same spectrums... Is the ...
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Why Fourier series and transform of a square wave are different?
Here is a square-wave presented by Fourier series perspective:
Above coefficients shows that a square-wave is composed of only its odd harmonics.
But here below a square-wave is presented by ...
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Periodicity of Fourier series's coefficients
Why exactly continious Fourier-series's coefficients aren't periodic like coefficients of discrete Fourier series (DFS)? $e^{-j2\pi}$ is periodic in both sequences.
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Discreteness and periodicity in Fourier transform
Why discreteness in time / frequency domain dictates periodicity in the other frequency / time domian?
For example the DTFT is perodic in frequency?
Why it doesn't contain all the frequencies?
Why ...
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Fourier series expansion of $\exp[-j2kA\sin(\omega t)]$
I've been going through this paper where we exploit the well known Fourier expansion of the model signal as shown in the image below.
I've never come across this well known fourier expansion before. ...
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Discrete Fourier Series - Time Index
I have been trying to understand Discrete Fourier Time Series (NOT Transform).
It is defined as
$$a_k = \sum_{n=0}^{N-1} x[n]e^{jk\frac{2\pi}{N}n}$$
where N is the Time period and an integer (by ...
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How we can obtain the time-integration of a sine signal?
could we calculate the below integral by the Fourier series or the Fourier transform properties?
$$\int_{-\infty}^t \sin(\omega_0\tau)d\tau=?$$
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What is the time-integration property in the Fourier series analysis?
In the continuous Fourier series properties for a periodic continuous-time signal, we have time-integration property:
$$
\int_{-\infty}^t x(\alpha)d\alpha \leftrightarrow \frac{a_k}{jk\omega_0}
$$
...
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Inverse fourier transform of Hermitian function, getting an imaginary part
in the cartoon below
It shows that if we take the inverse Fourier transform of a Hermitian function, real part even and imaginary part is odd we should get a purely real function in the time domain.
...
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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?
user18425
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Signal equation using signal waveform and Fourier series?
Part A: Obtain signal waveform from mathematical equation of the signal
Let a sinusoidal periodical signal is represented by an equation
$$y=f(t)=10+10\cos\left(\frac{2\pi f_1t}{T} +\frac{\pi}{6}\...
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Deriving the DFT magnitude of $A\cos(2\pi nk/N)$
Given that $$x(n) = A\cos(2\pi nk/N),$$ the $N$-point DFT of $x(n)$ can be expressed as follows—the derivation can be found in here: $$X(m) = \color{red}{\frac{A}{2}\sum_{n=1}^{N-1}e^{-j2\pi n(m-k)/N}}...
