Questions tagged [fourier-series]

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16 views

Which frequency bins give the best interpolation for the derivative of a function?

A function $u:[0,2\pi]\to\mathbb R$ sampled over $N$ equidistant points $\theta_j=(2\pi/N)j,\, j = 0, \dots, N-1,$ can be interpolated by a set of functions $\{u_{k_0}\}$ enumerated by integers $k_0\...
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26 views

Fourier series expansion of $ x_1(t) = \sum _{-\infty}^{\infty} \Delta (t-2n) $

I want to evaluate the Fourier series expansion of $ x_1(t) = \sum _{-\infty}^{\infty} \Delta (t-2n) $, where $ \Delta (t) $ is a triangular function defined as: I have done the following ...
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32 views

Fourier series representation of a full wave rectifier output

I am trying to compute the Fourier series representation of a full wave rectifier output. The equation of the signal is: $ x_8(t) = | \cos (2 \pi f_o t) $ | I have tried to find the Fourier series ...
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27 views

Fourier Transform: interpretation of continuous spectrum at specific frequencies

B. P. Lathi in his book "Principles of Linear Systems and Signals" mentions in the Fourier Transform: When $x(t)$ is periodic, the spectrum is discrete, and $x(t)$ can be expressed as a sum of ...
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149 views

Aliasing when interpolating with DFT?

I'm coming from an understanding of the continuous-time Fourier Transform, and the effects of doing a DFT and the inverse DFT are mysterious to me. I have created a noiseless signal as: ...
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1answer
447 views

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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659 views

Is Fourier series a sampled version of Fourier transform?

I recently learned about dtft and how dft/dfs is the sampled version of dtft. I was wondering if Fourier series is also obtainable by sampling Fourier transform? I am a noob in the subject so sorry if ...
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1answer
62 views

Proof of First Difference Property for Fourier Series

I am having trouble with deriving a proof for the first difference property for the Fourier Series. Here is my attempt at the derivation: $$ y[n] = x[n] - x[n-1] $$ Fourier Series Representation: ...
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812 views

How to get Fourier coefficients to draw any shape using DFT?

I'm teaching myself about Fourier Series and the DFT and trying to draw a stylised $\pi$ symbol by fourier epicycles as detailed by Mathologer on youtube (from 18:39 onwards), and the excellent ...
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47 views

How to find fundamental frequency of two signals?

I am facing difficulty with finding fundamental frequency of signals I mean by fundamental frequency=(1/Time period) Correct me if I am wrong consider two continuous time signals with Time period ...
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1answer
36 views

How do I obtain the fourier series coefficients for a signal obtained by multiplication of two signals of different frequency?

What i assume here is that LCM of time periods of the two taken signals exist that is signals periods are not like pi/2 and 1 but are rather like 1 and 2 (just an example) I am given fourier series ...
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6k views

How to Remove the Periodic Oscillations from a Signal

The task that I have is to remove the annual and semiannual oscillation from a set of temperature measurements, taken over several years, by means of least squares method. I found the method ...
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234 views

integration property of fourier series

Please help me sort this issue out. The integration property in Fourier series is as follows: So, for proving the above property, i took this approach: This is where my doubt is. Some books and ...
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97 views

Phase diagram of a rectangular pulse with Fourier Series - help understanding

I understand perfectly fine how to plot the magnitude of a Fourier series, but I'm having serious trouble understanding how to plot the phase spectrum. Below is a picture of a rectangular pulse. The ...
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1answer
55 views

fftshift in MATLAB with even number of data points in double sided spectrum

I have a question with reference to this Table. With even N, the frequency axis extremes should be $\pm$Fs/2, where Fs is the sampling frequency. However in the array we have only one value ...
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1answer
35 views

Is there a generalized method to get the input with the given output and the impulse response?

$y(t)=y(t+12), y(t) = x(t) \ast h(t)$ The continuous time signal output $y(t)$ is a periodic square wave, 50% duty cycle pulse. The impulse response is a box function.($A = 1, T = 2$) By using ...
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2k views

Intuition behind the scaling property of Fourier Transforms

The Fourier transform of $f(ax)$ is $\frac{1}{|a|}F(\frac{u}{|a|})$. So the frequencies are scaled horizontally but the magnitudes are also scaled when the graph of $f$ is scaled horizontally. On the ...
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9 views

2D Fourier Synthesis (IDFT) not yielding expected result

I am trying to recover a 2D signal using inverse DFT, to my understanding the IDFT outputs the coefficients of the fourier series of the original function up to the Nyquist frequency. So for example ...
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37 views

inner product zero?

I am studying about Fourier series from book"Signals and Systems Laboratory with MATLAB" I came across topic "Orthogonality of Complex Exponential Signals" I am confused in case when m=k, will the ...
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4k views

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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141 views

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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45 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 ...
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1answer
71 views

Fourier Transforms, symmetry, real/imaginary

I was hoping to clarify if the following was correct: -a real function (neither even nor odd) in time exhibits conjugate symmetry in frequency, so the real part of the frequency response is even, and ...
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181 views

The Fourier Transform of a periodic function and it's series

Let $X(f)$ be the Fourier transform of $x(t)$: $$ X(f) \triangleq \mathscr{F}\Big\{ x(t) \Big\} = \int\limits_{-\infty}^{\infty} x(t)\,e^{-j 2 \pi f t} \ \mathrm{d}t $$ $$ x(t) \triangleq \mathscr{F}...
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95 views

Why do waveforms that are symmetrical above and below their horizontal centerlines contain no even-numbered harmonics?

All About Circuits site states that waveforms that are symmetrical above and below their horizontal centerlines contain no even-numbered harmonics. Can somebody explain this mathematically, or point ...
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1answer
22 views

Sampling Frequency and Spectral Regrowth

Sampling a cosine wave of 10 Hz at Fs = 64 and number of samples Ns = 256 I setup my time vector for the cosine wave as n=(0:Ns-1)*(1/Fs) If I change the sampling frequency from 64 to 64.0005 I get ...
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1answer
494 views

How can I improve my fit of cosines to periodic data using Python?

I have a space-separated csv file containing a measurement. First column is the time of measurement, second column is the corresponding measured value, third column ...
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2answers
181 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 ...
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4answers
933 views

Formulas of the Fourier transform family

It has annoyed me that there doesn't seem to be a source online where the complete complex Fourier transform family is presented with every variable defined. The lack of definitions can be a nuisance ...
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2answers
99 views

What is the basic idea behind Fourier transform? [closed]

What is the basic idea behind (discrete and continuous) Fourier transform (FT)? In short, what is the difference between discrete and continuous FT? I have read multiple answers on the web related to ...
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239 views

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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Intuition behind FT of Dirac Comb [duplicate]

What is the intuitive explanation behind a dirac comb having a dirac comb as Fourier transform? How to calculate this is clear, but I cannot picture why this intuitively makes sense.
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118 views

What to do after this last step?

I am solving a question from book in which I have to use summation. It is as follows: $$ \frac{1}{10}\sum_{n=0}^{9} e^{-jk\omega_0n} $$ The value of $\omega_0$ is $\frac{2\pi}{10}$. What I ...
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1answer
29 views

Correct form of discrete-time Fourier series representation

As I see in this slides, Fourier series representation for discrete-time signal $s[n]$ with period $N$ is $\sum_{k = 0}^{N-1} c_k e^{j2\pi k n / N}$ According to Wiki, Fourier series representation ...
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44 views

How to find period of signal, when spectrum is known?

Here is signal in frequency domain .Division of frequencies does not give an integer number or real number. May it can be here another method of finding period for ths signal.
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1answer
50 views

How to do convolution in Fourier Series?

Two signals are given to me : $$x(t)=\cos(4\pi t)$$ $$y(t)=\sin(4\pi t)$$ I have founded their coefficients as follows: $$a_k = a_1=a_{-1}=\frac{1}{2} $$ $$b_k = b_1=b^*_{-1}=\frac{1}{2j}...
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62 views

How was this result on discrete Fourier series achieved?

I was trying to do the question 10, part b of the following document (https://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/assignments/MITRES_6_007S11_hw10.pdf) I was going through ...
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426 views

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) $\...
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1answer
105 views

Does the Fourier series coefficient of AC components remains same if DC component is subtracted form the given signal?

Suppose a signal is defined by $ x(t)= \begin{cases} t & 0\leq t \leq 1 \\ 2-t & 1\leq t\leq 2 \\ \end{cases} $ Since $x(t)$ has even symmetry, I can calculate fourier coefficient as $$ a_n = ...
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1answer
382 views

Spectrum of Cosine in Complex Form

The complex exponential form of cosine $$\cos(k \omega t) = \tfrac{1}{2} e^{i k \omega t} + \tfrac{1}{2} e^{-i k \omega t}$$ The trigonometric spectrum of $\cos(k \omega t)$ is single amplitude of ...
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1answer
36 views

Find Fourier series $f(t) = e^{jx t}$ , $−\pi < t < \pi$ [closed]

I need to find the Fourier series of the $f(t) = e^{jxt}$ , $− \pi < t < \pi$ What will be the first step to solve it?
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2k views

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 ...
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400 views

Why do frequencies of analog signals range from $-\infty$ to $\infty$ while frequencies of digital signals are restricted to $[0,2\pi]$?

In Fourier analysis while dealing with discrete-time signals, frequencies range from $0$ to $2\pi$ why? Intuitively how can i understand it?
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1k views

Scaling property of Fourier Transform

Problem 4.6(b) from Oppenheim, Wilsky & Nawab (2nd ed) reads: Given that $x(t)$ has the Fourier transform $X(j\omega)$, express the Fourier transform of $x(3t - 6)$ in terms of $X(j\omega)$. The ...
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1k views

Calculation of cosine in frequency domain instead of calculatin in time-domain followed by a FFT

I got an $N$, in my case 512, point FFT of a real-valued signal. Based on some calculation in my application I determine the parameters $k \in [1, N-1]$, the number of oscillations per period, $\phi \...
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58 views

Fourier series of $cos(\omega_0 t)$ in continuous time

Can any one please help me with understanding how we can calculate the Fourier series of Cos(w0t) using the formula: I saw that they did the following calculus, but I Don't really understand how we ...
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43 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 ...
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4answers
87 views

What is the moment when all oscillators aligned to make a jump called?

Say we have a square wave and its Fourier series. When the wave doesn't jump, its oscillators aren't aligned: But if they are aligned, the wave will jump: What is this moment called? They might not ...
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9k views

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 ...