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Questions tagged [fourier-series]

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4answers
973 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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1answer
113 views

Calculating original signal from Discerete Fourier Transform

I am trying to calculate the original equation using a DFT. I start with a equation, generate values from this equation and then get the dft of these values. The aim is to generate the original ...
3
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2answers
539 views

Convolution in frequency domain

Simple math question. The convolution theorem states that multiplication in time domain is equal to convolution in frequency domain and vice versa. There is a condition that the signal has to be ...
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4answers
7k views

About Fourier transform of periodic signal

In Fourier transform for periodic signal, I checked different books and I found a different explanation in each book. Let's take the explanation in Signals and Systems by Rajeshwari & Rao: The ...
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2answers
583 views

Finding Fourier Series Coefficients

I'm just beginning to learn about Fourier series and I'm trying to figure out how to find the Fourier series coefficients for $$x(t) = e^{j100\pi t}$$ I know that $$x(t) = \sum_{-\infty}^{\infty} a_{...
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2answers
6k views

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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1answer
214 views

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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1answer
2k views

Relation between a Fourier series harmonic component and its power

My question is about the meaning of power spectrum derived from the Fourier series coefficients. Fourier series is shown below: $$f(t)=a_0+\sum_{n=1}^{\infty}a_n\sin(\omega_n t)+\sum_{n=1}^{\infty}...
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4answers
518 views

Why Fourier series if Fourier transform can be calculated for both periodic and aperiodic?

While learning about Fourier Transform after Fourier Series, That we can calculate Fourier transform of periodic signals too. If we can take the Fourier transform of periodic signal too then my ...
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1answer
490 views

Determining Fourier Series coefficient for Discrete time

I am trying to solve the proof for determining the Fourier series representation of a periodic signal. I understand fourier series equation for Discrete time which is $$x[n] = \sum^{}_{k=<N>} ...
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1answer
4k views

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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3answers
1k views

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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1answer
156 views

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 ...
1
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1answer
111 views

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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1answer
51 views

Value of $A_k$ in Fourier series

Fourier series in continuous time domain while representing $a_k$ in rectangular form $$ a_k = B_k + jC_k$$ But when using the value of $a_k$ in the main equation: $$ x(t) = a_0 + 2\sum^{+\infty}_{k\...
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1answer
68 views

Fourier Series Proof

I want to ask Question about the Fourier series in continuous time domain while reading a book signals and systems Alan Oppenheim. I have confusion in understanding the statement on page 189 of its ...
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1answer
55 views

Fourier series in continous time domain

I want to ask Question about the Fourier series in continuous time domain. I am following signal and systems 2nd Edition by Alan Oppenheim. I have confusion in understanding the statement that ...
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0answers
70 views

the DFT of a periodic signal represented by a fourier series

If I have a signal represented by a Fourier series(like in the photo), which is sampled with $T_s$: $$x[n]=x(t=nT_s) = \sum_{m=-\infty}^{\infty}a[m]e^{j2\pi m(nT_s)/T_0} $$ How do I find its DFT? I ...
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6answers
3k views

When is the Fourier transform of a signal periodic?

I mean not the time-domain signal being periodic, but the Fourier transform being periodic.
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1answer
560 views

Bartlett's and Welch's Method for PSD

Lets say. I have an image of 100 samples, and I want to find the presence of smaller image of 24 sample using cross-correlation in the fourier domain. I use Bartlett's and Welch's method for PSD, ...
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1answer
235 views

Finding Correlation response in fourier domain

Lets say I have a system that is trying to find a small image (assume all images are grayscale) within in an image by using correlation. So this system has the baseline image, and I input 5 different ...
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1answer
86 views

Fourier series calculation [closed]

I have tried to solve, but do not know if the answer is correct or not. A person has a periodic voltage input to a circuit. The input repeats itself every 0.02 seconds i.e. the fundamental period is ...
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1answer
273 views

How to simplify the Fourier Series Using an Approximation?

I have a signal, $f(t)$. I know a function that can be used to generate this signal, such that I can determine its Fourier series. I want to express this Fourier series in simpler terms so that the ...
1
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1answer
176 views

What is difference between outputs of Fourier transform and Fourier series of a periodic square waveform?

We can use Fourier transform of an aperiodic signal and Fourier series of periodic signal. But we can use Fourier transform formula for periodic function also. Now, let us consider a periodic square ...
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1answer
255 views

How Fourier decomposition is performed?

The Fourier decomposition explains a time series entirely as a weighted sum of sinusoidal functions and with the Fourier series,it is possible to do it. But I have some doubts Suppose ,for any ...
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2answers
115 views

why fourier series gives lower amplitude for max value of signal

I want to approximate below signal using fourier series on Matlab. My code is below ...
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0answers
35 views

Approximating the fourier coefficients from a discrete time signal

Suppose I have a discrete time signal $x_t$ sampled with a frequency $f_s$. I know that if I take the discrete fourier transform of the signal and compute the power, I can easily (by visual ...
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2answers
168 views

Signal decompositon

I am not a good in writing algorithm but please follow below steps 1.There are 4 1D sinusoidal periodic signals.3 of them are given by \begin{cases} x(t)=4\sin(10\pi t) \\ y(t)=8\cos(20\pi t) \\ ...
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1answer
106 views

Change of discrete summation to definite integral

The Exponential Fourier Series for a signal is written as, $$x(t) = \sum_{n=-\infty}^{\infty} X_n e^{jnw_0t}\tag{1}$$ and, Fourier Coefficient, $X_n$, is written as, $$X_n = \frac{1}{T} \int_{t_0}^{...
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1answer
174 views

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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1answer
106 views

Exact formula for alias of Discrete Fourier transform for periodic sigals

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/...
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3answers
287 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 ...
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1answer
167 views

Signal Processing using Fourier Transform

How can I derive the fourier transform of ...
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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 ...
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1answer
100 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 $x(t) = > \cos4t +\sin8t+3$?
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1answer
168 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)...
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1answer
521 views

Deriving time-scaling property for Fourier Series

thanks for taking the time to help with this problem! I have to prove the time-scaling property: $$ x_{(m)}[n] = \begin{cases} x[n/m], & n=0,\pm m, \pm 2m,...\\ 0, & otherwise \end{cases} $$ ...
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1answer
236 views

DSP interview question: use of the identity in development of a significant transform

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 + ...
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1answer
72 views

Fourier synthesis

I know what I am attempting is not easy but I have a spectrum with peaks at 10Hz 20Hz and 30Hz. I also have various amplitudes at these peaks. I want to recreate my original signal. I initially ...
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1answer
3k views

Determine spectrum amplitudes for half-wave rectified sine

I am trying to learn how to solve a bunch of digital signal problems and I have trouble understanding the solutions provided by this book I'm using. Basically, this problem asks me to determine ...
3
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1answer
365 views

Estimate the Discrete Fourier Series of a Signal with Missing Samples

Assuming we have a discrete signal $ { \left\{ x \left[ n \right] \right\}}_{n = 1}^{N} $. Which has a Discrete Fourier Series. Now, assume I'd like to estimate its Discrete Fourier Series ...
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0answers
53 views

Power contained in a signal

Given a signal $$x(t)=16\cos(20\pi t+\frac\pi 4)+6\cos(30\pi t+\frac\pi 6)+4\cos(40\pi t+\frac\pi 3)$$how can I calculate the power contained in a frequency interval, say 12Hz to 22Hz. The total power ...
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1answer
75 views

Synthesis discrete time signal from fourier coefficients

Following information is given about a signal $x[n]$ $x[n]$ is real and even signal $x[n]$ has a period $N=10$ and Fourier coefficients $a_k$ $a_{11}=5$ $\frac1 {10}\sum_{n=0}^9 |x[n]|^2=50$ How can ...
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2answers
209 views

Trignometric Fourier series representation of a continous time signal

While learning Fourier series I read the definitions of representation for a continuous time signal $x(t)$ as: $$x(t)=A_0 + 2 \sum_{k=1}^{\infty} A_k \cos(k \omega_0 t) - B_k \sin(k \omega_0 t) \tag{...
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4answers
909 views

Why does the Fourier series not include phase information?

From my understanding, the Fourier series is a way to describe an arbitrary continuous signal in terms of sinusoids of varying frequencies and amplitudes, as shown here in Equation 1. Why are the ...
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1answer
663 views

Discrete Fourier Transform and Opposite Convolution Theorem

I am reading the Wiki for DFT. There is a part for circular convolution theorem which sounds a bit odd saying: $$ \mathcal{F} \left \{ \mathbf{x\cdot y} \right \}_k \ \stackrel{\mathrm{def}}{=} \sum_{...
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1answer
116 views

Why we need fourier transform of periodic signal although we have fourier series for periodic signal?

I was going through some of the basics of fourier series and fourier transform. And I came across one topic "Fourier transform of periodic signal". But I am not able to understnad why we need to go ...
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3answers
93 views

Fourier Transforms and Series for the NON mathematically inclined.

This would most likely be the opposite of this question ( Mathematically inclined Signal and Systems/Signal Processing book? ) I figured I'd ask here if there are any good books that while, ...
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1answer
84 views

Reason for bimodal behavior while low second fourier coefficient

If I have a time series (for eg. for 23 timestamps) and if I plot it and see that it is bimodal, that means it might be having high value of second fourier coefficient (with frequency = 2). But when I ...
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1answer
147 views

How to check if Fourier components are in phase of out of phase?

I have a time series (of 23 timestamps) of which I take the Fourier transform. Now the fourier transform has 23 imaginary values and each has an amplitude and a phase. When I get the phase angle, it ...