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Questions tagged [fourier-series]

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16 views

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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0answers
25 views

How to find “The complex Fourier coefficients $X_n$”

In the picture I have the question with my attempt , I could find everything without problem but the 3rd branch's asking about $X_n$ (Fourier coefficients ) since I have $X_3$ and $X_4$ are equal 0 , ...
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1answer
19 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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1answer
37 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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2answers
113 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
38 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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0answers
20 views

Why can't I directly apply difference property on this Fourier Series question?

I am given a discrete time signal, with fundamental period 10 and coefficients $a_k$, which is as follows: $$ x[n] = \left\{ \begin{array}{ll} 1 & \quad 0 \leq n \leq7 \\ ...
2
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1answer
73 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
99 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
54 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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1answer
34 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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2answers
100 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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2answers
388 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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2answers
47 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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0answers
29 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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0answers
205 views

Cosine Fourier Transform and Phase

If we do FT of cosine wave, then sine wave will be orthogonal. So, the imag parts of FT will be 0. This is my think. But, the result isn't show that. Like this. Figure 1. Real and Imag parts (Y label ...
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0answers
43 views

calculation of derivative using FFT

Here is my issue: I am trying an algorithm using FFT to fit and get a function's approximative in order and calculate it's derivative in an more easy way ( for any data), but I get some discontinuity....
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1answer
82 views

Fourier coefficients of sum of two functions with different fundamental periods?

If we assume $\quad x(t)\leftrightarrow a_k\:$ and it is periodic with fundamental period T. How can we determine the fourier coefficients of the sum $x(t-7)+x(-2t+3)$ I know that $x(t-7)\...
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1answer
47 views

Is there any special when all Fourier components have the same angle?

When a square wave doesn't jump, its oscillators aren't aligned: But if they are in sync, the wave will jump to its extrema: However this is just an example of square waves. The tool Understanding ...
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4answers
80 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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1answer
53 views

How to determine which Fouriers Series terms to use to approximate a signal?

I have a signal (a time-series of air temperature values) that I can approximate quite well with a Fourier series. However, the number of terms in the series grows rapidly, to the point that 30 - 40 ...
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1answer
130 views

characterization of DC component

Consider the following two statements: In time axis: A signal without a DC component is a signal which doesn't have the zero frequency (the DC frequency) A signal without a DC component is averaged ...
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1answer
703 views

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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0answers
27 views

Applying a filter described by differential equations to a signal with autoregressive noise

I have been trying to resovle a problem for quite some time now without success. It is a signal processing issure, so I hope here I can find the help I need. I use Scilab for my computations, but I am ...
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1answer
69 views

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 ...
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0answers
14 views

Fourier series coefficients if time period changes [duplicate]

We have to find the Fourier series coefficient of x(t) = cos(2πt) if we were to regard it as a signal with period=3 when it's fundamental period is 1. Now I have a doubt. Since it's fundamental ...
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1answer
60 views

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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1answer
72 views

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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1answer
62 views

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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1answer
62 views

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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1answer
118 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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2answers
1k 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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1answer
43 views

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 ...
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1answer
41 views

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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2answers
107 views

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 ...
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1answer
233 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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2answers
147 views

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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1answer
938 views

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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0answers
29 views

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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2answers
127 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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2answers
111 views

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 ...
4
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1answer
215 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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1answer
2k 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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1answer
97 views

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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1answer
329 views

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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2answers
102 views

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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3answers
4k 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 ...
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1answer
64 views

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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0answers
112 views

Problem on Complex Fourier Series

I am trying to solve this problem but I am stuck. Please I would appreciate if some one guide me on how to proceed
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1answer
389 views

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 ...