Questions tagged [fourier-series]

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3answers
200 views

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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4answers
737 views

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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1answer
296 views

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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0answers
157 views

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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1answer
921 views

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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1answer
898 views

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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1answer
5k views

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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1answer
46k views

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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3answers
3k views

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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2answers
550 views

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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2answers
646 views

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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3answers
562 views

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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1answer
83 views

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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1answer
995 views

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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1answer
1k views

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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3answers
3k views

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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2answers
2k views

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?
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1answer
231 views

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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1answer
2k views

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}}...
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1answer
73 views

Where does $\frac{N}{2}$ came from in approximating an N-point DFT?

I've came across the author saying that ... for a real cosine input having k cycles in the N-point input time sequence, the amplitude response of an N-point DFT bin in terms of the bin index m is ...
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1answer
285 views

Signal's Fundamental Frequency is different from Plotted Signal

I've been attempting to plot the following function using MATLAB: $$ x(k)=\sum_{n=11}^{50} \sqrt{n} \sin (2n\pi k) +\sum_{n=1}^{40}\sqrt[3]{n} \sin (3n\pi k) $$ Note that $k$ is a continuous variable....
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1answer
586 views

square wave frequency representation

New to this signals stuff and i'm confused about the frequency representation of the square wave. Correct me if i'm wrong, a periodic square wave is composed of odd harmonics sine waves which are ...
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1answer
154 views

Relation between Fourier Series & Fourier transform [duplicate]

So i was just revising some basic DSP concepts. Just wanted to verify this fact. Fourier series represents a periodic signal $\hat{x}(t)$ with period P as a countably infinite sum of sinusoids of ...
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1answer
369 views

How do I calculate the eigenvalues of a covariance matrix which contains harmonic functions

I have read that $ce^{-\frac{\phi_i-\phi_j}{\rho}}$ is a harmonic function of the form $e^{-in\phi_i}$, and therefore it's eigenvalues are one of the Fourier components of $(|\phi_i-\phi_j|)$. My ...
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1answer
774 views

Relation between sawtooth Fourier coefficients and its DFT

I'm having some trouble with understanding the DFT of a sawtooth single period signal and its relation with sawtooth Fourier coefficients. Let's say I have a signal $$ s(t) = \frac{At}{T} - \frac{A}{...
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2answers
3k views

How to find Fourier Series Coefficients

I saw many solved examples about this topic but again I coudn't come up with any solutions about this question. How can I find the Fourier Series coefficients of the following signal ? $x(t)=2 \cos(3\...
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2answers
1k views

FFT of SIN waves with different phase delays

I have come across a peculiarity of FFTs which has got me somewhat baffled. I've simply summed up 101 sine waves and taken the FFT using this matlab script : ...
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4answers
1k 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
200 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 ...
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2answers
1k 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
21k 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
755 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
9k 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
354 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
3k 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
866 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
932 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
6k 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
3k 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
164 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 ...
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1answer
128 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
61 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
85 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
62 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
73 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
5k 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
687 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
373 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
89 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
317 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 ...