Questions tagged [fourier-series]

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48 views

The magnitude spectrum of a sharpening filter

I'm trying to derive an expression for the magnitude spectrum of the following sharpening filter. $$ g(m,n) = \delta(m,n)+\lambda (\delta(m,n) - h(m,n)) $$ where $\lambda$ is some positive constant ...
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1answer
26 views

Why Does ASK Modulation Create Fourier Sidebands?

I know why analog amplitude modulation has side bands, it is related to (fc+fd) and (fc-fd). But what about DAM? ASK(DAM) is a type of digital modulation, and there are only two state: carrier signal ...
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0answers
14 views

Fourier series representation properties

Solve this problem part (c) and (d). Part (a) ans is a0=1 , a1=a-1=1/2 Part (b) ans is b1=b*-1=e^-jpi/4 /2
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1answer
64 views

Why is the continuous time Fourier series of DC signal an impulse?

In case of continuous time Fourier transform(CTFT), I can easily calculate the Fourier transform of DC signal by using Fourier duality or inverse CTFT. But I don't know how to calculate the continuous ...
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1answer
64 views

Fourier transform of time division

I know that Fourier transform of $t^n f(t)= i^n \frac{d}{d\omega^n} F(\omega)$. But does this work when $n<0$? Is there any direct relation to compute the Fourier transform of $\frac{f(t)}{t}$?
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1answer
34 views

multiplication of a function with a Fourier-transformed equals to Fourier-transformed with a function

I already showed b item using the fact that it is $h\left(0\right)=\int \:f\left(t\right)g\left(0-t\right)dt$ I struggle a lot of hours trying to find the trick in item C. Can anyone help please ?
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2answers
325 views

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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3answers
193 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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2answers
606 views

What does it mean for a function to have frequencies?

In a lecture, my professor mentioned that $\cos$ has two frequencies. I see that using the inverse Euler's formula we can express $\cos$ as a some linear combination of complex exponentials, each with ...
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1answer
49 views

How to find inverse Fourier transform of summ of delta functions?

I am practicing for my exam that I have this semester and I stumbled upon this one. How can i find inverse Fourier transform given: $$ X(j\omega) = \sum_{k=-\infty}^{\infty}\delta(\omega-2k+1) $$
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2answers
496 views

Fourier series of cycloid

What is the Fourier series representation of a cycloid? The parametric representation of the curve is as follows. $$ t=\dfrac{\theta-\sin\theta}{\pi}\\ x=\dfrac{1-\cos\theta}{\pi} $$ The period is $2$...
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1answer
24 views

Finding the discrete time Fourier series for signal

I think I did everything correctly here, but I must be missing something still. We have the following signal: My approach: We are told that the signal has period $N = 4$ We know $$Y[k] = \frac{1}{N}\...
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1answer
25 views

Solution verification for this Fourier series problem

We have a signal with period $T = 2$ We want to find the continuous time fourier series for this signal. Since $T = 2$, $\omega = \pi$. All we have to do know is find the frequency domain. $$x(t) = \...
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1answer
39 views

Finding the discrete time fourier coefficients to this problem

I'm trying to find the fourier series to this discrete time signal. $$x_1[n] =\begin{cases} +\frac72&\text{if }0\le n \le 4\\ -\frac72&\text{if }5\le n \le 9 \end{cases}$$ My approach: We see ...
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1answer
29 views

Represent DFT coefficients with respect to Continuous time-Fourier series coefficients

Does anyone know how to represent the Discrete Fourier transform (DFT) coefficient, $X[k]$, with respect to the Continuous time-Fourier series (CT-FT) coefficient, $X_k$? I come to the conclusion as $...
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1answer
38 views

Why is the Fourier Series a special case of the Fourier Transform and not the other way around?

I was reading a text book on the frequency domains in signal processing and my understanding is that the Fourier Transform considers signals that are a-periodic in time while the Fourier Series ...
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1answer
40 views

MIT 6.003 HW#8 Problem 4 - Fourier Coefficients of Triangle Wave

In the mentioned homework, part of the solution involves finding the Fourier coefficients of the triangle wave. The solution mentions that we can express this function as follows: What does that ...
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2answers
227 views

Problems computing the DFT of finite length sequence

I am having trouble finding the same answer as the solution manual for this sequence. The problem asks to compute the DFT of $$ x[n] = \begin{cases} 1 & \text{for even } n \in \{0\...
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1answer
36 views

Mistake or not - Fourier Series of x(2t+3)

I have a couple of resources I have from my university I had being checking and I found this: Find Fourier Series coefficients of x(2t+3). x(t) is continuous and periodic by T. I see this solution: ...
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0answers
35 views

How to compose a Discrete Prolate Spheroid (DPSS) dictionary?

I have a model of signal as $$ Y=AX + N $$ where $Y$ is received data in a linear array, $A$ is steering matrix, $X$ is data of sources and $N$ is noise. If $A$ has the form of $A=\exp(\alpha \sin(\...
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1answer
48 views

Output of LTI (in time and frequency $\omega$ domain) : when input goes through LPF

I would like to raise a mathematical question : Let's say we are been given : $$x(t) = \begin{cases} \cos(\pi t) & |t| \leq 0.5 \\ 0 & \textrm{otherwise} ...
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1answer
53 views

Fourier coefficients of discrete difference of a square wave

I have a discrete square wave $f(t)$ where $t \in \mathbb{N}$, of amplitude $A$, period $T$ and duty cycle $1/T$ $$ f(t) = \left\{\begin{matrix} A, & \mathrm{if}\;t=Tn\\ 0, & \mathrm{if}\;t\...
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0answers
45 views

Forecasting that FFT result

I have the following FFT result charts. Searching for a way to forecast the future AMPLITUDE steps from the Time Domain Plot if possible, or the next maximum/minimum deviation based on some previous ...
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2answers
7k views

The difference between DFT and DFS

In the literature, I've found that DFS and DFT are one and the same. If they are one and the same why to use two different names for them? If there is really a difference what is it and what is the ...
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1answer
59 views

If a time-series has odd number of samples does it have no energy at Nyquist frequency?

Suppose I have real time series A with n samples and time-spacing dt and I want to analyze ...
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1answer
235 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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5answers
8k 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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3answers
59 views

Sum of equidistant exponents

Consider the next sum \begin{equation} \sum_{k = 0}^{N - 1}e^{-j\frac{2\pi}{N}k} \end{equation} Its geometric meaning is the sum of uniformly distributed vectors on the unit circle. Thus, we can say ...
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0answers
29 views

How to get coefficient of the discrete fourier Series from the fourier transform

Given $X(v)$ the Discrete Fourier transform of a discrete periodic signal $x(n)$, it's possible to arrive to the $c_k$ of the Fourier series $$x(n)=\sum_{k=0}^{n-1} c_k \exp(2\pi i k t) $$ directly?
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21 views

Evaluate a complex exponential at negative infinity [duplicate]

I am learning about the properties of the Fourier Series (FS), which is defined by: $$x(t) = \sum_{k=-\infty}^{\infty}c_{x}[k]e^{j2\pi kt/T}\tag{1}$$ where $$c_{x}[k] = \frac{1}{T}\int_{T}x(t)e^{-j2\...
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5answers
238 views

Finding Fourier series coefficients for discrete time signal

Let $x[n]$ be a periodic sequence with period $N$ and Fourier series representation $$x[n] = \sum _{k=<N>}a_ke^{jk\frac{2\pi}{N}n}$$ Determine the Fourier series coefficients for $$y[n] = \...
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56 views

Rectangular Pulse Train and Sinc Function

I wanted to ask that in frequency domain the rectangular pulse is a sinc function, so is this sinc function periodic or aperiodic? Also if signals that are continuous in time domain then they are ...
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1answer
31 views

Why do we scale bins in FFT in this code?

Hi I am learning FFT I am confused about this bit of code: what is the reason for scaling the sampling frequency and what is bin scale and why and when do we use it? thank you ...
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2answers
190 views

Autocorrelation for periodic signals

Autocorrelation for power signals is defined by $$R_x(\tau)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^Tx(t)x^*(t-\tau)dt\tag{1}$$ Is it true that for periodic signals $(1)$ can be computed by $$R_x(\tau)=...
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89 views

Can different Discrete-Time-Fourier-Series(DTFS) coefficients have the same discrete sequence in the time domain?

Please, check the following discrete periodic sequence when the period $N=2$. $x[k]=\exp(j\frac{2\pi}{N}k), N=\text{period}$ $..., x[0]= 1, x[1]= -1, x[2]= 1, x[3]= -1, ... , N=2$ According to my ...
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2answers
102 views

Can I sample at Nyquist rate if I know that my samples are taken only at the signal's maxima or minima?

I know that in general the sampling rate, $f_s$, must be greater than twice the highest frequency of the signal, $f$. If I sample at the Nyquist rate, it can lead to the following: However, if the ...
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1answer
129 views

How to save Fourier series approximated signal to a WAV file

I changed this Matlab/Octave code to approximate square wave by using combination of Fourier series and Fejér taper: ...
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0answers
23 views

Redundancy when using Nyquist rate for a time series

I sample temperature at one sample per second (the hardware I am using takes the temperature at this rate, so this is the max I can efficiently sample.) The Nyquist rate for this signal would be 2 ...
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1answer
59 views

why does the additive synthesis method for a triangle wave require amplitude scaling by 8/pi^2?

I had to make a bunch of band limited digital triangle waves recently, so I went to (where else) wikipedia for the equations. I noticed that there is a constant amplitude scalar of ...
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1answer
100 views

Fourier transform of the sampled signal

I want to calculate Fourier transform of the sampled signal in two ways. Let $$s(t) = \sum_{k = -\infty}^{\infty}\delta(t - kT)$$And $z(t) = x(t)s(t)$. So we have $$z(t) = \sum_{k = -\infty}^{\infty}x(...
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7answers
6k views

What is the most lucid, intuitive explanation for the various FTs - CFT, DFT, DTFT and the Fourier Series?

Even after having studied these for quite sometime, I tend to forget (if I'm out of touch for a while) how they are related to each other and what each stands for (since they have such similar ...
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1answer
169 views

Duality in the discrete-time Fourier series

Suppose $g[n]$ is periodic with fundamental period $N$ and $f[k]$ being its Fourier coefficients i.e. $$ f[k] = \frac{1}{N}\sum_{n=<N>}g[n]e^{-j\frac{2\pi}{N}nk}$$ In more convenient notation $$...
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2answers
44 views

Where did the length of time’s period disappear in Periodic Fourier Series Discrete Time

In continuous time Periodic Fourier Series has smallest n as possible, since it is an integral and a length of the repeating time (period time) which is T0. In discrete time however we don’t have ...
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28 views

How to apply FT to real-life signal that labeled in seconds, not radians

In training examples we always do a transformation on signals which have t-scale in labeled in radians. I understand that Pi is just a number, but I still have some troubles to understanding how to ...
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2answers
29 views

How can it be that there is a series of integrals in Fourier Series if it’s a projection on a continuous basis?

If the process of finding Fourier coefficient is finding the projection of a signal on a member from an orthonormal basis, basis which is continuous in frequency. How can it be that Fourier ...
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34 views

frequencies in frequency spectrum with no correlation together

I have a lack of understanding of the following questions: If I have a signal from a motor that is recorded with an accelerometer. And the rotating speed of the motor is 150Hz(rpm 9000 ), I can see in ...
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2answers
36 views

Where did the k of $a_k$ disappear from Fourier Reverse Transform if $\omega=\omega_0k$?

Where did the $k$ of $a_k$ disappear from Fourier Reverse Transform if $\omega=\omega_0k$? We turn $\omega0$ to be $d\omega$, but $\omega=\omega_0k$, so shouldn’t there be a $k$ in the reverse ...
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1answer
46 views

harmonic waves as integer multiple in spectrum

i have a motor that is rotating with a certain frequency. If i check the frequency spectrum it contains a peak on 150 hz. Also i have peaks at 300,450,600 ... i guess that those peaks are harmonics. ...
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1answer
88 views

Fourier transform of t*rect(t)

In my previous post I asked for help for a Fourier transform of $$ t \text{rect} ( t- \frac{1}{2} ) $$ and I think I’ve understand the process. Now I’ve 2 another similar Fourier transform to do , I ...
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1answer
848 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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