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A “Fourier Phase” for (stationary) random processes?

Let $X_t$ be a real w.s.s. random process. Its spectrum is given by $S(f)=\mathcal{F}R_X(\tau)(f)$ where $R_X$ is the process autocorrelation. As $X_t$ is real, the spectrum will be real and ...
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23 views

How can I make the mean of samples be approximately equal to the mean of actual continuous signal?

Suppose there is signal f(t) that is continuous and periodic. It is known that this f is T-periodic. (but it's not necessarily a single cosine f(t).( I'd like to make the mean of samples be ...
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26 views

Frequency scaling property for Fourier series

For Fourier transform, there is an equation connecting time-scaling with frequency-scaling. (By scaling, I mean multiplying by constant for time or frequency) Is there such a relation for Fourier ...
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35 views

Help in understanding step function calculation

Dear community I would appreciate if you can help me understand these equations. I mean how did he jump from line 1 to line 2? How do u[n] get cancel? Then in the last line where did the "8" come ...
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How to check periodicity of $f(t)$ using samples

Suppose that we know that signal $f(t)$ is $T_1$-periodic. Let $f_1 = 1/T_1$. But we want to know whether signal is $T_2$-periodic also. Let $f_2 = 1/T_2$, and $f_2$ is positive integer multiples of ...
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29 views

If $f(t)$ is periodic, is there any $t$ that would equal to DC components?

Suppose $f(t)$ is periodic with period $T$. Would there be $t$ that would necessarily equal to DC component (it can be scaled)? By DC component, I mean $F(0)$ where $F$ is fourier coefficient of $f$. ...
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23 views

Is there anything similar to DTFT for Fourier series?

So if sampling condition is met well, with aperiodic signals we have discrete-time Fourier transform (DTFT) that allows us to get frequency-domain data that resemble continuous-time fourier transform. ...
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24 views

Convergence property of DTFT toward DFT when function is periodic

from Wikipedia: When the input data sequence $x[n]$ is $N$-periodic, DTFT can be computationally reduced to a discrete Fourier transform (DFT), because: $ X_{1/T}(f)$ converges to zero ...
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24 views

Is it possible for continuous fourier transform of a function to have values only on finite number of frequencies?

Is it possible for continuous fourier transform of a function to have values only on finite number of frequencies? Or do these values necessarily impulse values, not complex numbers?
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45 views

What is a window function with positive spectrum?

I need a real, symmetric window function $x(t) = x(-t)$ whose Fourier transform $\hat{x}(\omega)$ (also real and symmetric) is non-negative $\hat{x}(\omega) \ge 0$ for all $\omega$. The function does ...
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26 views

Is the DTFT of a sampled Gaussian a positive function?

I have an infinite sequence $x_{n}$ for $n \in \mathcal{Z}$ which is a sampled Gaussian function $x_{n} = \exp(-n^2/a)$ with a > 0. I need to check whether its DTFT $x(\theta) = \sum_{n \in ...
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35 views

What is the relationship between DTFT and continuous fourier transform?

As title says, what is the relationship between DTFT and continuous fourier transform? Let's say there is continious signal $f(t)$. Continuous Fourier transform convert this into $F(\omega)$. Now let ...
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30 views

Help understanding Wiener filtering formula

I would like some help interpreting the following formula, equation 1 from this paper: https://www.math.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/strela.pdf $\hat{X} = ...
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14 views

How to work with 4-1 multiplexer In digital logic?

Here is my image of multiplexer, http://d18khu5s3lkxd9.cloudfront.net//wp-content/uploads/2014/04/GATECS2014Q55.png and this one ...
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1answer
23 views

Understanding, Non-Negative Sparse Coding algorithm

I have a question regarding sparse coding, Non-negative sparse coding. Iterate until convergence: $ \mathbf{A_i} \leftarrow \arg \! \min_{A \geq 0} || \mathbf{X}_i - \mathbf{B}_i\mathbf{A}||_F^2 + ...
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1answer
64 views

Fourier transform and splitting frequency range into 4 channels

I have code example that divides audio frequency into 6 channels. It uses Fast Fourier Transform (FFT). Algorithm process the frequency range using 6 capture[x] samples based on the range of n between ...
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84 views

Discrete Fourier Transform question

Let $R_{M\times N}$ be a space of size $M\times N$. Define the 2D Discrete Fourier Transform of $f\in R_{M\times N}$ to be \begin{equation} ...
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146 views

About integrating product of two sinc function using Fourier transform

So the problem is which I think is pretty straight-foward by using Fourier transform and convolution property of two sinc functions and evaluating the convolution at 5. However, I got sinc(t) for ...
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1answer
43 views

Deriving the autocorrelation function for the ARMA model

Definitions The ARMA model $$x_n=-\sum_{p=1}^P a_px_{n-p}+\sum_{q=0}^Qb_qw_{n-q} \tag{1}$$ where $w_n$ is zero mean stationary white noise with unit variance. Question To derive the ...
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1answer
46 views

Yule walker equation limited matrix size

Definitions For an ARMA model $$x_n=-\sum_{p=1}^P a_px_{n-p}+\sum_{q=0}^Qb_qw_{n-q} \tag{1}$$ where $w_n$ is zero mean stationary white noise with unit variance. It is straightforward to show that ...
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32 views

If the signal's frequency is multiples of the first harmonic frequency, transform method similar to DFT but use less number of samples?

Suppose that a continuous signal $f(t)$ has the first harmonic frequency $f_1$. $f(t)$'s frequencies that are not integer multiples of $f_1$ are known to have zero signal magnitude $|F(\omega)|$. This ...
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1answer
44 views

DSP Time domain and frequency domain

I'm new here and wish to say hello to this great community. I'm starting to learn DSP, I don't have a lot of Maths background but I'm trying to learn. I am new to DSP too and I am reading this great ...
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1answer
33 views

Which of arithmetic, geometric or harmonic mean is the most appropriate in this case?

I have a software that periodically detects tempo out of an audio signal and I would like to compute the average tempo out of all the generated values. Example: ...
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1answer
89 views

$E[x_i^2 x_j^2]$ for white Gaussian noise

If $x_n$ is a discrete time random signal and is white Gaussian noise (ergodic and WSS) so $$E[x_n x_{n+l}]=\sigma ^2 \delta (l)$$ and $$E[x_n]=0$$ Where $n \in \mathbb{R}$ and $l\in\mathbb{R}$ ...
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1answer
114 views

matlab problem - removing frequencies after FFT, signal processing

I want to stress that this is not a coding problem, my problem is that i don't fully understand the mathematics surrounding the subject and that's why I believe I have a problem. I was given an ...
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1answer
52 views

convolution and associativity

Ok Let talk about this,... I am now so confused. 1-$$\mathcal{F}\Big\{c(x-x_0)b(x-x_0)\Big\}=\mathcal{F}\Big\{c(x-x_0)\Big\}\circ\mathcal{F}\Big\{b(x-x_0)\Big\}\\=\Bigg[e^{-2ix_0y}C(y) ...
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1answer
93 views

How one can show $P(ax+n|x)=P(n)$? [closed]

Let $x$ be a signal and $n$ be an independent noise. How one can show $P(ax+n|x)=P(n)$? Thanks. Well, let $y=ax+n$, so we have $n=y-ax$. Now if the probability density function (PDF) of $n$ for ...
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2answers
62 views

Convolution sum. Compute $y[n]=x[n]\ast h[n]$

Compute $y[n]=x[n]\ast h[n]$ $x[n]=(-\frac{1}{2})^2u[n-4]$ $h[n]=4^nu[2-n]$ In this question, when I try to calculate the convolution sum. I face with: ...
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33 views

Calculate Distance between Fourier Transforms

I'm working with signal data (specifically data from accelerators and gyroscopes), and I take their Fourier transforms to get a better idea of the dominant frequencies. I'd like to compare the ...
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1answer
24 views

DFT by $n$ samples of a continuous periodic signal with more than $n$ frequencies

It is known that if we only have $n$ samples and take DFT, we only get at most $n$ distinct frequency data. But let's say that there is a continuous periodic signal with more than $n$ frequencies, ...
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2answers
36 views

If a signal is periodic, can the error of approximation by Discrete Fourier Transform be avoided when using finite number of samples?

As title says, if a signal $f(t)$ is periodic, can approximation errors of approximation by discrete Fourier transform (DFT) be avoided when only finite number of samples are used?
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1answer
37 views

Why are discrete-time Fourier series and discrete Fourier transform only defined on integer $k$?

In ordinary Fourier series/transform of a continuous signal $f(t)$, fourier frequencies $\omega$ of series/transforms can be any of $\mathbb{C}$, not just $\mathbb{Z}$. But why is it the case that ...
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1answer
38 views

Using Discrete Fourier trasform of the samples of a continuous/periodic signal to obtain frequency data similar to FT of the original signal

Suppose we have a continuous and periodic real-valued 1D signal $f(t)$. Let us say we obtain finite number of samples $f(n)$ from $f(t)$. Is there a way to take discrete Fourier transform of $f(n)$ ...
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43 views

How to fit a stochastic matrix to given data.?

Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...
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28 views

Continuous second derivative over the support of a Daubechies4 wavelet

I can not entirely follow the proof from section 3.1.1 from the book "A primer on Wavelets" by Walker. After the first part (listed below), I can grasp the rest so if you could help I would greatly ...
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25 views

How to find Bilateral Laplace Transform of $e^{at}$ Using Changing of the Time Horizon

Ok, this has me a bit stumped. In my class the teacher "showed" us how to find the bilateral Laplace transform of x(t)=$e^{at}$ where $-\infty<t<\infty$. Breaking them into the two parts ...
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1answer
24 views

Can we obtain Fourier transform of a continuous signal using finite number of samples of the signal with known frequency cutoff?

Suppose that there is a continuous signal with highest frequency known. Is there a way so that we only sample the signal finite times and obtain the Fourier transform of the original signal (which ...
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25 views

Frequency spectrum of signal and is it real?

$x(t) = 2 + 5 cos(-t + \pi/4) - 2sin(3t + 5) + 3(cos(5 t + \pi/2).cos(4t) - e^je^t $ a) To find Fourier series coefficients of the following signal I need to use inverse Euler formula. But I need ...
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1answer
43 views

What is the connection between random variables and time series?

I always felt that there was a disconnect between random variable and time series. Clearly, random variable and time series can both be treated with statistical methods. First order, second order ...
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34 views

Two forms of cross-correlation

Wikipedia and MATLAB defines cross-correlation in this way. In time series analysis (P21), it defines cross-correlation upon cross-covariance: Let $\{X_t\}$ and $\{Y_t\}$ be two time series, ...
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1answer
27 views

Algorithm - return aliasing frequency

I posted this question on StackOverflow as well (link), but it is somewhere between a math question and a programming question (I'm looking for some formulas regarding aliasing frequencies and I want ...
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1answer
76 views

Fourier transform: noise and variance

I wrote a short program to generate $N$ samples of a sinusoid with some noise (ie: $$ f(t) = \cos(2\pi t) + 0.1 * \text{noise}(t) $$ where $\text{noise}(t)$ is chosen uniformly from $[-1 , 1]$. ...
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1answer
70 views

Orthonormal basis from Riesz basis

This question is with respect to Theorem 7.1 of Mallat's Wavelet Tour text. It is a follow-up of sorts to a previous question. Preliminaries Suppose I have a set $\{\theta(t-n)\}_{n \in \mathbb{Z}}$ ...
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68 views

Hessian Of Convolution's Quadratic Form

For the discrete inputs $\mathbf{x} \in \mathbb{C}^{M}$ and $\mathbf{y} \in \mathbb{C}^{N}$, I want to find the Hessian of $\Vert x \ast y \Vert_2^2$, where $\ast$ is the discrete convolution ...
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1answer
43 views

Discrete-time sinusoids with same frequency

I've read that sine waves of the form $x_n = \sin(w_{0}n)$, with frequencies $w_{0}$ and $w_{0} + 2\pi$, are indistinguishable from each other when considering discrete time. The book gives ...
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1answer
78 views

How to plot phasors of signals?

I have 3 singals and I'm trying to plot their phasors and their sum. I need to plot them end to end to demonstrate phasor addition. That is, the first phasor must start from the origin. The second ...
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1answer
22 views

A filter with frequency response $H(f)=\operatorname{sinc}(f).$

In signal processing, a sinc filter is an idealized filter that have the following frequency response $$H(f) = \mathrm{rect} \left( \frac{f}{2B} \right)$$ that is the rectangular function. In the real ...
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23 views

Alternative Function Definitions for the Square Wave signal

Are there any other function definitions for the Square Wave signal rather than the : and those referred to Wikipedia ?
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21 views

Time invariace of a linear system dependent on a particular time instant

$$y[n]=x[n]+35*x[n-1]+x[0]$$ Is this system time invariant? I am under the impression that $x[0]$ can be considered a constant. Am I right?
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22 views

how to find out how many Fourier coefficients there are (which are not zeros)

given a real periodic (with period $T_0$) signal $x(t)$ with fourier transform in which $$X(jw)=0\ \ \forall |w|\ge {6\pi \over T_0}$$ I know that the fourier series will have finite coefficients (5 ...