Questions on the mathematical aspects of signal processing. Please consider first if your question might be more suitable for http://dsp.stackexchange.com/

learn more… | top users | synonyms (1)

1
vote
1answer
95 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 ...
0
votes
2answers
66 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: ...
1
vote
0answers
36 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 ...
0
votes
1answer
26 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, ...
0
votes
2answers
39 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?
1
vote
1answer
38 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 ...
1
vote
1answer
40 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 $f(n)$ ...
0
votes
0answers
47 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) ...
1
vote
0answers
29 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 ...
1
vote
0answers
27 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 ...
0
votes
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 ...
0
votes
0answers
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 ...
0
votes
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 ...
1
vote
0answers
37 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, ...
0
votes
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 ...
1
vote
1answer
109 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 $[-1 , 1]$. ...
2
votes
1answer
76 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 $\{\theta(t-n)\}_{n \in \mathbb{Z}}$ ...
0
votes
0answers
69 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 ...
1
vote
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 ...
0
votes
1answer
89 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 ...
0
votes
1answer
24 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 ...
0
votes
0answers
30 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 ?
0
votes
2answers
22 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?
0
votes
1answer
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 ...
2
votes
1answer
50 views

Fourier transform of a certain equality / discrete time Fourier transform of Dirac delta?

This comes from Stephane Mallat's Wavelet Tour text; however, I will phrase my question independently of it. I apologize that this is sort of long-winded. We have a function $f$ which satisfies the ...
0
votes
0answers
23 views

How to estimate parameters of a parametric function if its values for a set of arguments are known?

Suppose we have a parametric function $F(\alpha_1, ..., \alpha_{N_p}, \mathbf{x})$. For a set of arguments $\mathbf{x_1}$ ... $\mathbf{x_N}$ it's values $F(\mathbf{x_1})$ .... $F(\mathbf{x_N})$ are ...
1
vote
0answers
29 views

Fourier transform of integral function

A function $s(t)$ is defined by $s(t)=\int_x p(t-cx)dx$ where $\tau = cx$ is a time variable and $t\neq \tau$. What is the Fourier transform, $S(\omega)$, of the function $s(t)$? I know that for a ...
0
votes
0answers
14 views

Is “quantum” a correct term for the subsets used by a quantization function?

A quantizer is a many-to-few map. Its domain then is sets. I've heard those sets referred to as quanta (the plural of quantum). That usage seems to agree with what I understand to be the ...
0
votes
0answers
16 views

What is the term for the value from the set of values in a quantum that is closest to the aliased value of the quantum?

Signal quantization results in aliasing of the quanta. Is there a term for the value in the quantized set that is closest to the aliased value of the quantum? Something like "nearest neighbor" or ...
2
votes
0answers
33 views

Regarding the unilateral Laplace transform of LTI systems

Consider an LTI system described by the following differential equation, $$ \sum_{k=0}^{N}a_k\frac{d}{dt^k}y(t) = \sum_{k=0}^{M}b_k\frac{d}{dt^k}x(t) $$ With initial conditions, $$ y(t)|_{t=0}, ...
9
votes
2answers
194 views

compare lines and recognize similar ones

how can I find similar patterns in a line if I got a "template-line"? In this example, if I got the template (red), how can I find out that there are two occurences in the green one? The lines ...
0
votes
1answer
65 views

Convolution Properties

I have a quick question about certain algebraic properties of convolution. If I have 3 functions $f(x)$, $g(x)$ and $h(x)$, is the following true? $\Big[ f(x) . g(x)\Big] \circ h(x) = \Big[f(x) \circ ...
1
vote
1answer
65 views

Frequency response of a linear, shift-variant system

I am working my way through recorded lectures and a textbook related to DSP, and have come across a question that I am not sure how to answer. This is probably just due to how new I am to these ...
1
vote
1answer
29 views

Determine a time signal from another time signal

The given time signal is: $$u(t) = -3\sigma(t+4) + 6\sigma(t) - 3\sigma(t-4)$$ $\sigma$ - unit step function The same signal can be describes with the following mathematical relation between ...
2
votes
0answers
96 views

Fourier spectrum reflected across origin and Nyquist frequency

Recently I've been trying to figure out what's the point of negative frequencies produced by the fourier transform. One answer was it's just there to make calculations more elegant. It could be ...
0
votes
1answer
38 views

integration and convolution

Please can some one help me on the following integration. $$ G(\nu)=\frac{1}{\Delta t}\int_{t_a - \frac{\Delta t}{2}}^{t_a + \frac{\Delta t}{2}} f(t_a -t)e^{-2\pi\nu it}dt $$ where ...
1
vote
0answers
27 views

Can wavelets be used for texture discrimination?

I've recently been studying wavelet analysis with a view to differentiating certain areas of texture images where the texture differs from the background pattern (which is quite random); for example a ...
0
votes
1answer
49 views

interpreting the multivariate Kalman filter update equations

consider a multi-dimensional Kalman filter model with these state transition and measurement probabilities: $P(x_{t+1} | x_{t}) = Normal(Fx_{t}, \Sigma_{x})$ $P(z_{t} | x_{t}) = Normal(Hx_{t}, ...
0
votes
0answers
32 views

Simple proof for a continuous-time linear system and impulse $\delta$?

From Schaum's Outlines of Signals & Systems: Let's work with continuous-time signals. Let $T$ be a linear time-invariant system (LTI). Input $x(t)$ can be expressed as $x(t) = ...
0
votes
0answers
20 views

Calculate f(t) if I have its power spectral density

I have a power spectral density of a function, which is S(w) = 1/(1+w²) + d(w-2) + d(w+2) W is omega (rad) d is an impulse I want to calculate f(t) which is the signal that has this power spectral ...
1
vote
1answer
37 views

Why is $\cos((\omega+\alpha\cos(\omega' t))t)$ the wrong model for frequency modulation?

So I was trying to program vibrato, or freqency modulation, naively using the model: $$\cos((\omega + \alpha\cos(\omega' t))t)$$ Where $\alpha \lt \omega$ and $\omega' \ll \omega$. For practical ...
0
votes
0answers
175 views

Nyquist–Shannon Sampling Theorem Counter Example?

I was learning about the Nyquist theorem regards signal processing the area of interest which I will rephrase below: Given a signal lasting infinitely long with a maximum frequency of f, then you can ...
3
votes
2answers
58 views

Discrete Time Fourier Transform of the signal represented by $x[n] = n^2 a^n u[n]$

I have a homework problem that I am just not sure where to start with. I have to take the Discrete Time Fourier Transform of a signal represented by: $$x[n] = n^2 a^n u[n]$$ given that $|a| < ...
3
votes
1answer
23 views

Is this function/series periodic?

$$f(t)=\sum_{k=-\infty}^{\infty}(-1)^kp_{0.5}(t-2k)$$ Recall: $$p_{\Delta}=\begin{cases}\frac{1}{\Delta},&0\leq t\leq\Delta\\0&\text{ otherwise.}\end{cases}$$ Is the function periodic? If ...
0
votes
0answers
15 views

EZW parent-child relation

I’m trying to learn the EZW principle. I’m having trouble understanding the parent-child relationship. In my case, I want to use it on a 1 dimensional signal. So, let’s say for example a signal of 4 ...
0
votes
1answer
40 views

How are sinusoids and roots of unity related to each other?

The discrete Fourier transform (DFT) is often teached as being a transform that decomposes a given signal or sequence of numbers into sinusoids with frequencies $\large\frac{k}{N}$ where $k \in [0, ...
1
vote
1answer
84 views

Absolute value in exponential, signal energy?

How can this give this result? Isn't the absolute of $(e^(-2*t))$ always 1?
1
vote
0answers
17 views

Detection theory - Sensitivity, Specificity - in Multi-Detection scenario

I am working in computer vision and have this scenario: For each frame of a video sequence I have the following: Image with a resolution of width * height discrete pixel locations. List of ...
0
votes
1answer
72 views

Which topics in maths should I know before I dive into programming for image processing?

I am a student who wants to start out with programming for Image processing but as I do not have a good mathematical background(I haven't studied A-level Maths) I would like to know what are the ...
1
vote
0answers
47 views

a window slides over a sinusoid, which calculation on window of length p/4 always returns a maximum value compared to other window lengths?

We have a set of discretely sampled points that are on a sinusoid, in this case its period is 40: If we have windows of different lengths that slide over this time series, like this little ...