Questions tagged [signal-processing]

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

Filter by
Sorted by
Tagged with
-1 votes
0 answers
18 views

Let $q\in\mathbb{Z} $. What is the smallest period of $\sin(2t)−\sin(qt)$?

Question: Let $q\in\mathbb{Z} $. What is the smallest period of $\sin(2t)−\sin(qt)$? My attempt: the period of $\sin(t)$ is $2\pi$, so the period of $\sin(2t)$ is $\pi$ and the period of $\sin(qt)$ is ...
user avatar
  • 143
0 votes
0 answers
5 views

Energy of a signal simplification

I'm learning about signals from the textbook Signals and Systems Second Edition, by Oppenheim and Willsky in order to learn more about Fourier Analysis and I'm having difficulty with the Energy ...
user avatar
0 votes
0 answers
23 views

What is the cubic expectation (third-order moment) of a complex gaussian vector (say, E[$aa^{T}a$])?

For a real gaussian vector, an explicit formula for the cubic expectation can be found in Matrix Reference Manual (search 'Cubic Expectations' in this link for detail), i.e., say $a$ is a complex ...
user avatar
-1 votes
0 answers
18 views

Time instant when signal reaches positive peak [closed]

I need to calcualte at what time a waveform x(t) reaches its positive peak where x(t) = 0.5cos(500t + 130). Please note that ...
user avatar
  • 1
0 votes
0 answers
22 views

Error in solution of 3-45 in Solution Manual of Signals and Systems Oppenheim 2nd edition

Hello I was solving the question 3-45 of the book Signals and Systems 2nd edition and I assume there is an error in the solution manual, kindly if someone confirm that am I right? or the solution ...
user avatar
0 votes
0 answers
10 views

Discrete fourier series from impulse train

Im struggling to derive the DFS from a periodic impulse train: let x(t) be a periodic signal over $NT_s$. As such, its sampled, impulse train version is: $x_p(t) = \sum_{n\in Z}x(nT_s)\delta(t-nT_s) = ...
user avatar
  • 45
0 votes
1 answer
25 views

Relationship between convolution for functions and for measures?

Is the definition of the convolution of two measures in any way analogous to the definition of convolution of two functions? I know almost nothing about measure theory and have trouble making sense of ...
user avatar
  • 167
0 votes
0 answers
20 views

Frequency peak always appearing at half of Nyquist frequency in Fourier transform

When taking the FT of a signal I always get a sharp peak at exactly half the Nyquist frequency. My signal is shown here: and its FT here: The Nyquist frequency is 36.7 KHz. As can been seen in the ...
user avatar
  • 1
1 vote
2 answers
63 views

Finding the righthand system from the Laplace transform $G(s) = \frac{1}{1- e^{-sT}}$

I want to find the impulse response $g(t) \in \mathbb{C}$, that it's two-sided Laplace transform is: $$\mathcal{L}\{ g(t)\} = G(s)=\frac{1}{1-e^{-sT}}$$ I tried to find $g(t)$, by finding the inverse ...
user avatar
0 votes
0 answers
48 views

On finding an upper bound on the error of a sparse approximation

$x \in R^n$ is a non-negative vector such that $ \Sigma_i^n x_i = 1$ ($\forall i, 0 \le x_i \le 1$). The components are ordered: $x_1 \ge x_2 \ldots \ge x_n$. We are also given : $ \Sigma_i^n x_i^2 \...
user avatar
1 vote
0 answers
25 views

The latest best approach to determine the order of the system model

The classical maximum likelihood estimation using Akaike's criteria is defined by $$\text{AIC}=-2\log^-\text{(maximum likelihood)} + 2 \text{(no. of independently adjusted parameters within the model)}...
user avatar
-2 votes
1 answer
81 views

Is there a formula for $\frac{\sin(xy)}{\sin(y)}$? [closed]

Can I do anything for any of the following? $$\sin(xy) \tag{1}$$ $$\sin^{2}(xy) \tag{2}$$ $$\frac{\sin(xy)}{\sin(y)} \tag{3}$$ $$\frac{\sin^{2}(xy)}{\sin^{2}(y)} \tag{4}$$
user avatar
0 votes
1 answer
37 views

How to argue that $\min_{u\in \mathbb{R}^n} \frac{1}{2}\|u-u_0\|^2+\alpha S_\mu(u)$ has a unique solution

I am stuck at the following exercise: Consider a signal $u^*$ and a noisy signal $u_0$. I need to argue that the following problem $$\min_{u\in \mathbb{R}^n} \frac{1}{2}\|u-u_0\|^2+\alpha S_\mu(u)$$ ...
user avatar
  • 2,452
1 vote
1 answer
43 views

Find solution for $\min_{u\in \mathbb{R}^n} \frac{1}{2}\|u-u_0\|^2+\alpha R(u)$

I have a signal $u^*$ and a noisy signal $u_0$. I want to minimize $$\min_{u\in \mathbb{R}^n} \frac{1}{2}\|u-u_0\|^2+\alpha R(u)$$ where $\alpha>0$ and $R(u) = \|Au\|^2$ with $A=\begin{pmatrix}-1&...
user avatar
  • 1,404
0 votes
2 answers
36 views

Derivation of a function's formulae from its graph

I am given the plot of a signal function. The expectation is to derive the full function given the graph's parameters. Here's what I've come up with so far for the negative side of the function: $\Pi(\...
user avatar
0 votes
0 answers
24 views

Verify solution to linear Fredholm integral equation of the second kind

Let $\int_a^b C_X(t, s)\psi_k(s)ds = \lambda_k\psi_k(t)$, which corresponds to a homogeneous linear (Fredholm) integral equation of the second kind. Where $C_X(t, s)$ is the covariance function and is ...
user avatar
  • 1,301
0 votes
0 answers
8 views

inverse Discrete-time fourier transform of $1 / (a+jw)$

I know that, in the continuous-time domain, by using Fourier transform relations, one can derive an ODE for the inverse Fourier transform of $1/(a+jw)$. This process is well-outlined in this question. ...
user avatar
  • 207
0 votes
1 answer
44 views

Finding out the frequency of a sine wave

I have a signal $x[n]$ which has $500$ samples, and I want to generate it such as it consists of two sinusoids added together. The first sinusoid will have an amplitude of $1$, and make $6$ complete ...
user avatar
0 votes
0 answers
57 views

On cascading linear systems

From chapter 5 of Steven W. Smith's The Scientist and Engineer's Guide to Digital Signal Processing, I understand that linearity is commutative according to Smith. If two matrices, $A$ and $B$, each ...
user avatar
1 vote
0 answers
19 views

Strange divergence at ends when recovering a continuous function from its Fourier Transform with DFT

I can sample a continuous function that is the Fourier transform of an unknown function, and I want to use the sampled data points from that function to do an inverse discrete Fourier transform (DFT) ...
user avatar
  • 11
1 vote
0 answers
12 views

Total time from Poisson process and exponential waiting time

Suppose the number of active calls on a switch is given by a Poisson process $M(t)$ with rate $\lambda$ calls per minute. Each call is then active for an amount of time $T\sim Exp(\frac{1}{\mu})$. We ...
user avatar
2 votes
0 answers
17 views

FFT energy calculation by frequency

For a FFT result in frequencies $k$ and associated amplitude magnitude $A(k)$ it is possible to calculate the signal energy by: $E=\sum_{k=1}^{N} A(k)^2 $ However, what does it mean when the energy ...
user avatar
  • 21
0 votes
0 answers
8 views

Find $E_\infty$ and $P_\infty$ of signal : $\cos(x)$?

$\int_{-\infty}^\infty \lvert \cos(x) \rvert ^2dx$ Do i just do: $$ E_\infty=\int_{-\infty}^\infty\lvert \cos(x) \rvert ^2 dx=\int_{-\infty}^\infty\lvert\cos^2(x)\rvert dx=\int_{-\infty}^\infty |{\...
user avatar
  • 1
0 votes
0 answers
14 views

Chained Kalman Filters

I have read the term “chaining Kalman filters” and I wanted to know precisely what the chained form of a Kalman filter is. I’ve seen also the term dual Kalman filter framed as a different concept ...
user avatar
  • 145
0 votes
0 answers
6 views

Why these operations calculates group delay?

In textbook, group delay is defined as negative derivative of phase (in frequency domain). And in discrete signal, derivative can be approximately calculated as differentiation. But when I browse some ...
user avatar
3 votes
1 answer
99 views

Relation of variance and energy of signal

The total energy of a signal $g(t)$ is $$\int_{-\infty}^{\infty} |g(t)|^2 dt$$ If the random variable $X$ has a probability density function $f(x)$, then its variance is $${\displaystyle \operatorname ...
user avatar
  • 7,278
0 votes
0 answers
10 views

iterative method for a singular matrix equation

I have a question when I tried to derive the MLE of variance of the i-th entry: I want to solve the following equation to get the MLE of the variance of the i-th entry $q_i^2$: $$\Phi_i \Phi_i^T=y y^T ...
user avatar
1 vote
0 answers
16 views

Understanding Quasi-Stationary Processes

Assume an Ergodic Process. A random process $s[n]$ (this notation refers to a sequence but is popular in the field of signal processing) is said to be quasi-stationary provided that it satisfies the ...
user avatar
  • 331
2 votes
0 answers
52 views

Spectral analysis of signal that is observed for a finite time ("philosophical" question)

We want to estimate the spectral density of a certain real signal $x(t)$, where time $t$ runs indefinitely over the real line (let's say that $x(t)$ may be the realization of a continuous stationary ...
user avatar
  • 1,904
0 votes
0 answers
9 views

Difference between techniques of period calculations? Continuous time vs discrete time?

For calculating period of continuous time signal,we simply divide 2pi by omega and get period value But in case of discrete time signal, procedure is not straight forward like continuous time, ...
user avatar
  • 143
0 votes
0 answers
53 views

Hot to get $e^{iwT} = 1$, $T = \frac{2\pi}{w}$?

Can someone please explain how to get to the above conclusion for T from the equation provided? I understand: $e^{iwT} = 1$ Can be written as: $\cos(wT) + i\sin(wT) = 1$ But I’m at a loss as how to ...
user avatar
0 votes
0 answers
8 views

Approximation of a function using a correlated function

Suppose we have two functions: $f_1(x)$ and $f_2(x)$ that are highly correlated, but with the exact correlation unknown. Both functions are sampled; $f_1$ with a large number of samples such that a ...
user avatar
1 vote
1 answer
64 views

Problem integrating with Dirac-delta functions

I'm currently studying signal and system, and the homework requires to find the impulse response of the system $$y(t)=\int_{-\infty}^t(t-\tau+2)x(\tau)d\tau$$ I want to plug in $x(\tau)=\delta(\tau)$: ...
user avatar
  • 15
2 votes
1 answer
34 views

Unable to prove the $L^2$ version of Nyquist–Shannon sampling theorem without an additional assumption

I want to prove the following version of Nyquist–Shannon sampling theorem for square-integrable functions: Let $f \in L^2(\mathbb{R})$ and $B > 0$. If $\hat{f}(\xi) = 0$ for almost every $|\xi| &...
user avatar
0 votes
0 answers
16 views

Simplification of real fourier coefficients (series)

I want to simplify the following expression for this fourier series : $$ g(t)= \alpha + \frac{1}{\pi} \sum_{n=1}^{\infty} \frac{\sin(2 \pi \alpha)}{n} cos(n\omega t)+ \frac{(1-\cos(2\pi \alpha))}{n}\...
user avatar
1 vote
0 answers
51 views

Is Fourier Transform relation a Convolution?

I was wondering if the Fourier transform relation can be viewed as evaluation in zero of the convolution between $x(t)$ and $e^{j\omega t}$, so if $*$ is the convolution operator $$X(f) = \left.x(t) * ...
user avatar
1 vote
0 answers
12 views

Finding the resolution of the discrete Fourier transform of an image

Suppose I have a 2d function defined on a square of length $L$, that is given as an $N \times N$ matrix of values. The resolution to which we know the function is $\delta L = L/N$. We may take the (...
user avatar
  • 433
0 votes
0 answers
22 views

How to do such questions in matlab?

t = 0 : 0.0001 : 0.1 fc = 1000; fo = 50; fmax = 100; I have range of |f| as [950, 1050]. And its used in this formula ...
user avatar
  • 113
1 vote
0 answers
14 views

How can I use variables with different units inside of a dirac delta function?

If I have the following model, $$\int_x\int_\tau F(x)\cdot\delta(t -d(x) - \tau)\: d\tau \: dx$$ where $t$ represents time in nanoseconds (ns), $d(x)$ is round trip time in ns and $\tau$ is total time ...
user avatar
  • 55
0 votes
0 answers
47 views

Fourier series and Toeplitz matrix

Assume the truncated discrete Fourier series of a function $f(x)$: $$ f(x)=\sum_{n=-N}^N C_n e^{2\pi n x i /L} $$ where $L$ is the period of $f(x)$. There is a rule for $f(x)$ which claims, $$ \forall ...
user avatar
0 votes
1 answer
116 views

Show that $f_n$ converges uniformly to $f$ on $[0, 1]$

Show that $f_n$ converges uniformly to $f$ on $[0, 1]$. Please help me. I have no idea.
user avatar
0 votes
0 answers
21 views

Given a vector $[a^0 a^1 a^2 \ldots a^{K-1}]$, for which matrix $M$ and scalar $s$ does the following expression hold?

Given a $K$-dimensional row-vector $[a^0 a^1 a^2 \ldots a^{K-1}]$, for which $K \times (K-1)$ matrix $M$ and scalar $s$ does it hold the following equality? $$[a^0 a^1 a^2 \ldots a^{K-1}] M R_2= s[a^0 ...
user avatar
  • 354
0 votes
2 answers
95 views

What is one sided Fourier transform?

Consider the function $$\phi (x) = \frac{1}{2 \pi i x} \{\text{exp}(2 \pi i x-1)$$ I know that the Fourier transform is \begin{align}\hat{\phi}(\omega) = \begin{cases} \frac{1}{2 \pi}, \ \ \ 0 \leq \...
user avatar
1 vote
0 answers
19 views

Name for series with subtracted base

I'm looking for a standard name in the math world for a sequence, which has some base subtracted, where the base is close to the average. There is an example: Assume there is a sequence with the first ...
user avatar
  • 111
0 votes
0 answers
28 views

Integration involving Fourier transform

Consider the Fourier transform of $g(t) = \psi (2^mt-k)$ is $$\hat{g}(\omega) = 2^{-m} e^{-\frac{i\omega k}{2^m}} \hat{\psi} (\omega 2^{-m})$$ where \begin{align} \hat{\psi} (\omega) = \begin{cases} ...
user avatar
0 votes
0 answers
23 views

Truncated Inverse convolution of triangular function

Define the triangular function with width $w$ to be $T_w(x) = \frac{1}{w}\text{max}(w-|x|, 0)$. Also, given a function $f(x)$, define the convolution $\displaystyle g_{\epsilon}(x) = (f * T_{\epsilon})...
user avatar
0 votes
1 answer
25 views

Identifying a raised sinusoid with uniformly spaced samples

You are given a model for an input signal of the form $y(t) = a \cos(\omega t) + b \sin(\omega t) + c $ where the constants $a,b,c$, and $\omega$ are unknown. You want to identify these unknowns from ...
user avatar
0 votes
0 answers
13 views

Back-calculating the octaves of 2D Perlin Noise

I am working with 2-dimensional fractal Perlin noise, and I am trying to find a way to back-calculate the octaves that were summed together. What I would like to do is, given a particular 2-...
user avatar
  • 1
0 votes
0 answers
67 views

How do I define a full set of orthogonal basis functions?

I'm doing an undergraduate-level course in media computing. I have some background in linear algebra but I can't seem to wrap my head around this. I'm given a signal = $3 + 2sin(6\pi t) + 7cos(12\pi t)...
user avatar
0 votes
0 answers
42 views

What is the Fourier transform of $x(t)=0.2+0.5\cdot \operatorname{sinc}^2(3t)?$

I want to calculate the Fourier transform for the signal $x(t)=0.2+0.5\cdot \operatorname{sinc}^2(3t)$. I know that the Fourier transform for $\operatorname{sinc}^2(3t)$ is $\frac13 \operatorname{tri}\...
user avatar
  • 69

1
2 3 4 5
39