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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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61 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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65 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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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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52 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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15 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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18 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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18 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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20 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 ...
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1answer
45 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 ...
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18 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 ...
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25 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 ...
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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 ...
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15 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 ...
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30 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}, ...
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186 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 ...
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1answer
47 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 ...
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1answer
49 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 ...
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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 ...
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82 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 ...
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1answer
34 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 ...
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24 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 ...
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1answer
34 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}, ...
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28 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) = ...
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18 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 ...
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1answer
34 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 ...
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129 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 ...
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2answers
42 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| < ...
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1answer
21 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 ...
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13 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 ...
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1answer
33 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, ...
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1answer
59 views

Absolute value in exponential, signal energy?

How can this give this result? Isn't the absolute of $(e^(-2*t))$ always 1?
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13 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 ...
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1answer
60 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 ...
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44 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 ...
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1answer
31 views

Show that for a real impulse response function the response to a sine input is …

Working on this problem on linear invariant systems in signal processing, but unsure if I've got the right answer: Show that for a real impulse response function of $H(\omega)$, the response to a ...
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2answers
89 views

Changing a sigmoid curve to have an adjustable point of inflection

I am trying to an implement an adjustable Sigmoid curve such as in the YouTube video here. I found a potentially good candidate: $$f_k(x) = \frac{\left(x-x\cdot k\right)}{k-\left|x\right|\cdot 2\cdot ...
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251 views

How to calculate wavelet energy?

Part of my assignment about signal processing says the following: Compute the Discrete Wavelet Transform for the input signals Group the wavelet coefficients in trees growing across scales ...
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21 views

How to reconstruct a sparsely sampled multiperiodic function?

I have $m$ oscillators, where $m$ is unknown, with periods $\vec p = p_1, p_2, \ldots, p_{m}$. Each of the oscillators $j$ has associated with it a vector of sine coefficients $\vec A_j$ and angle ...
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1answer
164 views

Implementing 1D Discrete Wavelet Transform in Matlab

I'm trying to write my own version of the Discrete Wavelet Transform using the bior4.4 filters. I think my implementation is not properly working yet, because whenever I input a signal and a number ...
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2answers
51 views

Why is the maximum value of cross-correlation achieved at similar section?

I'm a bit confused and probably need some sleep. When trying to find a short signal inside a long one (or the delay), it's almost a trivial fact that we should look for the maximal valued coefficient ...
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29 views

The probability that a process signals (simple conditional probability)

I have this problem (from Montgomery's Applied Probability and Statistics, 5th Edition, problem 2-145, if anyone wants to see the original problem) but it's long, so for the sake of brevity I'll give ...
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1answer
47 views

Is this signal band-limited?

I'm self-learning signal processing now, and I've run into this question about band-limited signals: Consider the signal $x(t) = 1$ for $0 \leq t \leq T$ and $0$ otherwise. I've found that its ...
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2answers
98 views

Laplace transform of $f(t)=te^{-t}\sin(2t)$

I was asked to find the Laplace transform of the function $\displaystyle f(t)=te^{-t}\sin(2t)$ using only the properties of Laplace transform, meaning, use clever tricks and the table shown at ...
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188 views

Improvement of Minimum description length (MDL) estimate.

I earnestly request apology if this question is inappropriate for the forum. The question has two parts one technical and the other is not technical. I would appreciate any response. Let me consider ...
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1answer
39 views

Why is a wave with high FM aperiodic?

I was playing with sound synthesis in a program I wrote and I had a wave of the form $\sin(2\cdot\pi\cdot(f_c+\sin(2\cdot\pi\cdot f_m \cdot t)) \cdot t) $ So, just simple frequency modulation. When ...
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1answer
92 views

From Orthogonal vectors to Useful Bivector

If we have set of orthogonal vectors (X) can we form a set of orthogonal bivectors from that set? I am trying to find if there is a way to get 'more information' from an orthogonal matrix by some ...
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1answer
37 views

Finding the coefficients of an MA(1) process given the expectation and variance.

The following is preparation of an exam I have coming up, any help would be appreciated. An MA(1) process is selected to model a stationary time series $\{ X_t \}$. We are given the lag one ...
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1answer
29 views

Minimum Phase Filter

Suppose we want to find the minimum phase filter of a causal system with system function $H(z)=z^{-1}−0.3$. The minimum phase filter is $H_1(z)=1−0.3z^{−1}$ (by taking the zero to its conjugate ...
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42 views

Sine wave from fos + simple signal

I have a first order system $\frac {1}{(s+c)}$ and a signal of the form $\sum_{k=0}^\infty (-1)^{k}e^{-2ks}a(\frac{1-e^{-2s}}{s}- be^{-s}(\frac{1-e^{-s}}{s^2})) $ i.e a periodic signal of a square ...