# Tagged Questions

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

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### Error analysis for a non-optimal nonuniform quantizer

I'm new to the theory on the subject of quantization. I'm wondering if there are any references that I can look at on error analysis for a non-optimal nonuniform quantizer. More specifically, I ...
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### Is $y[n]=x[n]-x[n-1]$ invertible system?

Well, the title says everything. I know I can find a z-transform, find $H(z)$ and then find a appropriate invert system and comment on that. How do I explain it to a person who does not know z-...
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### Sampling Theorem for Lattices

I am looking for a reference for an analogue of the Shannon sampling Theorem for more general lattices (in any dimension). Something along the lines of the theorem in this wikipedia article: https://...
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### Inverse z-transform of $z^4+1.827z^3+2.338z^2+1.827z+1$

I need to transform the following $H(z)$ back to time domain: $$H(z)=(z-e^{j\frac{8}{15}\pi})(z-e^{-j\frac{8}{15}\pi})(z-e^{j\frac{12}{15}\pi})(z-e^{-j\frac{12}{15}\pi})$$ I did the following steps ...
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### How to determine the states of ideal diodes in simple circuits with only DC sources and resistors [closed]

How can the states of ideal diodes be determined in simple circuits with only DC sources and resistors without a trial and error approach? I posted this question on Electronics SE and found out that ...
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### Is a signal summable when given a z-transform?

The output of a system is given as z-transform: $$Y(z)=\frac{1+z^{-2}}{(1+\frac{1}{4}z^{-1})(1-\frac{1}{2}z^{-1})}$$ I want to know if the signal in the time domain is summable, meaning that the ...
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### fourier series of unknown functions

I am confused in understanding use of fourier expansions of functions. This answer, for example says that we can write voice as a sum of sines and cosines of different frequencies and amplitudes, but ...
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### Is the power of a complex exponential signal always zero?

Is the power of a complex exponential signal always zero? For example say I have the function $f(t) = Ae^{i\omega t}$ Then, I think power is defined as: $P=\int_{-T/2}^{T/2} f^2(t) dt$ So is it ...
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### Do combined waves with non-rational frequencies have a common period?

I am facing a problem where I have two waves combined: $$y = A\sin(b_1x)+B\cos(b_2x)$$ Where $b_1$ and $b_2$ are non-rationals. i.e. \begin{align} & b_1 = \sqrt{3+\...
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### How do I compute the output of this LTI system?

2.11. Consider an LTI system with frequency response $$H(e^{j\omega})=\frac{1-e^{-j2\omega}}{1+\frac12e^{-j4\omega}},\quad-\pi<\omega\le\pi.$$ Determine the output $y[n]$ for all $n$ if the ...
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### Convolution between a discrete stochastic signal and a continuous function

I'm trying to find the convolution between a discrete, stochastic signal (for which I have data at each t) and an exponential decay function (which is continuous in t). Now I know that one can ...
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### Expressing a function in terms of sinc(t)

Given the function: $S(t) = sin(t/\Delta)/t$ How can one express this function in terms of: $S(t) = sin(t)/t$ Thanks!
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### Why do high-frequency dynamics quickly go away in a step response?

As we know, a step input hits all the frequencies of a dynamical system. However, my professor told me today that the high-frequency response is only present for a short time at the very start, and ...
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### How does hard clipping change the frequency of a pure sinusoidal signal?

Suppose that we decide to limit the magnitude of a real-valued signal $f(t)$ by maximum cutoff $V_s$. Thus, if $|f(t)| > V_s$, a transformed signal $g(t) = V_s$ or $g(t) = -V_s$ depending on the ...
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### Help with solving for difference equation coefficient terms

I am having trouble remembering how to solve for the $a_k$ and $b_k$ terms for a difference equation. (it has been some time since taking a signal processing course, where I first learned) I am trying ...
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### Using DTFT to find the sum of $\sum_{n=-\infty }^{\infty }\text{sinc}(n\alpha_1)\text{sinc}(n\alpha_2)$

I am trying to use DTFT (as asked in a problem) to find the following sum $$\sum_{n=-\infty }^{\infty }\text{sinc}(n\alpha_1)\text{sinc}(n\alpha_2)$$ for real $\alpha_1>0$ and $\alpha_2<1$. I ...
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### Express covariance function of a sampled process as a fourier transform (help with proof)

I'm wrestling with this theorem in my 'stationary stochastic processes' course.It's about sampling of a continous process and a way of rewriting the covariance function of the sample(I'm not entirely ...
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### Calculate Inverse Discrete Time Fourier Transform

Calculate Inverse Discrete Time Fourier Transform of the following where $|a| < 1$: $$X(e^{j\omega}) = \frac{1-a^2}{(1-ae^{-j\omega})(1-ae^{j\omega})}$$ Plugging this directly into the IDTFT ...
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### Derive Symmetry Properties of Discrete Fourier Transform

Using the standard definitions of IDFT and DFT: \begin{align*} x[n] &= \frac{1}{2\pi} \int_\pi^\pi X(e^{j\omega}) e^{j \omega n} d\omega \\ X(e^{j\omega}) &= \sum\limits_{n=-\infty}^\...
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### Simple Discrete Convolution Question

With the discrete step function $$u[n] = \begin{cases} 1, & n \ge 0 \\ 0, & n < 0 \\ \end{cases}$$ And the output $y[n]$ defined as a discrete convolution of the input $x[n]$ ...
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### Series of $\csc(x)$ or $(\sin(x))^{-1}$

In some cases I found that $$\csc(x)= \lim\limits_{k\rightarrow \infty}\sum_{n=-k}^{k}(-1)^{n}\frac{1}{x-n\pi}$$ Is anything to prove or disprove that?
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### Variance of estimating coefficients by correlating a sequence

I have a sequence $$r[n] = a_1.t_1[n] + a_2.t_2[n] + a_3.t_3[n] + ...$$ where $t_1, t_2, t_3,...$ are uncorrelated, two-level (+A/-A), zero mean, pseudo-random sequences. To estimate $a_1$, $r[n]$...
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### Sampling a Chebyshev polynomial with the discrete cosine transform

I have a Chebyshev polynomial $f$ of degree $n$ in point-value form \begin{align} f&=:S = \left( \left( x_i, y_i \right) \right)_{i=0}^n, \tag{1} \\ x_i &= \cos\left( \frac{i \pi}{n} \right), ...
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### DFT of subdomain of periodic domain

$f(t_i,x_j)$ is a solution of stochastic differential equation on grid. $j=[0,N+1]$, $i=[0,\infty]$ and boundary conditions are periodic: $f(t_i,x_0) = f(t_i,x_N)$ and $f(t_i,x_{N+1}) = f(t_i,x_1)$ ...
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### Inverse-Fourier transform of a function after non-linear frequency modulation

Suppose $g\in L^1(\mathbb{R})$ such that $\hat{g}\in L^1(\mathbb{R})$ too. So $\tilde{g}(x) = \int_{-\infty}^{\infty}e^{i\pi \xi^2}\hat{g}(\xi)e^{2\pi i \xi x}\,d\xi$ is well-defined. Question is: Is ...
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### How to apply a time shift to a pulse-shape, spanned with spline functions?

I have a sampled pulse shape: $h = [1, 0.5]$ and I do not know what is its real underlying continuous-time pulse. I want to compute the samples of $h(t-\Delta t)$. If I write the continuous pulse ...
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### How to decorrelate/Whiten a non-white additive random variable?

I have a signal processing problem where I have the Additive Noise Model (assume Gaussian noise). $$y = x + w$$ where, $y$ is corrupted signal, $x$ is original signal & $w$ is a non-white ...
I am looking for a robust way to represent and generate multiple stochastic processes that contain time and cross-correlations i.e. I am looking at stochastic processes $X_t^{1}$, $X_t^{2}$, $\ldots$, ...