# 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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### Result of sampling of function with too small fs and reconstruction with square function

everybody I have an exam in signal processing tomorrow and doing past exams to prepare myself however I'm on stuck on a particular problem. It is stated as: Considering an ideal sampling with f_s = ...
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### secret formula for the “sin” wave with variable rising/falling edge

My math is pretty much forgotten. I was wondering if someone can take a look at this and share what's the formula for creating something like this. https://drive.google.com/file/d/...
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### Find Discrete Time Fourier coefficients of $(-1)^n x[n]$

Given that $x[n]$ is an N-periodic sequence with Fourier coefficients $a_k$, I want to find the Fourier coefficients of $$(-1)^n x[n]$$ for the situation in which $N$ is odd. I'm also interested in ...
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### Lost power with apodizing mask

I previously asked a similar question on the signal processing community, but I think may be more easily solved from a mathematical perspective. I start out with an image, $I$. To be able to process ...
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### Fourier transforms of random processes

In the Wikipedia article on Brownian noise, the Fourier transform of Brownian noise is determined. How is that Fourier transform defined? It seems it is a non-random quantity there, so it is not ...
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### Derivative of unit step function

The ramp function is given by r(t)=tu(t) If we differentiate ramp ,we get unit step function. That is, u(t)=1 So the derivative of unit step function is definitely 0 since u(t) is constant over the ...
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### Amplitude vs. Amplitude Spectrum?

I feel like I keep getting confused about amplitudes when talking about signal processing. Maybe someone can help clear my confusion. Lets say I have a simple sinusoid $f(t)=Asin(\omega_0 t)$ The ...
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### Confusion with a function transformation

I got a HW problem wrong in my Signals and Systems class and am hoping someone can help me understand why. There's a discrete-time signal ...
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### Limits to Discrete Fourier Transforms for Spectral Analysis

I am trying to leverage DFT and IDFT for some noise filtering on some data I am collecting. I am trying to do this in a user friendly/cheap manner by coding this in excel (I know this is not the best ...
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### ICA obsession with gaussianity

There are two reasons to focus on "gaussianity". (1) Orthogonal transformations of gaussian distributions are again gaussian. (2) Mixing of signals tends to a gaussian distribution via Central Limit ...
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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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I've been studying for a signals and systems class coming this fall and can't figure out how the following change of variable is being applied according to standard definition: $$T[x(t-\sigma)] = \... 1answer 95 views ### confused with the FFT output I am taking some sensor output and doing fft on it. how to get the exact frequencies from the complex output? my understanding is that bin frequencies and the input frequencies are different. Please ... 0answers 49 views ### averaging of multiple curves for signal processing I have response (vibration amplitude over frequency steps) measured over various point on my structure. In simpler way: i have 5 response curves(amplitude vs frequency plot) from same structure is ... 1answer 72 views ### Find a, given y(n)=x(n)+ax(n-d), interesting question Me and two friends of mine are working on a project (scholarly purposes only). The goal of this project is to clean an audio signal (speech, a song, anything audio) of echo. Generally speaking, if x(... 0answers 25 views ### Express the sum of squares as a percentage of how well two signals match? So I am using matlab to compare two signals using the sum of squares. So the best possible match will be zero. eg \sum(y_2 - y_1) where y_2 = y_1 would be 0. The larger the sum of squared value ... 1answer 41 views ### How to calculate the partition function of a given distribution? As noted in A FULL BAYESIAN APPROACH FOR INVERSE PROBLEMS, let  y = Ax + n, where y is a m dimensional signal and n is white Gaussian noise with precision \beta, so we have:$$ y|x, \beta \...
A partial Fourier Series with no coefficients is equal to the closed form expression: {A \over n} \sum_{k=1}^n \cos(k\theta) = {A \over 2n} \left\{{\sin([2n + 1]\theta/2) \over \sin(\theta/2)} - 1\...