# 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/

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### Double Fourier transform of a single variable function

So what I'd like to achieve is the following: suppose we have a time dependent signal $f(t)$ - e.g. the wave of a song. This song contains a drumbeat that (for the sake of simplicity) has a single ...
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### A generalized differential equation for a convolved differential operator.

The solution to perhaps the world's very first differential equation $$f'(x) = f(x)$$ is the well known exponential functions $$f(x) = k\exp\left[x\right]$$ But what if we consider another ...
1answer
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### Sine wave - Calculate x(t) on 20KHz where t = 1

I'm trying to calculate a sine wave using the formula... $$x(t) = A\sin{(2 \cdot \pi \cdot f_q \cdot t)}$$ Where $t$ is time (seconds), $A$ is amplitude and $f_q$ is frequency (Hz) When I calculate ...
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### Energy spectral density in an LTI system

Consider the LTI system: $$x\left( t \right) \to LTI\, system \to y\left( t \right)$$ The graph of the $x(t)$ signal is the following: Remark: The graph when $t>0$ is a quarter of a circle. Q1:...
1answer
38 views

### What does tilde on top of functions mean in fourier series and fourier transforms?

For an example these in these equations for continuous time fourier series: Does it denote that the function is periodic? If it denotes that the function is periodic then shouldn't there be a tilde ...
1answer
71 views

### Frequency spectrum and unit impulse response

In the context of signal processing, consider the following system: $$x\left( t \right) \to System\, A \to z\left( t \right) \to System \, B \to y\left( t \right) .$$ Let the global system be ...
1answer
24 views

### What's the minimal time to distinguish 1 Hz difference in frequency?

Given a fragment of a single-frequency wave, what's the minimal length of it to calculate its frequency with error less than 1 Hz? Or if there are two fragments with frequency difference of 1 Hz, how ...
0answers
36 views

### Orthogonal functions definition

I know the condition for two functions to be orthogonal. It is $\int_{a}^{b}f_1(x)f_2(x)dx=0$. Also, on YouTube and websites this kind of formula is used, no other. I read from my book Signals and ...
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### Calculation of alarm level for a signal detection in Poisson distributed noise

I am trying out my luck with the first post here... Well, to start off I have a problem, and I am not really sure about the solution. I hope that someone might chime in with suggestions.. We have an ...
0answers
18 views

### Why does a shift over the $x$ axis of the sine/cosine function has not only a high value for the DC coefficient but also for $0$?

I can't understand why does a shift over the $x$ axis of the sine/cosine function has not only a high value/spike for the DC coefficient but also for zero? I saw this in a video but I couldn't get it. ...
1answer
52 views

### Sequence of polynomials approximating sines and cosines

Suppose that you create a sequence of polynomials by first stating that $$P_1(x) = \pi x$$ and then describing successive polynomials by: $P_{n+1}(x)$ is the indefinite integral of $2\pi P_n(x)$, ...
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### Boundary of a Function from Bessel's Inequality

From Bessel's inequality, we know that for a partial sum of Fourier Series represented by $$[a_0, \mathbf{a}, \mathbf{b}], \mathbf{a}, \mathbf{b} \in \mathbb{R}^N$$ of a periodic function that is ...
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### Approaching the Fourier transform of a damped cosine with a linear frequency chirp

I want to take the Fourier transform of the following transient signal, $$f(t) = e^{-t/\tau} \cos((\omega_0 + m t)t)$$, where $m$ is some gradient parameter in units of $\rm{Hz}/s$. I thought this ...
0answers
36 views

### Is there any condition for matched filter to be optimal?

The question is related to the optimallity of matched filter. The definition of matched filter is given in the link here: https://en.wikipedia.org/wiki/Matched_filter Cosider that the signal is ...
1answer
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### The system $y(n) = c\cdot x(n) +d$ is given. Is the system linear, time-invariant, stable and causal?

The system $y(n) = c\cdot x(n) +d$ is given. (Unfortunately, nothing more is given.) Is the system linear, time-invariant, stable and/or causal? I assumed, that $c,d\in\mathbb{R}$ Linearity We ...
1answer
50 views

### Posterior of a corrupted transformation

Let $z\sim\mathcal{N}(z|0,1)$ and let $p(x|z)=\mathcal{N}(x|f(z),\gamma)$ where $f:\mathbb{R}\to\mathbb{R}$ is a differentiable bijection. I am trying to approximate the posterior $p(z|x)$ with a ...
1answer
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### if $x[n] = (-1)^n$ and $y[n] = (-2)^n$, why the system is not LTI?

I am a student and I answered this question "it can be LTI" but my answer was incorrect and the right answer is that this system can't be LTI (Linear time-invariant system). I don't understand why. ...
0answers
27 views

### Calculate energy of signals

Consider the following two signals \begin{aligned} x(t) &= e^{-kt} u(t)\\\\ y(t) &= x(t) \,\frac{\sin(\theta t)}{\pi t}\end{aligned} Determine the relation between $k$ and \$\...
1answer
34 views