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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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What's the difference between $\mathbb{R}^2$ and the complex plane?

I haven't taken any complex analysis course yet, but now I have this question that relates to it. Let's have a look at a very simple example. Suppose $x,y$ and $z$ are the Cartesian coordinates and we have a function $z=f(x,y)=\cos(x)+\sin(y)$.…
Cancan
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How do I exactly project a vector onto a subspace?

I am trying to understand how - exactly - I go about projecting a vector onto a subspace. Now, I know enough about linear algebra to know about projections, dot products, spans, etc etc, so I am not sure if I am reading too much into this, or if…
Spacey
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Real world application of Fourier series

What are some real world applications of Fourier series? Particularly the complex Fourier integrals?
Namit Sinha
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Rapid approximation of $\tanh(x)$

This is kind of a signal processing/programming/mathematics crossover question. At the moment it seems more math-related to me, but if the moderators feel it belongs elsewhere please feel free to migrate it. I'm working on a project where I have…
MattyZ
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Correct way to calculate numeric derivative in discrete time?

Given a set of discrete measurements in time $x_t, t \in \{0,\Delta t, 2\Delta t,\ldots,T-\Delta t,T\}$, what is the correct way to compute the discrete derivative $\dot x_t$. Is it more correct to take the difference with the previous value: $$\dot…
CodeFusionMobile
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Fourier transform of $\left|\frac{\sin x}{x}\right|$

Is there a closed form (possibly, using known special functions) for the Fourier transform of the function $f(x)=\left|\frac{\sin x}{x}\right|$? $\hspace{.7in}$ I tried to find one using Mathematica, but it ran for several hours without producing…
Vladimir Reshetnikov
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What does the Fourier Transform mean in the context of images?

This is clearly a very important equation with tonnes of properties that I see come up a lot in image processing literature, but I don't understand why this equation is important, and what it is saying. What does it really mean and why is the…
water
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What is the sum over a shifted sinc function?

What is the sum of a shifted sinc function: $$g(y) \equiv \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \, ?$$
DanielSank
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Adjustable Sigmoid Curve (S-Curve) from $(0,0)$ to $ (1,1)$

I feel like this is such a simple question but I am at such a loss. I currently have a set of values that I would like to weigh by an S Curve. My data ranges from $0$ to $1$ and never leaves those bounds, but nearly every version of a Sigmoid I see…
pure_bordem
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Is the convolution an invertible operation?

I have a signal $f(x,y)$, which is discrete. I convolve this signal with a kernel $h(x,y)$: $y(x,y) = f(x,y) \star h(x,y)$ (where $\star$ is the convolution operator) Can I obtain $f(x,y)$ given only $y(x,y)$ and $h(x,y)$ ? Note: Even though this…
dynamic
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Is there a way to relate prime numbers and the Fourier transform?

According to what I know about Fourier transforms, any continuous periodic signal can be represented as a combination of sine and cosine functions. To me, this looks analogous to the fundamental theorem of arithmetic (every integer $\ge 2$ can be…
user63152
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Extracting exact frequencies from FFT output

Say I pass 512 samples into my FFT My microphone spits out data at 10KHz, so this represents 1/20s. (So the lowest frequency FFT would pick up would be 40Hz). The FFT will return an array of 512 frequency bins - bin 0: [0 - 40Hz) - bin 1: [40 -…
P i
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Prove of the Parseval's theorem for Discrete Fourier Transform (DFT)

If $x[k]$ and $X[r] $ are the pair of discrete time Fourier sequences, where $x[k]$ is the discrete time sequence and $X[r]$ is its corresponding DFT. Prove that the energy of the aperiodic sequence $x[k]$ of length $N$ can be expressed in terms of…
Cheung
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Waves of differing frequency are orthogonal - help me understand

I know that sinusoidal waves of different frequencies are orthogonal to each other. For instance: # Shows that 1Hz and 2Hz waves are orthogonal import numpy, scipy x = numpy.linspace(0, 1, 1000) wave_1hz = scipy.sin(1 * 2*scipy.pi*x) wave_2hz =…
caleb
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Matrix performing local differintegral analysis being its own inverse. Coincidence?

I found a curious matrix $$T = \begin{bmatrix}1&2&1\\1&0&-1\\1&-2&1\end{bmatrix}$$ This matrix (or actually $\frac 1 2 T$) performs Local mean value (integral) estimation. Local derivative estimation by central midpoint distance. Local second order…
mathreadler
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