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42
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10answers
4k views

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 ...
21
votes
8answers
6k views

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 ...
17
votes
3answers
1k views

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,...
17
votes
3answers
21k views

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 ...
13
votes
3answers
678 views

Fast computation/estimation of the nuclear norm of a matrix

The nuclear norm of a matrix is defined as the sum of its singular values, as given by the Singular Value Decomposition of the matrix itself. It is of central importance in Signal Processing and ...
12
votes
3answers
17k views

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 ...
10
votes
5answers
6k views

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 ...
10
votes
1answer
526 views

Creating intuition about Laplace & Fourier transforms

I've been reading up a bit on control systems theory, and needed to brush up a bit on my Laplace transforms. I know how to transform and invert the transform for pretty much every reasonable function, ...
10
votes
0answers
203 views

Sampling theorem.

Let us consider \begin{equation} \hat{f}(x)=\sum_{n\in \mathbb Z}\left\langle\hat{f},e^{i n x}\right\rangle_{L^2[-\pi,\pi]} e^{i n x} \ \ \ \ \ \ \ \ (1) \end{equation} where $\langle g, h\rangle_{L^2[...
9
votes
2answers
207 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 don'...
9
votes
1answer
158 views

Smooth sawtooth wave $y(x)=\cos(x-\cos(x-\cos(x-\dots)))$

Consider an infinite recursive function $$y(x)=\cos(x-\cos(x-\cos(x-\dots)))$$ $$y=\cos(x-y)$$ Plotting the function $y(x)$ implicitly we get a smooth sawtooth-like wave: Was this function ...
9
votes
3answers
767 views

Removing noise when the signal is not smooth

Suppose we have (an interval of) a time series of measurements: We assume it can be explained as a "simple" underlying signal overlaid by noise. I'm interested in finding a good algorithm to ...
8
votes
2answers
20k views

Integration of sawtooth, square and triangle wave functions

Context After a discussion about how to plot the results of a frequency modulation between two signals on Stack Overflow, I understood that I need to find the time-integral of the following wave ...
7
votes
2answers
556 views

Usage of inverse Laplace transform

At my current study level in college, use of inverse Laplace transform is not mentioned well - textbooks say "use tables." So, can anyone show me how to use inverse Lapalce transform? And also proof? ...
7
votes
3answers
1k views

Looking for a Calculus Textbook

I want to start signal processing and I need a book that satisfies my mathematical requirements: I am in the third grade of high school and I don't know any useful thing about limit, differential, ... ...
7
votes
2answers
4k views

What are the limitations /shortcomings of Fourier Transform and Fourier Series?

I am fond of Fourier series & Fourier transform. But every approach has some outcomes and some shortcomings. It's limitations lead to innovation of new approach. So, can anybody explain about ...
7
votes
2answers
946 views

Is Fourier series used always for periodic signals and Fourier transform for aperiodic signals only?

I want to ask basic question. In our mathematics classes ,while teaching the Fourier series and transform topic,the professor says that when the signal is periodic ,we should use Fourier series and ...
6
votes
3answers
719 views

Describing a Wave

I have this wave in front of me, and I am to describe this into a math description such as its function that is equivalent to representing this wave. I have no idea how to start and could use some ...
6
votes
2answers
13k views

FFT bins from exact frequencies

I'm trying to understand a few concepts about Fourier Transforms (mainly in the context of signal processing). Let's suppose a signal is sampled at 10kHz and that the FFT size is 1000. If 1000 ...
6
votes
2answers
708 views

Which time-frequency coefficients does the Wavelet transform compute?

(I asked this on Stack Overflow a while ago and didn't get a satisfying answer, so I'm trying again here.) The Fast Fourier Transform takes O(N log N) operations, while the Fast Wavelet Transform ...
6
votes
2answers
11k views

How to fit a curve to a sinusoidal wave

I am wondering how to fit a sinusoidal wave (approximation). I would like to fit it in the form: $y = A\sin(Bx + C) + D$ where $A,\,B,\,C$ and $D$ are constants. The only constants I really care about ...
6
votes
2answers
756 views

Image Reconstruction:Phase vs. Magnitude

Figure 1.(c) shows the Test image reconstructed from MAGNITUDE spectrum only. We can say that the intensity values of LOW frequency pixels are comparatively more than HIGH frequency pixels. $$ f(x,y)=...
6
votes
1answer
910 views

Zero-padding data for FFT

If I take a discrete Fourier transform of $\{ c_1, c_2, \ldots, c_n\}$ where $n$ is prime, I am rather limited in the FFT algorithms available to me and their performance. Additionally, having mixed-...
6
votes
3answers
807 views

Bases in compressed sensing (signal reconstruction)

I have been posting this kind of question in Cross Validated, but since this one deals almost entirely with mathematics, I will post it here. In signal reconstruction using compressed sensing, we ...
6
votes
1answer
306 views

Does this curve tend to a square wave?

I have put some Mathematica code here: http://pastebin.com/cY6r7skS that uses this algorithm: $$y1 = Sin[x];$$ $$y2 = Sin[y1];$$ $$y3 = Sin[y1 + y2];$$ $$y4 = Sin[y1 + y2 + y3];$$ $$y5 = Sin[y1 + y2 ...
6
votes
1answer
436 views

Qualitative interpretation of Hilbert transform

the well-known Kramers-Kronig relations state that for a function satisfying certain conditions, its imaginary part is the Hilbert transform of its real part. This often comes up in physics, where ...
5
votes
3answers
648 views

Period of a product of $\sin$ and $\cos$

I want to find the period of $\sin(t) \cos(\pi t)$. I started off by transforming that into $\frac{1}{2}\left [ \sin((\pi +1)t) - \sin((\pi - 1)t\right ]$, but then I get stuck. How do I find the ...
5
votes
6answers
652 views

How to generalise the Fourier transform

The Fourier transform approximates a signal using a bunch of sine and cosine waves. The inverse Fourier transform then reconstructs the original signal from this information. I am told that it's ...
5
votes
3answers
712 views

How can I interpret “energy” in signals?

I am learning about various signal processing methods in my university course, and I can't seem to grasp what 'energy' in signals represent. I mean, I know that it is the integral of the absolute ...
5
votes
2answers
3k views

Waves of differing frequency are orthogonal - help me understand

I know that sinusoidal waves of different frequencies are orthogonal to each other. For instance: ...
5
votes
2answers
4k views

Speech processing pre-emphasis: how does it work?

In speech processing, the original signal usually has too much lower frequency energy, and processing the signal to emphasize higher frequency energy is necessary. To perform pre-emphasis, we choose ...
5
votes
1answer
322 views

What is the sum over a shifted sinc function?

What is the sum of a shifted sinc function: $$g(y) \equiv \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \, ?$$
5
votes
2answers
117 views

How can I recover a sequence of numbers given a corrupted version of it?

I have an unknown sequence of real numbers $x_i$ and a known sequence of real numbers $y_i$; $y_i$ is a corrupted version of $x_i$, i.e., $$y_i=x_i+n_i$$ where $n_i$ is a random number distributed ...
5
votes
2answers
129 views

partially reconstruct information of function convoluted with boxcar kernel

the function (f) I want to reconstruct partially could look like this: The following properties are known: It consists only of alternating plateau (high/low). So the first derivation is zero ...
5
votes
3answers
134 views

What does it mean that a sine wave is unchanged when added to another sine wave?

From the wikipedia article on sine waves: The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and ...
5
votes
1answer
488 views

Fourier Transforms

I'm having a terrible time trying to understand Fourier transforms. I'm very visual so leaving the $X,Y,Z,t$ domain is not working form me :) I'm trying to figure out the basics at the moment. Like,...
5
votes
1answer
331 views

Mathematically inclined books on Signal Processing Theory

First off, i know this may seem off topic but i could not find help in signal processing communities so i was hoping there would be people here who both love mathematics and have interest in signal ...
5
votes
1answer
582 views

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 ...
5
votes
0answers
53 views

When is a mapping the proximity operator of some convex function?

Sorry for cross-posting from MO. It's been a few days and the question hasn't received any attention there. So, is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which ...
5
votes
0answers
137 views

Scale invariance and $1/f^2$ power spectrum

In the paper Occlusion Models for Natural Images : A Statistical Study of a Scale-Invariant Dead Leaves Model; Lee, A. B. Mumford, D. B. Huang, J.; International Journal of Computer Vision I read ...
4
votes
3answers
5k views

Proofs of dirac delta property

How would I formally prove this property of dirac delta? $$\int \delta(a-x) \delta(x-b) \,dx = \delta(a-b) $$ I attempted to use the definition of a dirac delta $$\int f(x)\delta(a-x)\delta(x-b)\,dx=...
4
votes
2answers
138 views

detect largest period in non-harmonic components

let us consider following sinusoidal components $\sin(2\pi 13.5t)+\sin(2\pi 13.99t)+\sin(2\pi 25.3t)+\sin(2\pi 26t)$, clearly this is not periodic in total,because frequencies or periods are not ...
4
votes
2answers
104 views

Reduce formula using Euler's?

I am performing a self-study, and I am lost as to a derivation that has taken place. I basically started with this equation: $$ \Upsilon(\phi) = e^{-j\frac{N-1}{2}\phi} \ \Big[ \frac{1 - e^{j N \...
4
votes
3answers
168 views

What the terms “basis functions” and “orthogonal” denote in the case of signals?

I am a beginer. I have read that any given signal whether it is simple or complex one,can be represented as summation of orthogonal basis functions. Here, what the terms Orthogonal and Basis function ...
4
votes
1answer
2k views

Derivative of a random variable w.r.t. a deterministic variable

I'm reading about time series and I thought of this procedure: can you differentiate a function containing a random variable. For example: $f(t) = a t + b + \epsilon$ where $\epsilon \sim N(0,1)$. ...
4
votes
2answers
340 views

Simple lowpass frequency response

Okay, so hopefully this isn't too hard or off-topic. Let's say I have a very simple lowpass filter (something that smooths out a signal), and the filter object has a position variable and a cutoff ...
4
votes
4answers
4k views

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 ...
4
votes
2answers
421 views

When does Discrete Fourier analysis fail to detect a frequency?

I'm using python to learn about Discrete Fourier Analysis. What I want to understand is when does the technique fail to recover some frequency of the signal? I understand how this can occur via the ...
4
votes
2answers
630 views

Is the convolution an invertible operation?

If I have a signal $f(x,y)$ (discrete) and I convolve this signal with a kernal $h(x,y)$: $y(x,y) = f(x,y) \star h(x,y)$ (where $\star$ is the convolution operator) Can I obtain $f(x,y)$ given only $...
4
votes
1answer
35 views

Equation for a sinusoidal wave with changing frequency

I would like to find a sinusoidal wave whose period or frequency change to half (or double) with every step, someting like this But I cant find the precise coefficients for the period to decrease (...