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23 votes

Replacing "e" in Euler's formula with another number

Say you're interested in $$M^{j2\pi f_0 t}. \tag{1}$$ Note that $$M = e^{\log M},$$ so $(1)$ can be written as \begin{align} M^{j2\pi f_0 t} &= \left( e^{\log M} \right) ^ {j2\pi f_0 t} \\ &= ...
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19 votes
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What is the first derivative of Dirac delta function?

If you imagine a Dirac delta impulse as the limit of a very narrow very high rectangular impulse with unit area centered at $t=0$, then it's clear that its derivative must be a positive impulse at $0^-...
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15 votes

What is the first derivative of Dirac delta function?

First of all the dirac delta is NOT a function, it's a distribution. See for example http://web.mit.edu/8.323/spring08/notes/ft1ln04-08-2up.pdf Treating it as a conventional function can lead to ...
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10 votes
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Why Does the Median Filter Minimize the Absolute Value Error $L_1$ Cost Function?

Given a set of values $ {\left\{ {s}_{i} \right\}}_{i = 1}^{N} $, we're basically after: $$ \arg \min_{x} \sum_{i = 1}^{N} \left| {s}_{i} - x \right| $$ One should notice that $ \frac{\mathrm{d} \left ...
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  • 40.3k
9 votes
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Books/resources for implementing various mathematical functions in fixed point arithmetic for DSP purposes

the general polynomial form is: $$\begin{align} f(u) &= \sum\limits_{n=0}^{N} \ a_n \ u^n \\ \\ &= a_{\small{0}} + \Bigg(a_{\small{1}} + \bigg(a_{\small{2}} + \Big(a_{\small{3}} + \,... \...
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9 votes
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Closed form expression for $\sum_{k=0}^{n} \alpha^{-k}$?

\begin{align} (1-x)\sum_{k=0}^{n}x^k &= \sum_{k=0}^{n}x^k - x\sum_{k=0}^{n}x^k \\ \\ &= \sum_{k=0}^{n}x^k - \sum_{k=1}^{n+1}x^k \\ \\ &=1 + x+x^2+ \dots+x^n \ \\ &...
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  • 5,760
9 votes

What is the first derivative of Dirac delta function?

Maybe a picture is worth a thousand words? Here's how a Gaussian pulse of variable width and its derivatives look like: As others have said, Dirac is a distribution, hence the Gaussian pulse, and its ...
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8 votes
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How does the assumption that symbols are equi-probable hold

If I draw a number uniformly between zero and one, what is the probability that they are equal? Mathematically, it should be zero but I don't recall why? Can somebody please help in explaining why ...
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7 votes

How to Calculate Gaussian Kernel for a Small Support Size?

Gaussian Kernel is made by using the Normal Distribution for weighing the surrounding pixel in the process of Convolution. Since we're dealing with discrete signals and we are limited to finite length ...
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  • 40.3k
6 votes
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Looking for an arcsin algorithm

Here's just a polynomial version: $$ \arcsin(x) = x + \frac{1}{2} \frac{x^3}{3} + \frac{1 \cdot 3}{2 \cdot 4} \frac{x^5}{5} + \frac{1\cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{x^7}{7} $$ ...
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  • 22.2k
6 votes

Upper Bound for the DFT (FFT) Coefficients of a Bounded Signal

The upper bound would be a coherent summation of all samples of $ x \left [ x \right ] $. Specifically: $$\begin{aligned} X \left[ k \right ] & = \sum_{n = 0}^{N - 1} x \left[ n \right] \exp^{-j ...
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6 votes
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Upper Bound for the DFT (FFT) Coefficients of a Bounded Signal

You can obtain a bound on the magnitude of the DFT of $x[n]$ ($|x[n]|\le 1$) as follows: $$\big|X[k]\big|=\left|\sum_{n=0}^{N-1}x[n]e^{-jkn2\pi/N}\right|\le\sum_{n=0}^{N-1}|x[n]|\le N\max_n|x[n]|=N\...
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  • 80.1k
6 votes
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Deriviation of the "Twiddle Sum" property

The most straightforward way to see this is to note that for $k=mN$ $$W_N^{kn}=e^{j2\pi mnN/N}=e^{j2\pi mn}=1$$ So the sum for the case $k=mN$ is simply $$\sum_{n=0}^{N-1}1=1+1+\ldots+1=N$$ Note ...
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6 votes
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What is the meaning of this notation?

It just means "the transformation that turns $x$ into $y$." You might also see $\mathbf{T}^{-1}$ which means the inverse: turning $y$ into $x$.
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6 votes

What is the first derivative of Dirac delta function?

Dirac's $\delta$ is a distribution. Distributions can be interpreted as limits of smooth functions under an integral or as operators acting on functions in ways which are defined by integrals. Both ...
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  • 161
5 votes
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Weighted Sum of Auto Correlation - Lower Bound

Defining $ a \left[ k \right] = {2}^{- \left| k \right|} $. Moreover, the Auto Correlation function of $ v $ defined as $ {r}_{vv} \left[ k \right] = \left \langle {v}^{\left( 0 \right)}, {v}^{\left( ...
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  • 40.3k
5 votes
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How to prove this norm inequality?

Although the question could belong to SE.math, mastering inequalities for $\ell_p$ norms (for $p\ge 1$) or quasinorms (for $0<p< 1$), and their norm ratios and powers, is quite important in ...
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5 votes
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Absolute value based AM envelope detection viewed in the frequency domain

$\DeclareMathOperator{\sgn}{sgn}$ The modulating signal in AM is $$s(t) = C + a(t)\text,$$ where $a(t)$ is the (audio) amplitude, and $C$ is a constant so that $s(t) \ge 0 \;\forall t$. (Otherwise, ...
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5 votes

What is the first derivative of Dirac delta function?

$\delta(t)$ is a distribution, which means it is represented by a limitng set of functions. To find $\delta'(t)$, start with a limiting set of functions for $\delta(t)$ that at least have a first ...
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  • 2,525
5 votes

DFT of pure sinusoidal wave

First of all, welcome to DSP SE. What you see in the image you have linked is termed (spectral) leakage. When you are dealing with the Fourier series you deal with a periodic continuous function which ...
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  • 654
4 votes
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What Is the Intuition of Convolution in The Signal Processing World

I think this MathOverflow post gives a lot of intuition about Convolution: What's convolution intuitively? In the Signal Processing world, an LTI (Linear and Time Invariant) system basically scales ...
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  • 40.3k
4 votes

Deriviation of the "Twiddle Sum" property

The key is in the last step of your work: $$ \frac{1 -e^{j2\pi k}}{1-e{\frac{j2\pi k}{N}}} $$ If $k$ is some integer multiple of $N$, then the exponents in the numerator and denominator are both ...
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  • 23.7k
4 votes
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Minimum Mean Square Estimator - Equivalent Expressions to Minimize

Since $ M \in \mathbb{S}^{N}_{++} $ (In other convention $ M \succ 0 $) by Cholesky Decomposition there is a Triangular Matrix $ R \in \mathbb{R}^{N \times N} $ such that $ M = {R}^{T} R $. Using ...
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  • 40.3k
4 votes
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How moving part pixel intensity values of video frames becomes dominant compared to stationary part intensities in reconstructed frames?

Q1. As shown by Oppenheim's experiment, the phase spectrum contains most of the structural information about the image. In 2D this are things like lines and edges. In 3D it is things like lines and ...
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  • 3,591
4 votes

Looking for an arcsin algorithm

i have a pretty good implementation of $\arctan()$ here. i think you can use the identity: $$ \arcsin(x) = \arctan\left( \frac{x}{\sqrt{1-x^2}} \right) $$ to get what you want.
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4 votes

Mathematical question that comes out of using bilinear transform

To complement my part to this question: Here is a somewhat shorted answer based upon a manual expansion of the odd function $f(x)$ \begin{align*} f(x)&=\ln\left(\arctan\left(\alpha e^x\right)\...
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4 votes
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What is this theorem in this formula?

i have no idea what the "reconstruction fidelity term" is or what it's about. Hermitian symmetry is a term usually applied to some form the Fourier Transform of a signal that is purely real. for ...
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4 votes

What is the meaning of this notation?

Mathematically, an electrical circuit is as an operator, i.e., a function that takes a function and returns another function. Let this operator be denoted by $\mathcal T$, let $x : \mathbb R \to \...
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4 votes
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Laplacian of Gaussian Approximation and Gaussian Blur as the Solution of Heat Equation

I'm not sure I fully understood what's the issue you're having. Yet I will show a simple property of the Gaussian filter which might make things clearer. For simplicity, I will use 1D Signal. Yet it ...
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  • 40.3k
4 votes

What is the first derivative of Dirac delta function?

Simply put, $\delta'$ picks the opposite of the derivative of $f$ at the origin. Let us imagine that I can forget for a moment about that $\delta$ is not a function, that it should be defined in a ...
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