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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} \\ &= ...
MBaz's user avatar
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22 votes
Accepted

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^-...
Matt L.'s user avatar
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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 ...
Hilmar's user avatar
  • 44.8k
12 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 ...
a concerned citizen's user avatar
9 votes
Accepted

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 \ \\ &...
AlexTP's user avatar
  • 6,595
9 votes

Why Hilbert Transform is terrible choice for amplitude demodulation of broadband signals?

It's not only a matter of "broadband or not": The Hilbert estimate degrades for multi-component signals - that is, whatever we can't draw without lifting our pen, left-to-right, in time-...
OverLordGoldDragon's user avatar
9 votes

Under what conditions does DFT(f(x)) = f(DFT(x)) hold?

The fft is an efficient computation of the DFT. So your question is about the DFT, not the fft. The DFT of a signal can be ...
Royi's user avatar
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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 ...
AlexTP's user avatar
  • 6,595
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 $x$.
Peter K.'s user avatar
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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 ...
tobi_s's user avatar
  • 161
6 votes
Accepted

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 ...
Royi's user avatar
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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 ...
Laurent Duval's user avatar
5 votes
Accepted

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, ...
Marcus Müller's user avatar
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 ...
Andy Walls's user avatar
  • 2,710
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 ...
ZaellixA's user avatar
  • 1,288
5 votes

Under what conditions does DFT(f(x)) = f(DFT(x)) hold?

One (almost trivial) function is the ifft. So fft(ifft(x))=ifft(fft(x)).
Mark's user avatar
  • 357
5 votes
Accepted

How to know if a continuous function can be represented by a finite sum of sinusoids?

I think a good rule of thumb is this: "If it isn't already written as a finite sum of sinusoids, then it probably can't be written as a finite sum of sinusoids." Most functions are not a ...
Tanner Swett's user avatar
4 votes

Derivation of fixed-point $\tt atan2$ with self-normalization

I had the exact question, nearly a decade later - and think I figured out the cool fixed-points tricks thanks to and edaboard thread and helpful write up in the IEEE Signal Processing Magazine. First, ...
Scott Howard's user avatar
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)\...
Markus Scheuer's user avatar
4 votes
Accepted

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 ...
robert bristow-johnson's user avatar
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 ...
Laurent Duval's user avatar
4 votes
Accepted

Why Cramér spectral representation and not DTFT for stochastic process

I will introduce some terminology and intuition that will be helpful when reading other references. It will be neither complete nor completely rigorous. The measures that we first encounter in real ...
Joe Mack's user avatar
  • 616
4 votes

Why Hilbert Transform is terrible choice for amplitude demodulation of broadband signals?

Amplitude extraction / AM demodulation criteria I shall prove, $y(t) = x(t) \cos(\omega_c t)$ demodulates perfectly to $|x(t)|$ if A) $x$'s highest frequency, $\omega^\text{max}_x$, is $<\omega_c$...
OverLordGoldDragon's user avatar
4 votes

How to know if a continuous function can be represented by a finite sum of sinusoids?

There are actually 4 different types or Fourier Transform. Which one to use depends on the signal properties: specifically whether a signal as periodic vs aperiodic and whether it is continuous vs ...
Hilmar's user avatar
  • 44.8k
3 votes

Mathematical question that comes out of using bilinear transform

The problem as posed in the question appears to have no closed-form solution. As mentioned in the question and shown in other answers, the result can be developed into a series, which can be ...
Matt L.'s user avatar
  • 90k
3 votes
Accepted

Mathematical question that comes out of using bilinear transform

okay, i promised to put up bounty and i will keep my promise. but i have to confess that i might renege a little bit on being satisfied with just the third derivative of $f(x)$. what i really want ...
robert bristow-johnson's user avatar
3 votes

Mathematical question that comes out of using bilinear transform

(Converting comment to answer.) Using Wolfram Alpha, $f'''(x)$ at $x=0$ evaluates to: $$\begin{align} \\ f'''(0) = & -\frac{6 \alpha^2}{(\alpha^2 + 1)^2 (\arctan(\alpha))^2} \ + \ \frac{2 \alpha}{...
Atul Ingle's user avatar
  • 4,134
3 votes

Mathematical question that comes out of using bilinear transform

so here are some quantitative results. i plotted spec'd bandwidth $bw$ for the digital filter on the x-axis and the resulting digital bandwidth on the y-axis. there are five plots from green to red ...
robert bristow-johnson's user avatar
3 votes

Looking for an arcsin algorithm

So we have two requirements: Use at most 5 multiplications. Errror is no more than 10%. Requirement 2 means that we want to optimize (minimze) the relative error of the approximation. Requiring 5 ...
emacs drives me nuts's user avatar

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