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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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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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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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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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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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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)).
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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 ...
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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, ...
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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 ...

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)\...

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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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 ...
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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 $<\omega_c$...
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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 ...
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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 ...
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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 ...
(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}{... • 4,134 3 votes What is this theorem in this formula? Let us writeg(\omega) = \hat{f}(\omega) - \sum_i \hat{u}_i(\omega)+\frac{\hat{\lambda}(\omega)}{2}\,. This term is a typical "reconstruction error" term: it denotes the error made (pointwise, i.e....
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 ...