6

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


6

You might be mixing two concepts, pertaining to (following Robert Bristow-Johnson) the "analog context" (more formally, the "continuous-time case"), the "digital context" (more formally the "discrete-time case"). The first concept corresponds to the continuous Fourier transform, for which you can use a form of normalized frequency cycles per second or ...


5

It means $x$ is an $m$-vector of real values. $\mathfrak{R}$ itself is the set of real numbers.


4

What a weird typographical error. It's a minus. Get a working copy of the PDF from http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.44.87.


4

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 $x : \mathbb R \to \mathbb R$ be the input signal and let $y := \mathcal T (x)$ be the output signal. Let $\mathcal D_{t_0}$ be the delay operator that delays its input by $t_0 > 0$....


3

Since the word did not appear in previous answers, I would suggest the meaning of "transfer" function. Systems theory is globally about relationship between inputs, outputs, and some process in-between, sometimes called an input–process–output (IPO) model: The input–process–output (IPO) model, or input-process-output pattern, is a widely used approach ...


2

i think it's because poles are more important than zeros. the location of the poles determine the stability of the system. when partial fraction expansion, it's the poles that survive in the partial fractions. it's the dominant poles that determine the decay rate of the impulse response (or any response after the input goes to zero). and it's in the ...


2

My suspicion is that this ordering comes from the difference equation, which in most texts precedes the $\mathcal Z$-transform: $$a_0 y[n] + a_1 y[n-1] + \cdots = b_0 x[n] + b_1 x[n-1] + \cdots$$


2

The matrices $\hat{\mathbf{M}}$ and $\tilde{\mathbf{M}}$ are constructed in such a way that the relation $\mathbf{M}\mathbf{x}=\mathbf{y}$ implies $\hat{\mathbf{M}}\hat{\mathbf{x}}=\hat{\mathbf{y}}$ and $\tilde{\mathbf{M}}\tilde{\mathbf{x}}=\tilde{\mathbf{y}}$. Consequently, for constructing the matrix $\tilde{\mathbf{M}}$, each element $m_{kl}$ of $\...


1

Your question raises many very present concerns about measuring differences, and optimizing, in signal/image processing. Measuring can help: to compare different outcomes from two different processes, to optimize processing per se (for denoising, restoration). Most use cost functions related to $p$-norms (when $p\ge 1$) or quasi-norms (when $0<p\le 1$)...


1

What you are looking for is called an Indicator Function which would assign a $1$ for all those elements of $E(t)$ that satisfy the given condition. The same could also be expressed via the use of Set Builder Notation as $E_{interested} = \{1|e \in E \land e>C\}$ translating to "Elements of $E_{interested}$ take the value of 1 for all elements $e$ in $E$ ...


1

Possibly a pseudo-Darwinian effect, related to the autoregressive or all-pole models, and the initial letter 'a', the first of the alphabet. Details follow. This question made me dig into early works related to the (re)-discovery and usage of the $z$-transform for the representation of systems. Apparently, the concept of the $z$-transform was known to ...


1

1) What notations to use for probability density function is it the one below: $\mathsf{P}_y(y_n|{\mathbf{u}_n})$ $\mathsf{P}_z(z_n|{\mathbf{u}_n})$ what goes in the subscript if I want to use $z$? One way to write the density function is to subscripted upper case letters for the random variables. So $p_{Y_n}$ denotes the density of random variable $Y_n$ ...


1

You observe a sequence $(s_1,\ldots, s_N)$ where random variables $S_k$ are i.i.d. and are drawn from a set (alphabet) $\mathcal{A} = \{a_1,\ldots, a_m\}$ with letter probabilities $P(a_i)=p_i$ for $1\leq i \leq m$ and $\sum_{i=1}^m p_i = 1$. The likelihood function can be written as: \begin{eqnarray*} l( p_1,\ldots, p_m; s_1, \ldots, s_N ) &=& P( ...


1

As stated in the text, it's called operational notation, i.e. $D$ is an operator being applied to a continuous-time signal: $$D^mx(t)=\frac{d^mx(t)}{dt^m}$$ In discrete time you have the operator $T$, defined by $$T^mx[n]=x[n+m]$$ (see e.g. here). What is often done by people is that they use the complex Laplace transform variable $s$ instead of $D$, or ...


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