即使以相同的『數學方法』，因為談論不同的事，有時還是得腦筋『急轉彎』。這個腦筋『急轉彎』的由來，也許是因為得用『非常不同』的觀點來考察事物。突然發現某種不期而遇之『共通處』。若問笛卡爾如何想出『解析幾何』，使得幾何得以用『座標系』與『座標』來處理？雖說不得而知，但是這套數學『思維方法』構成了牛頓力學的骨幹。就像問著『約瑟夫‧傅立葉』 Joseph Fourier 為什麼會將函數用『正弦級數』來作展開？？難道是受到笛卡爾的啟發！或是牛頓的影響！！可能很難定論。此處只能借著科學史上的一個小故事來暗示那種概念間冥冥的『聯繫』︰

……

或許能說明『正交函數族』之概念，並非一蹴而成的，所謂『正交』也不可講成『畢氏定理』自然的擴張。雖然說這些『象徵』表達概念可能的深層『融會』，提示讀者進入的門徑。究其實，依舊是例說寓言罷了。

就像『振幅調變』 $x(t) \cdot y(t)$ 之『頻譜』，可以用『頻譜摺積』來計算，然而什麼是『摺積』呢？

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated. Convolution is similar to cross-correlation. It has applications that include probability, statistics, computer vision, natural language processing, image and signal processing, engineering, and differential equations.

The convolution can be defined for functions on groups other than Euclidean space. For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 10 at DTFT#Properties.) A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing.

Computing the inverse of the convolution operation is known as deconvolution.

Visual comparison of convolution, cross-correlation and autocorrelation.

摺積

在泛函分析中，捲積、疊積、摺積或旋積，是通過兩個函數f和g生成第三個函數的一種數學算子，表徵函數f與經過翻轉和平移的g的重疊部分的面積。如果將參加摺積的一個函數看作區間的指示函數，摺積還可以被看作是「滑動平均」的推廣。

圖示兩個方形脈衝波的捲積。其中函數”g”首先對 $\tau=0$ 反射，接著平移”t”，成為 $g(t-\tau)$ 。那麼重疊部份的面積就相當於”t”處的捲積，其中橫坐標代表待積變量 $\tau$ 以及新函數 $f\ast g$ 的自變量”t”。

圖示方形脈衝波和指數衰退的脈衝波的捲積（後者可能出現於RC電路中），同樣地重疊部份面積就相當於”t”處的捲積。注意到因為”g”是對稱的，所以在這兩張圖中，反射並不會改變它的形狀。

簡單介紹

摺積是分析數學中一種重要的運算。設： $f(x)$ , $g(x)$ 是 $\mathbb{R}$ 上的兩個可積函數，作積分：

\int_{-\infty}^{\infty} f(\tau) g(x - \tau)\, \mathrm{d}\tau

可以證明，關於幾乎所有的 $x \in (-\infty,\infty)$ ，上述積分是存在的。這樣，隨著 $x$ 的不同取值，這個積分就定義了一個新函數 $h(x)$ ，稱為函數 $f$ 與 $g$ 的摺積，記為 $h(x)=(f*g)(x)$ 。我們可以輕易驗證： $(f * g)(x) = (g * f)(x)$ ，並且 $(f * g)(x)$ 仍為可積函數。這就是說，把摺積代替乘法， $L^1(R^1)$ 空間是一個代數，甚至是巴拿赫代數。雖然這裡為了方便我們假設 $\textstyle f, g\in L^1(\mathbb{R})$ ，不過捲積只是運算符號，理論上並不需要對函數 $f,g$ 有特別的限制，雖然常常要求 $f,g$ 至少是可測函數（measurable function）（如果不是可測函數的話，積分可能根本沒有意義），至於生成的卷積函數性質會在運算之後討論。

摺積與傅立葉變換有著密切的關係。例如兩函數的傅立葉變換的乘積等於它們摺積後的傅立葉變換，利用此一性質，能簡化傅立葉分析中的許多問題。

由摺積得到的函數 $f*g$ 一般要比 $f$ 和 $g$ 都光滑。特別當 $g$ 為具有緊支集的光滑函數， $f$ 為局部可積時，它們的摺積 $f * g$ 也是光滑函數。利用這一性質，對於任意的可積函數 $f$ ，都可以簡單地構造出一列逼近於 $f$ 的光滑函數列 $f_s$ ，這種方法稱為函數的光滑化或正則化。

摺積的概念還可以推廣到數列、測度以及廣義函數上去。

─《勇闖新世界︰ W!O《卡夫卡村》變形祭︰品味科學‧教具教材 ‧【專題】 PD‧箱子世界‧摺積》

為說明『摺積』的拉普拉斯變換，先請讀者注意『數學方法』講究『嚴謹』！『記號法』當慎選！務明確『概念』之『定義』也！

Definition

The convolution of f and g is written f∗g, using an asterisk or star. It is defined as the integral of the product of the two functions after one is reversed and shifted. As such, it is a particular kind of integral transform:

\displaystyle {\begin{aligned}(f*g)(t)&\,{\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau \\&=\int _{-\infty }^{\infty }f(t-\tau )g(\tau )\,d\tau .\end{aligned}}

While the symbol t is used above, it need not represent the time domain. But in that context, the convolution formula can be described as a weighted average of the function f(τ) at the moment t where the weighting is given by g(−τ) simply shifted by amount t. As tchanges, the weighting function emphasizes different parts of the input function.

For functions f, g supported on only $\displaystyle [0,\infty )$ (i.e., zero for negative arguments), the integration limits can be truncated, resulting in

\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau {\text{ for }}f,g:[0,\infty )\to \mathbb {R}

In this case, the Laplace transform is more appropriate than the Fourier transform below and boundary terms become relevant.

For the multi-dimensional formulation of convolution, see domain of definition (below).

Notation

A primarily engineering convention that one often sees is:^[1]

\displaystyle f(t)*g(t)\,{\stackrel {\mathrm {def} }{=}}\ \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(f*g)(t)},

which has to be interpreted carefully to avoid confusion. For instance, ƒ(t)* g(t − t₀) is equivalent to (ƒ* g)(t − t₀), but ƒ(t − t₀)* g(t − t₀) is in fact equivalent to (ƒ* g)(t − 2t₀).^[2]

Derivations

Convolution describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function). See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms.

認真對待所有推導『假設』！

Convolution

The convolution theorem states (if you haven’t studied convolution, you can skip this theorem)

note: we assume both f(t) and g(t) are causal.

We start our proof with the definition of the Laplace Transform

From there we continue:

	We can change the order of integration.
	Now, we pull f(λ) out because it is constant with respect to the variable of integration, t
	Now we make a change of variables
	Since g(u) is zero for u<0, we can change the lower limit on the inner integral to 0^–.
	We can pull e^-sλ out (it is constant with respect to integration).
	We can separate the integrals since the inner integral doesn’t depend on λ.
	We can change the lower limit on the first integral since f(λ) is causal.
	Finally we recognize that the two integrals are simply Laplace Transforms.
	The Theorem is proven

※ 註︰

『純』者，白賁无咎乎？用之於『數學』，思維奔馳於『抽象世界』中；用之於『物理』，想象悠遊於『宇宙自然』裡！

不知就『純邏輯』而言，它們是否都算『應用』耶？

『孤虛者』借物寓意，『數學』不必是『物理』也！？

所以在科學以及工程實務領域，鮮少談及

『兩邊‧拉普拉斯變換』哩？！

Two-sided Laplace transform

In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability‘s moment generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, and the ordinary or one-sided Laplace transform. If ƒ(t) is a real or complex valued function of the real variable t defined for all real numbers, then the two-sided Laplace transform is defined by the integral

\displaystyle {\mathcal {B}}\{f\}(s)=F(s)=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.

The integral is most commonly understood as an improper integral, which converges if and only if each of the integrals

\displaystyle \int _{0}^{\infty }e^{-st}f(t)\,dt,\quad \int _{-\infty }^{0}e^{-st}f(t)\,dt

exists. There seems to be no generally accepted notation for the two-sided transform; the $\displaystyle {\mathcal {B}}$ used here recalls “bilateral”. The two-sided transform used by some authors is

\displaystyle {\mathcal {T}}\{f\}(s)=s{\mathcal {B}}\{f\}(s)=sF(s)=s\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.

In pure mathematics the argument t can be any variable, and Laplace transforms are used to study how differential operators transform the function.

In science and engineering applications, the argument t often represents time (in seconds), and the function ƒ(t) often represents a signal or waveform that varies with time. In these cases, the signals are transformed by filters, that work like a mathematical operator, but with a restriction. They have to be causal, which means that the output in a given time t cannot depend on an output which is a higher value of t. In population ecology, the argument t often represents spatial displacement in a dispersal kernel.

When working with functions of time, ƒ(t) is called the time domain representation of the signal, while F(s) is called the s-domain (or Laplace domain) representation. The inverse transformation then represents a synthesis of the signal as the sum of its frequency components taken over all frequencies, whereas the forward transformation represents the analysis of the signal into its frequency components.

…

Relationship to other integral transforms

If u is the Heaviside step function, equal to zero when its argument is less than zero, to one-half when its argument equals zero, and to one when its argument is greater than zero, then the Laplace transform $\displaystyle {\mathcal {L}}$ may be defined in terms of the two-sided Laplace transform by

\displaystyle {\mathcal {L}}\{f\}={\mathcal {B}}\{fu\}.

On the other hand, we also have

\displaystyle {\mathcal {B}}\{f\}={\mathcal {L}}\{f\}+{\mathcal {L}}\{f\circ m\}\circ m,

where $\displaystyle m:\mathbb {R} \to \mathbb {R}$ is the function that multiplies by minus one ( $\displaystyle m(x):=-x\quad \forall x\in \mathbb {R}$ ), so either version of the Laplace transform can be defined in terms of the other.

The Mellin transform may be defined in terms of the two-sided Laplace transform by

\displaystyle {\mathcal {M}}\{f\}={\mathcal {B}}\{f\circ \exp \circ m\},

with $\displaystyle m$ as above, and conversely we can get the two-sided transform from the Mellin transform by

\displaystyle {\mathcal {B}}\{f\}={\mathcal {M}}\{f\circ m\circ \log \}.

The Fourier transform may also be defined in terms of the two-sided Laplace transform; here instead of having the same image with differing originals, we have the same original but different images. We may define the Fourier transform as

\displaystyle {\mathcal {F}}\{f(t)\}=F(s=i\omega )=F(\omega ).

Note that definitions of the Fourier transform differ, and in particular

\displaystyle {\mathcal {F}}\{f(t)\}=F(s=i\omega )={\frac {1}{\sqrt {2\pi }}}{\mathcal {B}}\{f(t)\}(s)

is often used instead. In terms of the Fourier transform, we may also obtain the two-sided Laplace transform, as

\displaystyle {\mathcal {B}}\{f(t)\}(s)={\mathcal {F}}\{f(t)\}(-is).

The Fourier transform is normally defined so that it exists for real values; the above definition defines the image in a strip $\displaystyle a<\Im (s)<b$ which may not include the real axis.

The moment-generating function of a continuous probability density function ƒ(x) can be expressed as $\displaystyle {\mathcal {B}}\{f\}(-s)$ .

……

Properties

It has basically the same properties of the unilateral transform with an important difference

**PROPERTIES OF THE UNILATERAL LAPLACE TRANSFORM**
	TIME DOMAIN	UNILATERAL-‘S’ DOMAIN	BILATERAL-‘S’ DOMAIN
DIFFERENTIATION	$\displaystyle f'(t)$	$\displaystyle sF(s)-f(0)$	$\displaystyle sF(s)$
SECOND DIFFERENTIATION	$\displaystyle f''(t)$	$\displaystyle s^{2}F(s)-sf(0)-f'(0)$	$\displaystyle s^{2}F(s)$

………

Causality

Bilateral transforms do not respect causality. They make sense when applied over generic functions but when working with functions of time (signals) unilateral transforms are preferred.

此處略提，不過是歸結耳。

小心『過程步驟』！

Laplace transform of convolution

※ 註︰

Order of integration (calculus)

In calculus, interchange of the order of integration is a methodology that transforms iterated integrals (or multiple integrals through the use of Fubini’s theorem) of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. In some cases, the order of integration can be validly interchanged; in others it cannot.

Problem statement

The problem for examination is evaluation of an integral of the form

\displaystyle \iint _{D}\ f(x,y)\ dx\,dy,

where D is some two-dimensional area in the xy–plane. For some functions f straightforward integration is feasible, but where that is not true, the integral can sometimes be reduced to simpler form by changing the order of integration. The difficulty with this interchange is determining the change in description of the domain D.

The method also is applicable to other multiple integrals.^[1]^[2]

Sometimes, even though a full evaluation is difficult, or perhaps requires a numerical integration, a double integral can be reduced to a single integration, as illustrated next. Reduction to a single integration makes anumerical evaluation much easier and more efficient.

Relation to integration by parts

Figure 1: Integration over the triangular area can be done using vertical or horizontal strips as the first step. This is an overhead view, looking down the z-axis onto the x-y plane. The sloped line is the curve y = x.

Consider the iterated integral

\int _{a}^{z}\,\int _{a}^{x}\,h(y)\,dy\,dx

which we will write using the prefix notation commonly seen in physics:

\displaystyle \int _{a}^{z}dx\,\int _{a}^{x}\,h(y)\,dy

In this expression, the second integral is calculated first with respect to y and x is held constant—a strip of width dx is integrated first over the y-direction (a strip of width dx in the x direction is integrated with respect to the y variable across the y direction), adding up an infinite amount of rectangles of width dy along the y-axis. This forms a three dimensional slice dx wide along the x-axis, from y=a to y=x along the y axis, and in the z direction z=f(x,y). Notice that if the thickness dx is infinitesimal, x varies only infinitesimally on the slice. We can assume that x is constant.^[3] This integration is as shown in the left panel of Figure 1, but is inconvenient especially when the function h ( y ) is not easily integrated. The integral can be reduced to a single integration by reversing the order of integration as shown in the right panel of the figure. To accomplish this interchange of variables, the strip of width dy is first integrated from the line x = y to the limit x = z, and then the result is integrated from y = a to y = z, resulting in:

\displaystyle \int _{a}^{z}\ dx\ \int _{a}^{x}\ h(y)\ dy\ =\int _{a}^{z}\ h(y)\ dy\ \ \int _{y}^{z}\ dx=\int _{a}^{z}\ \left(z-y\right)h(y)\,dy\ .

This result can be seen to be an example of the formula for integration by parts, as stated below:^[4]

\displaystyle \int _{a}^{z}f(x)g'(x)\,dx=\left[f(x)g(x)\right]_{a}^{z}-\int _{a}^{z}f'(x)g(x)\,dx

Substitute:

\displaystyle f(x)=\int _{a}^{x}\ h(y)\,dy{\text{ and }}g'(x)=1\ .

Which gives the result.

可以得

何謂『線性系統』？假使從『系統論』的觀點來看，一個物理系統 $S$ ，如果它的『輸入輸出』或者講『刺激響應』滿足

設使 $I_m(\cdots, t) \Rightarrow_{S} O_m(\cdots, t)$ ， $I_n(\cdots, t) \Rightarrow_{S} O_n(\cdots, t)$ ，

那麼 $\alpha \cdot I_m(\cdots, t) + \beta \cdot I_n(\cdots, t) \Rightarrow_{S} \alpha \cdot O_m(\cdots, t) + \beta \cdot O_n(\cdots, t)$

也就是說一個線線系統︰無因就無果、小因得小果，大因得大果，眾因所得果為各因之果之總計。

如果一個線性系統還滿足

$\left[I_m(\cdots, t) \Rightarrow_{S} O_m(\cdots, t)\right] \Rightarrow_{S} \left[I_m(\cdots, t + \tau) \Rightarrow_{S} O_m(\cdots, t + \tau)\right]$

，這個系統稱作『線性非時變系統』。系統中的『因果關係』是『恆常的』不隨著時間變化，因此『遲延之因』生『遲延之果』。

線性非時變 LTI Linear time-invariant theory 系統論之基本結論

是

任何 LTI 系統都可以完全祇用一個單一方程式來表示，稱之為系統的『衝激響應』。系統的輸出可以簡單表示為輸入信號與系統的『衝激響應』的『卷積』Convolution 。

雖然很多的『基礎現象』之『物理模型』可以用 LTI 系統來描述。即使已經知道一個系統是『非線性』的，將它在尚未解出之『所稱解』── 比方說『熱力平衡』時 ── 附近作系統的『線性化』處理，以了解這個系統在『那時那裡』的行為，卻是常有之事。

科技理論上偏好『線性系統』，並非只是為了『數學求解』的容易性，尤其是在現今所謂的『雲端計算』時代，祇是一般『數值解答』通常不能提供『深入理解』那個『物理現象』背後的『因果機制』的原由，所以用著『線性化』來『解析』系統『局部行為』，大概也是『不得不』的吧！就像『混沌現象』與『巨變理論』述說著『自然之大，無奇不有』，要如何『詮釋現象』難道會是『不可說』的嗎？？

─── 摘自《【SONIC Π】聲波之傳播原理︰原理篇《四中》》

矣◎

將不再懷疑前篇之『一般解』勒☆

Laplace domain

The series RLC can be analyzed for both transient and steady AC state behavior using the Laplace transform.^[16] If the voltage source above produces a waveform with Laplace-transformed $V (s)$ (where $s$ is the complex frequency $s = σ + jω$ ), the KVL can be applied in the Laplace domain:

\displaystyle V(s)=I(s)\left(R+Ls+{\frac {1}{Cs}}\right)\,,

where $I (s)$ is the Laplace-transformed current through all components. Solving for $I (s)$ :

\displaystyle I(s)={\frac {1}{R+Ls+{\frac {1}{Cs}}}}V(s)\,.

And rearranging, we have

\displaystyle I(s)={\frac {s}{L\left(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}\right)}}V(s)\,.

Laplace admittance

Solving for the Laplace admittance $Y (s)$ :

\displaystyle Y(s)={\frac {I(s)}{V(s)}}={\frac {s}{L\left(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}\right)}}\,.

Simplifying using parameters $α$ and $ω 0$ defined in the previous section, we have

\displaystyle Y(s)={\frac {I(s)}{V(s)}}={\frac {s}{L\left(s^{2}+2\alpha s+\omega _{0}^{2}\right)}}\,.

Poles and zeros

The zeros of $Y (s)$ are those values of $s$ such that $Y (s) = 0$ :

\displaystyle s=0\quad {\mbox{and}}\quad |s|\rightarrow \infty \,.

The poles of $Y (s)$ are those values of $s$ such that $Y (s) \to \infty$ . By the quadratic formula, we find

\displaystyle s=-\alpha \pm {\sqrt {\alpha ^{2}-\omega _{0}^{2}}}\,.

The poles of $Y (s)$ are identical to the roots $s 1$ and $s 2$ of the characteristic polynomial of the differential equation in the section above.

General solution

For an arbitrary $V (t)$ , the solution obtained by inverse transform of $I (s)$ is:

In the underdamped case, $ω 0 > α$ :

$\displaystyle I(t)={\frac {1}{L}}\int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\left(\cos \omega _{\mathrm {d} }\tau -{\frac {\alpha }{\omega _{\mathrm {d} }}}\sin \omega _{\mathrm {d} }\tau \right)\,d\tau \,,$
$\displaystyle I(t)={\frac {1}{L}}\int _{0}^{t}V(t-\tau )e^{-\alpha \tau }(1-\alpha \tau )\,d\tau \,,$
In the overdamped case, $ω 0 < α$ :

$\displaystyle I(t)={\frac {1}{L}}\int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\left(\cosh \omega _{\mathrm {r} }\tau -{\alpha \over \omega _{\mathrm {r} }}\sinh \omega _{\mathrm {r} }\tau \right)\,d\tau \,,$

where $ω r = \sqrt α 2 - ω 02$ , and $cosh$ and $sinh$ are the usual hyperbolic functions.

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STEM 隨筆︰古典力學︰轉子【五】《電路學》五【電感】 V‧中

Convolution

摺積

簡單介紹

Definition

Notation

Derivations

Convolution

Two-sided Laplace transform

Relationship to other integral transforms

Properties

Causality

Order of integration (calculus)

Problem statement

Relation to integration by parts

Laplace domain

Laplace admittance

Poles and zeros

General solution

輕。鬆。學。部落客