From 5651298d169c85b7376b42e2b6fd463e34b68dd7 Mon Sep 17 00:00:00 2001 From: Luc Date: Fri, 19 Jan 2024 13:26:24 +0100 Subject: [PATCH] Added signal analysis to mathematical physics section. --- config/en/mkdocs.yaml | 5 +- .../signal-analysis/signals.md | 219 ++++++++++++++++++ docs/en/physics/index.md | 13 +- 3 files changed, 235 insertions(+), 2 deletions(-) create mode 100644 docs/en/mathematics/mathematical-physics/signal-analysis/signals.md diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml index e47de3d..84148e8 100755 --- a/config/en/mkdocs.yaml +++ b/config/en/mkdocs.yaml @@ -112,7 +112,10 @@ nav: - 'Second order differential equations': mathematics/ordinary-differential-equations/second-order-ode.md - 'Systems of linear differential equations': mathematics/ordinary-differential-equations/systems-of-linear-ode.md - 'The Laplace transform': mathematics/ordinary-differential-equations/laplace-transform.md - + - 'Mathematical physics': + - 'Signal analysis': + - 'Signals': mathematics/mathematical-physics/signal-analysis/signals.md + - 'Physics': - physics/index.md diff --git a/docs/en/mathematics/mathematical-physics/signal-analysis/signals.md b/docs/en/mathematics/mathematical-physics/signal-analysis/signals.md new file mode 100644 index 0000000..27b641e --- /dev/null +++ b/docs/en/mathematics/mathematical-physics/signal-analysis/signals.md @@ -0,0 +1,219 @@ +# Signals + +## Definitions + +> *Definition*: a signal is a function of space and time. +> * Output can be analog or quantised. +> * Input can be continuous or discrete. + +
+ +> *Definition*: a signal can be sampled at particular moments $k T_s$ in time, with $k \in \mathbb{Z}$ and $T_s \in \mathbb{R}$ the sampling period. For a signal $f: \mathbb{R} \to \mathbb{R}$ sampled with a sampling period $T_s$ may be denoted by +> +> $$ +> f[k] = f(kT_s), \qquad \forall k \in \mathbb{Z}. +> $$ + +
+ +> *Definition*: signal transformations on a function $x: \mathbb{R} \to \mathbb{R}$ obtaining the function $y: \mathbb{R} \to \mathbb{R}$ are given by +> +> | Signal transformation | Time | Amplitude | +> | :-: | :-: | :-: | +> | Reversal | $y(t) = x(-t)$ | $y(t) = -x(t)$ | +> | Scaling | $y(t) = x(at)$ | $y(t) = ax(t)$ | +> | Shifting | $y(t) = x(t - b)$ | $y(t) = x(t) + b$ | +> +> for all $t \in \mathbb{R}$. + +For sampled signals similar definitions hold. + +### Symmetry + +> *Definition*: consider a signal $f: \mathbb{R} \to \mathbb{R}$ which is defined in an interval which is symmetric around $t = 0$, we define. +> +> * $f$ is *even* if $f(t) = f(-t)$, $\forall t \in \mathbb{R}$. +> * $f$ is *odd* if $f(t) = -f(-t)$, $\forall t \in \mathbb{R}$. + +For sampled signals similar definitions hold. + +> *Theorem*: every signal can be decomposed into symmetric parts. + +??? note "*Proof*:" + + Will be added later. + +### Periodicity + +> *Definition*: a signal $f: \mathbb{R} \to \mathbb{R}$ is defined to be periodic in $T$ if and only if +> +> $$ +> f(t + T) = f(t), \qquad \forall t \in \mathbb{R}. +> $$ + +For sampled signals similar definitions hold. + +> *Theorem*: a summation of two periodic signals with periods $T_1, T_2 \in \mathbb{R}$ respectively is periodic if and only if +> +> $$ +> \frac{T_1}{T_2} \in \mathbb{Q}. +> $$ + +??? note "*Proof*:" + + Will be added later. + +### Signals + +> *Definition*: the Heaviside step signal $u: \mathbb{R} \to \mathbb{R}$ is defined by +> +> $$ +> u(t) = \begin{cases} 1 &\text{ if } t > 0,\\ 0 &\text{ if } t < 0,\end{cases} +> $$ +> +> for all $t \in \mathbb{R}$. + +For a sampled function the Heaviside step signal is given by + +$$ + u[k] = \begin{cases} 1 \text{ if } k \geq 0, \\ 0 \text{ if } k < 0, \end{cases} +$$ + +for all $k \in \mathbb{Z}$. + +> *Definition*: the rectangular signal $\text{rect}: \mathbb{R} \to \mathbb{R}$ is defined by +> +> $$ +> \text{rect} (t) = \begin{cases} 1 &\text{ if } |t| < \frac{1}{2}, \\ 0 &\text{ if } |t| > \frac{1}{2},\end{cases} +> $$ +> +> for all $t \in \mathbb{R}$. + +The rect signal can be normalised obtaining the scaled rectangular signal $D: \mathbb{R} \to \mathbb{R}$ defined by + +$$ + D(t, \varepsilon) = \begin{cases} \frac{1}{\varepsilon} &\text{ if } |t| < \frac{\varepsilon}{2},\\ 0 &\text{ if } |t| > \frac{\varepsilon}{2},\end{cases} +$$ + +for all $t \in \mathbb{R}$. + +The following signal has been derived from the scaled rectangular signal $D: \mathbb{R} \to \mathbb{R}$ used on a signal $f: \mathbb{R} \to \mathbb{R}$ for + +$$ + \lim_{\varepsilon \;\downarrow\; 0} \int_{-\infty}^{\infty} f(t) D(t, \varepsilon)dt = \lim_{\varepsilon \;\downarrow\; 0} \frac{1}{\varepsilon} \int_{-\frac{\varepsilon}{2}}^{\frac{\varepsilon}{2}} f(t) dt = f(0), +$$ + +using the mean [value theorem for integrals](../calculus/integration.md#the-mean-value-theorem-for-integrals). + +> *Definition*: the Dirac signal $\delta$ is a generalized signal defined by the properties +> +> $$ +> \begin{align*} +> \delta(t - t_0) = 0 \quad \text{ for } t \neq t_0,& \\ +> \int_{-\infty}^\infty f(t) \delta(t - t_0) dt = f(t_0),& +> \end{align*} +> $$ +> +> for a signal $f: \mathbb{R} \to \mathbb{R}$ continuous in $t_0$. + +For sampled signals the $\delta$ signal is given by + +$$ + \delta[k] = \begin{cases} 1 &\text{ if } k = 0, \\ 0 &\text{ if } k \neq 0.\end{cases} +$$ + +## Signal sampling + +We already established that a signal $f: \mathbb{R} \to \mathbb{R}$ can be sampled with a sampling period $T_s \in \mathbb{R}$ obtaining $f[k] = f(kT_s)$ for all $k \in \mathbb{Z}$. We can also define a *time-continuous* signal $f_s(t)$ that represents the sampled signal using the Dirac signal, obtaining + +$$ + f_s(t) = f(t) \sum_{k = - \infty}^\infty \delta(t - k T_s), \qquad \forall t \in \mathbb{R}. +$$ + +> *Definition*: the sampling signal or impulse train $\delta_{T_s}: \mathbb{R} \to \mathbb{R}$ is defined as +> +> $$ +> \delta_{T_s}(t) = \sum_{k = - \infty}^\infty \delta(t - k T_s) +> $$ +> +> for all $t \in \mathbb{R}$ with a sampling period $T_s \in \mathbb{R}$. + +Then integration works out since we have + +$$ + \int_{-\infty}^\infty f(t) \delta_{T_s}(t) dt = \sum_{k = -\infty}^\infty \int_{-\infty}^\infty f(t) \delta(t - k T_s) dt = \sum_{k = -\infty}^\infty f [k], +$$ + +by definition. + +## Convolutions + +> *Definition*: let $f,g: \mathbb{R} \to \mathbb{R}$ be two continuous signals, the convolution product is defined as +> +> $$ +> f(t) * g(t) = \int_{-\infty}^\infty f(u)g(t-u)du +> $$ +> +> for all $t \in \mathbb{R}$. + +
+ +> *Proposition*: the convolution product is commutative, distributive and associative. + +??? note "*Proof*:" + + Will be added later. + +> *Theorem*: let $f: \mathbb{R} \to \mathbb{R}$ be a signal then we have for the convolution product between $f$ and the Dirac signal $\delta$ and some $t_0 \in \mathbb{R}$ +> +> $$ +> f(t) * \delta(t - t_0) = f(t - t_0) +> $$ +> +> for all $t \in \mathbb{R}$ + +??? note "*Proof*:" + + let $f: \mathbb{R} \to \mathbb{R}$ be a signal and $t_0 \in \mathbb{R}$, using the definition of the Dirac signal + + $$ + f(t) * \delta(t - t_0) = \int_{-\infty}^\infty f(u) \delta(t - t_0 - u)du = f(t - t_0), + $$ + + for all $t \in \mathbb{R}$. + +In particular $f(t) * \delta(t) = f(t)$ for all $t \in \mathbb{R}$; $\delta$ is the unity of the convolution. + +The average value of a signal $f: \mathbb{R} \to \mathbb{R}$ for an interval $\varepsilon \in \mathbb{R}$ may be given by + +$$ + f(t) * D(t, \varepsilon) = \frac{1}{\varepsilon} \int_{t - \frac{\varepsilon}{2}}^{t + \frac{\varepsilon}{2}} f(u)du. +$$ + +For sampled/discrete signals we have a similar definition for the convolution product, given by + +$$ + f[k] * g[k] = \sum_{m = -\infty}^{\infty} f[m]g[k - m], +$$ + +for all $k \in \mathbb{Z}$. + +## Correlations + +> *Definition*: let $f,g: \mathbb{R} \to \mathbb{R}$ be two continuous signals, the cross-correlation is defined as +> +> $$ +> f(t) \star g(t) = \int_{-\infty}^\infty f(u) g(t + u)du +> $$ +> +> for all $t \in \mathbb{R}$. + +Especially the auto-correlation of a continuous signal $f: \mathbb{R} \to \mathbb{R}$ given by $f(t) \star f(t)$ for all $t \in \mathbb{R}$ is useful, as it can detect periodicity without stating the proof. + +For sampled/discrete signals a similar definition exists given by + +$$ + f[k] \star g[k] = \sum_{m = -\infty}^{\infty} f[m]g[k + m], +$$ + +for all $k \in \mathbb{Z}$. \ No newline at end of file diff --git a/docs/en/physics/index.md b/docs/en/physics/index.md index 2928a39..b00ae27 100644 --- a/docs/en/physics/index.md +++ b/docs/en/physics/index.md @@ -1,3 +1,14 @@ # Physics -Welcome to the physics page. \ No newline at end of file +Welcome to the physics page. Some special physical environments that will be used in this seection are listed and explained below. + +* *Principles*: a fundamental rule or concept in physics serving as a basis for reasoning. +* *Assumptions*: a less fundamental rule or concept in physics that is taken to be true such that certain phenoma can be simplified. +* *Definitions*: a precise and unambiguous description of the meaning of a physical term. It characterizes the meaning of a word by giving all the properties an only those properties that must be true. +* *Laws*: a physical statement that is proved to be true, under the made assumptions and posed principles. In a physical text, the term law is often reserved for the most important results. +* *Propositions*: an often interesting result, but generally less important than a law. +* *Lemmas*: a minor result whose purpose is to help in proving a law. It is a stepping stone on the path to proving a law. +* *Corollaries*: a result in which the proof relies heavily on a given law. +* *Proofs*: a convincing argument that a certain physical statement is necessarily true. A proof generally uses deductive reasoning and logic but also contains some amount of ordinary language. + +The physics sections of this wiki are based on various books and lectures. A comprehensive list of references can be found below. \ No newline at end of file