diff --git a/docs/en/physics/mathematical-physics/signal-analysis/signal-filters.md b/docs/en/physics/mathematical-physics/signal-analysis/signal-filters.md index b23ee5e..ae80000 100644 --- a/docs/en/physics/mathematical-physics/signal-analysis/signal-filters.md +++ b/docs/en/physics/mathematical-physics/signal-analysis/signal-filters.md @@ -1,2 +1,116 @@ # Signal filters +## The impedance + +> *Proposition*: considering an ideal resistor $R \in \mathbb{R}$. When a voltage difference $v_i - v_o$ is applied to both ends a current $I: \mathbb{R} \to \mathbb{R}$ will be induced given by +> +> $$ +> v_i(t) - v_o(t) = R I(t), +> $$ +> +> for all $t \in \mathbb{R}$. By taking the Fourier transform of both sides of the expression we find +> +> $$ +> V_i(\omega) - V_o(\omega) = R I(\omega), +> $$ +> +> for all $\omega \in \mathbb{R}$ with $V_{i,o}: \mathbb{R} \to \mathbb{C}$ the fourier transforms of the voltage difference. + +??? note "*Proof*:" + + Will be added later. + +
+ +> *Proposition*: considering a load coil with inductance $L \in \mathbb{R}$. When a voltage difference $v_i - v_o$ is applied to both ends a current $I: \mathbb{R} \to \mathbb{R}$ will be induced given by +> +> $$ +> v_i(t) - v_o(t) = L I'(t), +> $$ +> +> for all $t \in \mathbb{R}$. By taking the Fourier transform of both sides of the expression we find +> +> $$ +> V_i(\omega) - V_o(\omega) = i \omega L I(\omega), +> $$ +> +> for all $\omega \in \mathbb{R}$ with $V_{i,o}: \mathbb{R} \to \mathbb{C}$ the fourier transforms of the voltage difference. + +??? note "*Proof*:" + + Will be added later. + +
+ +> *Proposition*: considering a capacitor with capacity $C \in \mathbb{R}$. When a voltage difference $v_i - v_o$ is applied to both ends a current $I: \mathbb{R} \to \mathbb{R}$ will be induced given by +> +> $$ +> v_i(t) - v_o(t) = \frac{1}{C} \int_{-\infty}^t I(t)dt, +> $$ +> +> for all $t \in \mathbb{R}$. By taking the Fourier transform of both sides of the expression we find +> +> $$ +> V_i(\omega) - V_o(\omega) = \bigg(\frac{1}{i \omega C} + \frac{\pi \delta(\omega)}{C} \bigg) I(\omega), +> $$ +> +> for all $\omega \in \mathbb{R}$ with $V_{i,o}: \mathbb{R} \to \mathbb{C}$ the fourier transforms of the voltage difference. + +??? note "*Proof*:" + + Will be added later. + +
+ +> *Definition*: the complex impedance $Z: \mathbb{R} \to \mathbb{C}$ is defined as +> +> $$ +> V_i(\omega) - V_o(\omega) = Z(\omega) I(\omega) +> $$ +> +> for all $\omega \in \mathbb{R}$. + +Therefore the complex impedance for the ideal resistor is given by $Z(\omega) = R$ and for the load coil $Z(\omega) = i \omega L$ for all $\omega \in \mathbb{R}$. + +> *Proposition*: the impedance elements $Z_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in series can be summed to obtain $Z$ +> +> $$ +> Z = Z_1 + \dots + Z_n. +> $$ + +??? note "*Proof*:" + + Will be added later. + +
+ +> *Proposition*: the impedance elements $Z_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in parallel can be inversely summed to obtain $Z$ +> +> $$ +> \frac{1}{Z} = \frac{1}{Z_1} + \dots + \frac{1}{Z_n}. +> $$ + +??? note "*Proof*:" + + Will be added later. + +
+ +## The transfer function + +> *Definition*: the relation between the input and output voltage in the frequency domain $V_{i,o}: \mathbb{R} \to \mathbb{C}$ can be written as +> +> $$ +> V_o(\omega) = H(\omega) V_i(\omega), +> $$ +> +> for all $\omega \in \mathbb{R}$ with $H: \mathbb{R} \to \mathbb{C}$ the transfer function. + +The transfer function may be interpreted as a frequency filter of the signal. + +Some ideal filters are given in the list below + +* a *low-pass* filter removes all frequency components $\omega > \omega_c$ with $\omega_c \in \mathbb{R}$ the cut-off frequency, +* a *high-pass* filter removes all frequency components $\omega < \omega_c$, +* a *band-pass* filter removes all frequency componets outside a particular frequency range, +* a *band-stop* filter removes all frequency compnents inside a particular frequency range. \ No newline at end of file diff --git a/docs/en/physics/mathematical-physics/signal-analysis/systems.md b/docs/en/physics/mathematical-physics/signal-analysis/systems.md index e69de29..0f8adb4 100644 --- a/docs/en/physics/mathematical-physics/signal-analysis/systems.md +++ b/docs/en/physics/mathematical-physics/signal-analysis/systems.md @@ -0,0 +1,213 @@ +# Systems + +> *Definition*: a system transforms signals. + +## Operators + +> *Definition*: let $x,y: \mathbb{R} \to \mathbb{R}$ be the input and output signal related to an operator $T$ by +> +> $$ +> y(t) = T[x(t)] +> $$ +> +> for all $t \in \mathbb{R}$. + +For example for a time shift of the signal $S_{t_0}: y(t) = x(t - t_0)$ we have $y(t) = S_{t_0}[x(t)]$ for all $t \in \mathbb{R}$. For an amplifier of the signal $P: y(t) = k(t) x(t)$ we have $y(t) = P[x(t)]$ for all $t \in \mathbb{R}$. + +> *Definition*: for systems $T_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in parallel we define operator addition by +> +> $$ +> T = T_1 + \dots + T_n, +> $$ +> +> such that for $x,y: \mathbb{R} \to \mathbb{R}$ the input and output signal obtains +> +> $$ +> y(t) = T[x(t)] = (T_1 + \dots + T_n)[x(t)] = T_1[x(t)] + \dots + T_n[x(t)], +> $$ +> +> for all $t \in \mathbb{R}$. + +
+ +> *Definition*: for systems $T_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in series we define operator multiplication by +> +> $$ +> T = T_n \cdots T_1, +> $$ +> +> such that for $x,y: \mathbb{R} \to \mathbb{R}$ the input and output signal obtains +> +> $$ +> y(t) = T[x(t)] =T_n \cdots T_1 [x(t)] = T_n[T_{n-1}\cdots T_1[x(t)]], +> $$ +> +> for all $t \in \mathbb{R}$. + +It may be observed that the operator product is not commutative. + +## Properties of systems. + +> *Definition*: a system $T$ with inputs $x_{1,2}: \mathbb{R} \to \mathbb{R}$ is linear if and only if +> +> $$ +> T[a x_1(t) + b x_2(t)] = a T_1[x_1(t)] + b T_2[x_2(t)] +> $$ +> +> for all $t \in \mathbb{R}$ with $a,b \in \mathbb{C}$. + +
+ +> *Definition*: a system $T$ is time invariant if and only if for all $t \in \mathbb{R}$ a shift in the input $x: \mathbb{R} \to \mathbb{R}$ results only in a shift in the output $y: \mathbb{R} \to \mathbb{R}$ +> +> $$ +> y(t) = T[x(t)] \iff y(t - t_0) = T[x(t - t_0)], +> $$ +> +> for all $t_0 \in \mathbb{R}$. + +
+ +> *Definition*: a system $T$ is invertible if distinct input $x: \mathbb{R} \to \mathbb{R}$ results in distinct output $y: \mathbb{R} \to \mathbb{R}$; the system is injective. The inverse of $T$ is defined such that +> +> $$ +> T^{-1}[y(t)] = T^{-1}[T[x(t)]] = x(t) +> $$ +> +> for all $t \in \mathbb{R}$. + +
+ +> *Definition*: a system $T$ is memoryless if the image of the output $y(t_0)$ with $y: \mathbb{R} \to \mathbb{R}$ depends only on the input $x(t_0)$ with $x: \mathbb{R} \to \mathbb{R}$ for all $t_0 \in \mathbb{R}$. + +
+ +> *Definition*: a system $T$ is causal if the image of the output $y(t_0)$ with $y: \mathbb{R} \to \mathbb{R}$ depends only on images of the input $x(t)$ for $t \leq t_0$ with $x: \mathbb{R} \to \mathbb{R}$ for all $t_0 \in \mathbb{R}$. + +It is commenly accepted that all physical systems are causal since by definition, a cause precedes its effect. But do not be fooled. + +> *Definition*: a system $T$ is bounded-input $\implies$ bounded-output (BIBO) -stable if and only if for all $t \in \mathbb{R}$ the output $y: \mathbb{R} \to \mathbb{R}$ is bounded for bounded input $x: \mathbb{R} \to \mathbb{R}$. Then +> +> $$ +> |x(t)| \leq M \implies |y(t)| \leq P, +> $$ +> +> for all $M, P \in \mathbb{R}$. + +## Linear time invariant systems + +Linear time invariant systems are described by linear operators whose action on a system does not expicitly depend on time; time invariance. + +> *Definition*: consider a LTI-system $T$ given by +> +> $$ +> y(t) = T[x(t)], +> $$ +> +> for all $t \in \mathbb{R}$. The impulse response $h: \mathbb{R} \to \mathbb{R}$ of this systems is defined as +> +> $$ +> h(t) = T[\delta(t)] +> $$ +> +> for all $t \in \mathbb{R}$ with $\delta$ the Dirac delta function. + +It may be literally interpreted as the effect of an impulse at $t = 0$ on the system. + +> *Theorem*: for a LTI-system $T$ with $x,y,h: \mathbb{R} \to \mathbb{R}$ the input, output and impulse response of the system we have +> +> $$ +> y(t) = h(t) * x(t), +> $$ +> +> for all $t \in \mathbb{R}$. + +??? note "*Proof*:" + + Will be added later. + +Therefore the system $T$ is completely characterized by the impulse response of $T$. + +> *Theorem*: for two LTI-systems in parallel given by $T = T_1 + T_2$ with $x,y,h_1,h_2: \mathbb{R} \to \mathbb{R}$ the input, output and impulse response of both systems we have +> +> $$ +> y(t) = (h_1(t) + h_2(t)) * x(t), +> $$ +> +> for all $t \in \mathbb{R}$. + +??? note "*Proof*:" + + Will be added later. + +
+ +> *Theorem*: for two LTI-systems in series given by $T = T_2 T_1$ with $x,y,h_1,h_2: \mathbb{R} \to \mathbb{R}$ the input, output and impulse response of both systems we have +> +> $$ +> y(t) = (h_2(t) * h_1(t)) * x(t), +> $$ +> +> for all $t \in \mathbb{R}$. + +??? note "*Proof*:" + + Will be added later. + +From the definition of convolutions we have $h_2 * h_1 = h_1 * h_2$ therefore the product of LTI-systems *is* commutative. + +For a causal system there is no effect before its cause, a causal LTI system therefore must have an impulse response $h: \mathbb{R} \to \mathbb{R}$ that must be zero for all $t \in \mathbb{R}^-$. + +> *Theorem*: for a LTI-system and its impulse response $h: \mathbb{R} \to \mathbb{R}$ we have +> +> $$ +> h(t) \overset{\mathcal{F}}\longleftrightarrow H(\omega), +> $$ +> +> for all $t, \omega \in \mathbb{R}$ with $H: \mathbb{R} \to \mathbb{C}$ the transfer function. + +??? note "*Proof*:" + + Will be added later. + +
+ +> *Theorem*: for a LTI system $T$ with $x,y,h: \mathbb{R} \to \mathbb{R}$ the input, output and its impulse if the inverse system $T^{-1}$ exists it has an impulse response $h^{-1}: \mathbb{R} \to \mathbb{R}$ such that +> +> $$ +> x(t) = h^{-1}(t) * y(t), +> $$ +> +> for all $t \in \mathbb{R}$ if and only if +> +> $$ +> h^{-1} * h(t) = \delta(t), +> $$ +> +> for all $t \in \mathbb{R}$. The transfer function of $T^{-1}$ is then given by +> +> $$ +> H^{-1}(\omega) = \frac{1}{H(\omega)}, +> $$ +> +> for all $\omega \in \mathbb{R}$. + +??? note "*Proof*:" + + Will be added later. + +
+ +Therefore a LTI-system is invertible if and only if $H(\omega) \neq 0$ for all $\omega \in \mathbb{R}$. + +> *Theorem*: the low pass filter $H: \mathbb{R} \to \mathbb{C}$ given by the transfer function +> +> $$ +> H(\omega) = \text{rect} \frac{\omega}{2\omega_b}, +> $$ +> +> for all $\omega \in \mathbb{R}$ with $\omega_b \in \mathbb{R}$ is not causal. Therefore assumed to be not physically realisable. + +??? note "*Proof*:" + + Will be added later. \ No newline at end of file