Added signal filters and systems to signal analysis.
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# Signal filters
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## The impedance
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> *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
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>
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> $$
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> v_i(t) - v_o(t) = R I(t),
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> $$
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>
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> for all $t \in \mathbb{R}$. By taking the Fourier transform of both sides of the expression we find
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>
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> $$
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> V_i(\omega) - V_o(\omega) = R I(\omega),
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> $$
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>
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> for all $\omega \in \mathbb{R}$ with $V_{i,o}: \mathbb{R} \to \mathbb{C}$ the fourier transforms of the voltage difference.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *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
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>
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> $$
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> v_i(t) - v_o(t) = L I'(t),
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> $$
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>
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> for all $t \in \mathbb{R}$. By taking the Fourier transform of both sides of the expression we find
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>
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> $$
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> V_i(\omega) - V_o(\omega) = i \omega L I(\omega),
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> $$
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>
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> for all $\omega \in \mathbb{R}$ with $V_{i,o}: \mathbb{R} \to \mathbb{C}$ the fourier transforms of the voltage difference.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *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
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>
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> $$
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> v_i(t) - v_o(t) = \frac{1}{C} \int_{-\infty}^t I(t)dt,
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> $$
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>
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> for all $t \in \mathbb{R}$. By taking the Fourier transform of both sides of the expression we find
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>
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> $$
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> V_i(\omega) - V_o(\omega) = \bigg(\frac{1}{i \omega C} + \frac{\pi \delta(\omega)}{C} \bigg) I(\omega),
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> $$
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>
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> for all $\omega \in \mathbb{R}$ with $V_{i,o}: \mathbb{R} \to \mathbb{C}$ the fourier transforms of the voltage difference.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Definition*: the complex impedance $Z: \mathbb{R} \to \mathbb{C}$ is defined as
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>
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> $$
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> V_i(\omega) - V_o(\omega) = Z(\omega) I(\omega)
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> $$
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>
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> for all $\omega \in \mathbb{R}$.
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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 $\omega \in \mathbb{R}$.
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> *Proposition*: the impedance elements $Z_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in series can be summed to obtain $Z$
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>
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> $$
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> Z = Z_1 + \dots + Z_n.
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> $$
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *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 $Z$
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>
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> $$
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> \frac{1}{Z} = \frac{1}{Z_1} + \dots + \frac{1}{Z_n}.
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> $$
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??? note "*Proof*:"
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Will be added later.
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<br>
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## The transfer function
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> *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
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>
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> $$
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> V_o(\omega) = H(\omega) V_i(\omega),
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> $$
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>
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> for all $\omega \in \mathbb{R}$ with $H: \mathbb{R} \to \mathbb{C}$ the transfer function.
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The transfer function may be interpreted as a frequency filter of the signal.
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Some ideal filters are given in the list below
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* a *low-pass* filter removes all frequency components $\omega > \omega_c$ with $\omega_c \in \mathbb{R}$ the cut-off frequency,
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* a *high-pass* filter removes all frequency components $\omega < \omega_c$,
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* a *band-pass* filter removes all frequency componets outside a particular frequency range,
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* a *band-stop* filter removes all frequency compnents inside a particular frequency range.
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# Systems
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> *Definition*: a system transforms signals.
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## Operators
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> *Definition*: let $x,y: \mathbb{R} \to \mathbb{R}$ be the input and output signal related to an operator $T$ by
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>
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> $$
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> y(t) = T[x(t)]
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> $$
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>
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> for all $t \in \mathbb{R}$.
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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 $t \in \mathbb{R}$.
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> *Definition*: for systems $T_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in parallel we define operator addition by
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>
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> $$
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> T = T_1 + \dots + T_n,
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> $$
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>
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> such that for $x,y: \mathbb{R} \to \mathbb{R}$ the input and output signal obtains
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>
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> $$
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> y(t) = T[x(t)] = (T_1 + \dots + T_n)[x(t)] = T_1[x(t)] + \dots + T_n[x(t)],
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> $$
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>
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> for all $t \in \mathbb{R}$.
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<br>
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> *Definition*: for systems $T_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in series we define operator multiplication by
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>
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> $$
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> T = T_n \cdots T_1,
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> $$
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>
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> such that for $x,y: \mathbb{R} \to \mathbb{R}$ the input and output signal obtains
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>
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> $$
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> y(t) = T[x(t)] =T_n \cdots T_1 [x(t)] = T_n[T_{n-1}\cdots T_1[x(t)]],
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> $$
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>
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> for all $t \in \mathbb{R}$.
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It may be observed that the operator product is not commutative.
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## Properties of systems.
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> *Definition*: a system $T$ with inputs $x_{1,2}: \mathbb{R} \to \mathbb{R}$ is linear if and only if
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>
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> $$
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> T[a x_1(t) + b x_2(t)] = a T_1[x_1(t)] + b T_2[x_2(t)]
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> $$
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>
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> for all $t \in \mathbb{R}$ with $a,b \in \mathbb{C}$.
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<br>
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> *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 $y: \mathbb{R} \to \mathbb{R}$
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>
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> $$
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> y(t) = T[x(t)] \iff y(t - t_0) = T[x(t - t_0)],
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> $$
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>
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> for all $t_0 \in \mathbb{R}$.
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<br>
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> *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
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>
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> $$
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> T^{-1}[y(t)] = T^{-1}[T[x(t)]] = x(t)
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> $$
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>
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> for all $t \in \mathbb{R}$.
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<br>
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> *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 $t_0 \in \mathbb{R}$.
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<br>
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> *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 $t_0 \in \mathbb{R}$.
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It is commenly accepted that all physical systems are causal since by definition, a cause precedes its effect. But do not be fooled.
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> *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
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>
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> $$
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> |x(t)| \leq M \implies |y(t)| \leq P,
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> $$
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>
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> for all $M, P \in \mathbb{R}$.
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## Linear time invariant systems
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Linear time invariant systems are described by linear operators whose action on a system does not expicitly depend on time; time invariance.
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> *Definition*: consider a LTI-system $T$ given by
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>
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> $$
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> y(t) = T[x(t)],
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> $$
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>
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> for all $t \in \mathbb{R}$. The impulse response $h: \mathbb{R} \to \mathbb{R}$ of this systems is defined as
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>
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> $$
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> h(t) = T[\delta(t)]
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> $$
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>
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> for all $t \in \mathbb{R}$ with $\delta$ the Dirac delta function.
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It may be literally interpreted as the effect of an impulse at $t = 0$ on the system.
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> *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
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>
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> $$
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> y(t) = h(t) * x(t),
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> $$
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>
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> for all $t \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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Therefore the system $T$ is completely characterized by the impulse response of $T$.
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> *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
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>
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> $$
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> y(t) = (h_1(t) + h_2(t)) * x(t),
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> $$
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>
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> for all $t \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *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
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>
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> $$
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> y(t) = (h_2(t) * h_1(t)) * x(t),
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> $$
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>
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> for all $t \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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From the definition of convolutions we have $h_2 * h_1 = h_1 * h_2$ therefore the product of LTI-systems *is* commutative.
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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 $t \in \mathbb{R}^-$.
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> *Theorem*: for a LTI-system and its impulse response $h: \mathbb{R} \to \mathbb{R}$ we have
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>
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> $$
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> h(t) \overset{\mathcal{F}}\longleftrightarrow H(\omega),
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> $$
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>
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> for all $t, \omega \in \mathbb{R}$ with $H: \mathbb{R} \to \mathbb{C}$ the transfer function.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *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
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>
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> $$
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> x(t) = h^{-1}(t) * y(t),
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> $$
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>
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> for all $t \in \mathbb{R}$ if and only if
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>
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> $$
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> h^{-1} * h(t) = \delta(t),
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> $$
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>
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> for all $t \in \mathbb{R}$. The transfer function of $T^{-1}$ is then given by
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>
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> $$
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> H^{-1}(\omega) = \frac{1}{H(\omega)},
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> $$
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>
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> for all $\omega \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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<br>
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Therefore a LTI-system is invertible if and only if $H(\omega) \neq 0$ for all $\omega \in \mathbb{R}$.
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> *Theorem*: the low pass filter $H: \mathbb{R} \to \mathbb{C}$ given by the transfer function
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>
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> $$
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> H(\omega) = \text{rect} \frac{\omega}{2\omega_b},
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> $$
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>
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> for all $\omega \in \mathbb{R}$ with $\omega_b \in \mathbb{R}$ is not causal. Therefore assumed to be not physically realisable.
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??? note "*Proof*:"
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Will be added later.
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