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 \(t \in \mathbb{R}\).

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 \(t \in \mathbb{R}\).

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 \(t \in \mathbb{R}\).

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.

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 \(\omega \in \mathbb{R}\).

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.

Proof:

Will be added later.