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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.