Linear time-invariant system

Systems are described as linear time-invariant (LTI) if they¹ are linear and invariant with respect to time. Analysis of LTI systems are important, because they describe many practical physical systems (i.e., circuits, mechanical systems). The output for any LTI system is given by:

$$y(t)=h(t)\ast x(t)$$ $$Y(s)=H(s)X(s)$$

where $h(t)$ is the impulse response of the system. The Laplace domain features prominently here as another means of analysis.

Note that only LTI systems will have transfer functions. Non-LTI systems can’t be neatly represented in the frequency domain with a transfer function.

System configuration

LTI systems in series are given by the output response:

$$y=h_2\ast (h_1\ast x(t))$$

i.e., it has an impulse response of $h_2\ast h_1$, by associativity. Note that this assumes right-sidedness, finite duration, and finite action (per the existence propositions for the convolution). For a parallel connection, we have:

$$y=(h_1\ast x)+(h_2\ast x)=(h_1+h_2)\ast x$$

by the linear property of the convolution. For the negative feedback combination, we have:

$$y=h_1\ast (x-h_2\ast y)=h_1\ast x-h_1\ast (h_2\ast y)$$ $$y=\delta \ast y=(\delta +h_1\ast h_2)\ast y=h\ast x$$

i.e., if the LTI system has an impulse response $\delta +h_1\ast h_2$ and is invertible, then $y=h\ast x$ where $h=(\delta +h_1\ast h_2{)}^{-1}\ast h_1$.

Differential equations

Under certain technical conditions, ODEs can define continuous-time systems, even if the ODE cannot be solved explicitly. In fact, linear, inhomogeneous, constant-coefficient ODEs (LICC-ODEs) can define CT systems that are causal and LTI — assuming the forcing function itself isn’t non-linear.

For an $n$-th order LICC-ODE that holds $\forall t\in \mathbb{R}$, $x$ is a CT signal, $y$ is an unknown CT signal, and the coefficients are real constants. We can also always assume $a_n=1$.

$$a_n\frac{d^{n}y(t)}{d{t}^n}+a_{n-1}\frac{d^{n-1}y(t)}{d{t}^{n-1}}+\cdots +a_{0}y(t)=b_m\frac{d^{m}x(t)}{d{t}^m}+\cdots +b_{0}x(t)$$

With some notational changes, we have:

$$Q(D)y(t)=P(D)x(t)$$

where $Q(D)$ and $P(D)$ refer to the derivative terms. Solid! We run into a few problems.

• Multiple solutions — due to constant terms, our systems may have infinitely many solutions.
• Initial conditions — sometimes an IVP may be used to set a unique solutions. But this isn’t always a silver bullet and may have solutions that aren’t well-defined.
• Causality — the ODE itself might not necessarily depend only on $\tau \le t$, i.e., we shouldn’t automatically expect a causal system.

These problems are resolved so long as inputs and outputs are all right-sided. By theorem, for each right-sided input $x(t)$, the LICC-ODE will have exactly one unique right-sided solution and therefore defines a system. Moreover, the system is LTI and causal. It’s also BIBO stable if and only if $m\le n$ and all roots of $Q(s)$ (where $s\in \mathbb{C}$) have a negative real part.

By roots, I mean in the high school maths polynomial root sense.

Okay, what if our LICC-ODEs are tedious as shit to analyse symbolically in the time domain? We can take the Laplace transform on both sides, then rearrange for $H(s)$ to get a rational function. Once we’ve done this, we can factor the denominator, compute a partial fraction decomposition, and then compute the inverse Laplace transform term by term. If $m\le n$, the expression is of the form:

$$h(t)=b_0\delta (t)+[\sum _{k=1}^n{b}_k{t}^{m_k}{e}^{p_{k}t}\sin ({\omega }_{k}t+{\phi }_k)]u(t)$$

where $b,p,m,\omega ,\phi$ are constants.

Stability

If $h(t)\in {\mathsf{L}}_1$, then $T$ is BIBO stable. The resulting output is given by:

$$y(t)=H(j{\omega }_0)x(t)-F(t)x(t)$$ $$F(t)={\int }_t^{\infty }h(\tau )e^{-j{\omega }_0\tau }d\tau$$

This is derivable from the time domain response of the system (i.e., through the convolution). Since $h\in {\mathsf{L}}_1$, $F(t)$ can be bounded such that we can prove:

$$\underset{t\to \infty }{\lim }F(t)=0$$

This is the transient response, and it dies down to zero.

Footnotes

1. Shocker! ↩