Linear Time-Invariant Systems (LTI)

Linear, time-invariant (LTI) systems form the basis for engineering design in many situations. They have the advantage that there is a rich and well-established theory for analysis and design of this class of systems. Furthermore, in many systems that are nonlinear, small deviations from some nominal steady operation are approximately governed by LTI models, so the tools of LTI system analysis and design can be applied incrementally around a nominal operating condition.

Impulse-Response Representation of LTI Systems

A very general way of representing an LTI mapping from an input signal $x$ to an output signal $y$ is through convolution of the input with system impulse response. In CT the relationship is

$$y(t)={\int }_{-\infty }^{\infty }x(\tau )h(t-\tau )$$

where $h(t)$ is the unit impulse response of the system, that is, the output of the system when the input is $\delta (t)$ In DT, we have

$$y[n]=\sum _{k=-\infty }^{\infty }x[k]h[n-k]$$

where $h[n]$ is the unit sample response of the system.

A common notation for this is written in a convolution way

$$\begin{array}{r}y(t)=x(t)\ast h(t)\\ y[n]=x[n]\ast h[n]\end{array}$$

This property is obtained by representing the input signal as a superposition of weighted impulses. In the DT case,

$$x[n]=\sum _{k=-\infty }^{\infty }x[k]\delta [n-k]$$

The response $y[n]$ to this input, by linearity and time-invariance, is the sum of the similarly scaled and shifted impulse responses, and is therefore given by the $y[n]=x[n]\ast h[n]$ above.

Tip

This mean a LTI system can be fully representation by the unit sample response $h[n]$ or the unit impulse response $h(t)$

Eigenfunction of LTI Systems

CT Case

Exponentials are eigenfunctions of LTI mappings, i.e., when the input is an exponential for all time, which we refer to as an "everlasting" exponential, the output is simply a scaled version of the input. Specifically, in the CT case, suppose

$$x(t)=e^{s_{0}t}$$

for some possibly complex value $s_0$ (termed the complex frequency). The from

$$\begin{array}{rl}y(t)& =h(t)\ast x(t)\\ & ={\int }_{-\infty }^{\infty }h(\tau )x(t-\tau )\mathrm{d}\tau \\ & ={\int }_{-\infty }^{\infty }h(\tau )\exp \left[s_0(t-\tau )\right]\mathrm{d}\tau \\ & =H(s_0)\exp (s_{0}t)\end{array}$$

where

$$H(s)={\int }_{-\infty }^{\infty }h(\tau )\exp (-s\tau )\mathrm{d}\tau$$

has a finite value for $s=s_0$ (otherwise the response to the exponential is not well defined), and is the corresponding eigenvalue

When $x(t)=e^{j\omega t}$, corresponding to having $s_0$ take the purely imaginary value $j\omega$ in $x(t)=e^{s_{0}t}$, the input is bounded for all positive and negative time, and the corresponding output is

$$y(t)=H(j\omega )e^{j\omega t}$$

where

$$H(j\omega )={\int }_{-\infty }^{\infty }h(t)e^{-j\omega t}\mathrm{d}t$$

We can see $H(j\omega )$ here is the continuous-time Fourier transform (CTFT) of the impulse response $h(t)$. The integral that defines the CTFT has a finite value if $h(t)$ is absolutely integrable, i.e. provided

$${\int }_{-\infty }^{+\infty }\left|h(t)\right|\mathrm{d}t<\infty$$

Tip

This condition is equivalent to the symptom being bounded-input, bounded-output (BIBO) stable.

DT Case

We can similarly examine the eigenfunction property in the DT case. A DT everlasting "exponential" is a geometric sequence of signal of the form

$$x[n]=z_0^n$$

for some possibly complex $z_0$ (termed the complex frequency). With this DT exponential input, the output of a convolution mapping is

$$y[n]=h[n]\ast x[n]=H(z_0)z_0^n$$

where

$$H(z)=\sum _{k=-\infty }^{\infty }h[k]z^{-k}$$

Again, an important case is when$x[n]=(e^{j\Omega }{)}^n=e^{j\Omega n}$, where $\Omega$, the (real) "frequency", denotes the angular position (in radians) around the unit circle in the $z$-plane. Such an $x[n]$ is bounded for all positive and negative time. The corresponding output is

$$y[n]=H(e^{j\Omega })e^{j\Omega n}$$

where

$$H(e^{j\Omega })=\sum _{n=-\infty }^{\infty }h[n]e^{-j\Omega n}$$

The function $H(e^{j\Omega })$ here is also the discrete-time Fourier transform (DTFT) of the impulse response.