3.2  Contrasting Signals and Systems

The previous discussion focused on signals, while this chapter discusses how to model and work with systems. System is a generic term that can be applied to any entity that converts one (or more) input signal(s) into one (or more) output signal(s). When there are multiple inputs and outputs, the system is called MIMO. This chapter will assume that the systems have a single input and a single output (SISO). It is conventional to call the input $x(t)$ or $x[n]$, for continuous and discrete-time signals, respectively, while the system output is denoted as $y(t)$ or $y[n]$. In general, the input to output mapping $y(t)=H\{x(t)\}$ will be described by an operator $H$ and depicted as

In the special case of linear and time-invariant (LTI) systems (these concepts will be further discussed in Section 3.4), a signal called impulse response $h(t)$ or $h[n]$ is capable of fully representing the behavior of the system (i. e., its output) to any input. In other words, the impulse response completely represents the LTI system. For this very special case of LTI systems, the input/output relation is represented by

in the case of discrete or continuous-time systems, respectively. Note that having a signal (the impulse response) representing a (LTI) system may be confusing.

The Laplace transform $H(s)$ of the continuous-time impulse response $h(t)$ is called system function or transfer function. In the discrete-time case, the system function is the Z transform $H(z)$ of $h[n]$. Similarly, the Fourier transforms $H(\omega )$ and $H(e^{j\mathrm{\Omega }})$ of the impulse responses are called frequency responses. Instead of the impulse response, the LTI system can be also represented via the system function as, for example in:

or via its frequency response $H(\omega )$ or $H(e^{j\mathrm{\Omega }})$. Table 3.1 summarizes these relations and nomenclature.

Both system function and frequency response relate the input to the output by a scaling factor (gain and phase) $\kappa$ that varies with the independent variable ($s$, $z$, $\mathrm{\Omega }$, etc.). For example, if the factor is $\kappa =3$ for a given situation, the output will be three times the input value. In other words, assume that $\kappa =H({\omega }_0)=3$ for a given frequency $\omega ={\omega }_0$ rad/s. If the Fourier transform $X(\omega )$ of the input has a value $X({\omega }_0)=7+j4$ at this frequency, the output at ${\omega }_0$ is $Y({\omega }_0)=H({\omega }_0)X({\omega }_0)=21+j12$. Note that, in general, both transfer functions and frequency responses are complex-valued.

The units for system functions and frequency responses depend on the corresponding units of the input and output signals. For example, $H(s)$ can be given in ohms if $x(t)$ and $y(t)$ are given in amperes and volts, respectively.

An important special case of continuous-time LTI systems $H(s)$ are the ones described by linear differential equations with constant coefficients (coefficients that do not change over time). Similarly, the discrete-time LTI system $H(z)$ described by a linear constant-coefficient difference equation (LCCDE) is widely used in DSP. The LCCDE is often simply called difference equation and can be found via the inverse Z transform of $H(z)$.

Most physical systems have non-linearities and a behavior that changes with time. In other words, most physical systems are not LTI. But even if the system is not strictly LTI, in many important applications it is useful to model the system as such, and benefit from the large number of tools that exist to design and analyze LTI systems.