Time-Domain Analysis of Continuous-Time Systems

Sources:

1. B. P. Lathi & Roger Green. (2021). Chapter 2: Time-Domain Analysis of Continuous-Time Systems. Signal Processing and Linear Systems (2nd ed., pp. 168-195). Oxford University Press.

For the purpose of analysis, we shall consider linear differential systems. This is the class of LTIC systems introduced in before chapter, for which the input $x(t)$ and the output $y(t)$ are related by linear differential equations of the form

Why this form?

$$\begin{array}{rl}& \frac{d^{N}y(t)}{d{t}^N}+a_1\frac{d^{N-1}y(t)}{d{t}^{N-1}}+\cdots +a_{N-1}\frac{dy(t)}{dt}+a_{N}y(t)\\ & =b_{N-M}\frac{d^{M}x(t)}{d{t}^M}+b_{N-M+1}\frac{d^{M-1}x(t)}{d{t}^{M-1}}+\cdots +b_{N-1}\frac{dx(t)}{dt}+b_{N}x(t)\end{array}$$ where all the coefficients $a_i$ and $b_i$ are constants. Using operator notation $D$ to represent $d/dt$, we can express this equation as $$\begin{array}{rl}& (D^N+a_1{D}^{N-1}+\cdots +a_{N-1}D+a_N)y(t)\\ & =(b_{N-M}{D}^M+b_{N-M+1}{D}^{M-1}+\cdots +b_{N-1}D+b_N)x(t)\end{array}$$ or $$Q(D)y(t)=P(D)x(t)$$ where the polynomials $Q(D)$ and $P(D)$ are $$\begin{array}{rl}& Q(D)=D^N+a_1{D}^{N-1}+\cdots +a_{N-1}D+a_N\\ & P(D)=b_{N-M}{D}^M+b_{N-M+1}{D}^{M-1}+\cdots +b_{N-1}D+b_N\end{array}$$ Theoretically, the powers $M$ and $N$ in the foregoing equations can take on any value. However, practical considerations make $M>N$ undesirable for two reasons.

In Sec. 6.3.3, we shall show that an LTIC system specified by Eq. (2.1) acts as an $(M-N)$ th-order differentiator. A differentiator represents an unstable system because a bounded input like the step input results in an unbounded output, $\delta (t)$. Second, noise is enhanced by a differentiator. Noise is a wideband signal containing components of all frequencies from 0 to a very high frequency approaching $\mathrm{\infty }$. ${}^{+}$Hence, noise contains a significant amount of rapidly varying components. We know that the derivative of any rapidly varying signal is high. Therefore, any system specified by Eq. (2.1) in which $M>N$ will magnify the high-frequency components of noise through differentiation. It is entirely possible for noise to be magnified so much that it swamps the desired system output even if the noise signal at the system's input is tolerably small.

Hence, practical systems generally use $M\le N$. For the rest of this text, we assume implicitly that $M\le N$. For the sake of generality, we shall assume $M=N$ in Eq. (2.1).

In Ch. 1 , we demonstrated that a system described by Eq. (2.2) is linear. Therefore, its response can be expressed as the sum of two components: the zero-input response and the zero-state response (decomposition property). ${}^{.}$Therefore, total response $=$ zero-input response + zero-state response

1. The zero-input response is the system output when the input $x(t)=0$, and thus it is the result of internal system conditions (such as energy storages, initial conditions) alone. It is independent of the external input $x(t)$.
2. In contrast, the zero-state response is the system output to the external input $x(t)$ when the system is in zero state, meaning the absence of all internal energy storages: that is, all initial conditions are zero.