2.4a Cascade

Cascade of Multiple Systems

In signals and systems, a cascade refers to the serial connection of two or more systems, where the output of one system serves as the input to the next system. For example, a cascade $H$ of two systems $F$ and $G$ can be graphically represented as

\begin{document}
\begin{tikzpicture}[scale=1.3]
 
% Input
\draw[->, thick] (-1,0) -- (0,0) node[at start, left]{$x(t)$};
 
% First block
\draw[draw, rectangle, minimum width=1.5cm, minimum height=1cm, thick] (0, -0.5) rectangle (1.5, 0.5);
\node at (0.75, 0) {$F$};
 
% Arrow between blocks
\draw[thick] (1.5,0) -- (2,0);
 
% Second block
\draw[draw, rectangle, minimum width=1.5cm, minimum height=1cm, thick] (2, -0.5) rectangle (3.5, 0.5);
\node at (2.75, 0) {$G$};
 
% Output
\draw[->, thick] (3.5,0) -- (4.5,0) node[at end, right]{$y(t)$};
 
% H
\draw[thick, dashed] (-0.5, -1) -- (4, -1) -- (4, 1) -- (-0.5, 1) -- cycle;
\node[thick, scale=1.3] at (-0.3, -0.8) {H};
\end{tikzpicture}
\end{document}

For LTI systems, the impulse response of a cascade is given as
$h(t)=(f\text{textasteriskcentered}g)(t),$
whereas the cascade’s frequency response is given as
$H(\omega )=F(\omega )G(\omega ).$

Note (Zero Cascade)

If

Proof

Examples

(Discrete-Time) Cascade

Consider a system $F$, governed by the LCCDE
$y(n)-y(n-1)=x(n)-x(n-3).$
Suppose we cascade $F$ with a new system $G$ ($F$ comes before $G$ as in the figure above), governed by the LCCDE
$y(n)-y(n-3)=x(n)-x(n-1).$
By plugging in the eigenfunction input $x(n)=e^{j\omega n}$, we find that the frequency responses of $F$ and $G$ are
$F(\omega )=\frac{1-e^{-j3\omega }}{1-e^{-j\omega }}\text{and}G(\omega )=\frac{1-e^{-j\omega }}{1-e^{-j3\omega }}.$
We see here that $F(\omega )$ is undefined for $\omega =2k\pi$, and $G(\omega )$ is undefined for $\omega =\frac{2k\pi }{3}$. We also know by definition that $H(\omega )=F(\omega )G(\omega )$. Since $F$ comes before $G$, any ${\omega }_0$ such that $F({\omega }_0)=0$ will lead to $H(\omega )=0$. Therefore, we can say that

\begin{cases} \text{Undefined} &\text{for } \omega = 2k\pi \\ 0 &\text{for } \omega = \frac{2k\pi}{3} \cap k \pmod{3} \not\equiv 0 \\ 1 &\text{otherwise}

\end{cases},$$

where $k(\mathrm{mod}3)\not\equiv 0$ becomes a condition since $H(\omega )$ is still undefined for $2k\pi$.