System function rational LTI

In this section we will derive the system function of a rational LTI function from the signal flow graph representation. This also leads to an alternative to find the frequency response of such an LTI system.

The signal flow graph of a rational LTI system is depicted in Fig. 1 and the Difference Equation is as follows: $$\begin{array}{ccc}y\left[n\right]& =& \sum _{k=0}^{M-1}{b}_{k}x[n-k]+\sum _{k=1}^{N-1}{a}_{k}y[n-k]\end{array}$$

As a first step we describe the signal flow graph in the $Z$-domain. For this we use the following property: $$h\left[n\right]=\delta [n-1]\stackrel{ZT}{\circ -\circ }H\left(z\right)=z^{-1}$$ Thus for the description of the signal flow graph in $Z$-domain, we interchange the time domain sequences $x\left[n\right]$ and $y\left[n\right]$ by their $Z$-transforms $X\left(z\right)$ and $Y\left(z\right)$ respectively and we use the symbol $z^{-1}$ to represent a delay. The result is depicted in Fig. 2. From this flow graph representation in the $Z$-domain we obtain the following algebraic equation between $X\left(z\right)$ and $Y\left(z\right)$: $$\begin{array}{}Y\left(z\right)=\sum _{k=0}^{M-1}{b}_k{z}^{-k}X\left(z\right)+\sum _{k=1}^{N-1}{a}_k{z}^{-k}Y\left(z\right)& \text{(1)}\end{array}$$ Furthermore by using the property that a convolution sum in time domain transforms in $Z$-domain to a product: $$y\left[n\right]=x\left[n\right]\star h\left[n\right]\stackrel{ZT}{\circ -\circ }Y\left(z\right)=X\left(z\right)\cdot H\left(z\right)$$ it follows that we can find the system function $H\left(z\right)$ by dividing $Y\left(z\right)$ by $X\left(z\right)$. Thus from equation ($\text{1}$) we simply can derive the rational system function $H\left(z\right)$ of the LTI system as follows: $$\begin{array}{}H\left(z\right)=\frac{Y\left(z\right)}{X\left(z\right)}=\frac{{\sum }_{k=0}^{M-1}{b}_k{z}^{-k}}{1-{\sum }_{k=1}^{N-1}{a}_k{z}^{-k}}& \text{(2)}\end{array}$$ Both numerator and denominator polynomials can be split as a product of first order terms: $$\begin{array}{ccc}H\left(z\right)& =& b_0\frac{{\prod }_{k=1}^M(1-{\beta }_k{z}^{-1})}{{\prod }_{k=1}^N(1-{\alpha }_k{z}^{-1})}\end{array}$$ Therefore, the system function $H\left(z\right)$ of an LTI system is defined, to within a scale factor $b_0$, by the location of the poles ${\alpha }_k$, and zeros ${\beta }_k$.

Finally the frequency response may be derived from the system function by evaluating the system function around the unit circle: $$H\left(e^{j\theta }\right)=H\left(z\right){|}_{z=e^{j\theta }}$$ which leads to an alternative way to find the frequency response of an LTI system: $$H\left(z\right)=\frac{Y\left(z\right)}{X\left(z\right)}\Rightarrow H\left(e^{j\theta }\right)=H\left(z\right){|}_{z=e^{j\theta }}$$