Question 9.3: System block diagram from a transfer function using the time......
System block diagram from a transfer function using the time-shifting property
A system has a transfer function
\mathrm{H}(z)={\frac{\mathrm{Y}(z)}{\mathrm{X}(z)}}={\frac{z-1/2}{z^{2}-z+2/9}},\ \ \ |z|\gt 2/3.
Draw a system block diagram using delays, amplifiers and summing junctions.
We can rearrange the transfer-function equation into
Y(z)(z^{2}-z+2/9)= X(z)(z-1/2)
or
z^{2}\mathbf{Y}(z)=z\mathbf{X}(z)-(1/2)\mathbf{X}(z)+z\mathbf{Y}(z)-(2/9)\mathbf{Y}(z).
Multiplying this equation through by z^{-2} we get
\mathbf{Y}(z)=z^{-1}\,\mathbf{X}(z)-(1/2)z^{-2}\,\mathbf{X}(z)+z^{-1}\,\mathbf{Y}(z)+(2/9)z^{-2}\,\mathbf{Y}(z).
Now, using the time-shifting property, if {\mathrm{if~}}x[n]\overset{\mathcal Z}{\longleftrightarrow } X(z) and \mathbf{y}[n]\overset{\mathcal Z}{\longleftrightarrow } Y(z), then the inverse z transform is
\mathbf{y}[n]=\mathbf{x}[n-1]-(1/2)\,\mathbf{x}[n-2]+\mathbf{y}[n-1]-(2/9)\mathbf{y}[n-2].
This is called a recursion relationship between x[ n] and y[n] expressing the value of expressing the value of y[n] at discrete time n as a linear combination of the values of both x[ n] and y[n] at discrete times n,n-1,n-2,\cdots .
From it we can directly synthesize a block diagram of the system (Figure 9.10). This system realization uses four delays, two amplifiers and two summing junctions. This block diagram was drawn in a “natural” way by directly implementing the recursion relation in the diagram. Realized in Direct Form II, the realization uses two delays, three amplifiers and three summing junctions (Figure 9.11). There are multiple other ways of realizing the system (see Chapter 14).
