System visualization

Basic building blocks LTI system

When realizing or implementing a discrete-time system, we need the following basic building blocks: A multiplier, which multiplies signal sample $x\left[n\right]$ by a scalar $\beta$ which results in a sample with value $y\left[n\right]=\beta \cdot x\left[n\right]$; An adder which adds two signal samples $x_1\left[n\right]$ and $x_2\left[n\right]$ together into $y\left[n\right]=x_1\left[n\right]+x_2\left[n\right]$ and finally an unit-delay operator which delays a signal sample $x\left[n\right]$ by one sample index into $y\left[n\right]=x[n-1]$. These basic building blocks are depicted in Fig. 1.

Realization scheme and Difference Equation LTI system

Fig. 2 shows the realization scheme, or flow diagram, of a basic LTI system.

The LTI systems that we consider in this reader can have both a feed forward path with coefficients $b_0$ until $b_{M-1}$ and feedback path with coefficients $a_1$ until $a_{N-1}$. From the figure it follows that we can describe such a basic LTI system with the following Difference Equation (DE): $$\begin{array}{}y\left[n\right]=\sum _{k=0}^{M-1}{b}_{k}x[n-k]+\sum _{k=1}^{N-1}{a}_{k}y[n-k]& \text{(1)}\end{array}$$