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 $x[n]$ by a scalar $\beta$ which results in a sample with value $y[n] = \beta \cdot x[n]$; An adder which adds two signal samples $x_1[n]$ and $x_2[n]$ together into $y[n]= x_1[n] + x_2[n]$ and finally an unit-delay operator which delays a signal sample $x[n]$ by one sample index into $y[n]=x[n-1]$. 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{equation} y[n] = \sum_{k=0}^{M-1} b_k x[n-k] + \sum_{k=1}^{N-1} a_k y[n-k] \end{equation}