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 β which results in a sample with value y[n]=β⋅x[n]; An adder which adds two signal samples x1[n] and x2[n] together into y[n]=x1[n]+x2[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.
Basic building blocks of a discrete-time system.
Realization scheme and Difference Equation LTI system
Fig. 2 shows the realization scheme, or flow diagram, of a basic LTI system.
Signal flow diagram or realization scheme of a basic LTI system.
The LTI systems that we consider in this reader can have both a feed forward path with coefficients b0 until bM−1 and feedback path with coefficients a1 until aN−1. From the figure it follows that we can describe such a basic LTI system with the following Difference Equation (DE):
y[n]=M−1∑k=0bkx[n−k]+N−1∑k=1aky[n−k](1)