Question

A causal, linear, and time-invariant system is given by the following difference equation: y[] =y[−1]+y[−2]+x[−1]

i. Find the transfer function H(z) for this system.

Answer 1

The transfer function H(z) for the given system is z^-1 / (1 - z^-1 - z^-2).

Step-by-step explanation:

The transfer function H(z) for the given system can be found by taking the z-transform of the difference equation. The z-transform of a sequence x[k] is defined as X(z) = Σ(x[k]z^-k), where k is the time index. Applying the z-transform to the difference equation, we have Y(z) = Y(z)z^-1 + Y(z)z^-2 + X(z)z^-1. Rearranging and factoring out the common terms, we get Y(z)(1 - z^-1 - z^-2) = X(z)z^-1. Dividing both sides by (1 - z^-1 - z^-2), we obtain the transfer function H(z) = Y(z)/X(z) = z^-1 / (1 - z^-1 - z^-2).