Question

A causal LTI discrete-time system is described by the following difference equation: y[n] - 0.8y[n-1] + 0.15y[n-2] = x[n] where x[n] and y[n] are, respectively the input and output sequences of the system. Determine the transfer function H(z) of the system.

Answer 1

The transfer function H(z) of the system described by the difference equation y[n] - 0.8y[n-1] + 0.15y[n-2] = x[n] is determined by taking the Z-transform of the equation and solving for Y(z)/X(z), yielding H(z) = 1 / (1 - 0.8z^{-1} + 0.15z^{-2}).

Step-by-step explanation:

To determine the transfer function H(z) of a causal LTI discrete-time system described by the difference equation y[n] - 0.8y[n-1] + 0.15y[n-2] = x[n], we apply the Z-transform to both sides of the equation, making use of the property that Z{y[n-k]} = z^{-k}Y(z). Taking the Z-transform of each term, we have:

• Y(z) - 0.8z^{-1}Y(z) + 0.15z^{-2}Y(z) = X(z).

Factor Y(z) out of the terms on the left side to isolate it:

• Y(z)(1 - 0.8z^{-1} + 0.15z^{-2}) = X(z).

Now, divide both sides by the factored polynomial to solve for H(z):

• H(z) = Y(z)/X(z) = 1 / (1 - 0.8z^{-1} + 0.15z^{-2}).

This expression represents the transfer function H(z) of the system in question.