Final answer:
The frequency response H(e^jω) of the system is found by taking the Z-transform of the system's difference equation, substituting Y(z) for y(n) and X(z) for x(n), and then evaluating H(z) = Y(z) / X(z) on the unit circle z = e^jω.
Step-by-step explanation:
The frequency response H(ejω) of a discrete-time system can be found by taking the Fourier transform of the system's difference equation. The given system equation is y(n) - 0.4y(n-1) = (x(n) + x(n-1))/31. Applying the Z-transform and assuming initial rest conditions (all past values of inputs and outputs are zero), we substitute Y(z) for y(n) and X(z) for x(n), and solve for H(z) = Y(z) / X(z). Then we evaluate H(z) on the unit circle z = ejω to get the frequency response H(ejω).
The process involves substituting z-1 for the delay elements and solving for H(z), which results in a rational function where the numerator corresponds to the Z-transform of the input and the denominator corresponds to the Z-transform of the output response. Evaluating on the unit circle (where |z|=1) simplifies the expression and gives us the frequency response in terms of ω, which is a function of frequency.