Question

Consider the system y(n) - 0.4y(n - 1) = x(n)+x(n-1)/3

Find the frequency response H(e) of this system.

Answer 1

The frequency response H(e^jω) of the system is found by applying the Z-transform to the difference equation and then replacing z with e^jω to obtain the expression representing H(e^jω).

Step-by-step explanation:

To find the frequency response H(ejω) of the given system, we apply the Z-transform to the difference equation of the system. The equation is given by:

y(n) - 0.4y(n - 1) = x(n) + x(n - 1)/3

Assuming zero initial conditions and applying the Z-transform to both sides of the equation yields:

Y(z) - 0.4Y(z)z-1 = X(z) + (1/3)X(z)z-1

Now we solve for the transfer function H(z) = Y(z)/X(z):

H(z) = (1 + z-1/3) / (1 - 0.4z-1)

Then, we replace z with ejω to find the frequency response:

H(ejω) = (1 + e-jω/3) / (1 - 0.4e-jω)

This expression represents the frequency response of the system, which can then be analyzed for its magnitude and phase characteristics over different frequencies.