Question

Consider the system y(n)−y(n−1)=x(n)+x(n−1)/2
Find the frequency response of this system.

Answer 1

The frequency response of the system described by the equation
`\(y(n) - y(n-1) = x(n) + (x(n-1))/(2)\)` can be found by taking the Z-transform of the given difference equation. The frequency response is given by
`\(H(z) = (Y(z))/(X(z)) = (1 + (1)/(2)z^(-1))/(1 - z^(-1))\).`

Step-by-step explanation:

The given difference equation
`\(y(n) - y(n-1) = x(n) + (x(n-1))/(2)\)` can be represented in the Z-domain using the Z-transform. Taking the Z-transform of both sides, we get:

`\[Y(z) - z^(-1)Y(z) = X(z) + (1)/(2)z^(-1)X(z)\]`

Rearranging terms and factoring out
`\(Y(z)\)` and
`\(X(z)\)`, we obtain the transfer function:

`\[H(z) = (Y(z))/(X(z)) = (1 + (1)/(2)z^(-1))/(1 - z^(-1))\]`

This expression represents the frequency response of the system. The numerator
`\(1 + (1)/(2)z^(-1)\)` corresponds to the coefficients of the current and past input samples, while the denominator
`\(1 - z^(-1)\)` corresponds to the coefficient of the past output sample.

Analyzing the frequency response provides insights into how the system responds to different frequencies. In this case, the system exhibits a pole at z = 1, indicating a potential resonance at that frequency.

In conclusion, the frequency response
`\(H(z) = (1 + (1)/(2)z^(-1))/(1 - z^(-1))\)` characterizes the behavior of the given system in the Z-domain.