Question

For a discrete-time system whose input-output relations is given below: y[n]=x[n]+1x[n−2]−2x[n−3]−1y[n−1]+2y[n−2], this system is at rest originally Determine the first 5 samples of its impulse response. You need to show your calculation result to receive the full credit although you can use Matlab to verify your solution.

Answer 1

To determine the first 5 samples of the impulse response of the given discrete-time system, set the input signal as an impulse function and calculate the output values using the provided input-output relation. Substitute the values of x[n] and y[n] into the equations to find the corresponding y[n] values.

Step-by-step explanation:

The given input-output relation of the discrete-time system is:

y[n] = x[n] + x[n-2] - 2x[n-3] - y[n-1] + 2y[n-2]

To determine the impulse response of the system, we can set the input signal, x[n], as an impulse function. The impulse function is defined as:

δ[n] = 1, if n = 0

δ[n] = 0, if n ≠ 0

By substituting δ[n] into the input signal, we can find the output values at different time instances. Let's calculate the first 5 samples of the impulse response:

1. y[0] = x[0] + x[-2] - 2x[-3] - y[-1] + 2y[-2]
2. y[1] = x[1] + x[-1] - 2x[-2] - y[0] + 2y[-1]
3. y[2] = x[2] + x[0] - 2x[-1] - y[1] + 2y[0]
4. y[3] = x[3] + x[1] - 2x[0] - y[2] + 2y[1]
5. y[4] = x[4] + x[2] - 2x[1] - y[3] + 2y[2]

By substituting the values of x[n] and y[n] into these equations, we can calculate the corresponding y[n] values.