Ts=0.18, H(s)=3/s+3 ,find hI[n],HI[e^jw},hs[n]
To find hI[n] and HI[e^jw], we first need to find hs[n].
Given H(s) = 3/(s + 3), we can find its inverse Laplace transform to get hs[n]:
1. Find the inverse Laplace transform of H(s):
- Express H(s) as a partial fraction: H(s) = A/(s + 3).
- Multiply both sides by (s + 3): 3 = A.
- Therefore, A = 3.
The inverse Laplace transform of H(s) is: h(t) = 3 * e^(-3t).
2. Obtain the discrete-time representation, hs[n], from the continuous-time representation, h(t):
- Discretize the continuous-time function h(t) by replacing t with n * Ts, where Ts is the sampling interval.
Therefore, hs[n] = 3 * e^(-3nTs).
To find hI[n], we can use the impulse response property of the system.
1. Determine the impulse response hI[n]:
- Take the inverse Laplace transform of H(s) to get the impulse response in the continuous-time domain, hi(t).
Therefore, hI[n] = 3 * e^(-3nTs), which is the same as hs[n].
To find HI[e^jw], we can use the frequency response property of the system.
1. Determine the frequency response, H(e^jw), by substituting s with jw in H(s):
- H(e^jw) = 3/(jw + 3).
Therefore, HI[e^jw] = 3/(jw + 3).