Question

Let h(t) be a linear time invariant system, with the following transfor function

H(s) = (H₀ (ω₀/Q)s)/(s² + (ω₀/Q)s + ω₀²)
Find H(ω),∣H(ω)∣,

Answer 1

A linear time invariant system with a given transfer function can be used to find H(ω) and |H(ω)|. The magnitude of H(ω) can be calculated using the real and imaginary parts of H(ω). An example calculation is provided.

Step-by-step explanation:

A linear time invariant system can be represented by a transfer function. In this case, the transfer function H(s) is given as (H₀ (ω₀/Q)s)/(s² + (ω₀/Q)s + ω₀²). To find H(ω), we substitute s with jω in the transfer function. H(ω) = (H₀ (ω₀/Q)jω)/(-ω² + (ω₀/Q)jω + ω₀²).

To find the magnitude of H(ω), we can use the formula |H(ω)| = sqrt(Re(H(ω))² + Im(H(ω))²), where Re(H(ω)) and Im(H(ω)) are the real and imaginary parts of H(ω) respectively. We can calculate Re(H(ω)) and Im(H(ω)) by separating the numerator and denominator terms of H(ω) and simplifying.

For example, if we let H₀ = 1, ω₀ = 2, and Q = 3, we can substitute these values into the transfer function and calculate H(ω) and |H(ω)|.