Question

The Laplace transform of a signal x(t) that is zero for t <0 is

X(s) = ((s³ + 2s² + 3s + 2) / ( s⁴ + 2s³ + 2s² + 2s + 2))
Determine the Laplace transform of the following signals:
(a) y(t) = 3x (t/3)

Answer 1

To find the Laplace transform of y(t) = 3x(t/3), apply the amplitude-scaling and time-scaling properties of the Laplace transform to the given Laplace transform X(s) of x(t), resulting in the Laplace transform Y(s) for y(t).

Step-by-step explanation:

The question is asking for the Laplace transform of the signal y(t) = 3x(t/3), given that the Laplace transform of x(t) is X(s) = (s³ + 2s² + 3s + 2) / (s⁴ + 2s³ + 2s² + 2s + 2). To find the Laplace transform of y(t), we can utilize the time-scaling and amplitude-scaling properties of the Laplace transform:

• Amplitude scaling property states that if L{x(t)} = X(s), then L{ax(t)} = aX(s) where a is a scalar.
• Time-scaling property states that if L{x(t)} = X(s), then L{x(at)} = (1/a)X(s/a) for a>0.

By applying both properties, we can first scale the amplitude by 3 and then scale the time by 1/3 which gives us the Laplace transform for y(t) as (1/3) * 3X(3s) = X(3s).

The final step is to replace all instances of s in the original X(s) with 3s, which leads to the Laplace transform of y(t):

Y(s) = ((3s)³ + 2(3s)² + 3(3s) + 2) / ((3s)⁴ + 2(3s)³ + 2(3s)² + 2(3s) + 2).