Odd and Even Components of a Signal:

In signal processing, a signal can be decomposed into its odd and even components. The odd component represents the part of the signal that is symmetric about the origin, while the even component represents the part of the signal that is anti-symmetric about the origin. The odd component is also known as the imaginary or sine component, while the even component is known as the real or cosine component.

The signal x(t) = e^(jt) can be represented in terms of its odd and even components as follows:

x(t) = x_odd(t) + x_even(t)

Odd Component:
The odd component of a signal can be obtained by subtracting its even component from the original signal. Therefore, to find the odd component of x(t) = e^(jt), we need to determine the even component first.

Even Component:
The even component of a signal can be obtained by taking the average of the original signal and its time-reversed version. In this case, the even component of x(t) = e^(jt) can be calculated as:

x_even(t) = (x(t) + x(-t))/2

Substituting x(t) = e^(jt) into the above equation:

x_even(t) = (e^(jt) + e^(-jt))/2

Odd Component:
Now that we have the even component, we can find the odd component by subtracting it from the original signal:

x_odd(t) = x(t) - x_even(t)

Substituting x(t) = e^(jt) and x_even(t) = (e^(jt) + e^(-jt))/2 into the above equation:

x_odd(t) = e^(jt) - (e^(jt) + e^(-jt))/2

Simplifying the expression:

x_odd(t) = (e^(jt) - e^(-jt))/2

Comparison with Given Options:
We are given the following options:
a) cos(t)
b) j*sin(t)
c) j*cos(t)
d) sin(t)

Comparing the odd component expression with the given options, we can see that option b) j*sin(t) matches the expression (e^(jt) - e^(-jt))/2.

Therefore, the correct answer is option b) j*sin(t).