Definition of a Signal
A signal is a function that conveys information about a phenomenon. In engineering, signals can be classified into different types based on their characteristics, such as continuous or discrete, periodic or aperiodic, and deterministic or random.
Even and Odd Components of a Signal
To determine the even and odd components of a signal, we apply the following definitions:
- Even Component (xe(t)): The even part of a signal is defined as:
xe(t) = (x(t) + x(-t)) / 2
- Odd Component (xo(t)): The odd part of a signal is defined as:
xo(t) = (x(t) - x(-t)) / 2
Now, let’s find the even and odd components for the given signals.
i) x(t) = cos t + sin t + sin t * cos t
- Step 1: Identify x(-t):
x(-t) = cos(-t) + sin(-t) + sin(-t) * cos(-t)
= cos t - sin t - sin t * cos t
- Step 2: Calculate even and odd components:
xe(t) = (x(t) + x(-t)) / 2
= [ (cos t + sin t + sin t * cos t) + (cos t - sin t - sin t * cos t) ] / 2
= cos t
xo(t) = (x(t) - x(-t)) / 2
= [ (cos t + sin t + sin t * cos t) - (cos t - sin t - sin t * cos t) ] / 2
= sin t + sin t * cos t
ii) x(t) = (1 + t^3) cos^3(10t)
- Step 1: Identify x(-t):
x(-t) = (1 - t^3) cos^3(10(-t))
= (1 - t^3) cos^3(10t)
- Step 2: Calculate even and odd components:
xe(t) = (x(t) + x(-t)) / 2
= [ (1 + t^3) cos^3(10t) + (1 - t^3) cos^3(10t) ] / 2
= cos^3(10t)
xo(t) = (x(t) - x(-t)) / 2
= [ (1 + t^3) cos^3(10t) - (1 - t^3) cos^3(10t) ] / 2
= (t^3) cos^3(10t)
In conclusion, the even and odd components provide insight into the symmetry properties of the signals.