Which of the following is false for a continuous time signal?

Even and odd property:

Any given signal g(t) can be written as the sum of its even part and odd part, i.e.

g(t) = ge(t) + go(t)

ge(t) = Even part of g(t) calculated as:

\(g_e(t)=\frac{g(t)+g(-t)}{2}\)

go(t) = Odd part of g(t) calculated as:

\(g_o(t)=\frac{g(t)-g(-t)}{2}\)

Periodicity of sum two periodic signals:

A signal x(t) is said to be periodic if for some positive constant T0;

x(t) = x (t + T0) for all t

The smallest value of T0 that satisfies the periodicity condition of the above equation is called the fundamental period of x(t).

The sum of two periodic functions is often periodic, but not always.

Suppose that x(t) is periodic with fundamental period P, and g(t) is periodic with fundamental period Q. Then x(t) + g(t) will be periodic with a period R, if P and Q both have a common multiple R.

Ex- sin x + sin πx

sin x is periodic with a period 2π 

sin πx is periodic with a period 2

But since 2π and 2 have no common multiple, the sum of the two signals will not be periodic.

Fourier Series:

Fourier series states that any periodic signal x(t) can be expressed as the sum of the sinusoidal signals.

Mathematically, it is defined as:

\(x\left( t \right) = {A_0} + \mathop \sum \limits_{n = 1}^{n = \infty } ({A_n}cosn{\omega _0}t + {B_n}\sin n{\omega _0}t)\)