Any continuous time signal can be expressed as the sum of an even signal and an odd signal:
x(t) = x_{e}(t) + x_{o}(t)
Even : x_{e}(t)  =  x_{e} (t) 
Odd: x_{o}(t)  =  x_{o}(t) 
An even signal is symmetric across the
vertical axis.
An odd signal is antisymmetric across the vertical axis.
x_{e}(t)  = 
(x(t) + x(t)) 

x_{o}(t)  = 
(x(t)  x(t)) 
Example, given the unit step function (a discontinuous continuoustime signal),
find u_{e}(t) and u_{o}(t)
How can we tell if a continuous time signal x(t) is periodic? That is, given t and T, is there some period T >0 such that
x(t) = x(t + T)?
If x(t) is periodic with period T, it is also periodic with period nT, that is:
x(t) = x(t + nT).
The minimum value of T that satisfies x(t) = x(t + T) is called the fundamental period of the signal and we denote it as T_{0}.
The fundamental frequency of the signal in hertz (cycles/second) is
and in radians/second, it is
If x_{1}(t) is periodic with period T_{1} and x_{2}(t) is periodic with period T_{2}, then the sum of the two signals x_{1}(t) + x_{2}(t) is periodic with period equal to the least common multiple(T_{1}, T_{2}) if the ratio of the two periods is a rational number, i.e.:
Let T ' = k_{1}T_{1} = k_{2}T_{2}, and z(t) = x_{1}(t) + x_{2}(t),
z(t + T ' ) = x_{1}(t + k_{1}T_{1}) + x_{2}(t + k_{2}T_{2}) = x_{1}(t) + x_{2}(t) = z(t)
Examples of periodic signals are infinite sine and cosine waves.
Examples: Given x_{1}(t) = cos(3t), and
x_{2}(t) = sin(5t).
find the period of x_{1}(t)+
x_{2}(t) or state that it is aperiodic.
Examples: Given
x_{1}(t) = cos(6t), and x_{2}(t) = sin(8t).
find the period of x_{1}(t)+
x_{2}(t) or state that it is aperiodic.
Examples: Given
x_{1}(t) = cos(t), and x_{2}(t) = sin(πt).
find the period of x_{1}(t)+
x_{2}(t) or state that it is aperiodic.
x(t) = C e^{at}, C and a can be complex.
a = σ + jω_{0}, σ = real part, ω_{0} = imaginary part.
These are very important because complex exponential are "eigenfunctions" of LTI systems:
As we will see, if you input x(t) = e^{at} to a linear timeinvariant system, we will get y(t) = H(a)e^{at} as the output.
*** EULER's FORMULA ***  Memorize:
e^{jθ} = cosθ + jsinθ
1.) Case 1, C and a real, x(t) = Ce^{at}
• a = σ > 0.
GROWING exponential,
Chemical reactions, uninhibited growth of bacteria (as might
be found if the potato salad is left out too long), human population
• a = σ < 0.
DECAYING exponential
Radioactive decay, RC circuit response, damped ME system.
• a = σ = 0 => x(t) is constant (DC) signal.
2.) Case 2, C complex, and a
imaginary,
a is purely imaginary (σ = 0),
a = jω_{0}
C = Ae^{jφ} where A and φ are
real
x(t) is a complex sinusoid.
If C is real (φ=0), then
x(t) = Acosω_{0}t + jAsinω_{0}t.
In both cases, x(t) is periodic, i.e.
x(t) = x(t + T) where T is the period.
Why is x(t) periodic?
x(t) 
= 

x(t + T) 
= 

A real sinusoid:
T is the fundamental period
3.) Case 3
In the most general case, C and a are complex:
C 
= 

x(t) 
= 

= 

= 

= 
The real part is Ae^{σt}cos(ω_{0}t + φ)
The imaginary part is Ae^{σt}sin(ω_{0}t + φ)
Example: Plot real part of this signal for σ > 0 and σ < 0