Signal Characteristics

Even and Odd Functions

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 :    xe(t)	=	xe (-t)
Odd:    xo(t)	=	-xo(-t)

An even signal is symmetric across the vertical axis.
An odd signal is anti-symmetric across the vertical axis.

xe(t)	=		(x(t) + x(-t))
xo(t)	=		(x(t) - x(-t))

Example, given the unit step function (a discontinuous continuous-time signal),

find u_e(t) and u_o(t)

Periodic Functions

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 T₀.

The fundamental frequency of the signal in hertz (cycles/second) is

and in radians/second, it is

If x₁(t) is periodic with period T₁ and x₂(t) is periodic with period T₂, then the sum of the two signals x₁(t) + x₂(t) is periodic with period equal to the least common multiple(T₁, T₂) if the ratio of the two periods is a rational number, i.e.:

Let   T '  =   k₁T₁  =  k₂T₂, and z(t) = x₁(t) + x₂(t),

z(t + T ' ) =  x₁(t +  k₁T₁) +  x₂(t +  k₂T₂) =  x₁(t) + x₂(t) =  z(t)

Examples of periodic signals are infinite sine and cosine waves.

Examples: Given x₁(t) = cos(3t), and x₂(t) = sin(5t).
                 find the period of x₁(t)+ x₂(t) or state that it is aperiodic.

Examples: Given x₁(t) = cos(6t), and x₂(t) = sin(8t).
                 find the period of x₁(t)+ x₂(t) or state that it is aperiodic.

Examples: Given x₁(t) = cos(t), and x₂(t) = sin(πt).
                 find the period of x₁(t)+ x₂(t) or state that it is aperiodic.

Common Signals in Engineering

Exponential Signals

x(t) = C e^at,  C and a can be complex.

a  = σ + jω₀, σ =  real part, ω₀ =  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 time-invariant 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 = jω₀

       C = Ae^jφ where A and φ are real

x(t) is a complex sinusoid.

If C is real (φ=0), then x(t) = Acosω₀t + jAsinω₀t.
In both cases, x(t) is periodic, i.e.

x(t) = x(t + T) where T is the period.

Why is x(t) periodic?

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^σtcos(ω₀t  +  φ)

The imaginary part is  Ae^σtsin(ω₀t  +  φ)

Example: Plot real part of this signal for σ > 0 and σ < 0