classification of signals

Classification of signals

Signals are classified into the following categories:

• Continuous Time and Discrete Time Signals
• Deterministic and Non-deterministic Signals
• Even and Odd Signals
• Periodic and Aperiodic Signals
• Energy and Power Signals
• Real and Imaginary Signals

Continuous Time and Discrete Time Signals

A signal is said to be continuous when it is defined for all instants of time.

A signal is said to be discrete when it is defined at only discrete instants of time/

Deterministic and Non-deterministic Signals

A signal is said to be deterministic if there is no uncertainty with respect to its value at any instant of time. Or, signals which can be defined exactly by a mathematical formula are known as deterministic signals.

A signal is said to be non-deterministic if there is uncertainty with respect to its value at some instant of time. Non-deterministic signals are random in nature hence they are called random signals. Random signals cannot be described by a mathematical equation. They are modelled in probabilistic terms.

Even and Odd Signals

A signal is said to be even when it satisfies the condition x(t) = x(-t)

Example 1: t2, t4… cost etc.

Let x(t) = t2
x(-t) = (-t)2 = t2 = x(t)
∴, t2 is even function

Example 2: As shown in the following diagram, rectangle function x(t) = x(-t) so it is also even function.

A signal is said to be odd when it satisfies the condition x(t) = -x(-t)

Example: t, t3 ... And sin t

Let x(t) = sin t
x(-t) = sin(-t) = -sin t = -x(t)
∴, sin t is odd function.

Any function ƒ(t) can be expressed as the sum of its even function ƒ_e(t) and odd function ƒ_o(t).

ƒ(t ) = ƒ_e(t ) + ƒ₀(t )
where
ƒ_e(t ) = ½[ƒ(t ) +ƒ(-t )]

Periodic and Aperiodic Signals

A signal is said to be periodic if it satisfies the condition x(t) = x(t + T) or x(n) = x(n + N).

Where

T = fundamental time period,
1/T = f = fundamental frequency.

The above signal will repeat for every time interval T₀ hence it is periodic with period T₀.

Energy and Power Signals

A signal is said to be energy signal when it has finite energy.

EnergyE=∫∞−∞x2(t)dt

A signal is said to be power signal when it has finite power.

PowerP=limT→∞12T∫T−Tx2(t)dt

NOTE:A signal cannot be both, energy and power simultaneously. Also, a signal may be neither energy nor power signal.

Power of energy signal = 0
Energy of power signal = ∞
Classification of Signals into Power and Energy Signals


Most signals can be classified into Energy signals or Power signals. A signal is classified into an energy or a power signal according to the following criteria


a)         Energy Signals:  an energy signal is a signal with finite energy and zero                                  average power  (0 ≤ E < ¥,  P = 0),
b)         Power Signals:   a power signal is a signal with infinite energy but finite                                 average power  (0 < P < ¥,  E ® ¥).


Comments:


1.         The square root of the average power of a power signal is what is usually defined as the RMS value of that signal.
2.         Your book says that if a signal approaches zero as t approaches ¥ then the signal is an energy signal. This is in most cases true but not always as you can verify in part (d) in the following example.
3.         All periodic signals are power signals (but not all non–periodic signals are energy signals).
4.         Any signal f that has limited amplitude (| f | < ¥) and is time limited
            (f  = 0   for    | t | > t₀  for some  t₀ > 0) is an energy signal as in part (g) in the following example.


Exercise 2–1: determine if the following signals are Energy signals, Power signals, or neither, and evaluate E and P for each signal (see examples 2.1 and 2.2 on pages 17 and 18 of your textbook for help).


            a)         ,




                        This is a periodic signal, so it must be a power signal. Let us prove it.


                        



                       
Notice that the evaluation of the last line in the above equation is infinite because of the first term. The second term has a value between –2 to 2 so it has no effect in the overall value of the energy.


Since a(t)  is periodic with period T = 2p/2p= 1 second, we get


                        





So, the energy of that signal is infinite and its average power is finite (9/2). This means that it is a power signal as expected. Notice that the average power of this signal is as expected (square of the amplitude divided by 2)


            b)         ,





                        Let us first find the total energy of the signal.


                       
                       



The average power of the signal is


                        





So, the signal b(t)  is definitely an energy signal.
So, the energy of that signal is infinite and its average power is finite (9/2). This means that it is a power signal as expected. Notice that the average power of this signal is as expected (the square of the amplitude divided by 2)


 c)    
                ,


d) 


                    
,



                        Let us first find the total energy of the signal.


                        



                       
                        So, this signal is NOT an energy signal. However, it is also NOT a power                            signal since its average power as shown below is zero.
                       
The average power of the signal is


                        





Using Le’hopital’s rule, we see that the power of the signal is zero. That is









So, not all signals that approach zero as time approaches positive and negative infinite is an energy signal. They may not be power signals either.


            e)        

,



            f)        

.





            g)        

.

Real and Imaginary Signals

A signal is said to be real when it satisfies the condition x(t) = x*(t)

A signal is said to be odd when it satisfies the condition x(t) = -x*(t)

Example:

If x(t)= 3 then x*(t)=3*=3 here x(t) is a real signal.
If x(t)= 3j then x*(t)=3j* = -3j = -x(t) hence x(t) is a odd signal.

Note: For a real signal, imaginary part should be zero. Similarly for an imaginary signal, real part should be zero.
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