Table of Contents

Introduction to Signal

Information converted into an electrical form suitable for transmission is called a signal. Schaum Series on Signals and systems states that A signal is a function representing a physical quantity or variable, and typically it contains information about the behaviour or nature of the phenomenon. For instance, in a RC circuit the signal may represent the voltage across the capacitor or the current through the resistor.

Mathematically, a signal is represented aas a function of an independent variable ‘t’. Usually t represent time. Thus a signal is denoted by x(t).

Types of Signals:

Continuous vs discrete signals
Analogvs digital signals
Real vs complex signals
Deterministic vs Random signals
Even vs Odd Signals

Continuous vs Discrete Signals

Continuous Signal	Discrete Signal
A signal is continuous-time if t is a continuous variable	A signal is discrete if x(t) is defined at discrete times
A continuous time signal is defined at each and every instant of time and there are no discontinuities	A discrete time signal is often identified as a sequence of numbers,
It is represented as x(t), where t is the continuous time base	It is represented as x[n] where n is integer.
A continous-time signal takes the following shape:	A discrete-time signal takes the following shape
It is x(t) = sin( 4t) for sine wave A square wave can be written as: x(t) = A; for 0	It is sampled at time t0,t1,t2…. and the samples asre x(t0), x(t1), ……,x(tn),….. Or x[0], x[1],……x[n],…. Or x0,x1,x2,….. Where we understand that xn = x[n] = x(tn)

Where xn’s are samples and time interval between them is called sampling interval.

When the sampling time are equal (Uniform sampling) the

Xn = x[n] = x(nTs) where the constant Ts is the sampling interval.

How to define discrete time signal:

There are two ways to define a discrete time signal x[n]

(a) We can specify a rule for calculating the nth value of a sequence. Example

X[n] = Xn= (1/2)ⁿ for n >= 0 and x[n] =0 for n<0

Or x[n] = { 1, ½, ¼, ……..(1/2)ⁿ,…..}

(b) List the values of the sequence for example.

We use arrows to denote the n=0, if arrow is not used then the first term is taken as at n=0

Sum and product of two sequences:

{c_n} = {a_n} + {b_n} à c_n = a_n + b_n
{c_n} = {a_n}.{b_n} à c_n = a_n . b_n
{c_n} =α {a_n} à c_n = α a_nwehere α is a constant

Analog vs Digital Signals

Analog Signals	Digital Signals
If the continuous signal x(t) can take on any values in the continuous interval from t -∞ to +∞ then the signal is analog signal.	If a discrete time signal can take on only a finite number of values then the signal is a digital signal

Real vs Complex Signals

Real Signals	Complex Signals
A signal x(t) is a real signal if its value is a real number	A signal x(t) is a complex signal if its value is a complex number. X(t) =x1(t) +jx2(t) X1(t) and x2(t) are real signals and j = root(-1)

Deterministic vs Random Signal

Deterministic Signal	Random Signal
A signal is deterministic if its value is completely specified for any given time.	A signal is random if it take random values at any given time and must be characterized statistically

Even vs Odd signal

Even signal	Odd signal
A signal c(t) or x[n] is even if: x(-t) = x(t) or x[-n] = x[n]	A signal c(t) or x[n] is odd if: x(-t) = – x(t) or x[-n] = – x[n]

Sum of two signals

Any signal can be expressed as sum of two signals one which is even and another as odd.

x(t) = x_e(t) + x_o(t)

x[n] = x_e [n] + x_o[n]

where

x_e(t) = ½{x(t) + x(-t)} and x_e [n] = 1/2{x[n] + x[-n]}

x_o(t) = ½{x(t) – x(-t) and x_o[n] = ½{x[n] – x[-n]}

Note that the product of two even or two odd signals is an even signal.

Periodic vs Non-Periodic signals :

Periodic	Non-Periodic
A continuous-time signal x(t) is said to be periodic with period T if there is a +ve non-zero value of T for which : x(t + T) = x(t) for all t	A continuous-time signal which is not periodic is called as non-periodic or aperiodic signal
A periodic discrete time signal is one where a sequence(discrete-time signal) x[n] is periodic with period N is there is a +ve integer for which: x[n + N] = x[n] for all n or from figure x[n+ mN] = x[n] for all n and any integer m	Any sequence which is not periodic is nonperiodic or aperiodic sequence If x[n + N] ≠ x[n] then sequence is non-periodic

**************NOTE*******************

A sequence obtained by uniform sampling of a periodic continuous-time signal may not be periodic
The sum of two continuous-time periodic signals may not be periodic but the sum of two periodic sequences is always periodic

****************************************

Energy and power signals

Consider v(t) to be the voltage across a resistor and the current flowing through it be the i(t) then :the instantaneous power p(t) per ohm is:

Sl	Power	Energy
1	the instantaneous power p(t) per ohm is: P(t) = v(t)i(t) / R = i(t)i(t) = i²(t) Watts/Ohm	E(t) = I²Rt Joules
2.	The total energy on a per-ohm basis is: Joules
3	The average power P on a per-ohm	P =
4	For an arbitrary continuous-time signal x(t), the normalized energy content E of x(t) is :	Similarly for a discrete time signal x[n], the normalized energy content E of x[n] is defined as:
5	The normalized average power P of x(t) is defined as :	The normalized average power P of x[n] is defined as: P =

Based on above equations, the following classes of signals are defined:

x(t) or x[n] is said to be energy signal (or sequence) if and only if 0<E<∞ and so P=0
x(t) or x[n] is said to be power signal (or sequence) if and only if 0<P<∞ and thus implying E = ∞
Signals that satisfy neither property are referred to as neither energy signals nor power signals.
*****NOTE**** note that a periodic signal is power signal if its energy content per period is finite, and then the average power of this signal need only be calculated over a period

Signals and its Classifications