Elementary signals

15EE55C – DIGITAL SIGNAL PROCESSING AND
ITS APPLICATIONS
ELEMENTARY SIGNALS
Dr. M. Bakrutheen AP(SG)/EEE
Mr. K. Karthik Kumar AP/EEE
DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING
NATIONAL ENGINEERING COLLEGE, K.R. NAGAR, KOVILPATTI – 628 503
(An Autonomous Institution, Affiliated to Anna University – Chennai)

ELEMENTARY SIGNALS
 The elementary signals are used for analysis of systems. Such signals
are,
 Step
 Impulse
 Ramp
 Exponential
 Sinusoidal

UNIT STEP SIGNAL
 The unit step signal has amplitude of 1 for positive value and amplitude
of 0 for negative value of independent variable
 It have two different parameter such as CT unit step signal u(t) and DT
unit step signal u(n).

UNIT STEP SIGNAL - CT
 The mathematical representation of CT unit step signal u(t) is given by

UNIT STEP SIGNAL - DT
 The mathematical representation of DT unit step signal u(n) is given by

UNIT IMPULSE SIGNAL
 Amplitude of unit impulse approaches 1 as the width approaches zero
and it has zero value at all other values.
 This is used to estimate the impulse response of LTI system

UNIT IMPULSE SIGNAL - CT
 The mathematical representation of unit impulse signal for CT is given
by,

UNIT IMPULSE SIGNAL - DT
 The mathematical representation of unit impulse signal for DT is given
by,

RAMP SIGNAL
 CT Ramp signal is denoted by r(t).
 It is defined as r(t)
 DT Ramp signal is denoted by r(n), and it is defined as r(n)

REAL SINUSOIDAL SIGNAL - CT
 A (CT) real sinusoid is a function of the form
x(t) = Acos(ωt +θ),
where A, ω, and θ are real constants.
 Such a function is periodic with fundamental period T = 2π/|ω|
and fundamental frequency |ω|.

REAL SINUSOIDAL SIGNAL - DT
 A (DT) real sinusoid is a sequence of the form x(n) = Acos(Ωn+θ),
 where A, Ω, and θ are real constants.
 A real sinusoid is periodic if and only if Ω/2π is a rational number, in which case
the fundamental period is the smallest integer of the form 2πk/|Ω| where k is a
positive integer.
 For all integer k, xk(n) = Acos([Ω+2πk]n+θ) is the same sequence.
 An example of a periodic real sinusoid with fundamental period 12 is shown
plotted below

REAL EXPONENTIAL SIGNAL - CT
 A (CT) complex exponential is a function of the form x(t) = Aeλt ,
 where A and λ are complex constants.
 A complex exponential can exhibit one of a number of distinct modes of
behavior, depending on the values of its parameters A and λ.
 For example, as special cases, complex exponentials include real
exponentials and complex sinusoids.

REAL EXPONENTIAL SIGNAL – CT – SPECIAL CASE
 A real exponential can exhibit one of three distinct modes of behavior, depending on
the value of λ, as illustrated below.
 If λ > 0, x(t) increases exponentially as t increases (i.e., a growing exponential).
 If λ < 0, x(t) decreases exponentially as t increases (i.e., a decaying exponential).
 If λ = 0, x(t) simply equals the constant A.

REAL EXPONENTIAL SIGNAL - DT
 A (DT) complex exponential is a sequence of the form x(n) = can,
 where c and a are complex constants.
 Such a sequence can also be equivalently expressed in the form x(n) =
cebn, where b is a complex constant chosen as b = lna. (This this form is
more similar to that presented for CT complex exponentials).
 A complex exponential can exhibit one of a number of distinct modes of
behavior, depending on the values of the parameters c and a.
 For example, as special cases, complex exponentials include real
exponentials and complex sinusoids.

REAL EXPONENTIAL SIGNAL – DT – SPECIAL CASE
 A real exponential can exhibit one of several distinct modes of behavior,
depending on the magnitude and sign of a.
 If |a| > 1, the magnitude of x(n) increases exponentially as n increases
(i.e., a growing exponential).
 If |a| < 1, the magnitude of x(n) decreases exponentially as n increases
(i.e., a decaying exponential).
 If |a| = 1, the magnitude of x(n) is a constant, independent of n.
 If a > 0, x(n) has the same sign for all n.
 If a < 0, x(n) alternates in sign as n increases/decreases.

REAL EXPONENTIAL SIGNAL – DT – SPECIAL CASE

OTHER SIGNALS – RECTANGULAR - CT
 The rectangular function (also called the unit-rectangular pulse
function), denoted rect, is given by
 Due to the manner in which the rect function is used in practice, the
actual value of rect(t) at t = ±1/2 is unimportant.
 Sometimes different values are used from those specified above. A plot
of this function is shown below.

OTHER SIGNALS – RECTANGULAR - DT
 A unit rectangular pulse is a sequence of the form
 where a and b are integer constants satisfying a < b.
 Such a sequence can be expressed in terms of the unit-step sequence as
p ( n) = u ( n − a ) − u ( n − b ). The graph of a unit rectangular pulse has
the general form shown below.

OTHER SIGNALS – TRIANGULAR
 The triangular function (also called the unit-triangular pulse function),
denoted tri, is defined as

OTHER SIGNALS – CARDINAL SINE FUNCTION
 The cardinal sine function, denoted sinc, is given by
 By l’Hopital’s rule, sinc 0 = 1.
 A plot of this function for part of the real line is shown below. [Note that
the oscillations in sinc(t) do not die out for finite t.]

OTHER SIGNALS – SIGNUM FUNCTION
 The signum function, denoted sgn, is defined as
 From its definition, one can see that the signum function simply
computes the sign of a number. A plot of this function is shown below.

Elementary signals

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Elementary signals