The document provides an overview of elementary signals used in digital signal processing, including unit step, impulse, ramp, sinusoidal, and exponential signals. It explains their mathematical representations and characteristics, detailing both continuous-time (ct) and discrete-time (dt) forms. Additionally, it includes definitions and properties of other signals like rectangular, triangular, cardinal sine, and signum functions.

1. 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)

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

3. 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).

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

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

6. 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

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

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

9. 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)

10. 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|.

11. 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

12. 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.

13. 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.

14. 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.

15. 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.

16. REAL EXPONENTIAL SIGNAL – DT – SPECIAL CASE

17. 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.

18. 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.

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

20. 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.]

21. 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.