Elementary signals

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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, x k (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) = Ae λt , 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) = c an , where c and a are complex constants. Such a sequence can also be equivalently expressed in the form x(n) = ce bn , 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.