A lacture on engineering

1. Classification of Signals
Lecture 2

2. Elementary Continuous Time Signals
 These signals include step, impulse, ramp, sinusoidal and
exponential functions.
 These signals can be used as building blocks for modelling of
more complex signals.

3. Unit step function
 The step function is an important signal used for analysis of many systems. For
example, when we turn on the ignition key of a car, we actually introduce a step
voltage (a step function) to start the motor.
 Likewise, when we apply brake to an automobile, we are applying a constant force
(a step function). If a step function has unity magnitude then it is called unit step
function.
 The unit step function is defined as
𝑢 𝑡 = 1 for 𝑡 ≥ 0
= 0 for 𝑡 < 0

4.  It is clear that when the argument 𝑡 in 𝑢(𝑡) is less than zero, then the unit step function
is zero, similarly when the argument 𝑡 in 𝑢(𝑡) is greater than or equal to zero then the
unit step function is unity.
 Similarly the shifted unit step function 𝑢(𝑡 − 𝑎) is zero if the argument 𝑡 − 𝑎 < 0 or 𝑡 <
𝑎 and 𝑢(𝑡 − 𝑎) is one if the argument 𝑡 − 𝑎 ≥ 0 or 𝑡 ≥ 𝑎.
That is
𝑢 𝑡 − 𝑎 = 1 for 𝑡 ≥ 𝑎
= 0 for 𝑡 < 𝑎

5. Unit ramp function
 The unit ramp function is defined as
𝑟 𝑡 = 𝑡 for 𝑡 ≥ 0
= 0 for 𝑡 < 0
Or
𝑟 𝑡 = 𝑡𝑢(𝑡)
 The ramp function can be obtained by applying unit step function to an integrator.
𝑟 𝑡 = 𝑢 𝑡 𝑑𝑡 = 𝑑𝑡 = 𝑡 (in the interval 𝑡 ≥ 0)
 In other words, the unit step function can be obtained by differentiating the unit ramp
function.
𝑢 𝑡 =
𝑑
𝑑𝑡
𝑟(𝑡)

6. Unit parabolic function
 The unit parabolic function is given by
𝑝 𝑡 =
𝑡2
2
for 𝑡 ≥ 0
= 0 for 𝑡 < 0
Or
𝑝 𝑡 =
𝑡2
2
𝑢(𝑡)
 The unit parabolic function can be obtained by integrating the ramp function.
𝑝 𝑡 = 𝑟 𝑡 𝑑𝑡 = 𝑡 𝑑𝑡 =
𝑡2
2
(for 𝑡 ≥ 0)
 In other words, the ramp function is derivative of the parabolic function.
 Thus, 𝑟 𝑡 =
𝑑𝑝(𝑡)
𝑑𝑡

7. Unit impulse function
 The impulse function occupies an important place in signal analysis. It is defined as
−∞
∞
𝛿 𝑡 𝑑𝑡 = 1 and 𝛿 𝑡 = 0 for 𝑡 ≠ 0
 That is the impulse function has zero amplitude everywhere except at 𝑡 = 0. At 𝑡 = 0, the
amplitude is infinity such that the area under the curve is equal to one.
 The delayed impulse function is defined as
−∞
∞
𝛿 𝑡 − 𝑎 𝑑𝑡 = 1 for 𝑡 = 𝑎
and 𝛿 𝑡 − 𝑎 = 0 for 𝑡 ≠ 𝑎

8. Rectangular pulse function
 The rectangular pulse function is defined as
Π 𝑡 = 1 for 𝑡 ≤
1
2
= 0 otherwise

9. Triangular pulse function
 The triangular pulse function is defined as
∆𝑎 𝑡 = 1 −
𝑡
𝑎
for 𝑡 ≤ 𝑎
= 0 for 𝑡 > 𝑎

10. Signum function
 The unit signum function is defined by
𝑠𝑔𝑛 𝑡 =
1, 𝑡 > 0
0, 𝑡 = 0
−1, 𝑡 < 0
 This function can be expressed in terms of unit step function as
𝑠𝑔𝑛 𝑡 = −1 + 2𝑢(𝑡)

11. Sinc function
 The sinc function can be defined by the expression
𝑠𝑖𝑛𝑐 𝑡 =
sin 𝑡
𝑡
− ∞ < 𝑡 < ∞

12. Sinusoidal signal
 A continuous time sinusoidal signal is given by
𝑥 𝑡 = 𝐴 sin(Ω𝑡 + Φ)
where 𝐴 is the amplitude, Ω is the frequency in radians per second and Φ is the phase
angle in radians.

13. Real exponential signal
 A real exponential signal is defined as
𝑥 𝑡 = 𝐴𝑒𝑎𝑡
where 𝐴 and 𝑎 are real. Depending upon the value of ‘𝑎’ we get different signals. If ‘𝑎’
is positive then the signal 𝑥 𝑡 is a growing exponential. If ‘𝑎’ is negative then the
signal 𝑥 𝑡 is decaying exponential. For 𝑎 = 0, 𝑥 𝑡 is constant.

14. Complex exponential signal
 The most general form of complex signal is given by
𝑥 𝑡 = 𝑒𝑠𝑡
where 𝑠 is a complex variable defined as
𝑠 = 𝜎 + 𝑗𝜔
 Depending upon the values of 𝜎 and 𝜔, we get different signals
1. If 𝜎 = 0 and 𝜔 = 0, then 𝑥 𝑡 = 1, that is the signal 𝑥 𝑡 is a pure DC signal.
2. If 𝜔 = 0, then 𝑠 = 𝜎, and 𝑥 𝑡 = 𝑒𝜎𝑡, which decays exponentially for 𝜎 < 0 and grows
exponentially for 𝜎 > 0.

15. Representation of Discrete-time Signals
 Graphical representation

16.  Functional representation
 Tabular representation
 Sequential representation

17. Elementary Discrete-Time Signals
 Unit Step Sequence

18. Unit Ramp Sequence

19. Unit Impulse Sequence