1. Signals & Systems (CNET - 221) Chapter-1
Mr. ASIF ALI KHAN
Department of Computer Networks
Faculty of CS&IS
Jazan University

2. Text Book & Online Reference
A.V. Oppenheim and R.W. Schafer, Digital Signal Processing, Prentice-hall
E-Learning:
General Reference:

3. Chapter Objective Following are the objectives of Chapter-I
Signal Definition, Classification, Representation of different signals including both Continuous and Discrete
Transformation of the independent variables
Signals Basic operation including Signals addition, subtraction and Multiplication

4. Course Description-Chapter-1
Introduction to Signals
1.1 Introduction to Signals
1.2 Continuous-Time & Discrete-Time Signals
1.2.l Examples and Mathematical Representation
Signals Energy and Power
1.3 Transformations of the Independent Variable
l.3.1 Time Shifting
Time Reversal
l.3.3 Time Scaling
1.3.4 Periodic and Aperiodic Signals
l.3.5 Even(Symmetric) and Odd(Anti-symmetric) Signals
1.4 Exponential and Sinusoidal Signals
1.5 Unit Impulse and Unit Step Function

5. Introduction to Signals
What is Signal?
SIGNAL is a function or set of information or data representing a physical quantity that varies with time, space or any other independent variable, usually time.
Examples:
Voltage/Current in a Circuit
Speech/Music
Any company’s industrial average
Force exerted on a shock absorber
A*Sin(ῳt+φ)

6. Introduction to Signals(Continued…..)
Example of various signals(Graphs) :

7. Introduction to Signals(Continued…..)
Measuring Signals

8. Classifications of Signals
Signals are classified into the following categories:
Continuous-Time Signals
Discrete-Time Signals
Even and Odd Signals
Periodic and Non-Periodic (aperiodic) Signals
Energy and Power Signals :

9. Continuous Time and Discrete Time Signals
A signal is said to be continuous when it is defined for all instants of time.
Example: Video, Audio
A signal is said to be discrete when it is defined at only discrete instants of time
Example: Stock Market, Temperature :

10. Even and Odd Signals
A signal is said to be even when it satisfies the condition
X(t) = X(-t) ; X[n] = X[-n]
A signal is said to be odd when it satisfies the condition
X(-t) = -X(t) ; X[-n] = -X[n] :

11. Periodic and Non-periodic (Aperiodic) Signals
A signal is said to be periodic if it satisfies the condition
x(t) = x(t + T) or x(n) = x(n + N) for all t or n
Where T = fundamental time period, 1/T = f = fundamental frequency. :

12. Energy and Power Signals
A signal is said to be energy signal when it has finite energy. or
A signal is said to be power signal when it has finite power.
NOTE:
A signal cannot be both, energy and power simultaneously. Also, a signal may be neither energy nor power signal. Power of energy signal = 0 & Energy of power signal = ∞ :

13. Signals Basics Operation
Example-1: For the given signals x1(t) and x2(t) plot the Signal z1(t) = x1(t) + x2(t)
Answer: As seen from the diagram above,
-10 < t < -3 amplitude of z(t) = x1(t) + x2(t) = = 2
-3 < t < 3 amplitude of z(t) = x1(t) + x2(t) = = 3
3 < t < 10 amplitude of z(t) = x1(t) + x2(t) = = 2

14. Signals Basics Operation (Continued……..)
Example-2: For the given signals x1(t) and x2(t) plot the Signal z1(t) = x1(t) - x2(t)
Answer: As seen from the diagram above,
-10 < t < -3 amplitude of z (t) = x1(t) - x2(t) = = -2
-3 < t < 3 amplitude of z (t) = x1(t) - x2(t) = = -1
3 < t < 10 amplitude of z (t) = x1(t) + x2(t) = = -2

15. Signals Basics Operation (Continued……..)
Example-3: For the given signals x1(t) and x2(t) plot the Signal z1(t) = x1(t) * x2(t)
Answer:
As seen from the diagram above,
-10 < t < -3 amplitude of z (t) = x1(t) ×x2(t) = 0 ×2 = 0
-3 < t < 3 amplitude of z (t) = x1(t) ×x2(t) = 1 ×2 = 2
3 < t < 10 amplitude of z (t) = x1(t) × x2(t) = 0 × 2 = 0

16. Transformations of the Independent Variable (Time Shifting )
x(t ± t0) is time shifted version of the signal x(t).
x (t + t0) →→ negative shift
x (t - t0) →→ positive shift

17. Transformations of the Independent Variable (Time Scaling)
x(At) is time scaled version of the signal x(t). where A is always positive.
|A| > 1 →→ Compression of the signal
|A| < 1 →→ Expansion of the signal
Note: u(at) = u(t) time scaling is not applicable for unit step function.

18. Transformations of the Independent Variable (Time Reversal)
x(-t) is the time reversal of the signal x(t).

19. Example: Time Scaling Solution:
For the given signal X(t) shown in figure below,
sketch each of the following signals.  
Time scaled signal X(2t).
Time shifted signal X(t+2).
X(t) + X(-t).
Solution:

20. Example: Time Scaling Solution:
For the given signal X(t) shown in Figure-6, sketch each of the following signals.
 a. Time reversed signal X(-t) .
b. Time scaled signal X(2t).
c. Time shifted signal X(t+2).
d. { X(t) + X(-t) }  e. { X(t) - X(-t) }
Solution:

21. Exponential and Sinusoidal Signals
Sinusoids and exponentials are important in signal and system analysis because they arise naturally in the solutions of the differential equations.
Sinusoidal Signals can expressed in either of two ways :
cyclic frequency form- A sin 2πfot = A sin(2π/To)t
radian frequency form- A sin ωot
ωo = 2πfo = 2π/To
To = Time Period of the Sinusoidal Wave

22. Exponential and Sinusoidal Signals(Continued…..)
x(t) = A sin (2πfot+ θ)
= A sin (ωot+ θ)
x(t) = Aeat Real Exponential
= Aejω̥t = A[cos (ωot) +j sin (ωot)] Complex Exponential
θ = Phase of sinusoidal wave
A = amplitude of a sinusoidal or exponential signal
fo = fundamental cyclic frequency of sinusoidal signal
ωo = radian frequency

23. The Unit Step Function
Definition: The unit step function, u(t), is defined as
That is, u is a function of time t, and u has value zero when time is negative (before we flip the switch); and value one when time is positive (from when we flip the switch).

24. The Unit Step Function-Example
Draw the waveform for the signal X(t) having the expression,
X(t) = 2 * {u (t + 0.5) – u (t – 0.5)}.
Answer

25. The Unit Impulse Function
So unit impulse function is the derivative of the unit step
function or unit step is the integral of the unit impulse function

26. Representation of Impulse Function
The area under an impulse is called its strength or weight. It is
represented graphically by a vertical arrow. An impulse with a
strength of one is called a unit impulse.

27. Unit Impulse Properties

28. Example: The impulse Function
Draw the waveform for the signal X[n] having the expression, X[n] = 𝑘=0 5 𝛿 [𝑛−𝑘].
Solution:

29. Videos
2. Signals & Systems Tutorial