EEE 223 Signals and Systems

Lecture 1Introduction, Complex Numbers, Basic Signal Classification

Dr. Shadan Khattak 1

Complex Numbers (1)

History of numbers: Natural (positive) numbers (number of children, number of animals etc.) Fractions (length of a field, weight of a quantity of butter etc.) Irrational numbers: (Pythagoras, diagonal of a unit square) Negative numbers:

(Hindus, 12th century, positive numbers have two square roots) (bankers, Florence, Venice, subtract 7 from 5 = -2) Provided solution to x + 5 = 0. But what about x2 + 1 = 0, x2 = -1?

What to do with numbers whose square equals -1? Imaginary (or complex) numbers (1777, Swiss mathematician Euler, i)

Electrical engineers use the notation j instead of i to avoid confusion with the notation i for current.

Dr. Shadan Khattak 2

Complex Numbers (2)

Representation of Complex Number (The Complex Plane)

Cartesian form + Where = , = Magnitude: = 2 + 2

Phase (Angle): = 1(

)

Polar form

Putting the values of a and b in Cartesian form, we get +

+ (Eulers formula: = + )

Magnitude: Phase (Angle):

Conversion between Cartesian and Polar forms Conjugate of a complex number ( )

Dr. Shadan Khattak 3

pronounced oyler

Complex Numbers - Numerical Problem

1. Express the following numbers in polar form:1. 2 + 32. 2 + 33. 2 34. 1 3

Answers:1. 2. 13123.7

0

3. 4.

Dr. Shadan Khattak 4

Complex Numbers (4)

Understanding some useful identities

+ = 2 = 2

= 1

2 = 1

/2 = = 1, 5, 9, 13,

/2 = = 3, 7, 11, 15,

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Complex Numbers (5)

Arithmetic Operations, Powers and Roots of Complex Numbers

Let Addition, Subtraction (in Cartesian form)

Multiplication, Division, Powers, Roots (in Polar form) Multiplication:

Division:

Power:

Root:

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Signals

A signal is a set of data or information e.g., Monthly sales of a corporation Daily closing down prices of a stock market Source and capacitor voltages Force applied on a car Automobile velocity Human vocal mechanism Brightness across the image

Mathematically, signals are represented as functions of one or more independent variable(s) e.g., Speech Signal: Acoustic pressure as a function of time

Image: Brightness as a function of two spatial variables

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Continuous Time (CT) vs Discrete Time (DT) Signals Continuous Time (CT)

The signal has a value at any instance in time Independent variable usually enclosed in

parenthesis () E.g., = 2 + 1

Discrete Time (DT) The signal has a value at discrete points in time Independent variable usually enclosed in

square brackets [] Sometimes referred to as a sequence E.g., = 2 + 1

The terms CT and DT qualify the nature of a signal along the time (horizontal) axis

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Signal classification Analogue and Digital Signals Analogue

Continuous value (CV)

Digital Discrete values (DV)

The terms Analogue and Digital qualify the nature of a signal amplitude (vertical axis)

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Signal classification Periodic and Aperiodic Signals Periodic

A signal () is said to be periodic if for some positive constant 0 = + 0 for all t

The smallest value of 0 that satisfies the periodicity condition of this equation is the fundamental period of ()

Aperiodic A signal that is not periodic, is aperiodic.

Signal classification Causal, Non-causal, and Anti-causal Signals Causal

A signal that does not start before = 0 i.e.,

= 0 < 0

Non-causal A signal that starts before = 0

Anti-causal A signal that is zero for all 0

Signal classification Energy and Power Signals Energy

A signal with finite energy and zero power.

Power Time average (over a very large interval) of energy.

A signal with finite power and infinite energy.

Take Home Messages!

1. While converting from Cartesian to Polar form, add or subtract 1800 in the angle of the result if the complex number lies in second or third quadrant.

2. A signal is generally represented as a function of one or more independent variable(s).

3. The terms CT and DT qualify the nature of a signal along the time (horizontal) axis

4. The terms Analogue and Digital qualify the nature of a signal amplitude (vertical axis)

5. A causal signal does not start before = 06. A signal cannot be both an energy signal and a power signal

7. But it is possible that a signal will neither be energy signal nor a power signal e.g., the ramp signal.

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Practice Problems

Exercise Problems:1.1, 1.2 (Oppenheim)

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Useful Readings

Section 1.1 (Oppenheim)

Sections 1.3 (Lathi)

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