Lecture 2Signal Size, Signal Operations, Periodic Signals, Even and Odd Signals

Size of a Signal (1)

Size of an entity determines its largeness or strength

Signal amplitude varies with time

Largeness should consider both amplitude and duration

How to measure the strength of such a signal? Naive way: Area under signal ()

But what about negative area?

More appropriate way: Area under signal 2() (Signal Energy)

Size of a Signal (2)

Signal Energy:CT =

() 2

DT = = |[]|2

For signal energy to be meaningful, it must be finite i.e.,CT , 0

DT , 0

If is finite, the signal is called energy signal

Size of a Signal (3)

But what if: CT a , 0

DT a , 0

measure signal power instead

Signal Power:CT = lim

1

2

() 2 ( , =1

0() 2)

DT = lim

1

2+1 |[]|2 ( , =

1

01 2)

If is finite and non-zero, the signal is called a power signal

Size of a Signal (4)

Examples

Signal with finite energy (zero power)

Signal with finite power (infinite energy)

Size of a Signal Numerical Problem

1. Determine the energy and power of the following signals:

a. 1 = (2+

4)

b. 2() = ()

c. [] = 0.5 0

0 < 0

Answers:

a. = , = 1

b. = , =1

2

c. = 1.582, = 0

(Parts (a) and (b): Problem 1.3, Oppenheim; part (c): Example 1.7, Mandal & Asif)

Size of a Signal (5)

Most periodic signals are typically power signals.

E.g., the power of a sinusoidal signal, (0 + ), is 2

2

And, the power of a complex exponential signal, 0 is given by 2

Signal Operations Time Shifting

A signal may be delayed by time T() = ( )

Or it may be advanced by time T() = ( + )

Signal Operations Time Scaling

A signal may be compressed in time() = (2t)

Or it may be expanded in time() = (/2)

Signal Operations Time Reversal

Reflection of a signal about the vertical axis

= ()

Signal Operations (4)

Combined operations

E.g.,

Two possible sequences of operations: Time shift, then time scale i.e.,

First, time shift () by b to obtain ( )

Then, time scale the above signal by a to obtain ( )

Time scale, then time shift i.e., First, time scale the signal () by a to obtained ()

Then, time shift the above signal by /

Signal Operations Numerical Problem (1)

1. For the signal shown below, find:1. ( + 1)2. ( + 1)

3. (3

2)

4. (3

2 + 1)

(Example 1.1, Oppenheim)

Signal Operations Numerical Problem (2)

2. For the CT signal shown at the right, find:1. ( 1)2. (2 )3. (2 + 1)4. (4

2)

3. For the DT sequence x[n] shown at the right, find:1. 42. 3 3. 34. 3 + 1

(Problem 1.2.1, 1.2.2, Oppenheim)

Periodic Signals (1)

CT:

A signal () is periodic if there is a positive value of for which() = ( + ) (1)

for all values of In other words, the signal is unchanged by a time shift of . In this case, is the period of the periodic signal (). Also,

= +

for all values of and for any integer The fundamental period 0 is the smallest positive value of for which equation

(1) holds.

Periodic Signals (2)

DT:[] = [ + ] (2)

for all values of

In other words, the signal is unchanged by a time shift of N.

In this case, is the period of the periodic signal [].

The fundamental period 0 is the smallest positive value of for which equation (2) holds.

Even and Odd Signals Even Signal

CT: () is even if = ()

DT: [] is even if =

An even signal: has the same value at the instants ( ) and ( ) for all values

of ( ).

is symmetrical about the vertical axis ( = 0, = 0).

Even and Odd Signals Odd Signal

CT: () is even if = ()

DT: [] is even if =

For an odd signal, the value at the instant (or ) is the negative of the value at the instant (or n).

An odd signal is antisymmetric about the vertical axis.

An odd signal must necessarily be 0 at = 0or = 0

Even and Odd Signals Properties

= = =

Even and Odd components of a signal Every signal can be expressed as a sum of even and odd functions

=1

2 + +

1

2[ ]

Take Home Messages!

For an energy signal as , 0

Power of a signal is the time average of its energy.

A signal will either be an energy signal, a power signal, or none.

The power of a signal of the form 0 is 2

The power of a signal of the form (0 + ) is 2

2

Considering a > 1, T > 0,

( ) is the signal () shifted to the right (delay) by T units

( + ) is the signal () shifted to the left (advance) by T units

() is the signal () compressed by factor a

(/) is the signal () expanded by factor a.

() is the signal () flipped over the vertical axis.

For combined operations (shift, scale, reversal), usually it is more convenient to first shift, then scale.

Practice Problems

Examples:1.1 1.5 (B. P. Lathi)

Problems:1.1.1 1.1.5, 1.2.1, 1.2.2 (B. P. Lathi)

1.3, 1.23, 1.24 (Oppenheim)

Useful Readings

Sections 1.1, 1.2, 1.3, 1.5, 3.1, 3.2 (Lathi)

Sections 1.1, 1.2 (Oppenheim)

Sections 1.1, 1.3 (Mandal & Asif)