Basic Signal Basic Signal Operations-2Operations-2

Recap..Recap..

• DefinitionDefinition• Size Size • Classification of SignalsClassification of Signals

• Analog and digital signalsAnalog and digital signals• Continuous-time and continuous-discrete Continuous-time and continuous-discrete

signalssignals• Periodic and aperiodic signalsPeriodic and aperiodic signals• Energy and power signalsEnergy and power signals• Causal and Non-causalCausal and Non-causal• Deterministic and probabilistic signalsDeterministic and probabilistic signals• Even and odd signalsEven and odd signals

Outline..Outline..

• Operations Operations • Unit Impulse FunctionUnit Impulse Function• Unit step functionUnit step function• Unit Impulse vs. Unit StepUnit Impulse vs. Unit Step

1.Operations1.Operations• Time ShiftingTime Shifting• Magnitude Shifting:Magnitude Shifting:• Time Scaling and Time InversionTime Scaling and Time Inversion• Magnitude Scaling and Mag. Magnitude Scaling and Mag.

InversionInversion

• a).Time Shiftinga).Time Shifting:: • given the signal given the signal ff((tt), ), • the signal the signal ff((tt––tt00)) is a time-shifted is a time-shifted

version of version of ff((tt) that is shifted ) that is shifted – to the to the leftleft if if tt00 is positive and is positive and – to the to the rightright if if tt00 is negative. is negative.

f(t) f(t-to)

to

f(t+to)

to

• b).Magnitude Shiftingb).Magnitude Shifting::• Given the signal Given the signal ff((tt), ), • the signal the signal c c ++ff((tt)) is a magnitude- is a magnitude-

shifted version of shifted version of ff((tt) that is ) that is – shifted shifted upup if if cc is positive and is positive and – shifted shifted downdown if if cc is negative. is negative.

f(t)

t t

f(t)

c

f(t)

tc

• c). i. Time Scaling c). i. Time Scaling :: – Given Given ff((tt), ), – the signal the signal ff((aatt)) is a time-scaled version is a time-scaled version

of of ff((tt), ), – where where aa is a constant, such that is a constant, such that

• ff((aatt) is an ) is an expandedexpanded version of version of ff((tt) if 0<|) if 0<|aa||<1, <1,

• and and ff((aatt) is a ) is a compressedcompressed version of version of ff((tt) if ) if ||aa|>1.|>1.

• Time scaling compresses or dilates a signal by multiplying the time variable by some quantity.

• If that– quantity is greater than one, the signal

becomes narrower and the operation is called compression, while if

– quantity is less than one, the signal becomes wider and is called dilation.

The recommended approach to sketching time-scaled signals isThe recommended approach to sketching time-scaled signals issimply to evaluate simply to evaluate y(t) for a selection of values of t until the result y(t) for a selection of values of t until the result becomes clear. Forbecomes clear. Forexample,example,

• Notice that in addition to compression or dilation, the `beginning Notice that in addition to compression or dilation, the `beginning time’ or `ending time’ of a pulse-type signal will be changed in time’ or `ending time’ of a pulse-type signal will be changed in the new time scale.the new time scale.

• ii.Time Inversion:ii.Time Inversion:

– Given Given ff((tt), ),

– the signal the signal ff((aatt)) is a time-scaled version of is a time-scaled version of ff((tt), where ), where aa is a constant, is a constant,

• If If aa is negative, the signal is negative, the signal ff((aatt) is also a ) is also a time-time-invertedinverted version of version of ff((tt).).

• What happens when the time variable is multiplied by a negative number?

• The answer to this is time reversal. • This operation is the reversal of the time

axis, or flipping the signal over the y-axis.

a= -1

• d). i. Magnitude Scalingd). i. Magnitude Scaling ::• Given Given ff((tt), ), • the signal the signal bbff((tt)) is a magnitude-scaled is a magnitude-scaled

version of version of ff((tt), where ), where bb is a constant is a constant, , such that such that – bbff((tt) is an ) is an attenuatedattenuated version of version of ff((tt) if 0<|) if 0<|

bb|<1, and |<1, and – bbff((tt) is an ) is an amplifiedamplified version of version of ff((tt) if |) if |bb|>1.|>1.

attenuated amplifiedff((tt))

• ii. Mag. Inversion:ii. Mag. Inversion:• Given Given ff((tt), ), • the signal the signal bbff((tt)) is a magnitude inversion is a magnitude inversion

of of ff((tt), where ), where bb is a constant, is a constant, – if bif b is negative is negative – the signal the signal bbff((tt) is also a magnitude-flipped ) is also a magnitude-flipped

version of version of ff((tt).).

ff((tt)) ff(-(-tt))

• Example:• Given f (t) we would like to plot

f (at-b)

(a)Begin with f (t) (b) Then replace t with ‘at’ to get f (at) (c) Finally, replace t with t-b/a to get f(a(t-b/a))

Ex. : Ex. : Given f(t), sketch 4–3f(–Given f(t), sketch 4–3f(–2t–6)2t–6)

-2

2

-1

6

f(t)

t

2.Unit Impulse Function 2.Unit Impulse Function ((Dirac delta function)Dirac delta function)

• Mathematical Definition: Mathematical Definition: The unit impulse function The unit impulse function ((tt) satisfies the ) satisfies the following conditions:following conditions:

1.1. ((tt) = 0 if ) = 0 if tt 0, 0,

2.2.

• The unit impulse is not defined in terms of its values, but is defined by how it acts inside an integral when multiplied by a smooth function f(t).

1)(

dtt

• Graphical Graphical Definition: Definition: The rectangular The rectangular pulse shape pulse shape approaches the approaches the unit impulse unit impulse function as function as approaches 0 approaches 0 (notice that the (notice that the area under the area under the curve is always curve is always equal to 1).equal to 1).

(t)

t

• Since it is quite Since it is quite difficult to draw difficult to draw something that is something that is infinitely tall, we infinitely tall, we represent the Dirac represent the Dirac with an arrow with an arrow centered at the point it centered at the point it is applied. is applied.

• The dirac delta The dirac delta function and unit function and unit impulse are shown in impulse are shown in Figure (a) and (b) Figure (a) and (b) respectively.respectively.

(t)

t

(a)

(b)

Unit Impulse (cont.)Unit Impulse (cont.) The unit impulse is a valuable idealization and is used widely in

science and engineering. Impulses in time are useful idealizations. Impulse of current in time delivers a unit charge instantaneously

to a network. Impulse of force in time delivers an instantaneous momentum to

a mechanical system. Impulse of mass density in space represents a point mass. Impulse of charge density in space represents a point charge. Impulse of light intensity in space represents a point of light.

Impulse of light intensity in space and time represents a briefflash of light at a point in space.

Properties of Delta Properties of Delta FunctionFunction

1.Multiplication of a Function by an Impulse1.Multiplication of a Function by an Impulse

• ΦΦ((tt))(t) = (t) = ΦΦ(0)(0)((tt))

• Similarly Similarly

2. Sampling property of the Unit Impulse function2. Sampling property of the Unit Impulse function

• It means area under the product of a function with an It means area under the product of a function with an

impulse impulse (t) (t) is equal to value of that function at the instant is equal to value of that function at the instant

where the impulse is located.where the impulse is located.

• It is known as It is known as samplingsampling or or siftingsifting property. property.

)0()()(

dttt

)()()()( TtTdtTtt

3.Unit Impulse as a generalized Function3.Unit Impulse as a generalized Function• non unique functionnon unique function (t)(t) is not a true function in ordinary sense is not a true function in ordinary sense• an ordinary function is specified by its values for all time an ordinary function is specified by its values for all time

t. t. • range is undefinedrange is undefined• defined not as an ordinary fn but as a defined not as an ordinary fn but as a generalizedgeneralized fn fn• defined in terms of the effect it has on the test function defined in terms of the effect it has on the test function

ΦΦ((tt)) . .

We define an unit impulse as a fn for which the area We define an unit impulse as a fn for which the area under its product with a function under its product with a function ΦΦ(t)(t) is equal to value of is equal to value of

the function the function ΦΦ(t)(t) at the instant where the impulse is at the instant where the impulse is located.located.

3.Unit step function3.Unit step function• u(t) = 1 t ≥ 0u(t) = 1 t ≥ 0 = 0 t < 0= 0 t < 0

• If it is desired to start a signal at If it is desired to start a signal at t=0, multiply it by u(t).t=0, multiply it by u(t).

• A signal that does start before t=0 is A signal that does start before t=0 is called called causal causal signal. e.g.signal. e.g.

• g(t) = 0 t < 0g(t) = 0 t < 0

1 u(t)u(t)

0 t

• area from –∞ to t of area from –∞ to t of (t)(t) = 0 if t < 0 = 0 if t < 0 and unity if t ≥ 0.and unity if t ≥ 0.

• consequentlyconsequently

)( 0,1

0,0)( tu

t

td

t

)()(

therefore, tdt

tdu

Definition

Integration of the unit impulse yields the unit step function

which is defined as

Unit Step (cont…)Unit Step (cont…)

As an example of the method for dealing with generalized functions consider the generalized function

Since u(t) is discontinuous, its derivative does not exist as an ordinary function, but it does as a generalized function. To see what x(t) means, put it in an integral with a smooth testing function

and apply the usual integration-by-parts theorem

4.Unit Impulse vs. Unit 4.Unit Impulse vs. Unit StepStep

The result is that

which, from the definition of the unit impulse, implies that

That is, the unit impulse is the derivative of the unit step in a generalized function sense.

Unit Impulse vs. Unit Unit Impulse vs. Unit Step (cont.)Step (cont.)

•““Love yourself; you are Love yourself; you are a wonderful creation of a wonderful creation of God”.God”.