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ELEC2400 Signals & SystemsChapter 2. Signal Types and Operations

Brett Ninnes

[email protected].

School of Electrical Engineering and Computer Science

The University ofNewcastle

Slides by Juan I. Yuz ([email protected]) - July 24, 2003 p.1/??

2. Signal Types and Operations

Outline

What is a signal?

Types of Signals

Operations on Signals

Concluding Summary

Chapter 2. Signal Types and Operations p.2/??

What is a signal?

It is the time evolution of a quantity, for example:

The price of a share in a publically listed company;

The level of water in a reservoir;

The speed of a car;

The temperature in a room;

The voltage driving the speaker in a mobile telephone;

many others . . .



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Signal f(t) is usually defined on the real line R:

0 t

f(t)

t

f(t)

Typically, the origin at time t = 0 is thought of as being now,with f(t) for t > 0 being thought of as the futuretimeevolution of the signal, and f(t) for t < 0 being the pasttimeevolution.


Types of signals

There are many different kind of signals, with time

evolutions qualitatively and quantitatively very different.

It is useful to be able to classify signals consideringsimplified but common signal types.


Types of signals

Constant signal

f(t) A R, t (,).

0

A f(t)

t



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Step signal1(t)

1 ; t 0

0 ; t < 0.

0

1 1(t)


Types of signals

Ramp signal

r(t) t ; t 0

0 ; t < 0.

0 t

r(t)

t

t


Ramp signal

The ramp signal r(t) may be derived from another, since itis the cumulative area under the step signal 1(t) up untiltime t:

r(t) =t

1() d.

0 t

r(t) =

t

1() d



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As a consequence, and using the fundamental principlethat differentiation is the inverse operation to integration, wealso have that:

d

dtr(t) = 1(t).

That is, the time rate of change of the ramp signal is thestep signal.Finally, r(t) and 1(t) are also related by the equation

r(t) = t 1(t).


Types of signals

(Dirac) Delta Function or (Unit) Impulse signal

(t)

0 ; t = 0

Undefined ; t = 0

It is useful to imagine that (0) = +:

t0

(t)


Impulse signal

Diracrefers to the Quantum Physics pioneer Paul Diracwho used this to represent electrons as units of chargeoccupying an infinitesimal amount of space.

Strictly, (t) is not a function. It may be characterised asa relationship for any continuous function f(t):

f(t) (t) d= f(0).

That is, (t) picks outthe value of f(t) at t = 0. This meansthat (t) is distributionrather than a function.



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(t) can be approximated by the function K(t) defined as:

(t) K(t) =

1/ ; |t| /2

0 ; |t| > /2

t

(t)

1/

K(t)

/2 /2

0


Impulse signal

Hence, we have that:

f(t) (t) d

f(t)K(t) dt =

/2/2

f(t)1

dt.

If is made small, then f(t) f(0) for all t (/2, /2) since

it is continuous. In this case:/2/2

f(t)1

dt

f(0)

/2/2

dt =f(0)

= f(0).

And, when 0:

lim0

/2/2

f(t) 1

dt = f(0).


Impulse signal

Considering

(t) = lim0

K(t)

then

f(t)(t) dt =

f(t) lim0

K(t) dt

= lim0

f(t)K(t) dt = f(0).

This is also why (t) is known as the unit impulse, since:

(t) dt = lim0

K(t) dt = lim0

/2/2

1

dt = 1.



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Finally, notice that there is an integral relationship betweenthe unit step 1(t) and the Dirac delta (t):

1(t) =

t

() d

And, by differentiating both sides, there is also a differentialrelationship:

d

dt1(t) = (t).


Types of Signals

Periodic signal

It is a signal which repeats the same wave-shapeevery Tseconds and infinitely often, i.e.:

f(t) = f(t + T), for all t.

For example:

0 t

f(t)

T


Periodic signal

The signal

f(t) = sin t

is periodic with period T= 2/, since:

f(t + T) = sin

t + 2

= sin(t + 2)

= sin t cos2 1

+sin2 0

cos t = sin t = f(t).

Also, if f(t) = f(t + T) and g(t) = g(t + T) then for any

, R:

h(t) = f(t) + g(t) = f(t + T) + g(t + T) = h(t + T).



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Even signal: symmetric about the y axis.

f(t) = f(t), for all t.

Odd signal: symmetric respect to the origin.

g(t) = g(t), for all t.

0 tf(t)

g(t)

y


Types of Signals

Exponential signal

f(t) = 1(t) Aet =

Aet ; t 0

0 ; t < 0

If > 0 then et is exponentially increasing,if < 0 then et is exponentially decreasing, and

if = 0 then et = 1 and it is the step signal 1(t).

t0

A

Aet, = 0

Aet, < 0

Aet, > 0


Types of Signals

Exponential signals: complex representation for

signals with oscillating components.

f(t) = 1(t)Aet cos(t+) = A et cos(t + ) ; t 00 ; t < 0

where:

the oscillation period is T= 2/,

the exponential envelope is A et, and

the co-sinusoidal shape has phase offset .



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It is useful to note that:

Aet+j(t+) = A et ej(t+)

= Aet [cos(t + ) + j sin(t + )]

Then

f(t) = Aet cos(t + ) = Real

Ae+j(t+)

.


Exponential signals (complex representation)

If < 0, then f(t) = Aet cos(t + )

is a decreasing oscilation:

t

2

Aet

Aet cos(t + )

A



If > 0, then f(t) = Aet cos(t + )is an increasing oscilation:

t

Aet cos(t + )

Aet

2

A



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If = 0, then:

f(t) = Aet cos(t + ) = A cos(t + )

A

A cos(t + )

t

2



For this puresinuosidal signals, it is common to use a

phasor representation:

f(t) = Real

Aej(t+)

= A cos(t + )

Real

Imaginary

Aej

0

A

Complex Representation (Phasor)



More complicated signals can be derived (orexpressed) via various fundamental operations overthe basic signals previously defined.



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Magnitude Scaling: It is the multiplication by aconstant K R:

g(t) = Kf(t).

f(t)

t

t

t

t

g(t) = Kf(t), K < 1g(t) = Kf(t), K < 0

g(t) = Kf(t), K > 1



Time Shifting (Translation):

g(t) = f(t T).

If T > 0, then g(t) = f(t T) is f(t) delayedby Tseconds:

0 t

T

f(t) g(t) = f(t T)


Time Shifting

Notice that, for Dirac delta signals:

g(t) = (t T) = 0 ; t = TUndefined ; t = T.We can think of (t T) as having all its massconcentratedat t = T:

t0

(t T)

T

g(t)



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And, for a continuous function f(t), we have that:

f(t)(t T) dt = f(T).

This principle will turn out to be of great importance in laterdevelopments.



Time Reversal (Flipping):

g(t) = f(t)

This operation produces a new signal g(t) by reversingthe time direction:

t0

f(t)g(t) = f(t)


Time Reversal

In some situations time reversal operation is combined withthe time shifting operation (e.g., in the convolution).For example, consider g(t) = f(T t):

0

f(t)

t

T

g(t) = f(T t)

Note that, if h(t) = f(t) (flipping)

then h(t T) = f((t T)) = f(T t) = g(t) (shifting)



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Time Scaling. The signal is stretched or compressedby a factor R+:

g(t) = f(t)

T

f(t)

t0

t0 t0T / T/

g(t) = f(t), < 1 g(t) = f(t), > 1


Time Scaling

Note that:

If < 0, there is a time reversal and scaling.

For the Dirac delta function (t), using the change ofvariable = t which implies that d = dt, we have

f(t)(t) dt =

f

()

1

d =

1

f(0)

so that

(t) =1

(t).

i.e., the time scaling decrease or increase the area

under the delta function.


Concluding Summary

We introduced some of the most fundamental ideas in thefield of Signals and Systems:

A signal is the time evolution of a quantity;It can be represented as a mathematical functionf(t) : R R;



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We introduced some of the most fundamental ideas in thefield of Signals and Systems:

There are some fundamental signal types,

1. Constant: f(t) = K;

2. Unit step: f(t) = 1(t);

3. Unit ramp: f(t) = r(t) = t 1(t);

4. Dirac Delta: f(t) = (t);

5. Periodic of period T: f(t) = f(t + T);

6. Even f(t) = f(t) and odd f(t) = f(t);7. Exponential: f(t) = Aet;

8. Generalised Exponential: f(t) = Aet cos(t + ).


Concluding Summary

We introduced some of the most fundamental ideas in the

field of Signals and Systems:

There are some fundamental signal operations,

1. Magnitude Scaling: g(t) = Kf(t);

2. Time shifting (translation): g(t) = f(t T);

3. Time reversal (flipping): g(t) = f(t);

4. Time scaling: g(t) = f(t).


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