2

Signals

and

Systems:

Part

I

In

this lecture,

we

consider

a

number

of

basic

signals

that

will

be

important

building

blocks

later

in

the

course.

Specifically,

we

discuss

both

continuous-

time

and discrete-time

sinusoidal

signals as

well

as real

and

complex expo-

nentials.

Sinusoidal signals

for

both

continuous

time

and

discrete

time

will

be-

come

important

building

blocks

for

more

general signals,

and

the

representa-

tion

using

sinusoidal

signals

will

lead

to

a

very

powerful

set

of

ideas

for

repre-

senting signals

and

for

analyzing

an

important

class

of

systems.

We

consider

a

number

of

distinctions between

continuous-time and discrete-time

sinusoidal

signals.

For

example,

continuous-time

sinusoids

are

always

periodic.

Further-

more,

a

time

shift

corresponds

to

a

phase

change

and

vice

versa.

Finally,

if

we

consider

the

family

of

continuous-time

sinusoids

of the

form

A

cos

wot

for

dif-

ferent

values

of

wo,

the

corresponding

signals

are

distinct.

The

situation

is

considerably

different

for

discrete-time

sinusoids.

Not

all

discrete-time sinu-

soids

are

periodic.

Furthermore,

while

a

time

shift

can

be

related

to

a

change

in

phase,

changing

the phase

cannot

necessarily

be

associated

with

a

simple

time

shift

for

discrete-time

sinusoids.

Finally,

as

the

parameter

flo

is

varied

in

the discrete-time

sinusoidal sequence

Acos(flon

+

4),

two

sequences

for

which

the

frequency

flo

differs

by

an

integer

multiple

of

27r

are

in

fact indistin-

guishable.

Another

important

class

of

signals

is

exponential

signals.

In

continuous

time,

real

exponentials

are

typically

expressed

in

the

form

cet,

whereas

in

dis-

crete

time

they

are

typically

expressed

in

the

form

ca".

A

third important

class

of

signals

discussed

in

this

lecture

is

continuous-

time

and

discrete-time

complex

exponentials.

In

both

cases the

complex ex-

ponential

can

be

expressed

through

Euler's

relation

in

the

form

of

a

real

and

an imaginary

part,

both

of

which are

sinusoidal

with

a

phase

difference

of

'N/2

and with

an

envelope

that

is

a

real

exponential.

When

the

magnitude

of

the

complex

exponential

is

a

constant,

then

the

real and

imaginary

parts

neither

grow

nor

decay

with

time;

in

other

words,

they

are purely

sinusoidal.

In

this

case

for

continuous

time,

the

complex

exponential

is

periodic.

For

discrete