3

Signals

and

Systems:

Part

II

In

addition

to

the

sinusoidal

and

exponential

signals

discussed

in

the

previous

lecture,

other

important

basic

signals

are

the

unit step

and

unit

impulse.

In

this

lecture,

we

discuss

these

signals

and

then proceed

to

a

discussion

of

sys-

tems,

first

in

general

and

then

in

terms of

various

classes

of

systems

defined

by

specific

system

properties.

The

unit

step,

both

for

continuous

and discrete

time,

is

zero

for

negative

time

and

unity

for positive

time.

In

discrete

time

the unit step

is

a

well-defined

sequence,

whereas

in

continuous

time

there

is

the

mathematical

complication

of

a discontinuity

at

the

origin.

A

similar

distinction

applies

to

the unit

im-

pulse.

In

discrete

time

the

unit

impulse

is

simply

a

sequence

that

is

zero

ex-

cept

at

n

=

0,

where

it

is

unity.

In

continuous

time,

it

is

somewhat

badly be-

haved mathematically,

being

of

infinite

height

and

zero

width

but

having

a

finite

area.

The

unit

step

and

unit

impulse are

closely

related.

In

discrete

time

the

unit

impulse

is

the

first

difference

of the unit

step,

and

the

unit

step

is

the run-

ning

sum

of

the

unit

impulse.

Correspondingly,

in

continuous

time

the

unit

im-

pulse

is

the

derivative

of

the

unit step,

and

the

unit step

is

the

running

integral

of

the

impulse.

As

stressed

in

the lecture, the

fact

that

it

is

a

first difference

and a running

sum

that

relate

the

step

and

the

impulse

in

discrete

time

and

a

derivative

and running

integral

that

relate

them

in

continuous

time

should

not

be

misinterpreted

to

mean

that

a

first difference

is

a

good

"representation"

of

a

derivative

or

that

a running

sum

is

a

good

"representation"

of

a

running

inte-

gral.

Rather, for

this particular

situation

those

operations

play

corresponding

roles

in

continuous

time

and

in

discrete

time.

As

indicated

above,

there

are a

variety

of

mathematical

difficulties

with

the

continuous-time

unit

step and

unit

impulse

that

we do

not

attempt

to

ad-

dress

carefully

in

these lectures.

This

topic

is

treated

formally

mathematically

through the

use

of

what

are

referred

to

as generalized

functions,

which

is

a

level

of

formalism

well

beyond

what

we

require

for

our

purposes.

The

essen-

tial

idea, however,

as

discussed

in

Section

3.7

of

the

text,

is

that

the

important

aspect

of these

functions,

in

particular

of

the

impulse,

is

not

what

its value

is

at

each

instant

of

time

but

how it behaves

under

integration.