1

IntroductiontoSignalsandSystems

Continuous‐timeandDiscrete‐timeSignals

There are two classes of signals:

continuous-time signals ( ), x t t

discrete-time signals [ ], , 2, 1,0,1, 2, .x n n

or is called the .t n independent variable

Source of Discrete-time Signals

Sampling ( )x t at integer time instants

Scanning an image at successive pixel locations

Recoding the price of a stock daily

2

SignalEnergyandPower

Assume ( ) is a voltage applied to a 1 ohm resistor.x t R

2

1

2

1

2

21 2

2

2 1

2

( )

( )

1( )

1lim ( )

2

t

t

t

t

T

T T

instantaneous power x t

total energy expended during t t t x t dt

average power over the time interval x t dtt t

average power over the entire time interval x t dtT

2

In general, ( ) can take on complex values.

In that case,

( )

x t

instantaneous power x t

2

2

2

Similarly, for discrete-time signals ,

[ ]

[ ]

1lim [ ]

2 1

n

N

Nn N

x n

instantaneous power x n

total energy x n

average power over the entire time interval x nN

3

TransformationoftheIndependentVariable

Time Shift

Time Reversal

Time Scaling

Time Shift

For continuous-time signals,

0

0

0

( ) is a version of ( );

if 0

if 0

x t t time - shifted x t

delayed t

advanced t

Similarly, for discrete-time signals.

0

0

is a time-shifted version of [ ].

is an integer.

n nx x n

n

Time Reversal

( ) is a of ( ) 0.

Similarly,

[ ] is a reflection of [ ] about 0.

x t reflection x t about t

x n x n n

4

Time Scaling

( ) is a version of ( );

1

1

x t x t

if

if

time scaled

compressed

stretched

For discrete-time signals,

[ ] is defined unless is an integer

for all values of .

x n not n

n

5

Combined Transformation

( ) is a time scaled and shifted version of ( ).

To draw ( ),

shift ( ) by to get and then

scale by a factor of

Or, making use of ( ) ,

scale by a factor of to get

x t x t

x t

x t first x t

x t x t

x t

and then

shift ( ) by x t

Example @1.3

32

32

3 22 3

To draw ( 1), either

shift ( ) first by 1, .( ), and then scale by a factor of , .( ); or

scale by a factor of , .( ), and then shift by , .( ).

x t

x t fig b fig e

fig d fig e

6

Example

x t

t0 3

1x t

t0 1

1x t

t01

x t

t03

4

2

x t

t03

1x t

t01

1x t

t02

x t

t0

4

1

3

x t

t0 3

Time Reversal

7

PeriodicSignals

A continuous-time signal ( )x t is said to be periodic with period T if

( ) ( ) for all values of .x t x t T t

If ( ) is periodic with period , then ( ) is periodic with period 2 , 3 , .x t T x t T T

The of ( ) is the smallest positive period with which

( ) is periodic.

x t

x t

fundamental period

A signal that is not periodic is referred to as an signal.aperiodic

Similarly, a discrete-time signal x n is said to be periodic with period N if

[ ] [ ] for all values of .x n x n N n

Even and Odd Signals

A signal ( ) or x t x n is referred to as an even signal

if it is identical to its time-reversed counterpart, i.e., to its reflection about the origin.

( ) ( ) [ ] [ ]x t x t or x n x n

A signal is referred to as odd if

( ) ( ) [ ] [ ]x t x t or x n x n .

Properties

Any signal can be broken into a sum of an even and an odd signal:

( ) ( ) ( ) ( )( ) .

2 2Note the first term is an even signal and the second term is an odd signal.

x t x t x t x tx t

Conti

Euler’

jre

Pola

is pj

j

e

Period

Consid

where

When a

Note

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In this c

We wil

inuous‐

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cor

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periodic wit

dic Comp

er ( )

is, in gen

ax t e

a

a is purely im

0

0

( )

is call

Average P

j

a j

x t e

0

ay write the

convention,

ll use .

fun

x

fun

‐TimeC

on

os sinj r

artesian

th period of

lex Expon

,

neral, a com

at

maginary,

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led the

Power 1.

t

fund

0

00

signal as (

1, ,

( ) .j t

x

Tf

ndamenral p

x t e

ndamenral p

Complex

n

f 2 .

nential

mplex numbe

dic with the

damental fr

02

0

0

( )

, which says

j f tt e

period T

period T

xExpon

er.

e fundamen

requency.

.

s

t

fundament

fundament

ential

ntal period T

tal frequenc

tal frequenc

00

2.T

0 = 1. cy f

0= 2 .cy

Im

r

.

8

Re

9

Sinusoidal Signals

0

0

( ) cos( )

2is periodic with period of .

x t A t

With seconds as the units of t,

0fundamental frequency

radians per second

phase radians

Harmonically Related Complex Exponentials

For a given fundamental frequency 0 ,

0( ) is called the th jk tk t e k harmonic,

k = 0, 1, 2,

0

0

0

( )

2has the fundamental period of ;

2all are periodic with period of .

jk tk t e

k

10

Damped Sinusoids

0

Consider

( ) ,

where and .

at

j

x t Ce

C e a r jC

0

0

0 0

( )

with Euler's relation

cos sin

r j tj

j trt

rt rt

x t e eC

e eC

t te j eC C

0 0 acts as an envelope for sinusoids cos and sin .

When 0, the envelope decays in time and

the signals are called .

rt t teC

r

damped sinusoids

Discr

Periodic

Fre

Ind

Period

Consid

0 ij ne

The

Gen

the d

We nee

Examp

Consid

As the

0For

0For

0For

0For

0For

rete‐Tim

city in

equency

dependent V

dicity in F

der [ ] jx n e

s periodic in

0 2

signal at fr

neralizing th

discrete-tim

je

0 0,

0

ed to consid

0 2

ple

0

er

angle r

x n e

n

0,

Re x n

2

,

Re x n

,

Re x n

32

,

Re x n

2 ,

Re x n

meCom

Variable

Frequency

0 . j n

n frequency

2

0

=

equency

he above arg

me complex

n j ne

02 , 4

der the signa

or

0 cos

rotates, Re

j ne

1, 1,1, 1, 1,

1, 0, 1, 0,

1, 1, 1, 1,

1, 0, 1, 0,

1, 1, 1, 1, 1,

mplexEx

y

y with the pe

0

0

=

2 is the

gument,

exponential

j n jne e

, and so on

0

al only in frq

.

0 sin

follo

n j

x n

as n

1, as n

, as n

1, as n

as n

xponent

eriod of 2

0 .

e same as th

l is the sam

n

n.

quencies wi

0

0ows cos

n

n

0,1,2, .

0,1,2, .

0,1,2, .

0,1,2, .

0,1,2, .

tial

.

he signal at

me signal at f

ithin any pe

.n

Highes

same as

frequency

frequencie

s

eriod of 2

st Oscillatio

0 0s

Im

0.

s

:

on

0w n

11

Re

12

Another view of the above example is

Duality

0

0

0

With a continuous-time complex expontial ( ) ,

the signal oscillates more rapidly as increases.

On the contrary, with a discrete-time complex expontial ,

the signal is periodic in freque

j t

j n

x t e

x n e

ncy with the period of 2 .

13

Periodicity in the independent variable

0Consider [ ] and

suppose [ ] is periodic with period of .

j nx n e

x n N

0 0 0 0

0

0

0

0

Then

[ ] [ ]

= = .

Therefore must equal 1.

must be an integer multiple of 2 .

=2 for some integer .

2 .

j j n j N j nn N

j N

x n N x n

e e e e

e

N

N m m

m

N

0

0

[ ] is periodic if and only if

2 for some integer and .

j nx n e

mm N

N

When and are relatively prime,

the fundamental period is , and

2the fundamental frequency is .

m N

N

N

14

Problem @1.35

2

0Let . Show the fundamental period is = .gcd ,

mj n

N Nx n e N

m N

Example @1.6

2 3

3 4

1 32 2

3 8

Consider [ ] .

[ ] + .

In the first term, , (1,3) are relatively prime. So 3.

In the second tgerm, , (3,8) are relatively prime. So 8.

Therefore [ ] has (3,8)

j n j n

j n j n

x n e e

x n e e

m N FP

m N FP

x n FP lcm

24.

15

Harmonically Related Complex Exponentials

0

2

2For a given frequency ,

is called the th .

jk nN

k

N

e k = 0, 1, 2,n

k harmonic

Properties

Although the harmonics are defined for , they repeat after the .k = 0, 1, 2, Nth

2 2

2

0

12

1

22

2

12

1

=

= .

There are only distinct harmonics:

1

jk n jN nN N

k N

jk nN

k

j nN

j nN

Nj n

NN

e en

e

n

N

n

en

en

en

Duality

2

2

2

2

With a continuous-time complex expontial ( ) ,

all harmonics are distinct.

On the contrary, with a discrete-time complex expontial [ ] ,

there are distinct harmonics , 0,1

j tT

jk tT

j nN

jk nN

x t e

e

x n e

N e k

, , 1.N

16

A Note on Signals vs. Functions

is a continuous-time signal

defined for .

We can plot in terms of time .

On the other hand, 5 is simply a constant,

the value of at 5.

In that sense, we can regard as function

of the

x t

t

x t t

x

x t t

x t

variable .

The similarity between signals and functions

is particularly useful

in understanding and working with the unit impulse and the unit step signals.

t

17

TheUnitImpulseandUnitStepFunctions

Discrete-Time Unit Impulse and Unit Step Signals

The discrete-time unit impulse is

defined as

1, 00, 0

nn

n

The discrete-time unit step is

defined as

1, 0,1,0, 1, 2,

nu n

n

Properties

1 .n u n u n Regard as a signal .

0

.

k

u n n k Regard as a signal

.

.n

k

u n k Regard as a function

.

00 0

For any sequence , x n

n nx n n n x n Regard as a signal .

18

Continuous-Time Unit Impulse and Unit Step Signals

Consider the function .u t

0

The continuous-time unit step signal

is defined as

lim .

u t

u t u t

Therefore

0 01 0

0

tu t t

undefined t

The continuous-time unit impulse signal

is defined as a signal such that

. @(1.71)t

t

d u t

Differentiating (1.71),

,dt u t

dt

which can be interpreted as

0

lim ,d

t u tdt

0

lim .t t

The continuous-time unit impulse can be viewed as a very narrow boxcar function.

0

0 0 0 0 0 00 0

For any function continuous at ,

lim lim

x t t t

x t t t x t t t x t t t x t t t

u t

t

1

19

Continuous‐TimeandDiscrete‐TimeSystems

( ) ( )x t y t

x n y n

Example @1.5.1

Let

( ) the force applied to the car with mass

( ) the velocity of the car.

Then

the net force ( ) ( ),

where is the coefficient of kinetic friction.

f t m

v t

f t v t

net forceApplying the model, acceleration ,

mass

( ) ( ) ( ).

Rearranging terms,

( ) 1( ) ( ).

Replacing and with ( ) and ( ) as the input and the output respectively,

( ) ( )

dv t f t v t

dt m

dv tv t f t

dt m m

f t v t x t y t

dy t a y t

dt

( ) @(1.85)

which is a first-order linear differential equation.

bx t

20

Example @1.11

Digital Simulation of eq. (1.85), ( ) ( ) ( )d

y t a y t bx tdt

.

Resolve time into discrete interval of length , and make following transformation:

( ) ( ),

( ) ( ),

( ) (( 1) )( ) .

x t x n

y t y n

d y n y ny t

dt

Then (1.85) is written as

1 .

Rearranging terms,

1 ( ) ( ( 1) ) ( ).

1Introducing new parameters and ,

1 1

( ) (( 1) ) ( ).

Letting ( ) and ( ),

y n y n a y n b x n

a y n y n b x n

bc d

a a

y n c y n d x n

x n x n y n y n

y n c y

1 , @(1.89)

which is a first-order linear difference equation.

n d x n

21

Interconnection of Systems

22

Example – RC Circuit @1.5.2

1 2

1

2 2

( ) ( ) ( ).

1For the capacitor, ( ) ( ) .

( )( ) induces ( ) through the resistor, ( )

t

i t i t i t

v t i drC

v tv t i t i t

R

23

BasicSystemProperties

Memoryless

Invertible

Causal

Stable

Time Invariant

Linear

Systems with and without Memory

A system is memoryless if the output at any time is dependent on the input only at that same time.

Examples

( ) ( ) is memoryless.y t R x t

A delay 1 is with memory.y n x n

An accumulator is with memory.n

k

y n x k

Invertibility and Inverse Systems

A systen is invertible if an inverse system exists.

In an invertible system, distinct inputs lead to distinct outputs.

Examples

An accumulator [ ] [ ] is invertible with

the inverse system [ ] [ ] [ 1].

n

k

y n x k

w n y n y n

2 is not invertible.y n x n

24

Causality

A system is causal if the output at any time depends on values of the input at only the present and past times.

Examples

1 is not causal.y n x n x n

is not causal,

because 5 5 .

y xn n

y x

( ) ( )cos( 1) is causal,

because cos( 1) has no relation to the input.

y t x t t

t

Stability

A system is stable if the output does not diverge whenever the input is bounded.

Examples

( ) ( ) is not stable.

For the bounded input ( ) 1, ( ) diverges.

y t t x t

x t y t t

( )( ) is stable.

If , then ( ) .( )

x t

B B

y t e

B e y t ex t

An accumulator is not stable,

beacuse for a bounded input such as 1, diverges.

n

k

y n x k

x n y n

1A running average is stable.

2 1

M

k M

y n x n kM

25

Time Invariance

Consider a discrete-time system: [ ] [ ]x n y n .

The system is is said to be time invariant if 0 0 0[ ] [ ] for any .x n n y n n time shift n

Similarly, in a time invariant continuous-time system: ( ) ( )x t y t ,

0 0 0( ) ( ) for any .x t t y t t time shift t

Examples

0 0 0

( ) sin ( ) is time invariant.

( ) sin ( ), which is identical to ( ).

y t x t

x t t x t t y t t

0 0

[ ] [ ] is not time invariant.

[ ] [ ] 0 [ ] 0, which implies 0 for any

[ 2] [ 2] 2 [ 2], which does not equal 2

y n n x n

x n n y n n y n n n

x n n n y n

0 0 0 0

( ) (2 ) is not time invariant because the time shift is also compressed.

( ) (2 ) which is not equal to ( ) (2( )).

y t x t

x t t x t t y t t x t t

26

Linearity

A system is linear if it possesses the superposition property:

If the input is a weighted sum of several signals, then the output is the superposition—that is, the weighted sum—of the responses of the system to each of those signals.

1 1 2 2

1 2 1 2

Let ( ) ( ) and ( ) ( ).

Then the system is linear if

( ) ( ) ( ) ( ) for any complex scalars and .

x t y t x t y t

a x t b x t a y t b y t a b

The same definition applies to discrete-time systems.

Properties

1 2 1 2

Linear systems have the

) property: ( ) ( ) ( ) ( ), and the

) or property: ( ) ( ).

i additive x t x t y t y t

ii scaling homogeneity ax t a y t

Let [ ] [ ] for 1,2, . Then

[ ] [ ].

k k

k k k kk k

x n y n k

a x n a y n

For linear systems, the property holds

because for any linear system [ ] [ ],

0 [ ] 0 [ ].

zero - in / zero - out

x n y n

x n y n

Examples - Linear

1 2 1 2 1 2

1 2

( ) ( ) is linear.

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

y t t x t

ax t bx t t ax t btx t a t x t b t x t

a y t b y t

2( ) ( ) is not linear. y t x t

27

1 1 1 2 2 2

1 1 1 2 2 2

3 1 2

3 3 1 2 1

[ ] Re [ ] is not linear.

Let [ ] [ ] [ ] and [ ] [ ] [ ].

Then [ ] [ ] [ ] and [ ] [ ] [ ].

Define a new input [ ] [ ] [ ].

Then [ ] [ ] [ ] [n] which equals

y n x n

x n r n js n x n r n js n

x n y n r n x n y n r n

x n x n x n

x n y n r n r y

2[ ] [ ].

The system has the additive propoerty.

Let [ ] [ ] [ ].

Then [ ] [ ] [ ].

However [ ] [ ] which does not equal [ ].

The system violates the homogeneity property.

n y n

x n r n js n

x n y n r n

jx n s n jy n

[ ] 2 [ ] 3 is not linear.

Let [ ] 0.

Then [ ] [ ] 3 which does not equal 0.

The system violates the zero-in/zero-out property.

However the system belongs to a class of systems.

y n x n

x n

x n y n

incrementally linear