1 Introduction to Signals and Systems Continuous‐time and Discrete‐time Signals There are two classes of signals: continuous-time signals ( ), xt t discrete-time signals [ ], , 2, 1, 0,1, 2, . xn 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
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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
We ma
In this c
We wil
inuous‐
’s Relatio
cor
ar Ca
1
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,
0 is perio
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
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