Signal and System Ch#1

7/30/2019 Signal and System Ch#1

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1 Signals and Systems

introducing language for describing signals and systems

Outline

1.1 ContinuousTime and DiscreteTime Signals

1.2 Elementary Signals

1.3 ContinuousTime and DiscreteTime Systems

1.4 Basic System Properties

Lampe, Schober: Signals and Communications


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1.1 ContinuousTime and DiscreteTime Signals

Unied representation of physical phenomena bysignals

Signal: Function or sequence that represents information.

One or more independent variables Continuous or discrete independent variables Examples: time , location, etc.

1.1.1 Mathematical RepresentationContinuoustime signals

Symbolt for independent variable Use parentheses()

Continuoustime signal:x(t)

Graphical representation

0 t

x(t)



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Discretetime signals

Symboln for independent variable Use brackets[]

Discretetime signal:x[n]

Graphical representation

x [0]

x [1]

x [ 1]

0 321

x [2]x [ 2]

n 3 2 154

x[n]



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1.1.2 Signal Energy and Power

Often classication of signals according toenergy and power

Terminologyenergy and power used for any signalx(t), x[n] Need not necessarily have a physical meaning

Signal energy

Energy of a possibly complex continuoustime signalx(t) in

interval t1 t t2

E (t1, t2) =

t2

t1

|x(t)|2 dt

Energy of a possibly complex discretetime signalx[n] in intervaln1 n n2

E (n1, n 2) =n

2

n= n1

|x[n]|2

Total energy

E = E ( , ) =

|x(t)|2 dt

E = E ( , ) =

n= |x[n]|2



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Example:

Total energy of the discretetime signal

x[n] =an n 00 n < 0

with |a | < 1.

E =

n= |x[n]|2 =

n=0

(|a |2)n =1

1 | a |2

Signal power Consider thetimeaveraged signal power

Average power of x(t) in intervalt1 t t2

P (t1, t2) =1

t2 t1

t2

t1

|x(t)|2 dt

Average power of x[n] in intervaln1 n n2

P (n1, n2) =1

n2 n1 + 1

n2

n= n1

|x[n]|2

Analogously

P = P ( , ) = limT

12T

T

T |x(t)|2 dt

P = P ( , ) = limN

12N + 1

N

n= N

|x[n]|2



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Classication of signals based on their energy and power

Signals with nite total energyE < Zero average powerP = 0Examples: example above, any signal with nite duration

Signals with nite average powerP < Innite total energyE = if P > 0Examples: periodic signals, e.g.x(t) = cos(t), x[n] = sin(5n)

Signals with innite powerP = and innite energyE =

Not desirable in engineering applicationsExamples: x(t) = et , x[n] = n10



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1.1.3 Transformations of the Independent Variable

Time shift

Replace t t t0 x(t) x(t t0)n n n0 x[n] x[n n0] Delay: t0, n0 > 0, Advance: t0, n0 < 0

Time reversal

Replace t t x(t) x( t)n n x[n] x[ n]

Time scaling

Replace t t , IR x(t) x(t )n n , ZZ x[n] x[n ]

Continuoustime case: | | < 1 : signal is linearly stretched| | > 1 : signal is linearly compressed

Time shift, time reversal, and time scaling operations arise naturallyin the processing of signals



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Example:

n

nt

n

n

t

t

Time-scaled signals

Time-reversed signals

Time-shifted signals

Signals

t

x[2n]

x[ n]

x[n 4]

x(2/ 3t)

x( t)

x(t t0 )

x(t) x[n]

t0



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1.1.4 Periodic Signals

Periodic continuoustime signal

x(t) = x(t + T ) , t T > 0: Period

x(t) periodic withT x(t) also periodic withmT , m IN

Smallest period of x(t): Fundamental period T 0.

Example (T 0 = T ):

0

x(t)

t4T 3T 2T 3T 2T T T

Periodic discretetime signal

x[n] = x[n + N ] , n

Integer N > 0: Period x[n] periodic withN x[n] also periodic withmN , m IN

Smallest period of x[n]: Fundamental period N 0.

Example(N 0 = 4):

n

3 6

x[n]

0 12

54

A signal that is not periodic is referred to asaperiodic .



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1.1.5 Even and Odd Signals

Even signal

x( t) = x(t) or x[ n] = x[n] Example:

x(t)

t

Odd signal

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

Example:

n

x[n]

Necessarily:x(0) = 0 or x[0] = 0

Decomposition of any signal into an even and odd part:

x(t) = Ev{x(t)} + Od{x(t)} or x[n] = Ev{x[n]} + Od{x[n]}

with

Ev{x(t)} =1

2(x(t) + x( t)) or Ev{x[n]} =

1

2(x[n] + x[ n])

and

Od{x(t)} =12

(x(t) x( t)) or Od{x[n]} =12

(x[n] x[ n])



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1.2 Elementary Signals

Several classes of signals play prominent role

model many physical signals serve as building blocks for many other signals serve for system analysis

1.2.1 ContinuousTime Complex Exponential and SinusoidalSignals

Complex exponential signal

x(t) = C eat

In general, complex numbersC and a (C, a C)

Real exponential signal if botha and C real (C, a IR)

Periodic complex exponential signal if a = j0With C = Ae j (A, IR):

x(t) = Ae j(0 t+ )

Signal is periodic:

Ae j(0 t+ ) = Ae j(0 (t+ T )+ ) = Ae j(0 t+ )e j0 T

where e j0 T != 1Excluding the trivial solution0 = 0 Fundamental period

T 0 =2|0|



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Set of harmonically related complex periodic exponentials

k(t) = e jk0 t , k = 0, 1, 2, . . .

k = 0: 0(t) constantk = 0: k(t) periodic with fundamental frequencyk0 andfundamental periodT 0/ |k|Sets of harmonically related complex exponentials used to

represent many other periodic signals

General complex exponential signal

With C = Ae j

(A, IR)and a = r + j0 (r, 0 IR)

C eat = Aert e j(0 t+ ) = Aert cos(0t + ) + jAert sin(0t + )

r > 0: exponentially growing signalr < 0: exponentially decaying signal

Sinusoidal signals

xc(t) = Acos(0t + ) = Re{Ae j(0 t+ )}

and

xs(t) = Asin(0t + ) = Im{Ae j(0 t+ )}

xc(t) and xs(t) also have periodT 0 = 2/ |0|, of course.



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Periodic signals have innite total energy but nite average power.

Exponentialx(t) = e j0 t

Energy over one periodT 0

E period =

T 0

0

|e j0 t |2 dt = T 0

Average power per period

P period =E period

T 0

= 1

Average power

P = limT

12T

T

T

|e j0 t |2 dt = 1



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1.2.2 DiscreteTime Complex Exponential and Sinusoidal Sig-nals

Complex exponential signal

x[n] = C n (= C en , = e )

Real exponential signal if bothC and real

General complex exponential signalWith C = Ae j and = | |e j0 (A,,0 IR)

x[n] = A| |ne j(0 n+ )= A| |ncos(0n + ) + jA| |nsin(0n + )

| | > 1: exponentially growing signal| | < 1: exponentially decaying signal | | = 1:

x[n] = Ae j(0 n+ ) = Acos(0n + ) + jAsin(0n + )

Sinusoidal signal

xc[n] = Acos(0n + ) = Re{Ae j(0 n+ )}

and

xs[n] = Asin(0n + ) = Im{Ae j(0 n+ )}

Both Ae j(0 n+ ) and Acos(0n + ) are discretetime signals withnite average power but innite total energy.



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Certain properties Important differences between discretetime and continuoustimecomplex exponentials

1. Increase frequency0 by integer multiples of 2

e j(0 + m2)n = e j0 ne jm2n = e j0 n

Observation: same exponential for frequency0 and frequencies0 2, 0 4, . . .Conclusion: sufficient to consider frequency interval of length2

Usually: intervals0 0 < 2 and 0 < usedExample: Fig. 1.27 in text book

2. Periodicity: periodN > 0

e j0 N != 1 0N = 2m,

or02

=m

N where m is integerObservation: e j0 n is periodic if 0/ (2) is a rational number,and is aperiodic otherwise.

3. Fundamental periodN 0:

N 0 = m20

for 0 = 0 and gcd(N 0, m) = 1 (N 0 and m have no factors incommon)



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Example:

x(t) = cos(8t/ 31)0 =Fundamental periodT 0 = 2/ 0 =

x[n] = cos(8n/ 31) (= x(t = n))0 =Periodic?Fundamental periodN 0 = m(2/ 0) =for m =

x[n] = cos(n/ 6)0 =Periodic?Fundamental periodN 0 = m(2/ 0) =for m =

Set of harmonically related discretetime periodic exponentialsk[n] = e jk(2/N )n, k = 0, 1, . . .

Common periodN Observation:

k+ N [n] = e j(k+ N )(2/N )n = e jk(2/N )ne j2n = k[n]

Only N distinct complex exponentials0[n], 1[n], . . .,N 1[n].



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1.2.3 The DiscreteTime Unit Impulse and Unit Step Se-quences

Unit impulse sequence (or unit impulse or unit sample)

[n] = 1, n = 00, n = 0

(also referred to asKronecker delta function)

n

[n ]

0

Unit step sequence (unit step)

u[n] = 1, n 00, n < 0

0 n

u[n ]



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Relation between [n] and u[n] First order difference

[n] = u[n] u[n 1] Running sum

u[n] =n

m= [m]

Sampling property of unit impulse

x[n] [n n0] = x[n0] [n n0]

1.2.4 The ContinuousTime Unit Impulse and Unit StepFunctions

Unit step function(unit step)

u(t) =1, t > 00, t < 0

u(t)

t0

1

Note: discontinuity att = 0

Unit impulse function(unit impulse, Dirac delta impulse)

(t) = ?, t = 00, t = 0



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Remark:

We use the short-hand notation:dx(t)

dt= x(t)

Relation between (t) and u(t) First order derivative

(t) = u(t) Running integral

u(t) =t

( ) d

Formal difficulty:u(t) is not differentiable in the conventional sensebecause of its discontinuity att = 0.

Some more thoughts on (t) Consider functionsu (t) and (t) instead of u(t) and (t):

u (t)

t

(t)1

t

1

where (t) = u (t)

u (t) =t

( ) d



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Limit 0u(t) = lim

0u (t)

(t) :

t

1 (t)

3 (t )

2 (t)

3 2 1

1

1

1 3

1 2

Observe: Area under (t) always 1

(t) is an innitesimally narrow impulse with area 1.

(t) = lim 0 (t)

( ) d = 1

Representation

a

t

a (t)

t 0

1

t

(t t 0 )

1

t

(t )



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PropertiesSampling property (x(t) continuous at t = t0)

x( ) ( t0) d = x(t0)x(t) (t t0) = x(t0) (t t0)

Linearity

(a ( ) + b ( ))x( ) d =

a ( )x( ) d +

b ( )x( ) d

= ( a + b)x(0)

a (t) + b (t) = (a + b) (t)

Time scaling (a IR)

(a )x( ) d =

1|a |

( )x(/a ) d =1

|a |x(0)

(at ) = 1|a | (t)



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Differentiation and derivative

( )x( ) d = (t)x(t)

( )x( ) d = x(0)

( )x( ) d = x(0)

t (t) = (t)

Remark:

More formal discussion of the unit impulse (t) in text books ongeneralized functions or distributions .



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1.3 ContinuousTime and DiscreteTime Systems

Unied representation of physical processes bysystems

System: Entity that transforms input signals into new output signals

One or more input and output signals Continuoustime system transforms continuoustime signals Discretetime system transforms discretetime signals

Formal representation of inputoutput relation Continuoustime system

x(t) y(t)

Discretetime system

x[n] y[n]

Remark: Another popular notation that you may nd in books isy(t) = S{x(t)}, whereS{} represents the system operator.

Pictorial representation of systems

Continuoustimesystemx (t) y(t)

systemDiscretetimex[n ] y[n ]



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1.3.1 Simple Examples of Systems

Quadratic system

y(t) = (x(t))2

System represented by a rst order differential equation

y(t) + ay(t) = bx(t)

with constants a and b

Delay system

y[n] = x[n 1]

System described by a rst order difference equation

y[n] = ay[n 1] + bx[n]

with constants a and b

1.3.2 Interconnections of Systems

Often convenient: break down a complex system into smaller subsystems

Series (cascade) interconnection

System 1 System 2Input Output

Examples: Communication channel and receiver, detector and de-coder in communications



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Parallel interconnection

System 1

System 2

OutputInput

Example: Diversity transmission:transmission of the same signalover two antennas and receiving it with one antenna

Feedback interconnection

OutputInput

System 2

System 1

Examples: Closed-loop frequency/phase/timing synchronization incommunications, human motion control

1.4 Basic System Properties

Simple mathematical formulation of basic (physical) system proper-ties

Classication of systemsFor conciseness: only denitions for continuous-time systemsReplacing (t) by [n] denitions for discrete-time systems



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1.4.1 Linearity

Let x1(t) y1(t) and x2(t) y2(t)

Linear system if 1. Additivity

x1(t) + x2(t) y1(t) + y2(t)

2. Homogeneity

ax 1(t) ay1(t) , a C

Linear systems possess property of superpositionLet xk(t) yk(t), then

K

k=1

akxk(t) = x(t) y(t) =K

k=1

akyk(t)

Not linear systems are referred to asnonlinear .

Example:

1. Systemy(t) = tx (t) is linear.To see this let

x1(t) y1(t) = tx 1(t)

x2(t) y2(t) = tx 2(t)and

x3(t) = ax 1(t) + bx2(t) ,



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and check

y3(t) = tx 3(t) = tax 1(t) + tbx2(t) = ay1(t) + by2(t) ,

i.e.,ax 1(t) + bx2(t) ay1(t) + by2(t)

2. Systemy[n] = (x[n])2 is nonlinear.To see this let

x1[n] y1[n] = (x1[n])2

x2[n] y2[n] = (x2[n])2

and check additivity for inputx3[n] = x1[n] + x2[n]y3[n] = (x3[n])2 = y1[n] + y2[n] + 2x1[n]x2[n]

= y1[n] + y2[n]

3. Systemy(t) = (x(t)) is ?

1.4.2 Time Invariance

Time invariant system if behavior and characteristics are time-invariant,i.e., identical response to same input signal no matterwhen inputsignal is applied

x(t t0) y(t t0)

Example:

1. The systemy(t) = (x(t))2 is ?2. The systemy[n] = nx [n] is ?



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Remark:

Linear and time-invariant (linear time-invariant (LTI)) systems playa prominent role for system modeling and analysis. The importance

of complex exponentials derives from the fact that they are eigen-functions of LTI systems.

1.4.3 Systems with and without Memory

Memoryless system if output signal depends only on present valueof input signal

Otherwise, a system is said to possess memory or to be dispersive.

Example:

Memoryless systems

1. Limiter:y[n] =x[n] , A x[n] A A , x[n] < A

A , x[n] > A2. Amplier:y(t) = A x(t)

Systems with memory

1. Accumulator:y[n] =n

k=

x[k] = x[n] + y[n 1]

2. Delay: y(t) = x(t t0)

3. Capacitor: y(t) =1C

t

x( ) d



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1.4.4 Invertibility and Inverse Systems

Invertible system if bijective transformationx(t) y(t) frominput to output

In this case aninverse systemy(t) w(t) = x(t) exists

InversesystemSystem

y(t)w(t) = x(t )x (t)

Example:

Invertible systems1. Amplier:y(t) = Ax(t), A = 0

Inverse system:w(t) = 1Ay(t) (=Amplier)

2. Accumulator:y[n] = y[n 1] + x[n]Inverse system:w[n] = y[n] y[n 1] (=Differentiator)

Noninvertible systems

1. Limiter:y[n] =x[n] , A x[n] A A , x[n] < AA , x[n] > A

2. Slicer:y[n] = 1 , x[n] 0 1 , else



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1.4.5 Causality

Causal system if output at any time depends only on past andpresent values of the input

If x1(t) = x2(t) for t t0 then y1(t) = y2(t) for t t0 , t0

Implication:

Causal+Linear:If x1(t) = 0 for t t0 then y1(t) = 0 for t t0, t0

Example:

Causal system

Accumulator:y[n] =n

k=

x[k] = x[n] + y[n 1]

Noncausal system

Averager: y[n] = 12N + 1

N

k= N x[n k]

All memoryless systems are causal

Causal systems in real-time processing



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1.4.6 Stability

Considerboundedinput boundedoutput (BIBO)stability

Stable system if for any bounded input signal|x(t)| Bx < , t

the output signal is bounded

|y(t)| By < , t

Example:

Stable system

Averager: y[n] =1

2N + 1

N

k= N

x[n k]

Bounded input|x[n]| < B x bounded output |y[n]| < B y =Bx

Instable system

Integrator: y(t) =t

x( ) d

E.g. bounded inputx(t) = u(t) unbounded outputy(t) = t

System stability is important in engineering applications, unstablesystems need to be stabilized.



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Example: The rst Tacoma Narrows suspension bridge collapsed dueto wind-induced vibrations, November 1940.

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k / r e s e a r c h

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Signal and System Ch#1