discrete time signal

8/9/2019 Discrete Time Signal

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Discrete Time Signal

Time-Domain Representation



Discrete Time Signal: Time-Domain

Representation• Signals represented as sequences of

numbers, called samples

• Sample value of a typical signal or

integer in the range

• x [n] defined only for integer values of n

and undefined for non-integer values of n

• Discrete-time signal represented by { x [n]}




Representation• Discrete-time signal may also be written

as

{ } ,...9.2,7.3,2.0,1.1,2.2,2.0...,][ −−=↑

n x

a sequence o num ers ns e races:

• In the above,

etc.• The arrow is placed under the sample at

time index n = 0

,1.1]1[,2.2]0[,2.0]1[ ==−=− x x x




Representation• Some alternative representation of

discrete time signal

– Graphical representation – unc ona represen a on

– Tabular representation

– Sequence representation



Graphical representation

• Graphical representation of a discrete-time

signal with real-valued samples:



Functional representation

=n 3,1,.........1

==

elsewhere,..0

.,.........



Tabular representation

n … -2 -1 0 1 2 3 4 5 …

x n … 0 0 0 1 4 1 0 0 …



Sequence representation

=

↑

,...0,1,4,1,0,0,0...,)(n x (Infinite form)

(Finite form)

=

↑

0,1,4,1,0)(n x



Some Elementary Discrete-Time

Signals• Unit sample sequence (Unit Impulse) is

denoted as δ(n) and is defined as

• It is a signal that is zero everywhere,

except at n = 0 where its value is unity

≠≡

0,0

,)(

nnδ




Signals• Unit Step Signal is denoted as u(n) and is

defined as

<

≥≡

0,0

0,1)(

n

nnu




Signals• Unit Ramp Signal is denoted as ur (n) and

is defined as

<

≥≡

0,0

0,)(

n

nnnur




Signals• Exponential Signal is a sequence of the

form x(n) = an for all n

• If the parameter a is real, then x(n) is areal si nal.

• If the parameter a is complex valued, itcan be expressed as a = re jθ, where r andθ

are now the parameters. Hence, x(n),can be express as x(n) = r ne jθn = r n(cosθn+ jsinθn)




Signals• Exponential Signal



Classification of Discrete Time

Signals• Energy and Power Signals

• Periodic and Aperiodic Signals

• Symmetric (even) and Antisymmetric (odd)gna s



Energy and Power Signals

• Energy is defined as ,if E

is finite, i.e., o < E < ∞ , then x(n) is called

∑+∞

∞−

= 2

)(n x E

nergy gna . owever, many s gna s

that have an infinite energy, have a finite

average power. Average power is defined

as

∑−

∞→ +=

N

N N

ave n x N

P 2

)(12

1lim




• If the signal energy of x(n) is define over

the interval (-N, N ) as

=

N

n x E 2

)(

• then , (subscript N to E

emphasize that EN

is the energy of the

signal x(n))

• and therefore,

− N

N N

E E ∞→

= lim

N N

ave E N

P 12

1lim

+=

∞→




• Note: if E is finite, then Pave = 0 and on

the other hand, if E is infinite, the average

power P may be either finite or infinite. If P

,

a power signal.




Example: Consider the following finite

discrete signals

1. x(n) = –1δ(n – 0) + 2δ(n – 1) – 2δ(n – 2) .

Find the energy and power in both signals.

,, −=↑

n x




Example: Consider the following discrete

periodic and nonperiodic signals. For the

periodic signal find the period N and also

.

signals, find the total energy.




(a) (b) (c)



Periodic and Aperiodic Signals

• A signal is periodic with period N (N>0 ), if andonly if x (n + N ) = x (n) for all n

• The smallest value of N for which x(n+N ) holds

is called the (fundamental) period. If there is no,

called nonperiodic or aperiodic.

• Energy of periodic signals is infinite but it might

be finite over a period. On the other hand, theaverage power at the periodic signal is finite andis equal to the P ave over a single period. Hence,periodic signals are power signals.



Symmetric (even) and

Antisymmetric (odd) Signals• A real-valued signal x(n) is called

symmetric (even) if

x (n) = x (− n)




Antisymmetric (odd) Signals• On the other hand, a signal x(n) is called

antisymmetric (odd) if

x (n) = − x (− n)




Antisymmetric (odd) Signals• Any signal can be written as:

x(n) = xe(n)+ xo(n)

−= 1

[ ]

−−= )()(

2

1)(

2

n xn xn x

Where

o

e



Block Diagram Representation of

Discrete-Time Systems• An Adder – performs the addition

operation (memoryless) of two signal

sequences to form another (the sum)

, .




Discrete-Time Systems• Constant multiplier – applying a scale

factor on the input x(n), this operation is

memoryless.




Discrete-Time Systems• Signal multiplier – multiplication of two

signal sequences to form another (the

product) sequence, denoted as y(n). This

.




Discrete-Time Systems• Unit delay element – is a special system

that simply delays the signal passing

through it by one sample.




Discrete-Time Systems• Unit advance element – is a special

system that simply moves the input signal

passing through it ahead by one sample.



Classification of Discrete-Time

Systems• Static versus Dynamic Systems

• Time invariant versus Time-variant

Systems • near versus non near

• Causal versus anticausal

• Stable versus Unstable Systems



Static versus Dynamic Systems

• Static Systems ≡ memory less ≡ the

output doesn’t depend on past or future

values of the input.

either infinite or finite memory.

Example:2)()(2)( n xn xn y +=

∑=

−= N

k

k n xn y

0

)()(

∑∞

=

−=0

)()(k

k n xn y



Time invariant versus Time-variant

Systems• Time-invariant – its input-output

characteristics do not change with time.

• Time-variant – its input-outputcharacteristics chan e with res ect totime.

• A relaxed system Γ is time-invariant if

x (n)→ y (n) x (n − k )→y (n − k )

Example:)1()()( −−= n xn xn y )()( nnxn y =



Causal versus anticausal

• Causal - if the output at any time depends only

on present and past values of the inputs and not

on future values of the input.

• Anti-Causal - if the output at any time dependsonly on future values of the input and not on the

present and past values of the inputs.

Example

y(n) = x(-n)



Linear versus nonlinear

• Linear system – is one that satisfies thesuperposition principle (additive andhomogenous).

• Nonlinear system – does not satisfy the.

• Superposition Principle – The response of thesystem to a weighted sum of signals be equal tothe corresponding weighted sum of theresponses (outputs) of the system to each of theindividual input signals.



Linear versus nonlinear

• A relaxed Γ system is linear if and only if

Γ[a1x1(n) + a2x2(n)] = a1Γ[x1(n)] + a2Γ[x2(n)]

• For any arbitrary input sequences x1(n)an x2 n , an any ar rary cons an s a1

and a2

Example:

y(n) = nx(n) y(n) = x2(n)



Stable versus Unstable Systems

• A system is Stable if any bounded input

produces bounded output (BIBO).

Otherwise, it is unstable.

Example:

Consider the nonlinear system described

by the input-output equation

y(n) = y2(n-1) + x(n) where: x(n) = Cδ(n)



Impulse and Step Responses

• The response of a discrete-time system toa unit sample sequence {δ[n]} is called theunit sample response or simply, the

impulse response, and is denoted by{h[n]}

• The response of a discrete-time system toa unit step sequence {µ[n]} is called the

unit step response or simply, the stepresponse, and is denoted by {s[n]}



Impulse Response

• Example - The impulse response of the

system

y[n] =α1

x[n]+α2

x[n −1]+α3

x[n − 2]+α4

x[n − 3]

h[n] =α1δ[n]+α2δ[n −1]+α3δ[n − 2]+α4δ[n − 3]

The impulse response is thus a finite-length

sequence of length 4 given by

{ h[n]} = {α1, α2, α3, α4}



Time-Domain Characterization

of LTI Discrete-Time System• Input-Output Relationship -

A consequence of the linear, timeinvariance property is that an LTI discrete

time system is completely characterizedby its impulse response.

Note: Knowing the impulse response onecan compute the output of the system forany arbitrary input




of LTI Discrete-Time System• Let h[n] denote the impulse response of a

LTI discrete-time system

• Compute its output y [n] for the input:

= − − −. . 0.75δ[n − 5]

• As the system is linear, compute its

outputs for each member of the inputseparately and add the individual outputsto determine y [n].




of LTI Discrete-Time System• Since the system is time-invariant

input output

δ[n + 2]→h[n + 2] δ[n − 1]→h[n −1]

δ[n − 2]→h[n − 2]

δ[n − 5]→h[n − 5]




of LTI Discrete-Time System• Likewise, as the system is linear

input output

0.5δ[n + 2]→0.5h[n + 2]

1.5δ[n −1]→1.5h[n −1] − δ[n − 2]→−h[n − 2]

0.75δ[n − 5]→0.75h[n − 5]

• Hence because of the linearity property:

y [n] = 0.5h[n + 2]+1.5h[n −1] − h[n − 2]+ 0.75h[n− 5]




of LTI Discrete-Time System• Any arbitrary input sequence x [n] can be

expressed as a linear combination of delayed

and advanced unit sample sequences in the form

∞

• The response of the LTI system to an input x [k ]δ[n − k ] will be x [k ]h[n − k ]

∑−∞=

−=k

k nk xn x ][][][ δ




of LTI Discrete-Time System• Hence, the response y [n] to an input

∑∞

−∞=

−=k

k nk xn x ][][][ δ

which can be alternately written as

∑∞

−∞=

−=k

k nhk xn y ][][][

∑∞

−∞=

−=k

k hk n xn y ][][][




of LTI Discrete-Time System• The summation

∑∑ ∞

−∞=

∞

−∞=

−=−=k k

k hk n xk nhk xn y ][][][][][

x [n] and h[n] and represented compactly as

][][][ nhn xn y ⊗=



Convolution Sum

• Properties :

• Commutative property:

•

][][][][ n xnhnhn x ⊗=⊗

• Distributive property:

])[][(][][])[][( n ynhn xn ynhn x ⊗⊗=⊗⊗

])[][][][])[])[(][( n yn xnhn xn ynhn x ⊗+⊗=+⊗



Convolution Sum

1) Time-reverse h[k ] about k = 0 to form h[-k ]

2) Shift to the right by n sampling periods if n

> 0 or shift to the left by n sampling-

3) Form the product v[k] = x[k]h[n-k ]

4) Sum all samples of v [k ] to develop the n-

th sample of y [n] of the convolution sum



Convolution Sum

The impulse response of a linear time-invariant

system is

}1,1,2,1{)( −=

↑nhDetermine the response of the system to the

input signal

Answer:

}1,3,2,1{)( ↑

=n x

...}0,0,1,2,3,8,8,4,1,0,0{....)( −−=↑

n y



Structure for the Realization of

Linear Time-Invariant Systems• Consider the first order system

y(n) = -a1y(n-1) + b0x(n) + b1x(n-1)

which is realized as




Linear Time-Invariant Systems• This realization uses separate delays

(memory) for both the input and output

signal samples and it is called a direct

.

Note: this system can be viewed as two

linear time-invariant systems in cascade.




Linear Time-Invariant Systems• The first is a nonrecursive, system

described by the equation

1−+= n xbn xbnvwhereas the second is a recursive system

described by the equation

)()1()( 1 nvn yan y +−−=




Linear Time-Invariant Systems• These structure can be generalize by

∑ ∑= =

−+−−= N

k

M

k

k k k n xbk n yan y1 0

)()()(

where:

∑=−=

M

k

k k n xbnv0

)()(

∑=

+−−= N

k

k nvk n yan y1

)()()(

(Nonrecursive System)

(Recursive System)




Linear Time-Invariant Systems• Interchanging the order of the cascaded

linear time-invariant systems, the overallsystem response remains the same.

• The first s stem described b the e uation

whereas the second system is describedby the equation

)1()()( 10 −+= n xbn xbn y

)()1()( 1 n xn yanw +−−=




Linear Time-Invariant Systems




Linear Time-Invariant Systems• Combining the two common delay of the

two system, this new realization requires

only one delay for the auxiliary quantity

,

terms of memory requirements. It is called

direct form II structure and it is used

extensively in practical applications




Linear Time-Invariant SystemsExample:

Draw the equivalent direct form I and

direct form II of the given difference

1.

2.

)(866.0)2(64.0)1(8.0)( n xn yn yn y +−−−=

)1()()2(4)1(4)( −−=−+−− n xn xn yn yn y




Linear Time-Invariant SystemsExample:

Write the equivalent difference equation

based from the given direct form II.a.




Linear Time-Invariant Systemsb.




Characterized by Constant Coefficient

Difference Equations• Consider the first order difference equation

∑ ∑= =

−+−−= N

k

M

k

k k k n xbk n yan y1 0

)()()(

or equivalently to,

1)()( 0

1 0

≡−=−∑ ∑= =

ak n xbk n ya N

k

M

k

k k





Difference Equations• The integer N is called the order of the

difference equation or the order of thesystem.

• The general solution of the N-order

considered by the following:

– Linearity

– Time invariance – Stability

• A recursive system may be relaxed ornonrelaxed, depending on the initial

conditions.





Difference Equations• A system is linear if it satisfies the

following three requirements:

1. the total response is equal to the sum of

the zero-in ut and zero-state res onses[i.e. y(n) =yzi(n) +yzs(n)}.

2. The principle of superposition applies tothe zero-state response (zero-state linear),yzi(n) = 0, (relaxed system)

3. The principle of superposition applies tothe zero-input response (zero-input linear).





Difference Equations• Since

1)()( 0

1 0

≡−=−∑ ∑= =

ak n xbk n ya N

k

M

k

k k

• The system described is time invariantsince the coefficients ak and bk are

constants and independent on time.• Therefore the recursive system described

by a linear constant coefficient differenceequation is linear and time invariant.



Solution of Linear Constant

Coefficient Difference Equations• Direct method

• Indirect method (z-transform)



Solution of Linear Constant

Coefficient Difference Equations

• The direct solution method assumes that

the total solution is the sum of two parts:

y(n) = yh(n) + y

p(n)

w ere:

yh(n) = homogeneous or complementary

solution

yp(n) = particular solution



Homogenous Solution of a

Difference equation

• Given

1)()( 0

1 0

≡−=−∑ ∑= =

ak n xbk n ya N

k

M

k

k k

assume x(n) = 0

homogeneous solution

0)(1

∑=

=− N

k

k k n ya

∑=

−= N

k

k h k n yan y

1

)()(




Difference equation

( ) 0...

0...

2

2

1

1

2

2

1

1

=++++

=++++

−−−

−−−

N

N N N N n

N n

N

nnn

aaa

aaa

λ λ λ λ

λ λ λ λ

• In general, it has N roots which denote asλ1, λ2, … λN. The roots can be real orcomplex valued. In practice thecoefficients a1, a2, …, aN are usually real.Complex valued roots occur as complexconjugate pairs. Some of the N roots maybe identical, in which case multiple roots




Difference equation

• Roots are real and distinct

n

N N

nn

h C C C n y λ λ λ +++= ...)( 2211

1, 2,…, N coefficients.

• Since the input, x(n) = 0, the

homogeneous solution can be used toobtain the zero-input response of thesystem.




Difference equation

Example:

Determine the homogeneous solution of

the system described by the first-order

)()1()( 1 n xn yan y =−+




Difference equation

Example:

Determine the zero-input response of the

system described by the homogeneous

-

0)2(4)1(3)( =−−−− n yn yn y



Particular Solution of the Difference

Equation

• The particular solution, yp(n), is a form that

depends on the form of the input x(n), that

required to satisfy the difference equation

1)()( 0

1 0

≡−=−∑ ∑= =

ak n xbk n yak k

k k




EquationTABLE 2.1 General form of the Particular Solution forseveral types of input signals

Input Signal,x(n) Particular Solution, yp(n)

A constant K

AMn KMn

AnM K0nM + K1n

M-1 + … +KM

AnnM An(K0nM + K1n

M-1 + … +KM)

Acosω0n K1cosω0n + K2sinω0n Asinω0n K1cosω0n + K2sinω0n




Equation

Example:

Determine the particular solution of the

first order difference equation

y n + a1y n- = x n a1 <

when the input x(n) = u(n)




Equation

Example:

Determine the particular solution of the

difference equation

y n = y n- – y n- + x n

when the forcing function x(n) = 2n, n ≥ 0

and zero elsewhere



Total Solution of the difference

equation

• The linearity property of the linear

constant-coefficient difference equation

allow to add the homogeneous solution

obtain the total solution.

y(n) = yh(n) + yp(n) or y(n) = yzi(n) + yzs(n)

where:yzi(n) = zero input response

yzs(n) = zero state response or force

res onse




equation

• The resultant sum y(n) contains the

constant parameters {Ci} embodied in the

homogeneous solution component, yh(n).

satisfy the initial conditions.




equation

Example:

Determine the response, y(n), n ≥ 0, of the

system described by the second-order

when the input sequence is

x(n) = 4nu(n)

)1(2)()2(4)1(3)( −+=−−−− n xn xn yn yn y




equation

Example:

Determine the response, y(n), n ≥ 0, of thesystem described by the second-order

difference e uation


x(n) = 2n

u(n)and the initial conditions are y(0) =-1 andy(1) = 1

)1()()2(9)1(6)( −−=−+−− n xn xn yn yn y



Impulse response of a Linear Time-

Invariant Recursive System

• The impulse response of a linear time-

invariant system was defined as the

response of the system to a unit sample

=. ., .

• In the case of a recursive system, h(n) is

simply equal to the zero-state response of

the system when the input x(n) = δ(n) andthe system is initially relaxed.





• Consider a linear time-invariant recursivesystem, the zero-state response isexpressed in terms of the convolution

summation n

• When the input is an impulse, x(n) = δ(n),it reduces to

and

0

−=k

zs

)()( nhn y zs = 0)( =n y p





Example:

Determine the impulse response, h(n), for

the system described by the second-order

)1(2)()2(4)1(3)( −+=−−−− n xn xn yn yn y



Crosscorrelation and

Autocorrelation Sequences

• Consider a two real signal sequences x(n)

and y(n) each of which has finite energy.

The crosscorrelation of x(n) and y(n) is a

xy ,

or, equivalently as

∑∞

−∞=

±±=−=n

xy l l n yn xl r ,...2,1,0)()()(

∑∞

−∞=

±±=+=n

xy l n yl n xl r ,...2,1,0)()()(





where:

l – is the time shift parameter

xy(subscripts) – indicate the sequencese ng corre a e . e or er o e

subscripts, with x preceding y, indicates

the direction in which one sequence is

shifted, relative to the other.





• In some special case where y(n) = x(n),

the autocorrelation of x(n), is defined as

the sequence

or, equivalently as

∑−∞=

±±=−=n

xx l l n xn xl r ,...2,1,0)()()(

∑∞

−∞=

±±=+=n

xx l n xl n xl r ,...2,1,0)()()(

C





Example:

Determine the crosscorrelation sequence

of the given sequences

Ans.

,...0,0,3,2,1,7,3,1,2,0,0....)( −−=↑

n x

,...0,0,5,2,1,4,2,2,1,1,0,0....)( −−−=↑

n y

3,5,7,16,18,13,7,0,33,14,36,19,9,10)( −−−−−=↑

l r xy

C l i d





Example:

Determine the autocorrelation of the given

sequence

1,3,3,1↑

=n x



Seatwork

Draw the equivalent direct form I and

direct form II of the given difference

equation

.

b.

−−−−

)4(3)2(2)1()()( −−−+−−= n xn xn xn xn y



Seatwork

Example:

Write the equivalent difference equation

based from the given direct form II.a.



Seatwork

b.



Seatwork



difference equation


x(n) = (-1)nu(n)

−−=−−−



Seatwork



difference equation


x(n) = (-1)nu(n)

and the initial conditions are y(0) =1 and

y(1) = 2

−−=−−−



Seatwork

Determine the impulse response, h(n), for

the system described by the second-order

difference equation

)1()()2(4)1(4)( −−=−+−− n xn xn yn yn y



Seatwork

Determine the crosscorrelation sequence

of the given sequences

= ,,,↑

1,2,1,1)(↑

=n y



Seatwork

Determine the autocorrelation of the given

sequence1,4,6,4,1)(

↑

=n x

discrete time signal

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