Topic 2: LTI Systems and Convolution Response of LTI Systems - - PowerPoint PPT Presentation

Response of LTI Systems Impulse response and unit response Characterization of LTI systems Properties using the system response Convolution Integral Convolution Integral Properties Convolution Sum Convolution Sum Properties Systems and Difference/Differential Equations Block Diagrams of Systems Summary

ELEC361: Signals And Systems

Topic 2: LTI Systems and Convolution

• Dr. Aishy Amer

Concordia University Electrical and Computer Engineering

Figures and examples in these course slides are taken from the following sources:

• A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997
• M.J. Roberts, Signals and Systems, McGraw Hill, 2004
• J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003

2

Response of LTI Systems

Linearity Time Invariance

] [ ] [ ] [ ] [ ] [ ] [ then ] [ ] [ if n n y n n x n y n x n y a n y n x a n x

k k k k k k

− → − ⇒ → = =

∑ ∑

3

Response of LTI Systems

) ( ) ( ) ( ) ( signals indivual those

• f

each to system the

• f

responses the

• f

sum weighted The ) ( ), ( ), ( signals several

• f

sum Weighted ) ( If

2 1 2 1 2 1

t by t ay t bx t ax t y t x t x t x + ⇒ + = ⇒ = L

Linear Combination

1

If we represent x(t) to LTI system as a linear combination

• f a set of BASIC signals

we can use the superposition to compute y(t) as the response to those basic signals

BASIC signals:

L ], sin[ ], cos[ , ], [ ], [ n n e n n u

jn

δ

4

Response of LTI Systems

Once the response to a unit impulse is known, the

response of any discrete-time LTI system to any arbitrary excitation can be found

Any arbitrary excitation is a sequence of amplitude-

scaled and time-shifted DT impulses

Therefore the response is a sequence of amplitude-

scaled and time-shifted DT impulse responses

This we call convolution

• The impulse response is conventionally designated by

the symbol, h[n] or h(t)

∑

∞ ∞ −

− = + − + + + − + = ] [ ] [ ] 1 [ ] 1 [ ] [ ] [ ] 1 [ ] 1 [ ] [ k n k x n x n x n x n x δ δ δ δ L L

5

Response of LTI Systems: Example

6

Response of LTI Systems: Example

7

Response of LTI Systems

] [ ]} [ { n y n x S =

S n h n h n S n x S n y n n x

• f

response Impulse : ] [ ] [ ]} [ { ]} [ { ] [ ] [ ] [ If = = = ⇒ = δ δ

1 2

Question: if h[n] known, how to find y[n]?

8

Response of LTI System: Identity System

4 4 4 3 4 4 4 2 1

n k at x[k] the ONLY Preserves Property Shifting

] [ ] [ ] [ ] [ ] [ ] [ ] [ impulse unit the is response impulse Unit & input the to equal Output System Identity then 1 K if ] [ ] [ constant a ] [ with ] [ ] [ if

=

= − = ∗ =

• ⇒

= = = = =

∑

n x k n k x n y n n x n y x[n] y[n] n Kx n y h K n K n h

k

δ δ δ

9

Response of LTI System: Example

] 1 [ ] [ ] 1 [ ] 1 [ ] [ ] [ ] [ ] [ ] [ − + = − + − = − =∑ n x n x n x h n x h k n x k h n y

If the system LTI Can be the response of a non- linear system but it is not possible to determine y[n] used

• n h[n]

10

Response of LTI System: Example

11

Response of LTI System: Example

4 4 4 3 4 4 4 2 1

n k (shifts) at x[n]

• nly the

Preserves

] [ ] [ ] [ ] [ ] [ ] [

=

∑

= − = ∗ =

k

n x k n k x n n x n y δ δ

12

Outline

• Response of LTI systems
• Impulse response and unit response
• Characterization of LTI systems
• Properties using the system response
• Convolution Integral
• Convolution Integral Properties
• Convolution Sum
• Convolution Sum Properties
• Systems and Difference/Differential Equations
• Block Diagrams of Systems
• Summary

13

Relationship: DT impulse response and step response

In any discrete-time LTI system let an excitation, x[n],

produce the response, y[n]

Then the excitation x[n] - x[n - 1] will produce the response

y[n] - y[n - 1]

It follows then that the unit impulse response is the first

backward difference of the unit step response and, conversely that the unit sequence (step) response is the accumulation of the unit impulse response

∑

−∞ =

= − − =

n k

k h n s n s n s n s n h ] [ ] [ response step the is ] [ where ] 1 [ ] [ ] [

14

DT impulse response and step response:Example

Suppose that the step response is given by What is the impulse response h[n] ?

] [ 5 4 4 5 ] [ n u n y

n

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − =

] 1 [ 5 4 4 ] [ 5 4 4 ] [ 5 ] 1 [ 5 4 4 5 ] [ 5 4 4 5 ] 1 [ ] [ ] [

] [ 1

− ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − − =

−

n u n n n u n u n y n y n h

n n n n n

4 4 4 8 4 4 4 7 6

δ

δ δ

15

Relationship: CT impulse response and step response

In any continuous-time LTI system let an excitation,

x(t), produce the response, y(t). Then the excitation will produce the response

It follows then that the unit impulse response is the

first derivative of the unit step response and, conversely that the unit step response is the integral

• f the unit impulse response

)) ( ( t x dt d )) ( ( t y dt d

16

Outline

• Response of LTI systems
• Impulse response and unit response
• Characterization of LTI systems
• Properties using the system response
• Convolution Integral
• Convolution Integral Properties
• Convolution Sum
• Convolution Sum Properties
• Systems and Difference/Differential Equations
• Block Diagrams of Systems
• Summary

17

Characterization of LTI System

The impulse response h[n] completely

characterizes an LTI system DT LTI Systems:

Use the unit impulse to construct any signal A DT signal is a sequence of individual weighted impulses The response of the system is the sum of delayed h[n]

18

Characterization of LTI System

DT LTI systems are described

mathematically by difference equations

CT LTI systems are described

mathematically by differential equations

19

Characterization of LTI System

• For any excitation, x[n] or x(t)
• the response, y[n] or y(t) can be found by

1.

finding the response to x[n] or x(t) as the only forcing function on the right-hand side and

2.

then adding scaled and time-shifted versions of that response to form y[n] or y(t)

• If x[n] or x(t) is a unit impulse,
• the response to it as the only forcing function is

simply the homogeneous solution of the difference

• r differential equation with initial conditions applied

20

Outline

• Response of LTI systems
• Impulse response and unit response
• Characterization of LTI systems
• Properties using the system response
• Convolution Integral
• Convolution Integral Properties
• Convolution Sum
• Convolution Sum Properties
• Systems and Difference/Differential Equations
• Block Diagrams of Systems
• Summary

21

Properties using the system response:

system Stability

It can be shown that a BIBO-stable DT

system has an impulse response that is absolutely summable

Proof

∞ <

∑

∞ −∞ = n

n h ] [

∑ ∑ ∑

∞ −∞ = ∞ −∞ = ∞ −∞ =

≤ − ≤ − =

n k k

n h B k h k n x k h k n x n y ] [ ] [ ] [ ] [ ] [ ] [

22

Properties using the system response:

system Stability

A CT system is BIBO stable if its impulse

response is absolutely integrable

∞ <

∫

∞ ∞ −

) (t h

23

Properties using the system response:

Invertible Systems

systems inverse are ] 1 [ ] [ ] [ : difference backward The ] [ ] [ : r accumulato The − − = = ∑

−∞ =

n y n y n w k x n y

n k

1 1

24

Properties using the system response:

Differentiators & Integrators

Differentiators are: Difficult to implement Sensitive to noise and errors Alternatives : Integrators Integrators : amplifiers

finite) (to ) ( 2 ) ( ) ( ) ( ) ( 2 ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( assume ) ( 2 ) ( ) (

∫ ∫ ∫ ∫

∞ − ∞ − ∞ − ∞ −

− + = − = − = ⇒ = −∞ − =

t t t t

d y x t y t y d y x t y d y x t y t y dt dt t dy y t y t x dt t dy τ τ τ τ τ τ τ τ τ

25

26

Outline

• Response of LTI systems
• Impulse response and unit response
• Characterization of LTI systems
• Properties using the system response
• Convolution Integral
• Convolution Integral Properties
• Convolution Sum
• Convolution Sum Properties
• Systems and Difference/Differential Equations
• Block Diagrams of Systems
• Summary

27

Convolution of Two Signals

A signal x[n] can be represented as linear

combination of DELAYED Impulses

If the system is LINEAR

∑ ∑ ∑

= ⇒ − = = − = ⇒ − =

∞ −∞ = k k k k k k k

n h k x n y k n y k h k n n y n y k x n y k n k x n x ] [ ] [ ] [ ] [ to response [n] ] [ ] [ to system the

• f

response ] [ with ] [ ] [ ] [ ] [ ] [ ] [ δ δ δ

28

Convolution of Two Signals

If the system is Time Invariant

n Convolutio ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ with Omit the ]] [ ] [ ] [ OR [ ] [ ] [ then ] [ to response ] [ if n h n x n y k n h k x n y n h k x n y n h n h n h k n h n h k n n h

k k k k k

∗ = − = ⇒ = = − = − = −

∑ ∑

δ

29

The Convolution Integral

)} ( { ) ( )}. ( { ) ( )} ( { ) ( m switch the can We

• peartor)

linear a also is (

• perator

linear a is } ). ( ) ( { )} ( { ) ( ). ( ) ( ) ( t S t h d t S x t x S t y S d t x S t x S t y d t x t x δ τ τ δ τ τ τ δ τ τ τ δ τ = − = = ⇒ − = = − =

∫ ∫ ∫ ∫

∞ ∞ − ∞ ∞ − ∞ ∞ −

30

The Convolution Integral

t t h d h t x d t h x t y t S t h S parameter with

• f

Function : ) ( ). ( ) ( ). ( ) ( ) ( )} ( { ) ( invariant

• time

a is τ τ τ τ τ τ τ τ τ δ τ − − = − = − = − ⇒

∫ ∫

∞ ∞ − ∞ ∞ −

31

The Convolution Integral

If a continuous-time LTI system is excited by an

arbitrary excitation, the response could be found approximately by approximating the excitation as a sequence of contiguous rectangular pulses.

32

The Convolution Integral

Approximating the excitation as a pulse train can be

expressed mathematically by

The excitation can be written in terms of pulses of

width, and unit area

33

The Convolution Integral

Let the response to an un-shifted pulse of unit area and

width, be the “unit pulse response”

Then, invoking linearity, the response to the overall

excitation is a sum of shifted and scaled unit pulse responses of the form

As approaches zero, the unit pulses become unit

impulses, the unit pulse response becomes the unit impulse response, h(t), and the excitation and response become exact

) (t hp

∑

∞ −∞ =

− ≅

n p p p p

nT t h nT x T t y ) ( ) ( ) (

34

35

The Convolution Integral

As approaches zero, the expressions for the

approximate excitation and response approach the limiting exact forms

Notice the similarity of the forms of the convolution

integral for CT systems and the convolution sum for DT systems

36

Convolution Integral: Graphical Illustration

The convolution integral is defined by For illustration purposes let the excitation, x(t),

and the impulse response, h(t), be the two functions below

37

Convolution Integral: Graphical Illustration

In the convolution integral there is a

factor

We can begin to visualize this quantity

in the graphs below

38

Convolution Integral: Graphical Illustration

The functional transformation in going

from to is

39

Convolution Integral: Graphical Illustration

The convolution value is the area under the

product of x(t) and

This area depends on what t is. First, as an

example, let t = 5

For this choice of t the area under the

product is zero

y(5) then ) ( ) ( ) ( if = ∗ = t h t x t y

40

Convolution Integral: Graphical Illustration

Now let t=0 Therefore y(0) = 2, the area under the

product

41

Convolution Integral: Graphical Illustration

The process of convolving to find y(t)

is illustrated below

42

43

44

Properties of Convolution

Integral

The properties help to solve convolution of

complex signals in term of operations (e.g., associative) on another signal for which the convolution is known

Example:

tion multiplica and delay t u a t y

t

→ − = ) 5 ( ) (

45

Convolution Integral Properties

46

Convolution Integral Properties

47

Convolution Integral Properties

48

Convolution Integral Properties: Example

49

Convolution Integral: Response to a Complex Exponential

• Let a continuous-time LTI system be excited by a complex

exponential of the form,

• The response is the convolution of the excitation with the impulse

response or

• The quantity

will later be designated the Laplace transform of the impulse response and will be an important transform method for CT system analysis

50

CT System Interconnections

The system-interconnection properties for CT

systems are exactly the same as for DT systems.

51

Outline

• Response of LTI systems
• Impulse response and unit response
• Characterization of LTI systems
• Properties using the system response
• Convolution Integral
• Convolution Integral Properties
• Convolution Sum
• Convolution Sum Properties
• Systems and Difference/Differential Equations
• Block Diagrams of Systems
• Summary

52

The Convolution Sum

The response, y[n], to an arbitrary excitation,

x[n], is of the form h[n] is the impulse response

This can be written in a more compact form,

called the convolution sum

L L + − + + + − + = ) 1 ( ) 1 ( ) ( ) ( ) 1 ( ) 1 ( ) ( n h x n h x n h x n y

∑

∞ −∞ =

− =

k

k n h k x n y ] [ ] [ ] [

53

Proof

54

Convolution Sum: Example 1

55

Convolution Sum: Example 1

56

Convolution Sum: Example 1

57

Convolution Sum: Example 1

58

Convolution Sum: Example 2

• Consider an LTI system with input x[n] and unit impulse h[n] response
• shown. Find the output of this system
• Solution: the output of the system

59

Convolution Sum: Analytical Example 1

60

Convolution Sum: Analytical Example 2

61

Convolution Sum: Analytical Example 2

62

Convolution Sum: Analytical Example 2

63

Convolution Sum: Analytical Example 2

64

Convolution Sum: Analytical Example 2

65

Convolution Sum Properties

1 The following properties can be proven

from the definition:

2 3

] [ ] [ ] [ n n Ax n n A n x − = − ∗ δ

] [ ] [ ] [ n h n x n y ∗ = ]) 1 [ ] [ ( ] [ ] 1 [ ] [ ] [ ] [ ] [ ] [ ] [ − − ∗ = − − ∗ − = − ∗ = − n h n h n x n y n y n h n n x n n h n x n n y

66

67

Convolution Sum Properties: Example

)) 5 ( exp( ) 5 ( ) 5 ( ) ( : have we property, n convolutio By the ANSWER ). 5 ( ) ( Compute ) exp( ) ( Let

5

− = − = − ∗ − ∗ =

−

n a n x n n x n n x n a n x

n n

π δ δ π

68

Convolution Sum Properties

69

Convolution Sum Properties

70

Convolution Sum Properties

71

Convolution Sum: Response to Complex Exponential

Let a discrete-time LTI system be excited by

a complex exponential of the form,

The response is the convolution of the

excitation with the impulse response or

which can be written as

n

z n x = ] [

4 3 4 2 1

constant complex

] [ ] [

∑

∞ −∞ = −

=

k k n

z k h z n y

∑ ∑

∞ −∞ = − ∞ −∞ =

= − =

k k n k k

k h z k n h z n y ] [ ] [ ] [

72

Convolution Sum: Response to Complex Exponential

The response of a discrete-time LTI system to a

complex exponential excitation is another complex exponential of the same functional form but multiplied by a complex constant

That complex constant is Later this will be called the z transform of the

impulse response and will be one of the important transform methods

∑

∞ −∞ = − n n

z n h ] [

73

DT System Interconnections

If the response of one system is the excitation of another

system the two systems are said to be cascade connected

The cascade connection of two systems can be viewed as a

single system whose impulse response is the convolution of the two individual system impulse responses

This is a direct consequence of the associativity property of

convolution

74

DT System Interconnections

If two systems are excited by the same signal and their

responses are added they are said to be parallel connected.

The parallel connection of two systems can be viewed as a

single system whose impulse response is the sum of the two individual system impulse responses. This is a direct consequence of the distributivity property of convolution.

75

Outline

• Response of LTI systems
• Impulse response and unit response
• Characterization of LTI systems
• Properties using the system response
• Convolution Integral
• Convolution Integral Properties
• Convolution Sum
• Convolution Sum Properties
• Systems and Difference and Differential Equations
• Block Diagrams of Systems
• Summary

76

LTI Systems: Difference/differential equations

DT LTI systems are described mathematically by

difference equations

CT LTI systems are described mathematically by

differential equations

77

LTI Systems: Difference/differential equations

• For any excitation, x[n] or x(t)
• the response, y[n] or y(t) can be found by

1.

finding the response to x[n] or x(t) as the only forcing function on the right-hand side and

2.

then adding scaled and time-shifted versions of that response to form y[n] or y(t)

• If x[n] or x(t) is a unit impulse,
• the response to it as the only forcing function is

simply the homogeneous solution of the difference

• r differential equation with initial conditions applied

78

LTI Systems: Differential Equations

General Nth-order linear constant-coefficient

differential equation, ak and bk are real constants

Differential equations play a central role in describing input-output

relationships in electrical systems

The general solution is given by: y(t) = yp(t) + yh(t) yp(t) is a particular solution yh(t) is the homogeneous solution satisfying To get yh(t), N auxiliary conditions are required Auxiliary conditions are the values of:

At some point in time

k k N k k k k N k k

dt t x d b dt t y d a ) ( ) (

∑ ∑

= =

=

) ( =

∑

= k k N k k

dt t y d a

1 1

) ( , , ) ( ), (

− − N N

dt t y d dt t dy t y L

79

LTI Systems: Differential

Equations

80

LTI Systems: Differential

Equations

Let a CT system be described by and let the excitation be a unit impulse at time, t = 0. Then

the response, y, is the impulse response, h.

Since the impulse occurs at time, t = 0, and nothing has

excited the system before that time, the impulse response before time, t = 0, is zero. After time, t = 0, the impulse has

• ccurred and gone away. Therefore there is no excitation

and the impulse response is the homogeneous solution

• f the differential equation.

) ( ) ( ) ( ) (

1 2

t x t y a t y a t y a = + ′ + ′ ′

) ( ) ( ) ( ) (

1 2

t t h a t h a t h a δ = + ′ + ′ ′

81

LTI Systems: Differential

Equations

What happens at time t = 0? The equation must

be satisfied at all times. So the left side of the equation must be a unit impulse. We already know that the left side is zero before time, t = 0 because the system has never been excited. We know that the left side is zero after time, t = 0, because it is the solution of the homogeneous equation whose right side is zero. This is consistent with an

• impulse. The impulse response might have in it an

impulse or derivatives of an impulse since all of these occur only at time, t = 0. What the impulse response does have in it depends on the equation.

) ( ) ( ) ( ) (

1 2

t t h a t h a t h a δ = + ′ + ′ ′

82

LTI Systems: Differential

Equations

83

LTI Systems: Differential

Equation: Example 3.1

84

LTI Systems: Differential

Equation: Example 3.1

85

LTI Systems: Differential

Equation: Example 3.1

86

LTI Systems: Differential

Equation: Example 3.1

87

LTI Systems: Difference

Equations

An LTI System can be described by a

difference equation

recursive)

• (non

memoryless is System , and 1 if ]} [ { ]} [ { ]} [ { ]} [ { , 1 ], [ ] [ ⇒ ∀ = = ∗ = ∗ ≥ = − = −

∑ ∑

= =

m a a n b n x n a n y n a m n x b m n y a

m M m M m m m

88

LTI System: Difference Equation Example

] 1 [ ] [ ] 1 [ ] [ ] 1 [ ] [ ] 1 [ ] [

1 1 1 1

− + + − − = − + = − + n x b n x b n y a n y n x b n x b n y a n y

A first order LTI system:

89

LTI Systems: Solving Difference

Equations: Example 1

90

LTI Systems: Solving Difference

Equations: Example 2

Let a DT system be described by The eigenfunction is the DT complex exponential,

y[n] must be of the form

Substituting into the homogeneous difference

equation,

Dividing through by Solving,

] [ ] 2 [ ] 1 [ 2 ] [ 3 n x n y n y n y = − + − +

n

α

n

A n y α = ] [

2 3

2 1

= + +

− − n n n

α α α

2 − n

α 1 2 3

2

= + + α α

47 . 33 . j ± − = α

n

j A n y ) 47 . 33 . ( ] [ ± − =

91

LTI Systems: Solving Difference

Equations: Example

• The homogeneous solution is then of the form
• The constants can be found by applying initial conditions
• Let at time n=0

n n

j K j K n h ) 47 . 33 . ( ) 47 . 33 . ( ] [

2 1

− − + + − =

12 . 17 . , 12 . 17 . 9 / 2 ) 47 . 33 . ( ) 47 . 33 . ( ] 1 [ 3 / 1 ) 47 . 33 . ( ) 47 . 33 . ( ] [ 9 / 2 ] 1 [ ] 1 [ ] 2 1 [ ] 1 1 [ 2 ] 1 [ 3 3 / 1 ] [ 1 ] [ ] 2 [ ] 1 [ 2 ] [ 3

2 1 2 1 2 1 2 1 3 / 1

j K j K j K j K h K K j K j K h h x h h h h x h h h − = + = − = − − + + − = = + = − − + + − = − = ⇒ = = − + − + = ⇒ = = − + − +

= = = =

3 2 1 4 3 4 2 1 3 2 1 4 3 4 2 1

] [ ] [ n n x δ =

92

Outline

• Response of LTI systems
• Impulse response and unit response
• Characterization of LTI systems
• Properties using the system response
• Convolution Integral
• Convolution Integral Properties
• Convolution Sum
• Convolution Sum Properties
• Systems and Difference/Differential Equations
• Block Diagrams of Systems
• Summary

93

System Block Diagrams

A very useful method for describing and analyzing

systems is the block diagram

A block diagram can be drawn directly from the

difference or differential equation which describes the system

• For example, if the system is described by

It can also be described by

the block diagram below in which “D” represents a delay

• f one in discrete time

94

Block Diagram Elements Discrete-Time

95

Block Diagram Elements Continuous-Time

96

Outline

• Response of LTI systems
• Impulse response and unit response
• Characterization of LTI systems
• Properties using the system response
• Convolution Sum
• Convolution Sum Properties
• Convolution Integral
• Convolution Integral Properties
• Difference and Differential Equations
• Block Diagrams of Systems
• Summary

97

Summary

98

Summary

99

Summary: a quiz

• Problem: a discrete-time LTI system has impulse response
• Find the output y[n] due to input

x[n] = u[n + 1] – u[n - 1] + 2δ[n - 2], where u[n] is the discrete time unit step function

• Suggestions:
• Use convolution properties
• Plot the functions of h[n] and x[n]
• In other problems: you may be
• Given y(t) = integral (..); find h(t) analytically or graphically
• Given x(t) and h(t) ; find y(t) analytically or graphically
• Pay attention that you may need to do variable substitution, e.g.,

integral(e^(t-p) h(p-5) dp) –inf to t p' = p-5 p=p'+5 integral(e^(t-p'-5) h(p') dp') -inf to t-5

• Solution: the simplest way to solve for the output y[n] would be to first plot

the functions of h[n] and x[n]

] 1 [ 2 ] [ 3 ] [ − − = n n n h δ δ

100

Summary: a quiz

The sequence h[n] consists of two samples. Therefore, convolving x[n] and h[n] can be simplified by

convolving x[n] with h[n] one sample at a time.

For example, can be convolved by

convolving x[n] first with and then convolving x[n] with

Finally, the convolution sum (y[n]) can be then obtained by

adding the two sequences (adding sample by corresponding sample).

In doing this, the output y[n] is The same can be represented graphically which is just as

good.

] 1 [ 2 ] [ 4 ] [ − − = n n n h δ δ

] [ 4 ] [

1

n n h δ =

] 1 [ 2 ] [

2

− − = n n h δ

] 3 [ 4 ] 2 [ 6 ] 1 [ ] [ ] 1 [ 3 ] [ − − − + − − + + = n n n n n n y δ δ δ δ δ