SPFirst-L11_EV0

Si l Lecture 11Signal Processing

Lecture 11Linearity Processing

FirstTime‐InvarianceConvolutionConvolution

Copyright © Monash University 2009 1

READING ASSIGNMENTSREADING ASSIGNMENTS

• This Lecture:– Chapter 5, Sections 5‐5 and 5‐6p 5, 5 5 5

• Other Reading:R i i Ch S 6 & 8– Recitation: Ch. 5, Sects 5‐6, 5‐7 & 5‐8

• CONVOLUTION

– Next Lecture: start Chapter 6


LECTURE OBJECTIVESLECTURE OBJECTIVES

GENERAL PROPERTIES of FILTERSLLINEARITY LTI SYSTEMSLLINEARITYTTIME-IINVARIANCE→ CONVOLUTIONCONVOLUTION

– BLOCK DIAGRAM REPRESENTATION• Components for Hardwarep• Connect Simple Filters Together to Build More Complicated Systems


OVERVIEWOVERVIEW

• IMPULSE RESPONSE, FIR

][nh

}{b– FIR case: same as

• CONVOLUTION

}{ kb

– GENERAL:

GENERAL CLASS f SYSTEMS

[ ] [ ] [ ] [ ] [ ]k

y n h n x n h k x n k= ∗ = −∑– GENERAL CLASS of SYSTEMS

– LINEAR and TIME‐INVARIANT

• ALL LTI systems have h[n] & use convolution


DIGITAL FILTERINGDIGITAL FILTERING

FILTER D-to-AA-to-Dx(t) y(t)y[n]x[n]

• CONCENTRATE on the FILTER (DSP)( )

• DISCRETE‐TIME SIGNALSf h d– FUNCTIONS of n, the “time index”

– INPUT x[n]– OUTPUT y[n]


GENERAL FIR FILTERGENERAL FIR FILTER

• FILTER COEFFICIENTS {bk}– DEFINE THE FILTER MDEFINE THE FILTER(k=0 →CAUSALM FIR)

∑=

−=k

k knxbny0

][][

M<∞→ FIR)– For example, }1,2,1,3{ −=kb

][][3

−= ∑ knxbny k Difference Equation

]3[]2[2]1[][3

][][0

−+−+−−=

∑=

nxnxnxnx

knxbnyk

k Difference Equation

Copyright © Monash University 2009 3/23/2011 6

][][][][

IMPULSE SIGNALIMPULSE SIGNAL

• x[n] has only one NON‐ZERO VALUE

⎪⎨⎧ =

=01

][n

nδ

UNIT-IMPULSE

⎪⎩⎨

≠ 00][

nnδ

1UNIT IMPULSE

n


FIR IMPULSE RESPONSEFIR IMPULSE RESPONSE

• Convolution = Filter Definition– Filter Coeffs = Impulse Responsep p

∑M

kb∑M

kbh δUse SHIFTED

∑=

−=k

k knxbny0

][][ ∑=

−=k

k knbnh0

][][ δ IMPULSES to write h[n]


MATH FORMULA for h[n]MATH FORMULA for h[n]• Use SHIFTED IMPULSES to write h[n]Use SHIFTED IMPULSES to write h[n]

]4[]3[]2[2]1[][][ −+−−−+−−= nnnnnnh δδδδδ

1

2][nh

11

1 3

1

n

–1

0 42

–1{ 1, 1, 2, 1, 1 }

[ ] [ ] [ 1] 2 [ 2] [ 3] [ 4]kb

y n x n x n x n x n x n= − −= − − + − − − + −


[ ] [ ] [ 1] 2 [ 2] [ 3] [ 4]y n x n x n x n x n x n+ +

Convolution SumConvolution Sum•• Output = Convolution of x[n] & h[n]Output = Convolution of x[n] & h[n]•• Output = Convolution of x[n] & h[n]Output = Convolution of x[n] & h[n]

– NOTATION: ][][][ nxnhny ∗=– Here is the FIR case:

FINITE LIMITS* is convolution not multiplication

*M

h k k h∑

FINITE LIMITS not multiplication

0[ ] [ ] [ ] : [ ] * [ ]

ky n h k x n k h n x n

=

= − =∑FINITE LIMITS

Same as bk


CONVOLUTION Examplep

0[ ] [ ] [ ] [0] [ ] [1] [ 1] [ ] [ ]

M

ky n h k x n k h x n h x n h M x n M

=

= − = + − + + −∑ L

Finite

Finite y[n]Copyright © Monash University 2009 3/23/2011 11

Finite y[n]

CONVOLUTION Commutes[ ] [ ] [ ] [0] [ ] [1] [ 1] [ ] [ ]

M

y n h k x n k h x n h x n h M x n M= − = + − + + −∑ L[ ] [ ] [ ] [0] [ ] [1] [ 1] [ ] [ ]M

y n x k h n k x h n x h n x M h n M= − = + − + + −∑ L0

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]k

y=∑

0[ ] [ ] [ ] [0] [ ] [ ] [ ] [ ] [ ]

ky n x k h n k x h n x h n x h n

=∑

[ ] [ ] [ ] [ ]* [ ]M

y n h k x n k h n x n= − =∑x[0]h[n-0] | 6 -2 4 2

0[ ] [ ] [ ] [ ] [ ]

ky n h k x n k h n x n

=

= =∑[ ] [ ] |

x[1]h[n-1] | 12 -4 8 4x[2]h[n-2] | 18 -6 12 6 x[3]h[n-3] | 2 -4 8 4[ ] [ ] [ ]* [ ]

M

x k h n k x n h n= − =∑x[3]h[n 3] | 2 4 8 4x[4]h[n-4] | 6 -2 4 20

[ ] [ ] [ ] [ ]k

x k h n k x n h n=∑

Copyright © Monash University 2009

3/23/2011

12

CONVOLUTION Example[ ] [ ] [ 1] 2 [ 2] [ 3] [ 4]h n n n n n nδ δ δ δ δ= − − + − − − + −CONVOLUTION Example

4

[ ] [ ]

[ ] [ ]* [ ] [ ] [ ]

x n u n

y n h n x n h k u n k

=

= = −∑0

[ ] [ ] [ ] [ ] [ ]k

y n h n x n h k u n k=

= = ∑n −1 0 1 2 3 4 5 6 7

I fi ix[n] 0 1 1 1 1 1 1 1 ...h[n] 0 1 −1 2 −1 1 0 0 0

Infinitex[n]

0 1 1 1 1 1 1 1 10 0 −1 −1 −1 −1 −1 −1 −1]1[]1[

][]0[−nxh

nxh

0 0 0 2 2 2 2 2 20 0 0 0 −1 −1 −1 −1 −1]3[]3[

]2[]2[]1[]1[

−−

nxhnxhnxh

0 0 0 0 1 1 1 1 10 0 0 0 0 1 1 1 1

y[n] 0 1 0 2 1 2 2 2Infinite y[n]

]4[]4[]3[]3[

−nxhnxh


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y[n] 0 1 0 2 1 2 2 2 ... y[n]

GENERAL FIR FILTERGENERAL FIR FILTER• SLIDE a Length‐L WINDOW over x[n]g

L=M+1L=M+1

x[n]x[n-M]


MATLAB for FIR FILTERMATLAB for FIR FILTER• yy = conv(bb xx)• yy = conv(bb,xx)

– VECTOR bb contains Filter Coefficients

( )– SP‐First: yy = firfilt(bb,xx)

• Previous example Previous example

bb=[3 -2 2 1]; xx=[2 4 6 4 2];[ ] [ ]

yy = firfilt(bb,xx)

ans = [6 10 18 16 18 12 8 2]


Polynomial product and convolutiony p2( ) 1 2 deg( ) 2

( ) 1 d ( ) 1p x x x p= − − + =

2 3

( ) 1 deg( ) 1( ) ( ) 1 3 deg( ) 3

q x x pp x q x x x x pq

= − =

= − − + − =

{ } 1 1[ ] ..., 0,0, 1, 2,1,0,0,... 3 ( 2)x n L M= − − = =

{ }

{ }

2 2

1

[ ] ..., 0,0,1, 1,0,0,... 2 ( 2)

[ ] [ ]* [ ] [ ] [ ] 0 0 1 1 3 1 0 0

h n L M

h h k k

= − = =

∑ { }0

1 2 1 2

[ ] [ ]* [ ] [ ] [ ] ..., 0,0,1, 1, 3, 1,0,0,...

1 4 ( 1 3)k

y n h n x n h k u n k

L L M M=

= = − = − − −

+ − = + − =

∑1 2 1 21 4 ( 1 3)L L M M+ +

>> conv([-1 -2 1],[1 -1]) => [ -1 -1 3 -1]


([ ],[ ]) [ ]

DCONVDEMO: MATLAB GUIDCONVDEMO: MATLAB GUI


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POP QUIZPOP QUIZ• FIR Filter is “FIRST DIFFERENCE”• FIR Filter is FIRST DIFFERENCE

– y[n] = x[n] ‐ x[n‐1]• INPUT is “UNIT STEP”

⎧

⎪⎩

⎪⎨⎧

<

≥=

00

01][

nnu

⎪⎩ < 00 n

]1[][][ −−= nununyFind y[n] ][]1[][][ nnununy δ=−−=


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HARDWARE STRUCTURESHARDWARE STRUCTURES

FILTER y[n]x[n] ∑ −=M

k knxbny ][][

S C f

FILTER ∑=k

k0

• INTERNAL STRUCTURE of “FILTER”– WHAT COMPONENTS ARE NEEDED?– HOW DO WE “HOOK” THEM TOGETHER?

• SIGNAL FLOW GRAPH NOTATION• SIGNAL FLOW GRAPH NOTATION


HARDWARE ATOMSHARDWARE ATOMS• Multiply, Store & Add

∑=M

knxbny ][][ ∑=

−=k

k knxbny0

][][

][][ β][][][ 21 nxnxny +=

][][ nxny β=][][][ 21y

]1[][ −= nxnyCopyright © Monash University 2009 20

][][y

FIR STRUCTURES UC U• Direct Form

∑=M

knxbny ][][SIGNALFLOW GRAPH

∑=

−=k

k knxbny0

][][

Stored sequence


FIR TRANSPOSE STRUCTURES OS S UC U

∑M

kb ][][ ∑=

−=k

k knxbny0

][][

x[n]

X XXX b0b3 b2 b1

UnitDelay

+ ++ y[n]UnitDelay

UnitDelay


SYSTEM PROPERTIESSYSTEM PROPERTIES

SYSTEMy[n]x[n]

•• MATHEMATICAL DESCRIPTIONMATHEMATICAL DESCRIPTION•• TIMETIME‐‐INVARIANCEINVARIANCE•• LINEARITYLINEARITYLINEARITYLINEARITY• CAUSALITY

– “No output prior to input”


TIME INVARIANCETIME-INVARIANCE

• IDEA:– “Time‐Shifting the input will cause the sameg ptime‐shift in the output”

• EQUIVALENTLY,– We can prove that

• The time origin (n=0) is picked arbitrary


TESTING Time InvarianceTESTING Time-Invariance


© 2003, JH McClellan & RW Schafer1, 0[ ] ,

0 0n

nn

δ=⎧

= ⎨ ≠⎩Q

0, 0[ ] 0

nn n for all nδ

≠⎩∴ =


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LINEAR SYSTEMLINEAR SYSTEM

• LINEARITY = Two Properties• SCALINGSCALING

– “Doubling x[n] will double y[n]”

• SUPERPOSITION:– “Adding two inputs gives an output that is the sum of the individual outputs”sum of the individual outputs


TESTING LINEARITYTESTING LINEARITY

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Ex: time-flip system y[n]=x[-n] is a linear system


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LTI SYSTEMSLTI SYSTEMS

L T I• LTI: Linear & Time‐Invariant• COMPLETELY CHARACTERIZED by• COMPLETELY CHARACTERIZED by:

– IMPULSE RESPONSE h[n]

– CONVOLUTION: y[n] = x[n]*h[n]• The “rule” defining the system can ALWAYS be re‐The rule defining the system can ALWAYS be rewritten as convolution

• FIR Example: h[n] is same as b• FIR Example: h[n] is same as bkExample transparence


LTI system and convolutionsyste a d co o ut o• y[n] = LTI(x[n]) h[n] = LTI(δ[n])y[n] LTI(x[n]) h[n] LTI(δ[n])

• y[n] = LTI(∑ x[k]δ[n‐k]) impulse repres.of x[n]y ∑ p p

• y[n] = ∑ LTI(x[k] δ[n‐k]) linearity• y[n] = ∑ x[k] LTI(δ[n‐k]) linearity• y[n] = ∑ x[k] h[n‐k] time invariance

• y[n] = x[n]*h[n]

Copyright © Monash University 2009 3/23/2011 31

CASCADE SYSTEMSCASCADE SYSTEMS

• Does the order of S1 & S2 matter?– NO, LTI SYSTEMS can be rearranged !!!, g– WHAT ARE THE RESULTING FILTER COEFFS? {bk} OR h[n]{bk} OR h[n]

S SS1 S2


CASCADE EQUIVALENTC SC QU– Find “overall” h[n] for a cascade ?

S SS1 S2

1 2[ ] [ ]* [ ]h n h n h n=

2 1[ ]* [ ]h n h n=

3/23/2011S1S2


⎧Example

≤⎧⎨⎩

1

1 0 3h [n]

0 otherwise⎩≤⎧

⎨⎩

2

1 1 3h [n]

0 h i⎨⎩

2

1 2

[ ]0 otherwise

h[n]=h [n]*h [n]

∑1 2

6

[ ] [ ] [ ]

y[n]= h[k]x[n-k]∑k=0

or5 4

∑ ∑3 3

k=0 k=1w[n]= x[n-k] y[n]= w[n-k]

5 + only no x

5 +, 4 x


k=0 k=1

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5 + only, no x

BUILDING BLOCKSBUILDING BLOCKS

FILTERy[n]x[n]

FILTER +OUTPUT

+INPUT

• BUILD UP COMPLICATED FILTERSFILTER

BUILD UP COMPLICATED FILTERS– FROM SIMPLE MODULES– Eg

FILTER = 3x[n]-x[n-1] +2x[n-2]+x[n-3]


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SPFirst-L11_EV0