Chapter 14 Finite Impulse Response (FIR) Filters ... - EE 301

Chapter 14Chapter 14Finite Impulse Response (FIR) FiltersFinite Impulse Response (FIR) Filters


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Chapter 14, Slide 2

Learning ObjectivesLearning Objectives

uu Introduction to the theory behind FIRIntroduction to the theory behind FIRfilters:filters:ww Properties (including aliasing).Properties (including aliasing).ww Coefficient calculation.Coefficient calculation.ww Structure selection.Structure selection.

uu Implementation in Matlab, C, assemblyImplementation in Matlab, C, assemblyand linear assembly.and linear assembly.

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004

Learning ObjectivesLearning Objectives

Introduction to the theory behind FIRIntroduction to the theory behind FIR

Properties (including aliasing).Properties (including aliasing).Coefficient calculation.Coefficient calculation.Structure selection.Structure selection.

Implementation in Matlab, C, assemblyImplementation in Matlab, C, assemblyand linear assembly.and linear assembly.



Chapter 14, Slide 3

IntroductionIntroduction

uu Amongst all the obvious advantages thatAmongst all the obvious advantages thatdigital filters offer, the FIR filter candigital filters offer, the FIR filter canguarantee linear phase characteristics.guarantee linear phase characteristics.

uu Neither analogue or IIR filters can achieveNeither analogue or IIR filters can achievethis.this.

uu There are many commercially availableThere are many commercially availablesoftware packages for filter design.software packages for filter design.However, without basic theoreticalHowever, without basic theoreticalknowledge of the FIR filter, it will beknowledge of the FIR filter, it will bedifficult to use them.difficult to use them.


IntroductionIntroduction

Amongst all the obvious advantages thatAmongst all the obvious advantages thatdigital filters offer, the FIR filter candigital filters offer, the FIR filter canguarantee linear phase characteristics.guarantee linear phase characteristics.Neither analogue or IIR filters can achieveNeither analogue or IIR filters can achieve

There are many commercially availableThere are many commercially availablesoftware packages for filter design.software packages for filter design.However, without basic theoreticalHowever, without basic theoreticalknowledge of the FIR filter, it will beknowledge of the FIR filter, it will bedifficult to use them.difficult to use them.



Chapter 14, Slide 4

Properties of an FIR FilterProperties of an FIR Filter

uu Filter coefficients:Filter coefficients:

[] ∑−

=

=1

0

N

k

bny

x[n]x[n] representsrepresents thethe filterfilterbbkk represents the filter coefficients,represents the filter coefficients,y[n]y[n] representsrepresents thethe filterfilterNN isis thethe numbernumber ofof

(order(order ofof thethe filter)filter)



[ ]−⋅k knxb

filterfilter input,input,represents the filter coefficients,represents the filter coefficients,

filterfilter output,output,filterfilter coefficientscoefficients

filter)filter)..



Chapter 14, Slide 5



[] ∑−

=

=1

0

N

k

bny

z-1

+

z-1

+

x(n)

xxxb0 b1

Filter structureFilter structureDr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004


[ ]−⋅k knxb

z-1

+

z-1

y(n)

xb2 bN-1

FIR equationFIR equation

Filter structureFilter structure



Chapter 14, Slide 6



[] ∑−

=

=1

0

N

k

bny

uu If the signal x[n] is replaced by an impulseIf the signal x[n] is replaced by an impulseδδ[n] then:[n] then:

[] ∑−

=

=1

0

N

k

bny

[] [] [bby −+= δδ 00 10



[ ]−⋅k knxb

If the signal x[n] is replaced by an impulseIf the signal x[n] is replaced by an impulse

[ ]−k knb δ

] [ ]Nbk −++− δL1



Chapter 14, Slide 7



[] ∑−

=

=1

0

N

k

bny


[] ∑−

=

=1

0

N

k

bny

[] [] [nbnbny −+= δδ 10



[ ]−⋅k knxb


[ ]−k knb δ

] [ ]Nnbk −++− δL1



Chapter 14, Slide 8



[] ∑−

=

=1

0

N

k

bny


[] ∑−

=

=1

0

N

k

bny

[ ]

=−01

knδ



[ ]−⋅k knxb


[ ]−k knb δ

≠=

knfor0knfor1



Chapter 14, Slide 9



[] ∑−

=

=1

0

N

k

bny

uu FinallyFinally::

b

b

b

k =

=

=

M

1

0



[ ]−⋅k knxb

[][]

[]kh

h

h

=

=

=

M

1

0



Chapter 14, Slide 10



[] ∑−

=

=1

0

N

k

bny

WithWith:: []khbk =

uu The coefficients of a filter are the same asThe coefficients of a filter are the same asthe impulse response samples of the filter.the impulse response samples of the filter.



[ ]−⋅k knxb

The coefficients of a filter are the same asThe coefficients of a filter are the same asthe impulse response samples of the filter.the impulse response samples of the filter.




Frequency Response of an FIR FilterFrequency Response of an FIR Filter

uu By taking the zBy taking the z--transform of h[n], H(z):transform of h[n], H(z):

uu Replacing z by eReplacing z by ejjωω in order to find thein order to find thefrequency response leads to:frequency response leads to:

()Τϕ/Φ4 31.648 Τφ1 0 0 1 255.84 511.41 Τµ ()∑=

=N

nzH

()Τϕ/Φ4 31.547 Τφ1 0 0 1 143.28 313.41 Τµ ()()Τϕ/Φ4 38.25 Τφ1 0 0 1 276.72 313.17 Τµ ()=

== jez

eHzH jω

ω



transform of h[n], H(z):transform of h[n], H(z):

in order to find thein order to find thefrequency response leads to:frequency response leads to:

[]∑−

=

−1

0

Nnznh

[]∑−

=

−=1

0

N

n

jnenh ω





uu Since eSince e--j2j2ππkk = 1 then:= 1 then:

uu Therefore:Therefore:

()Τϕ/Φ4 31.547 Τφ1 0 0 1 164.4 516.45 Τµ ()[∑−

==

=+

1

02

N

nez

hzH πω

()Τϕ/Φ4 36.391 Τφ1 0 0 1 267.6 355.41 Τµ ()πω kjeH =+2

uu FIR filters have a periodic frequencyFIR filters have a periodic frequencyresponse and the period is 2response and the period is 2



= 1 then:= 1 then:

[] ()Τϕ/Φ4 18.227 Τφ1 0 0 1 393.84 527.01 Τµ ()[]∑−

=

−+− =1

0

2N

n

jnjn enhen ωπω

()Τϕ/Φ4 36.391 Τφ1 0 0 1 362.88 355.41 Τµ ()ωjeH

FIR filters have a periodic frequencyFIR filters have a periodic frequencyresponse and the period is 2response and the period is 2ππ..





uu FrequencyFrequencyresponse:response:

FIRFIRx[n]x[n]

FFss/2/2FreqFreq

x[n]x[n]




uu Solution: Use an antiSolution: Use an anti

ADCADCAnalogueAnalogue

AntiAnti--AliasingAliasing

x(t)x(t)

FFss/2/2FreqFreq

x(t)x(t)



Solution: Use an antiSolution: Use an anti--aliasing filter.aliasing filter.

FIRFIR y[n]y[n]x[n]x[n]ADCADC

FFss/2/2FreqFreq

y[n]y[n]

PDF Crea

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Phase Linearity of an FIR FilterPhase Linearity of an FIR Filter

uu A causal FIR filter whose impulseA causal FIR filter whose impulseresponse is symmetrical is guaranteed toresponse is symmetrical is guaranteed tohave a linear phase response.have a linear phase response.

0n

h(n)

1 n n+1 2n+12n

N = 2n + 2

Even symmetryEven symmetry



A causal FIR filter whose impulseA causal FIR filter whose impulseresponse is symmetrical is guaranteed toresponse is symmetrical is guaranteed tohave a linear phase response.have a linear phase response.

0

h(n)

1 n n+1 2n2n-1n-1

N = 2n + 1

Odd symmetryOdd symmetry





uu A causal FIR filter whose impulseA causal FIR filter whose impulseresponse is symmetrical (ie h[n] = h[Nresponse is symmetrical (ie h[n] = h[Nfor n = 0, 1, …, Nfor n = 0, 1, …, N--1) is guaranteed to have1) is guaranteed to havea linear phase response.a linear phase response.

Condition Phase

−

−=2

1Nk

[] [ ]1−−= nNhnh

Positive Symmetryωk



A causal FIR filter whose impulseA causal FIR filter whose impulseresponse is symmetrical (ie h[n] = h[Nresponse is symmetrical (ie h[n] = h[N--11--

1) is guaranteed to have1) is guaranteed to havea linear phase response.a linear phase response.

Phase Property Filter Type

Linear phaseOdd Symmetry – Type 1

Even Symmetry – Type 2





uu Application of 90Application of 90°°

SignalSignalseparationseparation

9090oo

delaydelay

9090oo

delaydelay

II

QQ

tBt rf ωω sincos +

tBtA rf ωω cossin + tA

tAIH

f

f

ω

ω

sin

cos

+−=

+=



°° linear phase shift:linear phase shift:++

++

++

--

ReverseReverse

ForwardForward

IHIH

QHQH

tB

tB

r

r

ω

πω

π

cos2

sin2

++

BQIH cos2=+

BIQH sin2=−




Design ProcedureDesign Procedure

uu To fully design and implement a filter fiveTo fully design and implement a filter fivesteps are required:steps are required:

(1)(1) Filter specification.Filter specification.(2)(2) Coefficient calculation.Coefficient calculation.(3)(3) Structure selection.Structure selection.(4)(4) Simulation (optional).Simulation (optional).(5)(5) Implementation.Implementation.


Design ProcedureDesign Procedure

To fully design and implement a filter fiveTo fully design and implement a filter fivesteps are required:steps are required:

Filter specification.Filter specification.Coefficient calculation.Coefficient calculation.Structure selection.Structure selection.Simulation (optional).Simulation (optional).Implementation.Implementation.




Filter SpecificationFilter Specification

(a)

1

fc : cut-off frequency

pass-band

pass-band transition band

fpb : pass-band frequency

fsb : stop-band frequency

(b)

s∆

p∆0

-3

fc : cut-off frequency

|H(f)|(dB)

|H(f)|


Filter SpecificationFilter Specification -- Step 1Step 1

f(norm): cut-off frequency

stop-band

stop-band

1

sδ

pass-bandripple

stop-bandripple

: pass-band frequency

: stop-band frequencyf(norm)

p1 δ+

p1 δ−

fs/2

: cut-off frequency

fs/2

|H(f)|(linear)




Coefficient CalculationCoefficient Calculation

uu There are several different methodsThere are several different methodsavailable, the most popular are:available, the most popular are:ww Window method.Window method.ww Frequency sampling.Frequency sampling.ww ParksParks--McClellan.McClellan.

uu We will just consider the window method.We will just consider the window method.


Coefficient CalculationCoefficient Calculation -- Step 2Step 2

There are several different methodsThere are several different methodsavailable, the most popular are:available, the most popular are:

Frequency sampling.Frequency sampling.

We will just consider the window method.We will just consider the window method.




Window MethodWindow Method

uu First stage of this method is to calculateFirst stage of this method is to calculatethe coefficients of thethe coefficients of the

uu This is calculated as follows:This is calculated as follows:

()Τϕ/Φ4 27.105 Τφ1 0 0 1 192.72 398.37 Τµ ()()Τϕ0 0 0 ΡΓΕΤ218.88 331.17 µ244.32 331.17 λΣ0 ΓΒΤ/Φ4 27.105 Τφ1 0 0 1 286.8 270.45 Τµ 27.105 ΤΛ(()Τϕ0 0 0 ΡΓΕΤθ228.76 261.85 66.4 0.16 ρεΩ ν228.72 262.05 µ333.84 262.05 λΣΘ0 ΓΒΤ/Φ4 19.98 Τφ1 0 0 1 218.4 221.73 Τµ 19.98 ΤΛ()Τϕ/Φ4 19.98 Τφ1 0 0 1 218.4 212.13 Τµ ()Τϕ/Φ4 19.98 Τφ1 0 0 1 218.4 255.81 Τµ ()Τϕ/Φ4 19.98 Τφ1 0 0 1 218.4 240.69 Τµ ()Τϕ/Φ4 19.98 Τφ1 0 0 1 218.4 269.25 Τµ ()Τϕ/Φ4 19.98 Τφ1 0 0 1 202.56 244.05 Τµ (=)Τϕ/Φ4 19.98 Τφ1 0 0 1 276 326.37 Τµ (⋅)Τϕ/Φ4 19.98 Τφ1 0 0 1 202.56 326.37 Τµ (=)Τϕ/Φ4 19.98 Τφ1 0 0 1 202.56 398.13 Τµ (=)Τϕ/Φ4 31.98 Τφ1 0 0 1 255.6 318.21 Τµ 31.98 ΤΛ(∫)Τϕ/Φ4 31.98 Τφ1 0 0 1 252 390.21 Τµ (∫)Τϕ/Φ4 13.996 Τφ1 0 0 1 247.92 301.41 Τµ 13.996 ΤΛ(−)Τϕ/Φ4 13.996 Τφ1 0 0 1 247.92 373.17 Τµ (−)Τϕ/Φ2 19.98 Τφ1 0 0 1 266.64 217.89 Τµ 19.98 ΤΛ(2)Τϕ/Φ2 19.98 Τφ1 0 0 1 262.56 270.21 Τµ (σ)Τϕ1 0 0 1 270.24 270.21 Τµ (ι)Τϕ1 0 0 1 275.76 270.21 Τµ (ν)Τϕ/Φ2 19.98 Τφ1 0 0 1 230.4 270.21 Τµ (2)Τϕ/Φ2 19.98 Τφ1 0 0 1 265.44 326.37 Τµ (1)Τϕ/Φ2 19.98 Τφ1 0 0 1 220.56 310.53 Τµ (2)Τϕ/Φ2 19.98 Τφ1 0 0 1 226.56 339.09 Τµ (1)Τϕ/Φ2 19.98 Τφ1 0 0 1 220.56 382.53 Τµ (2)Τϕ/Φ2 19.98 Τφ1 0 0 1 226.56 411.09 Τµ (1)Τϕ/Φ5 13.996 Τφ1 0 0 1 287.76 212.85 Τµ 13.996 ΤΛ(χ)Τϕ/Φ5 13.996 Τφ1 0 0 1 289.68 236.37 Τµ (χ)Τϕ/Φ5 13.996 Τφ1 0 0 1 251.52 265.41 Τµ (χ)Τϕ/Φ5 13.996 Τφ1 0 0 1 295.44 335.25 Τµ (ϕ)Τϕ/Φ5 13.996 Τφ1 0 0 1 164.64 393.33 Τµ (δ)Τϕ/Φ5 19.98 Τφ1 0 0 1 280.8 217.89 Τµ 19.98 ΤΛ(φ)Τϕ/Φ5 19.98 Τφ1 0 0 1 264.48 241.41 Τµ (ν)Τϕ/Φ5 19.98 Τφ1 0 0 1 292.8 270.21 Τµ (ν)Τϕ/Φ5 19.98 Τφ1 0 0 1 244.56 270.21 Τµ (φ)Τϕ/Φ5 19.98 Τφ1 0 0 1 282.72 326.37 Τµ (ε)Τϕ/Φ5 19.98 Τφ1 0 0 1 264.72 398.13 Τµ (Η)Τϕ/Φ5 19.98 Τφ1 0 0 1 181.68 398.13 Τµ (ν)Τϕ/Φ5 19.98 Τφ1 0 0 1 154.8 398.13 Τµ (η)Τϕ/Φ5 9.988 Τφ1 0 0 1 261.6 348.93 Τµ 9.988 ΤΛ(χ)Τϕ/Φ5 9.988 Τφ1 0 0 1 265.92 297.81 Τµ (χ)Τϕ/Φ4 19.98 Τφ1 0 0 1 274.56 241.41 Τµ 19.98 ΤΛ(ω)Τϕ/Φ4 19.98 Τφ1 0 0 1 229.2 310.53 Τµ (π)Τϕ/Φ4 19.98 Τφ1 0 0 1 286.32 398.13 Τµ (ω)Τϕ/Φ4 19.98 Τφ1 0 0 1 229.2 382.53 Τµ (π)Τϕ/Φ4 13.996 Τφ1 0 0 1 251.04 352.29 Τµ 13.996 ΤΛ(ω)Τϕ/Φ4 13.996 Τφ1 0 0 1 255.36 301.41 Τµ (ω)Τϕ/Φ4 13.996 Τφ1 0 0 1 251.04 422.61 Τµ (π)Τϕ/Φ4 13.996 Τφ1 0 0 1 255.36 373.17 Τµ (π)ΤϕΕΤ296.16 152.61 295.92 539.76 ρε0.016 0.165 0.643 ργφ0.996 0.608 0.012 ργ0.996 0.608 0.012 ΡΓΒΤ/Φ2 12 Τφ1 0 0 1 327.84 155.25 Τµ 12 ΤΛ(∆)Τϕ1 0 0 1 336.48 155.25 Τµ (ρ)Τϕ1 0 0 1 339.84 155.25 Τµ (.)Τϕ1 0 0 1 345.6 155.25 Τµ (Ν)Τϕ1 0 0 1 354.24 155.25 Τµ (α)Τϕ1 0 0 1 359.52 155.25 Τµ (ι)Τϕ1 0 0 1 362.4 155.25 Τµ (µ)Τϕ1 0 0 1 375.36 155.25 Τµ (∆)Τϕ1 0 0 1 384 155.25 Τµ (α)Τϕ1 0 0 1 389.28 155.25 Τµ (η)Τϕ1 0 0 1 395.04 155.25 Τµ (ν)Τϕ1 0 0 1 400.8 155.25 Τµ (ο)Τϕ1 0 0 1 407.04 155.25 Τµ (υ)Τϕ1 0 0 1 413.04 155.25 Τµ (ν)Τϕ1 0 0 1 418.8 155.25 Τµ (,)Τϕ1 0 0 1 425.28 155.25 Τµ (Β)Τϕ1 0 0 1 433.2 155.25 Τµ (ρ)Τϕ1 0 0 1 437.28 155.25 Τµ (ι)Τϕ1 0 0 1 440.16 155.25 Τµ (σ)Τϕ1 0 0 1 444.72 155.25 Τµ (τ)Τϕ1 0 0 1 448.32 155.25 Τµ (ο)Τϕ1 0 0 1 454.56 155.25 Τµ (λ)Τϕ1 0 0 1 461.04 155.25 Τµ (Υ)Τϕ1 0 0 1 469.68 155.25 Τµ (ν)Τϕ1 0 0 1 475.44 155.25 Τµ (ι)Τϕ1 0 0 1 478.32 155.25 Τµ (ϖ)Τϕ1 0 0 1 484.32 155.25 Τµ (ε)Τϕ1 0 0 1 489.6 155.25 Τµ (ρ)Τϕ1 0 0 1 493.68 155.25 Τµ (σ)Τϕ1 0 0 1 498.48 155.25 Τµ (ι)Τϕ1 0 0 1 501.6 155.25 Τµ (τ)Τϕ1 0 0 1 505.2 155.25 Τµ (ψ)Τϕ1 0 0 1 510 155.25 Τµ (,)Τϕ1 0 0 1 520.08 155.25 Τµ (()Τϕ1 0 0 1 524.16 155.25 Τµ (χ)Τϕ1 0 0 1 529.44 155.25 Τµ ())Τϕ1 0 0 1 536.16 155.25 Τµ (Τ)Τϕ1 0 0 1 542.64 155.25 Τµ (ε)Τϕ1 0 0 1 547.92 155.25 Τµ (ξ)Τϕ1 0 0 1 553.68 155.25 Τµ (α)Τϕ1 0 0 1 558.96 155.25 Τµ (σ)Τϕ1 0 0 1 567.12 155.25 Τµ (Ι)Τϕ1 0 0 1 571.2 155.25 Τµ (ν)Τϕ1 0 0 1 576.96 155.25 Τµ (σ)Τϕ1 0 0 1 581.52 155.25 Τµ (τ)Τϕ1 0 0 1 585.12 155.25 Τµ (ρ)Τϕ1 0 0 1 589.2 155.25 Τµ (υ)Τϕ1 0 0 1 595.2 155.25 Τµ (µ)Τϕ1 0 0 1 604.08 155.25 Τµ (ε)Τϕ1 0 0 1 609.36 155.25 Τµ (ν)Τϕ1 0 0 1 615.12 155.25 Τµ (τ)Τϕ1 0 0 1 618.72 155.25 Τµ (σ)Τϕ1 0 0 1 626.88 155.25 Τµ (2)Τϕ1 0 0 1 632.88 155.25 Τµ (0)Τϕ1 0 0 1 638.88 155.25 Τµ (0)Τϕ1 0 0 1 644.88 155.25 Τµ (4)Τϕ/Φ0 36 Τφ1 0 0 1 163.68 651.33 Τµ 36 ΤΛ(Ω)Τϕ1 0 0 1 199.68 651.33 Τµ (ι)Τϕ1 0 0 1 209.76 651.33 Τµ (ν)Τϕ1 0 0 1 229.68 651.33 Τµ (δ)Τϕ1 0 0 1 249.6 651.33 Τµ (ο)Τϕ1 0 0 1 267.6 651.33 Τµ (ω)Τϕ1 0 0 1 302.4 651.33 Τµ (Μ)Τϕ1 0 0 1 336.48 651.33 Τµ (ε)Τϕ1 0 0 1 352.56 651.33 Τµ (τ)Τϕ1 0 0 1 364.56 651.33 Τµ (η)Τϕ1 0 0 1 384.48 651.33 Τµ (ο)Τϕ1 0 0 1 402.48 651.33 Τµ (δ)Τϕ/Φ0 36 Τφ1 0 0 1 163.68 651.33 Τµ (Ω)Τϕ1 0 0 1 199.68 651.33 Τµ (ι)Τϕ1 0 0 1 209.76 651.33 Τµ (ν)Τϕ1 0 0 1 229.68 651.33 Τµ (δ)Τϕ1 0 0 1 249.6 651.33 Τµ (ο)Τϕ1 0 0 1 267.6 651.33 Τµ (ω)Τϕ1 0 0 1 302.4 651.33 Τµ (Μ)Τϕ1 0 0 1 336.48 651.33 Τµ (ε)Τϕ1 0 0 1 352.56 651.33 Τµ (τ)Τϕ1 0 0 1 364.56 651.33 Τµ (η)Τϕ1 0 0 1 384.48 651.33 Τµ (ο)Τϕ1 0 0 1 402.48 651.33 Τµ (δ)Τϕ1 γ1 Γ/Φ0 31.922 Τφ1 0 0 1 40.8 581.49 Τµ 31.922 ΤΛ(Φ)Τϕ1 0 0 1 60.24 581.49 Τµ (ι)Τϕ1 0 0 1 69.12 581.49 Τµ (ρ)Τϕ1 0 0 1 83.28 581.49 Τµ (σ)Τϕ1 0 0 1 95.76 581.49 Τµ (τ)Τϕ1 0 0 1 114.24 581.49 Τµ (σ)Τϕ1 0 0 1 126.72 581.49 Τµ (τ)Τϕ1 0 0 1 137.28 581.49 Τµ (α)Τϕ1 0 0 1 153.36 581.49 Τµ (γ)Τϕ1 0 0 1 169.44 581.49 Τµ (ε)Τϕ1 0 0 1 191.76 581.49 Τµ (ο)Τϕ1 0 0 1 207.84 581.49 Τµ (φ)Τϕ1 0 0 1 226.32 581.49 Τµ (τ)Τϕ1 0 0 1 236.88 581.49 Τµ (η)Τϕ1 0 0 1 254.64 581.49 Τµ (ι)Τϕ1 0 0 1 263.52 581.49 Τµ (σ)Τϕ1 0 0 1 284.16 581.49 Τµ (µ)Τϕ1 0 0 1 310.32 581.49 Τµ (ε)Τϕ1 0 0 1 324.48 581.49 Τµ (τ)Τϕ1 0 0 1 335.04 581.49 Τµ (η)Τϕ1 0 0 1 352.8 581.49 Τµ (ο)Τϕ1 0 0 1 368.88 581.49 Τµ (δ)Τϕ1 0 0 1 395.04 581.49 Τµ (ι)Τϕ1 0 0 1 403.92 581.49 Τµ (σ)Τϕ1 0 0 1 424.32 581.49 Τµ (τ)Τϕ1 0 0 1 434.88 581.49 Τµ (ο)Τϕ1 0 0 1 459.12 581.49 Τµ (χ)Τϕ1 0 0 1 473.28 581.49 Τµ (α)Τϕ1 0 0 1 489.36 581.49 Τµ (λ)Τϕ1 0 0 1 498.24 581.49 Τµ (χ)Τϕ1 0 0 1 512.4 581.49 Τµ (υ)Τϕ1 0 0 1 530.16 581.49 Τµ (λ)Τϕ1 0 0 1 539.04 581.49 Τµ (α)Τϕ1 0 0 1 555.12 581.49 Τµ (τ)Τϕ1 0 0 1 565.68 581.49 Τµ (ε)Τϕ/Φ0 31.922 Τφ1 0 0 1 40.8 581.49 Τµ (Φ)Τϕ1 0 0 1 60.24 581.49 Τµ (ι)Τϕ1 0 0 1 69.12 581.49 Τµ (ρ)Τϕ1 0 0 1 83.28 581.49 Τµ (σ)Τϕ1 0 0 1 95.76 581.49 Τµ (τ)Τϕ1 0 0 1 114.24 581.49 Τµ (σ)Τϕ1 0 0 1 126.72 581.49 Τµ (τ)Τϕ1 0 0 1 137.28 581.49 Τµ (α)Τϕ1 0 0 1 153.36 581.49 Τµ (γ)Τϕ1 0 0 1 169.44 581.49 Τµ (ε)Τϕ1 0 0 1 191.76 581.49 Τµ (ο)Τϕ1 0 0 1 207.84 581.49 Τµ (φ)Τϕ1 0 0 1 226.32 581.49 Τµ (τ)Τϕ1 0 0 1 236.88 581.49 Τµ (η)Τϕ1 0 0 1 254.64 581.49 Τµ (ι)Τϕ1 0 0 1 263.52 581.49 Τµ (σ)Τϕ1 0 0 1 284.16 581.49 Τµ (µ)Τϕ1 0 0 1 310.32 581.49 Τµ (ε)Τϕ1 0 0 1 324.48 581.49 Τµ (τ)Τϕ1 0 0 1 335.04 581.49 Τµ (η)Τϕ1 0 0 1 352.8 581.49 Τµ (ο)Τϕ1 0 0 1 368.88 581.49 Τµ (δ)Τϕ1 0 0 1 395.04 581.49 Τµ (ι)Τϕ1 0 0 1 403.92 581.49 Τµ (σ)Τϕ1 0 0 1 424.32 581.49 Τµ (τ)Τϕ1 0 0 1 434.88 581.49 Τµ (ο)Τϕ1 0 0 1 459.12 581.49 Τµ (χ)Τϕ1 0 0 1 473.28 581.49 Τµ (α)Τϕ1 0 0 1 489.36 581.49 Τµ (λ)Τϕ1 0 0 1 498.24 581.49 Τµ (χ)Τϕ1 0 0 1 512.4 581.49 Τµ (υ)Τϕ1 0 0 1 530.16 581.49 Τµ (λ)Τϕ1 0 0 1 539.04 581.49 Τµ (α)Τϕ1 0 0 1 555.12 581.49 Τµ (τ)Τϕ1 0 0 1 565.68 581.49 Τµ (ε)Τϕ/Φ0 31.922 Τφ1 0 0 1 40.8 550.77 Τµ (τ)Τϕ1 0 0 1 51.36 550.77 Τµ (η)Τϕ1 0 0 1 69.12 550.77 Τµ (ε)Τϕ1 0 0 1 91.2 550.77 Τµ (χ)Τϕ1 0 0 1 105.36 550.77 Τµ (ο)Τϕ1 0 0 1 121.44 550.77 Τµ (ε)Τϕ1 0 0 1 135.6 550.77 Τµ (φ)Τϕ1 0 0 1 146.16 550.77 Τµ (φ)Τϕ1 0 0 1 156.72 550.77 Τµ (ι)Τϕ1 0 0 1 165.6 550.77 Τµ (χ)Τϕ1 0 0 1 179.76 550.77 Τµ (ι)Τϕ1 0 0 1 188.64 550.77 Τµ (ε)Τϕ1 0 0 1 202.8 550.77 Τµ (ν)Τϕ1 0 0 1 220.56 550.77 Τµ (τ)Τϕ1 0 0 1 231.12 550.77 Τµ (σ)Τϕ1 0 0 1 252 550.77 Τµ (ο)Τϕ1 0 0 1 268.08 550.77 Τµ (φ)Τϕ1 0 0 1 286.8 550.77 Τµ (τ)Τϕ1 0 0 1 297.36 550.77 Τµ (η)Τϕ1 0 0 1 315.12 550.77 Τµ (ε)Τϕ/Φ0 31.922 Τφ1 0 0 1 40.8 550.77 Τµ (τ)Τϕ1 0 0 1 51.36 550.77 Τµ (η)Τϕ1 0 0 1 69.12 550.77 Τµ (ε)Τϕ1 0 0 1 91.2 550.77 Τµ (χ)Τϕ1 0 0 1 105.36 550.77 Τµ (ο)Τϕ1 0 0 1 121.44 550.77 Τµ (ε)Τϕ1 0 0 1 135.6 550.77 Τµ (φ)Τϕ1 0 0 1 146.16 550.77 Τµ (φ)Τϕ1 0 0 1 156.72 550.77 Τµ (ι)Τϕ1 0 0 1 165.6 550.77 Τµ (χ)Τϕ1 0 0 1 179.76 550.77 Τµ (ι)Τϕ1 0 0 1 188.64 550.77 Τµ (ε)Τϕ1 0 0 1 202.8 550.77 Τµ (ν)Τϕ1 0 0 1 220.56 550.77 Τµ (τ)Τϕ1 0 0 1 231.12 550.77 Τµ (σ)Τϕ1 0 0 1 252 550.77 Τµ (ο)Τϕ1 0 0 1 268.08 550.77 Τµ (φ)Τϕ1 0 0 1 286.8 550.77 Τµ (τ)Τϕ1 0 0 1 297.36 550.77 Τµ (η)Τϕ1 0 0 1 315.12 550.77 Τµ (ε)Τϕ/Φ0 31.922 Τφ1 0 0 1 337.2 550.77 Τµ (ι)Τϕ1 0 0 1 346.08 550.77 Τµ (δ)Τϕ1 0 0 1 363.84 550.77 Τµ (ε)Τϕ1 0 0 1 378 550.77 Τµ (α)Τϕ1 0 0 1 394.08 550.77 Τµ (λ)Τϕ1 0 0 1 411.12 550.77 Τµ (φ)Τϕ1 0 0 1 421.68 550.77 Τµ (ι)Τϕ1 0 0 1 430.56 550.77 Τµ (λ)Τϕ1 0 0 1 439.44 550.77 Τµ (τ)Τϕ1 0 0 1 450 550.77 Τµ (ε)Τϕ1 0 0 1 464.16 550.77 Τµ (ρ)Τϕ/Φ0 31.922 Τφ1 0 0 1 337.2 550.77 Τµ (ι)Τϕ1 0 0 1 346.08 550.77 Τµ (δ)Τϕ1 0 0 1 363.84 550.77 Τµ (ε)Τϕ1 0 0 1 378 550.77 Τµ (α)Τϕ1 0 0 1 394.08 550.77 Τµ (λ)Τϕ1 0 0 1 411.12 550.77 Τµ (φ)Τϕ1 0 0 1 421.68 550.77 Τµ (ι)Τϕ1 0 0 1 430.56 550.77 Τµ (λ)Τϕ1 0 0 1 439.44 550.77 Τµ (τ)Τϕ1 0 0 1 450 550.77 Τµ (ε)Τϕ1 0 0 1 464.16 550.77 Τµ (ρ)ΤϕΕΤ337.44 545.97 140.88 1.44 ρε1 1 1 ργφ337.44 545.97 140.88 1.44 ρεφ1 γΒΤ/Φ0 31.922 Τφ1 0 0 1 478.32 550.77 Τµ 31.922 ΤΛ(.)Τϕ/Φ0 31.922 Τφ1 0 0 1 478.32 550.77 Τµ (.)Τϕ/Φ0 31.922 Τφ1 0 0 1 40.8 504.69 Τµ (Τ)Τϕ1 0 0 1 62.16 504.69 Τµ (η)Τϕ1 0 0 1 79.92 504.69 Τµ (ι)Τϕ1 0 0 1 88.8 504.69 Τµ (σ)Τϕ1 0 0 1 108.96 504.69 Τµ (ι)Τϕ1 0 0 1 117.84 504.69 Τµ (σ)Τϕ1 0 0 1 138.48 504.69 Τµ (χ)Τϕ1 0 0 1 152.64 504.69 Τµ (α)Τϕ1 0 0 1 168.72 504.69 Τµ (λ)Τϕ1 0 0 1 177.6 504.69 Τµ (χ)Τϕ1 0 0 1 191.76 504.69 Τµ (υ)Τϕ1 0 0 1 209.52 504.69 Τµ (λ)Τϕ1 0 0 1 218.4 504.69 Τµ (α)Τϕ1 0 0 1 234.48 504.69 Τµ (τ)Τϕ1 0 0 1 245.04 504.69 Τµ (ε)Τϕ1 0 0 1 259.2 504.69 Τµ (δ)Τϕ1 0 0 1 285.12 504.69 Τµ (α)Τϕ1 0 0 1 301.2 504.69 Τµ (σ)Τϕ1 0 0 1 321.6 504.69 Τµ (φ)Τϕ1 0 0 1 332.16 504.69 Τµ (ο)Τϕ1 0 0 1 348.24 504.69 Τµ (λ)Τϕ1 0 0 1 357.12 504.69 Τµ (λ)Τϕ1 0 0 1 366 504.69 Τµ (ο)Τϕ1 0 0 1 382.08 504.69 Τµ (ω)Τϕ1 0 0 1 405.6 504.69 Τµ (σ)Τϕ1 0 0 1 418.08 504.69 Τµ (:)Τϕ/Φ0 31.922 Τφ1 0 0 1 40.8 504.69 Τµ (Τ)Τϕ1 0 0 1 62.16 504.69 Τµ (η)Τϕ1 0 0 1 79.92 504.69 Τµ (ι)Τϕ1 0 0 1 88.8 504.69 Τµ (σ)Τϕ1 0 0 1 108.96 504.69 Τµ (ι)Τϕ1 0 0 1 117.84 504.69 Τµ (σ)Τϕ1 0 0 1 138.48 504.69 Τµ (χ)Τϕ1 0 0 1 152.64 504.69 Τµ (α)Τϕ1 0 0 1 168.72 504.69 Τµ (λ)Τϕ1 0 0 1 177.6 504.69 Τµ (χ)Τϕ1 0 0 1 191.76 504.69 Τµ (υ)Τϕ1 0 0 1 209.52 504.69 Τµ (λ)Τϕ1 0 0 1 218.4 504.69 Τµ (α)Τϕ1 0 0 1 234.48 504.69 Τµ (τ)Τϕ1 0 0 1 245.04 504.69 Τµ (ε)Τϕ1 0 0 1 259.2 504.69 Τµ (δ)Τϕ1 0 0 1 285.12 504.69 Τµ (α)Τϕ1 0 0 1 301.2 504.69 Τµ (σ)Τϕ1 0 0 1 321.6 504.69 Τµ (φ)Τϕ1 0 0 1 332.16 504.69 Τµ (ο)Τϕ1 0 0 1 348.24 504.69 Τµ (λ)Τϕ1 0 0 1 357.12 504.69 Τµ (λ)Τϕ1 0 0 1 366 504.69 Τµ (ο)Τϕ1 0 0 1 382.08 504.69 Τµ (ω)Τϕ1 0 0 1 405.6 504.69 Τµ (σ)Τϕ1 0 0 1 418.08 504.69 Τµ (:)ΤϕΕΤ296.16 200.61 149.76 239.76 ρε1 1 1 ργφθ297.16 200.41 148.96 239.44 ρεΩ ν145.68 440.85 µ446.88 440.85 λ446.88 199.65 λ145.68 199.65 λ145.68 440.85 λη146.88 200.37 µ146.16 200.85 λ446.16 200.85 λ445.68 200.37 λ445.68 440.37 λ446.16 439.65 λ146.16 439.65 λ146.88 440.37 λ146.88 200.37 ληφΘ0 γ0 ΓΒΤ/Φ4 27.105 Τφ1 0 0 1 302.64 398.37 Τµ 27.105 ΤΛ())Τϕ/Φ4 27.105 Τφ1 0 0 1 327.84 270.45 Τµ ()

=

≠

0nfor

0nforc

nj

nj

n

d

de

ω

ω

ωω

ω

ω




Window MethodWindow Methoduu Second stage of this method is to select a windowSecond stage of this method is to select a window

function based on the passband or attenuationfunction based on the passband or attenuationspecifications, then determine the filter length based onspecifications, then determine the filter length based onthe required width of the transition band.the required width of the transition band.

W ind o w T yp e N orm a lised T ra nsit ionW id th (∆ f (H z))

R ec tan gularN

9.0

H an nin gN

1.3

H am m ingN

3.3

B lackm anN

5.5

K aise r54.493.2

=→ βN

96.871.5=→ β

N

Using the HammingUsing the HammingWindow:Window:


Window MethodWindow MethodSecond stage of this method is to select a windowSecond stage of this method is to select a windowfunction based on the passband or attenuationfunction based on the passband or attenuationspecifications, then determine the filter length based onspecifications, then determine the filter length based onthe required width of the transition band.the required width of the transition band.

N orm a lised T ra nsit ion P assba nd R ip p le (d B ) Sto pb an d A tten ua tio n(dB )

0.7 41 6 21

0.0 54 6 44

0.0 19 4 53

0.0 01 7 74

0.0 27 4

0 .0 0 02 7 5

50

90

()Τϕ/Φ4 27.121 Τφ1 0 0 1 467.52 189.57 Τµ ()13284.12.13.33.3

=⋅−

=∆

= kHzkHzf

NUsing the HammingUsing the Hamming





uu The third stage is to calculate the set ofThe third stage is to calculate the set oftruncated or windowed impulse responsetruncated or windowed impulse responsecoefficients, h[n]:coefficients, h[n]:

()Τϕ/Φ4 27.121 Τφ1 0 0 1 99.12 426.45 Τµ ()()Τϕ/Φ4 27.121 Τφ1 0 0 1 161.76 426.45 Τµ ()()Τϕ/Φ4 27.121 Τφ1 0 0 1 211.68 426.45 Τµ ()nWnhnh d ⋅=

()Τϕ/Φ4 27.164 Τφ1 0 0 1 202.56 298.53 Τµ ()

46.054.0

46.054.0

+=

+=nW

forfor

Where:Where:



The third stage is to calculate the set ofThe third stage is to calculate the set oftruncated or windowed impulse responsetruncated or windowed impulse response

evenNforoddNfor

22

21

21

==

≤≤−

−≤≤

−−

NnN

NnN

1332cos46

2cos46

nN

n

π

π

6666 ≤≤− nforfor




Window MethodWindow Methoduu Matlab code for calculating coefficients:Matlab code for calculating coefficients:

close all;clear all;

fc = 8000/44100; % cutN = 133; % number of tapsn = -((N-1)/2):((N-1)/2);n = n+(n==0)*eps; % avoiding division by zero

[h] = sin(n*2*pi*fc)./(n*pi); % generate sequence of ideal coefficients[w] = 0.54 + 0.46*cos(2*pi*n/N); % generate window functiond = h.*w; % window the ideal coefficients

[g,f] = freqz(d,1,512,44100); % transform into frequency domain for plotting

figure(1)plot(f,20*log10(abs(g))); % plot transfer functionaxis([0 2*10^4 -70 10]);

figure(2);stem(d); % plot coefficient valuesxlabel('Coefficient number');ylabel ('Value');title('Truncated Impulse Response');

figure(3)freqz(d,1,512,44100); % use freqz to plot magnitude and phase responseaxis([0 2*10^4 -70 10]);


Window MethodWindow MethodMatlab code for calculating coefficients:Matlab code for calculating coefficients:

% cut-off frequency% number of taps

% avoiding division by zero

% generate sequence of ideal coefficients% generate window function% window the ideal coefficients

% transform into frequency domain for plotting

% plot transfer function

% plot coefficient values

% use freqz to plot magnitude and phase response





0 0.5 1-6000

-4000

-2000

0

Frequency (Hz)

Pha

se(d

egre

es)

0 0.2 0.4 0.6 0.8

-60

-40

-20

0

Frequency (Hz)

Mag

nitu

de(d

B)

0 20 40 60-0.1

0

0.1

0.2

0.3

0.4

Coefficient number, n

h(n)

Truncated Impulse Response



1.5 2

x 104Frequency (Hz)

1 1.2 1.4 1.6 1.8 2

x 104Frequency (Hz)

80 100 120 140Coefficient number, n

Truncated Impulse Response




Realisation Structure SelectionRealisation Structure Selection

()Τϕ/Φ4 26.984 Τφ1 0 0 1 262.32 524.61 Τµ ()∑=N

k

zH

()Τϕ/Φ4 27.121 Τφ1 0 0 1 46.8 444.45 Τµ ()()Τϕ/Φ4 27.121 Τφ1 0 0 1 107.04 444.45 Τµ ()()Τϕ/Φ4 27.121 Τφ1 0 0 1 155.76 444.45 Τµ ()zXzHzY ⋅= ()Τϕ/Φ4 27.121 Τφ1 0 0 1 228.48 445.89 Τµ ()()Τϕ/Φ2 12 Τφ1 0 0 1 263.28 440.85 Τµ 12 ΤΛ(0)Τϕ/Φ4 19.996 Τφ1 0 0 1 238.32 445.89 Τµ 19.996 ΤΛ(=)Τϕ/Φ5 19.996 Τφ1 0 0 1 286.8 445.89 Τµ (ν)Τϕ/Φ5 19.996 Τφ1 0 0 1 271.44 445.89 Τµ (ξ)Τϕ/Φ5 19.996 Τφ1 0 0 1 253.68 445.89 Τµ (β)Τϕ/Φ5 19.996 Τφ1 0 0 1 217.68 445.89 Τµ (ν)Τϕ/Φ5 19.996 Τφ1 0 0 1 201.84 445.89 Τµ (ψ)ΤϕΕΤ44.16 218.61 251.76 185.76 ρε1 1 1 ργφθ44.68 218.41 250.48 185.44 ρεΩ ν43.68 404.85 µ548.88 404.85 λ548.88 217.65 λ43.68 217.65 λ43.68 404.85 λη44.88 218.37 µ44.16 218.85 λ548.16 218.85 λ547.68 218.37 λ547.68 404.37 λ548.16 403.65 λ44.16 403.65 λ44.88 404.37 λ44.88 218.37 ληφΘ1.68 ω1 ϕ0 0 0 ΡΓ261.84 244.29 µ261.84 240.93 λ263.04 237.57 λ265.2 234.69 λ267.36 232.29 λ270.72 230.61 λ274.08 229.41 λ277.44 229.17 λ280.8 229.41 λ284.4 230.61 λ287.28 232.29 λ289.44 234.69 λ291.6 237.57 λ292.8 240.93 λ293.04 244.29 λ292.8 247.89 λ291.6 251.01 λ289.44 253.89 λ287.28 256.29 λ284.4 258.21 λ280.8 259.17 λ277.44 259.65 λ274.08 259.17 λ270.72 258.21 λ267.36 256.29 λ265.2 253.89 λ263.04 251.01 λ261.84 247.89 λ261.84 244.29 ληβ261.84 244.29 µ261.84 240.93 λ263.04 237.57 λ265.2 234.69 λ267.36 232.29 λ270.72 230.61 λ274.08 229.41 λ277.44 229.17 λ280.8 229.41 λ284.4 230.61 λ287.28 232.29 λ289.44 234.69 λ291.6 237.57 λ292.8 240.93 λ293.04 244.29 λ292.8 247.89 λ291.6 251.01 λ289.44 253.89 λ287.28 256.29 λ284.4 258.21 λ280.8 259.17 λ277.44 259.65 λ274.08 259.17 λ270.72 258.21 λ267.36 256.29 λ265.2 253.89 λ263.04 251.01 λ261.84 247.89 λ261.84 244.29 ληφ∗261.84 244.29 µ261.84 240.93 λ263.04 237.57 λ265.2 234.69 λ267.36 232.29 λ270.72 230.61 λ274.08 229.41 λ277.44 229.17 λ280.8 229.41 λ284.4 230.61 λ287.28 232.29 λ289.44 234.69 λ291.6 237.57 λ292.8 240.93 λ293.04 244.29 λ292.8 247.89 λ291.6 251.01 λ289.44 253.89 λ287.28 256.29 λ284.4 258.21 λ280.8 259.17 λ277.44 259.65 λ274.08 259.17 λ270.72 258.21 λ267.36 256.29 λ265.2 253.89 λ263.04 251.01 λ261.84 247.89 λ261.84 244.29 ληΣ122.4 351.57 µ166.8 351.57 λ166.8 397.41 λ122.4 397.41 λ122.4 351.57 ληβ∗122.4 351.57 µ166.8 351.57 λ166.8 397.41 λ122.4 397.41 λ122.4 351.57 ληφ∗122.4 351.57 µ166.8 351.57 λ166.8 397.41 λ122.4 397.41 λ122.4 351.57 ληΣ0 γ0 ΓΒΤ/Φ1 13.664 Τφ1 0 0 1 141.36 371.49 Τµ 13.664 ΤΛ(ζ)Τϕ/Φ1 13.664 Τφ1 0 0 1 149.52 379.17 Τµ (−)Τϕ1 0 0 1 154.08 379.17 Τµ (1)Τϕ0 0 0 ΡΓΕΤ1 1 1 ργ210.96 351.57 µ255.12 351.57 λ255.12 397.41 λ210.96 397.41 λ210.96 351.57 ληβ∗210.96 351.57 µ255.12 351.57 λ255.12 397.41 λ210.96 397.41 λ210.96 351.57 ληφ∗210.96 351.57 µ255.12 351.57 λ255.12 397.41 λ210.96 397.41 λ210.96 351.57 ληΣ0 γ0 ΓΒΤ/Φ1 13.664 Τφ1 0 0 1 229.92 371.49 Τµ 13.664 ΤΛ(ζ)Τϕ/Φ1 13.664 Τφ1 0 0 1 238.32 379.17 Τµ (−)Τϕ1 0 0 1 242.88 379.17 Τµ (1)Τϕ0 0 0 ΡΓΕΤ78.24 374.37 µ114 374.37 λΣ122.4 374.37 µ111.36 379.89 λ112.56 377.25 λ112.56 374.37 λ112.56 371.73 λ111.36 368.85 λ122.4 374.37 λη0 0 0 ργφ∗166.8 374.37 µ202.8 374.37 λΣ210.96 374.37 µ199.92 379.89 λ200.88 377.25 λ201.36 374.37 λ200.88 371.73 λ199.92 368.85 λ210.96 374.37 ληφ∗166.8 374.37 µ202.8 374.37 λΣ210.96 374.37 µ199.92 379.89 λ200.88 377.25 λ201.36 374.37 λ200.88 371.73 λ199.92 368.85 λ210.96 374.37 ληφ∗188.88 374.37 µ188.88 336.69 λΣ188.88 328.53 µ194.4 339.57 λ191.76 338.61 λ188.88 338.13 λ186 338.61 λ183.36 339.57 λ188.88 328.53 ληφ∗1 1 1 ργ173.04 244.29 µ173.28 240.93 λ174.48 237.57 λ176.64 234.69 λ178.8 232.29 λ182.16 230.61 λ185.28 229.41 λ188.88 229.17 λ192.24 229.41 λ195.84 230.61 λ198.72 232.29 λ200.88 234.69 λ202.8 237.57 λ204.24 240.93 λ204.48 244.29 λ204.24 247.89 λ202.8 251.01 λ200.88 253.89 λ198.72 256.29 λ195.84 258.21 λ192.24 259.17 λ188.88 259.65 λ185.28 259.17 λ182.16 258.21 λ178.8 256.29 λ176.64 253.89 λ174.48 251.01 λ173.28 247.89 λ173.04 244.29 ληβ∗173.04 244.29 µ173.28 240.93 λ174.48 237.57 λ176.64 234.69 λ178.8 232.29 λ182.16 230.61 λ185.28 229.41 λ188.88 229.17 λ192.24 229.41 λ195.84 230.61 λ198.72 232.29 λ200.88 234.69 λ202.8 237.57 λ204.24 240.93 λ204.48 244.29 λ204.24 247.89 λ202.8 251.01 λ200.88 253.89 λ198.72 256.29 λ195.84 258.21 λ192.24 259.17 λ188.88 259.65 λ185.28 259.17 λ182.16 258.21 λ178.8 256.29 λ176.64 253.89 λ174.48 251.01 λ173.28 247.89 λ173.04 244.29 ληφ∗173.04 244.29 µ173.28 240.93 λ174.48 237.57 λ176.64 234.69 λ178.8 232.29 λ182.16 230.61 λ185.28 229.41 λ188.88 229.17 λ192.24 229.41 λ195.84 230.61 λ198.72 232.29 λ200.88 234.69 λ202.8 237.57 λ204.24 240.93 λ204.48 244.29 λ204.24 247.89 λ202.8 251.01 λ200.88 253.89 λ198.72 256.29 λ195.84 258.21 λ192.24 259.17 λ188.88 259.65 λ185.28 259.17 λ182.16 258.21 λ178.8 256.29 λ176.64 253.89 λ174.48 251.01 λ173.28 247.89 λ173.04 244.29 ληΣ0 γ0 ΓΒΤ/Φ1 20.398 Τφ1 0 0 1 182.4 237.33 Τµ 20.398 ΤΛ(+)Τϕ0 0 0 ΡΓΕΤ1 1 1 ργ177.84 328.53 µ188.88 305.49 λ199.92 328.53 λ177.84 328.53 ληβ∗177.84 328.53 µ188.88 305.49 λ199.92 328.53 λ177.84 328.53 ληφ∗177.84 328.53 µ188.88 305.49 λ199.92 328.53 λ177.84 328.53 ληΣ188.88 305.49 µ188.88 268.05 λΣ188.88 259.65 µ194.4 270.69 λ191.76 269.73 λ188.88 269.49 λ186 269.73 λ183.36 270.69 λ188.88 259.65 λη0 0 0 ργφ∗θ255.16 374.17 40 0.16 ρεΩ∗ ν255.12 374.37 µ299.52 374.37 λΣΘ277.44 374.37 µ277.44 336.69 λΣ277.44 328.53 µ282.96 339.57 λ280.32 338.61 λ277.44 338.13 λ274.56 338.61 λ271.92 339.57 λ277.44 328.53 ληφ∗0 γ0 ΓΒΤ/Φ1 20.398 Τφ1 0 0 1 271.44 237.33 Τµ 20.398 ΤΛ(+)Τϕ0 0 0 ΡΓΕΤ1 1 1 ργ266.16 328.53 µ277.44 305.49 λ288.48 328.53 λ266.16 328.53 ληβ∗266.16 328.53 µ277.44 305.49 λ288.48 328.53 λ266.16 328.53 ληφ∗266.16 328.53 µ277.44 305.49 λ288.48 328.53 λ266.16 328.53 ληΣ277.44 305.49 µ277.44 268.05 λΣ277.44 259.65 µ282.96 270.69 λ280.32 269.73 λ277.44 269.49 λ274.56 269.73 λ271.92 270.69 λ277.44 259.65 λη0 0 0 ργφ∗100.32 374.37 µ100.32 336.69 λΣ100.32 328.53 µ105.6 339.57 λ102.96 338.61 λ100.32 338.13 λ97.44 338.61 λ94.8 339.57 λ100.32 328.53 ληφ∗1 1 1 ργ89.28 328.53 µ100.32 305.49 λ111.36 328.53 λ89.28 328.53 ληβ∗89.28 328.53 µ100.32 305.49 λ111.36 328.53 λ89.28 328.53 ληφ∗89.28 328.53 µ100.32 305.49 λ111.36 328.53 λ89.28 328.53 ληΣ100.32 305.49 µ100.32 244.29 λ169.44 244.29 λΣ172.56 244.05 µ161.52 249.57 λ162.48 246.93 λ162.72 244.05 λ162.48 241.17 λ161.52 238.53 λ172.56 244.05 λη0 0 0 ργφ∗204.48 244.29 µ258 244.29 λΣ261.6 244.05 µ250.56 249.57 λ251.52 246.93 λ252 244.05 λ251.52 241.41 λ250.56 238.77 λ261.6 244.05 ληφ∗θ295.24 244.09 −0.08 0.16 ρεΩ∗ ν295.2 244.29 µ332.88 244.29 λΣΘ0 γ0 ΓΒΤ/Φ1 13.664 Τφ1 0 0 1 118.08 316.29 Τµ 13.664 ΤΛ(β)Τϕ/Φ1 10.199 Τφ1 0 0 1 123.6 309.33 Τµ 10.199 ΤΛ(0)Τϕ/Φ1 13.664 Τφ1 0 0 1 295.2 316.29 Τµ 13.664 ΤΛ(β)Τϕ/Φ1 13.664 Τφ1 0 0 1 50.4 371.49 Τµ (ξ)Τϕ1 0 0 1 57.12 371.49 Τµ (()Τϕ1 0 0 1 61.68 371.49 Τµ (ν)Τϕ1 0 0 1 69.36 371.49 Τµ ())Τϕ/Φ1 13.664 Τφ1 0 0 1 210.48 316.29 Τµ (β)Τϕ/Φ1 10.199 Τφ1 0 0 1 216 309.33 Τµ 10.199 ΤΛ(1)ΤϕΕΤ0.996 0.608 0.012 ργ0.996 0.608 0.012 ΡΓΒΤ/Φ3 24 Τφ1 0 0 1 −2.64 581.49 Τµ 24 ΤΛ(υ)Τϕ/Φ3 24 Τφ1 0 0 1 −2.64 581.49 Τµ (υ)Τϕ1 γ1 Γ/Φ0 31.922 Τφ1 0 0 1 40.8 581.49 Τµ 31.922 ΤΛ(∆)Τϕ1 0 0 1 63.84 581.49 Τµ (ι)Τϕ1 0 0 1 72.72 581.49 Τµ (ρ)Τϕ1 0 0 1 86.4 581.49 Τµ (ε)Τϕ1 0 0 1 100.56 581.49 Τµ (χ)Τϕ1 0 0 1 114.72 581.49 Τµ (τ)Τϕ1 0 0 1 133.44 581.49 Τµ (φ)Τϕ1 0 0 1 144 581.49 Τµ (ο)Τϕ1 0 0 1 160.08 581.49 Τµ (ρ)Τϕ1 0 0 1 174.24 581.49 Τµ (µ)Τϕ1 0 0 1 208.8 581.49 Τµ (σ)Τϕ1 0 0 1 221.28 581.49 Τµ (τ)Τϕ1 0 0 1 231.84 581.49 Τµ (ρ)Τϕ1 0 0 1 246 581.49 Τµ (υ)Τϕ1 0 0 1 263.76 581.49 Τµ (χ)Τϕ1 0 0 1 277.92 581.49 Τµ (τ)Τϕ1 0 0 1 288.48 581.49 Τµ (υ)Τϕ1 0 0 1 306.24 581.49 Τµ (ρ)Τϕ1 0 0 1 319.92 581.49 Τµ (ε)Τϕ1 0 0 1 342.48 581.49 Τµ (φ)Τϕ1 0 0 1 353.04 581.49 Τµ (ο)Τϕ1 0 0 1 369.12 581.49 Τµ (ρ)Τϕ1 0 0 1 390.72 581.49 Τµ (α)Τϕ1 0 0 1 406.8 581.49 Τµ (ν)Τϕ1 0 0 1 432.48 581.49 Τµ (Φ)Τϕ1 0 0 1 451.92 581.49 Τµ (Ι)Τϕ1 0 0 1 464.4 581.49 Τµ (Ρ)Τϕ1 0 0 1 495.6 581.49 Τµ (φ)Τϕ1 0 0 1 506.16 581.49 Τµ (ι)Τϕ1 0 0 1 515.04 581.49 Τµ (λ)Τϕ1 0 0 1 523.92 581.49 Τµ (τ)Τϕ1 0 0 1 534.48 581.49 Τµ (ε)Τϕ1 0 0 1 548.64 581.49 Τµ (ρ)Τϕ1 0 0 1 562.8 581.49 Τµ (:)Τϕ/Φ0 31.922 Τφ1 0 0 1 40.8 581.49 Τµ (∆)Τϕ1 0 0 1 63.84 581.49 Τµ (ι)Τϕ1 0 0 1 72.72 581.49 Τµ (ρ)Τϕ1 0 0 1 86.4 581.49 Τµ (ε)Τϕ1 0 0 1 100.56 581.49 Τµ (χ)Τϕ1 0 0 1 114.72 581.49 Τµ (τ)Τϕ1 0 0 1 133.44 581.49 Τµ (φ)Τϕ1 0 0 1 144 581.49 Τµ (ο)Τϕ1 0 0 1 160.08 581.49 Τµ (ρ)Τϕ1 0 0 1 174.24 581.49 Τµ (µ)Τϕ1 0 0 1 208.8 581.49 Τµ (σ)Τϕ1 0 0 1 221.28 581.49 Τµ (τ)Τϕ1 0 0 1 231.84 581.49 Τµ (ρ)Τϕ1 0 0 1 246 581.49 Τµ (υ)Τϕ1 0 0 1 263.76 581.49 Τµ (χ)Τϕ1 0 0 1 277.92 581.49 Τµ (τ)Τϕ1 0 0 1 288.48 581.49 Τµ (υ)Τϕ1 0 0 1 306.24 581.49 Τµ (ρ)Τϕ1 0 0 1 319.92 581.49 Τµ (ε)Τϕ1 0 0 1 342.48 581.49 Τµ (φ)Τϕ1 0 0 1 353.04 581.49 Τµ (ο)Τϕ1 0 0 1 369.12 581.49 Τµ (ρ)Τϕ1 0 0 1 390.72 581.49 Τµ (α)Τϕ1 0 0 1 406.8 581.49 Τµ (ν)Τϕ1 0 0 1 432.48 581.49 Τµ (Φ)Τϕ1 0 0 1 451.92 581.49 Τµ (Ι)Τϕ1 0 0 1 464.4 581.49 Τµ (Ρ)Τϕ1 0 0 1 495.6 581.49 Τµ (φ)Τϕ1 0 0 1 506.16 581.49 Τµ (ι)Τϕ1 0 0 1 515.04 581.49 Τµ (λ)Τϕ1 0 0 1 523.92 581.49 Τµ (τ)Τϕ1 0 0 1 534.48 581.49 Τµ (ε)Τϕ1 0 0 1 548.64 581.49 Τµ (ρ)Τϕ1 0 0 1 562.8 581.49 Τµ (:)ΤϕΕΤ296.16 152.61 295.92 539.76 ρε0.016 0.165 0.643 ργφ0.996 0.608 0.012 ργ0.996 0.608 0.012 ΡΓΒΤ/Φ2 12 Τφ1 0 0 1 327.84 155.25 Τµ 12 ΤΛ(∆)Τϕ1 0 0 1 336.48 155.25 Τµ (ρ)Τϕ1 0 0 1 339.84 155.25 Τµ (.)Τϕ1 0 0 1 345.6 155.25 Τµ (Ν)Τϕ1 0 0 1 354.24 155.25 Τµ (α)Τϕ1 0 0 1 359.52 155.25 Τµ (ι)Τϕ1 0 0 1 362.4 155.25 Τµ (µ)Τϕ1 0 0 1 375.36 155.25 Τµ (∆)Τϕ1 0 0 1 384 155.25 Τµ (α)Τϕ1 0 0 1 389.28 155.25 Τµ (η)Τϕ1 0 0 1 395.04 155.25 Τµ (ν)Τϕ1 0 0 1 400.8 155.25 Τµ (ο)Τϕ1 0 0 1 407.04 155.25 Τµ (υ)Τϕ1 0 0 1 413.04 155.25 Τµ (ν)Τϕ1 0 0 1 418.8 155.25 Τµ (,)Τϕ1 0 0 1 425.28 155.25 Τµ (Β)Τϕ1 0 0 1 433.2 155.25 Τµ (ρ)Τϕ1 0 0 1 437.28 155.25 Τµ (ι)Τϕ1 0 0 1 440.16 155.25 Τµ (σ)Τϕ1 0 0 1 444.72 155.25 Τµ (τ)Τϕ1 0 0 1 448.32 155.25 Τµ (ο)Τϕ1 0 0 1 454.56 155.25 Τµ (λ)Τϕ1 0 0 1 461.04 155.25 Τµ (Υ)Τϕ1 0 0 1 469.68 155.25 Τµ (ν)Τϕ1 0 0 1 475.44 155.25 Τµ (ι)Τϕ1 0 0 1 478.32 155.25 Τµ (ϖ)Τϕ1 0 0 1 484.32 155.25 Τµ (ε)Τϕ1 0 0 1 489.6 155.25 Τµ (ρ)Τϕ1 0 0 1 493.68 155.25 Τµ (σ)Τϕ1 0 0 1 498.48 155.25 Τµ (ι)Τϕ1 0 0 1 501.6 155.25 Τµ (τ)Τϕ1 0 0 1 505.2 155.25 Τµ (ψ)Τϕ1 0 0 1 510 155.25 Τµ (,)Τϕ1 0 0 1 520.08 155.25 Τµ (()Τϕ1 0 0 1 524.16 155.25 Τµ (χ)Τϕ1 0 0 1 529.44 155.25 Τµ ())Τϕ1 0 0 1 536.16 155.25 Τµ (Τ)Τϕ1 0 0 1 542.64 155.25 Τµ (ε)Τϕ1 0 0 1 547.92 155.25 Τµ (ξ)Τϕ1 0 0 1 553.68 155.25 Τµ (α)Τϕ1 0 0 1 558.96 155.25 Τµ (σ)Τϕ1 0 0 1 567.12 155.25 Τµ (Ι)Τϕ1 0 0 1 571.2 155.25 Τµ (ν)Τϕ1 0 0 1 576.96 155.25 Τµ (σ)Τϕ1 0 0 1 581.52 155.25 Τµ (τ)Τϕ1 0 0 1 585.12 155.25 Τµ (ρ)Τϕ1 0 0 1 589.2 155.25 Τµ (υ)Τϕ1 0 0 1 595.2 155.25 Τµ (µ)Τϕ1 0 0 1 604.08 155.25 Τµ (ε)Τϕ1 0 0 1 609.36 155.25 Τµ (ν)Τϕ1 0 0 1 615.12 155.25 Τµ (τ)Τϕ1 0 0 1 618.72 155.25 Τµ (σ)Τϕ1 0 0 1 626.88 155.25 Τµ (2)Τϕ1 0 0 1 632.88 155.25 Τµ (0)Τϕ1 0 0 1 638.88 155.25 Τµ (0)Τϕ1 0 0 1 644.88 155.25 Τµ (4)Τϕ/Φ0 36 Τφ1 0 0 1 −7.44 651.33 Τµ 36 ΤΛ(Ρ)Τϕ1 0 0 1 18.48 651.33 Τµ (ε)Τϕ1 0 0 1 34.56 651.33 Τµ (α)Τϕ1 0 0 1 52.56 651.33 Τµ (λ)Τϕ1 0 0 1 62.64 651.33 Τµ (ι)Τϕ1 0 0 1 72.72 651.33 Τµ (σ)Τϕ1 0 0 1 86.64 651.33 Τµ (α)Τϕ1 0 0 1 104.64 651.33 Τµ (τ)Τϕ1 0 0 1 116.64 651.33 Τµ (ι)Τϕ1 0 0 1 126.72 651.33 Τµ (ο)Τϕ1 0 0 1 144.72 651.33 Τµ (ν)Τϕ1 0 0 1 173.28 651.33 Τµ (Σ)Τϕ1 0 0 1 193.2 651.33 Τµ (τ)Τϕ1 0 0 1 205.2 651.33 Τµ (ρ)Τϕ1 0 0 1 221.28 651.33 Τµ (υ)Τϕ1 0 0 1 241.2 651.33 Τµ (χ)Τϕ1 0 0 1 257.28 651.33 Τµ (τ)Τϕ1 0 0 1 269.28 651.33 Τµ (υ)Τϕ1 0 0 1 289.2 651.33 Τµ (ρ)Τϕ1 0 0 1 305.28 651.33 Τµ (ε)Τϕ1 0 0 1 330.24 651.33 Τµ (Σ)Τϕ1 0 0 1 350.16 651.33 Τµ (ε)Τϕ1 0 0 1 366.24 651.33 Τµ (λ)Τϕ1 0 0 1 376.32 651.33 Τµ (ε)Τϕ1 0 0 1 392.4 651.33 Τµ (χ)Τϕ1 0 0 1 408.48 651.33 Τµ (τ)Τϕ1 0 0 1 420.48 651.33 Τµ (ι)Τϕ1 0 0 1 430.56 651.33 Τµ (ο)Τϕ1 0 0 1 448.56 651.33 Τµ (ν)Τϕ/Φ0 36 Τφ1 0 0 1 −7.44 651.33 Τµ (Ρ)Τϕ1 0 0 1 18.48 651.33 Τµ (ε)Τϕ1 0 0 1 34.56 651.33 Τµ (α)Τϕ1 0 0 1 52.56 651.33 Τµ (λ)Τϕ1 0 0 1 62.64 651.33 Τµ (ι)Τϕ1 0 0 1 72.72 651.33 Τµ (σ)Τϕ1 0 0 1 86.64 651.33 Τµ (α)Τϕ1 0 0 1 104.64 651.33 Τµ (τ)Τϕ1 0 0 1 116.64 651.33 Τµ (ι)Τϕ1 0 0 1 126.72 651.33 Τµ (ο)Τϕ1 0 0 1 144.72 651.33 Τµ (ν)Τϕ1 0 0 1 173.28 651.33 Τµ (Σ)Τϕ1 0 0 1 193.2 651.33 Τµ (τ)Τϕ1 0 0 1 205.2 651.33 Τµ (ρ)Τϕ1 0 0 1 221.28 651.33 Τµ (υ)Τϕ1 0 0 1 241.2 651.33 Τµ (χ)Τϕ1 0 0 1 257.28 651.33 Τµ (τ)Τϕ1 0 0 1 269.28 651.33 Τµ (υ)Τϕ1 0 0 1 289.2 651.33 Τµ (ρ)Τϕ1 0 0 1 305.28 651.33 Τµ (ε)Τϕ1 0 0 1 330.24 651.33 Τµ (Σ)Τϕ1 0 0 1 350.16 651.33 Τµ (ε)Τϕ1 0 0 1 366.24 651.33 Τµ (λ)Τϕ1 0 0 1 376.32 651.33 Τµ (ε)Τϕ1 0 0 1 392.4 651.33 Τµ (χ)Τϕ1 0 0 1 408.48 651.33 Τµ (τ)Τϕ1 0 0 1 420.48 651.33 Τµ (ι)Τϕ1 0 0 1 430.56 651.33 Τµ (ο)Τϕ1 0 0 1 448.56 651.33 Τµ (ν)Τϕ/Φ0 36 Τφ1 0 0 1 477.36 651.33 Τµ (−)Τϕ/Φ0 36 Τφ1 0 0 1 477.36 651.33 Τµ (−)Τϕ/Φ0 36 Τφ1 0 0 1 498.24 651.33 Τµ (Σ)Τϕ1 0 0 1 518.16 651.33 Τµ (τ)Τϕ1 0 0 1 530.16 651.33 Τµ (ε)Τϕ1 0 0 1 546.24 651.33 Τµ (π)Τϕ1 0 0 1 575.28 651.33 Τµ (3)Τϕ/Φ0 36 Τφ1 0 0 1 498.24 651.33 Τµ (Σ)Τϕ1 0 0 1 518.16 651.33 Τµ (τ)Τϕ1 0 0 1 530.16 651.33 Τµ (ε)Τϕ1 0 0 1 546.24 651.33 Τµ (π)Τϕ1 0 0 1 575.28 651.33 Τµ (3)ΤϕΕΤ296.16 500.61 65.76 59.76 ρε1 1 1 ργφθ297.16 500.41 64.96 59.44 ρεΩ ν223.68 560.85 µ362.88 560.85 λ362.88 499.65 λ223.68 499.65 λ223.68 560.85 λη224.88 500.37 µ224.16 500.85 λ362.16 500.85 λ361.68 500.37 λ361.68 560.37 λ362.16 559.65 λ224.16 559.65 λ224.88 560.37 λ224.88 500.37 ληφΘ0 γ0 ΓΒΤ/Φ4 31.828 Τφ1 0 0 1 288 519.33 Τµ 31.828 ΤΛ(∑)Τϕ/Φ4 11.941 Τφ1 0 0 1 299.04 545.73 Τµ 11.941 ΤΛ(−)Τϕ/Φ4 11.941 Τφ1 0 0 1 296.16 504.45 Τµ (=)Τϕ/Φ4 11.941 Τφ1 0 0 1 340.8 533.49 Τµ (−)Τϕ/Φ2 11.941 Τφ1 0 0 1 305.28 545.73 Τµ (1)Τϕ/Φ2 11.941 Τφ1 0 0 1 303.6 504.45 Τµ (0)Τϕ/Φ5 11.941 Τφ1 0 0 1 288.96 545.73 Τµ (Ν)Τϕ/Φ5 11.941 Τφ1 0 0 1 348.24 533.49 Τµ (κ)Τϕ/Φ5 11.941 Τφ1 0 0 1 322.32 519.57 Τµ (κ)Τϕ/Φ5 19.891 Τφ1 0 0 1 330.96 524.61 Τµ 19.891 ΤΛ(ζ)Τϕ/Φ5 19.891 Τφ1 0 0 1 312.72 524.61 Τµ (β)ΤϕΕΤ296.16 428.61 281.76 41.76 ρε1 1 1 ργφθ297.16 428.41 280.96 41.44 ρεΩ ν187.68 470.85 µ578.88 470.85 λ578.88 427.65 λ187.68 427.65 λ187.68 470.85 λη188.88 428.37 µ188.16 428.85 λ578.16 428.85 λ577.68 428.37 λ577.68 470.37 λ578.16 469.65 λ188.16 469.65 λ188.88 470.37 λ188.88 428.37 ληφΘ0 γΒΤ/Φ4 27.121 Τφ1 0 0 1 297.84 445.89 Τµ 27.121 ΤΛ()()Τϕ/Φ4 27.121 Τφ1 0 0 1 386.4 445.89 Τµ ()()Τϕ/Φ4 27.121 Τφ1 0 0 1 563.52 445.89 Τµ ()1....1 11 +−++−+ − Nnxbnxbn N

z-1

+

b2

bN-1

y(n)

Direct form structure for an FIR filter:Direct form structure for an FIR filter:





()Τϕ/Φ4 26.984 Τφ1 0 0 1 262.32 524.61 Τµ ()∑=N

k

zH

uu Linear phase structures:Linear phase structures:

ww N even:N even:

ww N Odd:N Odd:

()Τϕ/Φ4 27.074 Τφ1 0 0 1 279.6 364.29 Τµ ()=zH

()Τϕ/Φ4 27.148 Τφ1 0 0 1 280.08 246.45 Τµ ()=zH

uu Direct form structure for an FIR filter:Direct form structure for an FIR filter:


Realisation Structure SelectionRealisation Structure Selection -- Step 3Step 3

∑−

=

−1

0

Nk

k zb

Linear phase structures:Linear phase structures:

()Τϕ/Φ4 33.434 Τφ1 0 0 1 451.44 364.53 Τµ ()∑−

=

−−− +=1

2

0

1

N

k

kNkk zzb

()Τϕ/Φ4 33.539 Τφ1 0 0 1 451.68 246.45 Τµ ()∑−

=

−−

−−−− ++=

21

0

21

21

1

N

k

N

NkNk

k zbzzb






(a) N even.(a) N even.(b) N odd.(b) N odd.

z-1

z-1

x(n)z

-1z

-1



+

b+

+

+

b

b

b

y(n)

(a)

z-1

z-1

z-1

+

b0+

+

+

b1

b2

b(N-3)/2

y(n)

b(N-1)/2

+

(b)

z-1

z-1

z-1

z-1





()Τϕ/Φ4 26.984 Τφ1 0 0 1 262.32 524.61 Τµ ()∑=N

k

zH

uu Cascade structures:Cascade structures:

()Τϕ/Φ4 27.121 Τφ1 0 0 1 118.08 371.73 Τµ ()

()Τϕ/Φ4 31.98 Τφ1 0 0 1 161.04 239.25 Τµ 31.98 ΤΛ(∏)Τϕ/Φ4 31.98 Τφ1 0 0 1 143.76 366.45 Τµ (∑)Τϕ/Φ4 12 Τφ1 0 0 1 172.32 224.13 Τµ 12 ΤΛ(=)Τϕ/Φ4 12 Τφ1 0 0 1 254.88 253.65 Τµ (−)Τϕ/Φ4 12 Τφ1 0 0 1 229.2 317.25 Τµ (−)Τϕ/Φ4 12 Τφ1 0 0 1 294.48 380.61 Τµ (−)Τϕ/Φ4 12 Τφ1 0 0 1 154.8 392.85 Τµ (−)Τϕ/Φ4 12 Τφ1 0 0 1 152.16 351.33 Τµ (=)Τϕ/Φ4 12 Τφ1 0 0 1 196.8 380.61 Τµ (−)Τϕ/Φ4 19.996 Τφ1 0 0 1 271.92 244.77 Τµ 19.996 ΤΛ(+)Τϕ/Φ4 19.996 Τφ1 0 0 1 204.48 244.77 Τµ (+)Τϕ/Φ4 19.996 Τφ1 0 0 1 127.92 244.77 Τµ (=)Τϕ/Φ4 19.996 Τφ1 0 0 1 162 303.81 Τµ ()Τϕ/Φ4 19.996 Τφ1 0 0 1 162 287.01 Τµ ()Τϕ/Φ4 19.996 Τφ1 0 0 1 162 322.77 Τµ ()Τϕ/Φ4 19.996 Τφ1 0 0 1 246.48 308.37 Τµ (+)Τϕ/Φ4 19.996 Τφ1 0 0 1 179.52 308.37 Τµ (+)Τϕ/Φ4 19.996 Τφ1 0 0 1 127.92 308.37 Τµ (=)Τϕ/Φ4 19.996 Τφ1 0 0 1 254.88 371.73 Τµ (+)Τϕ/Φ4 19.996 Τφ1 0 0 1 217.92 371.73 Τµ (=)Τϕ/Φ4 19.996 Τφ1 0 0 1 127.92 371.73 Τµ (=)Τϕ/Φ5 12 Τφ1 0 0 1 168.48 266.13 Τµ 12 ΤΛ(Μ)Τϕ/Φ5 12 Τφ1 0 0 1 165.36 224.13 Τµ (κ)Τϕ/Φ5 12 Τφ1 0 0 1 295.92 239.73 Τµ (κ)Τϕ/Φ5 12 Τφ1 0 0 1 228.24 239.73 Τµ (κ)Τϕ/Φ5 12 Τφ1 0 0 1 144.72 392.85 Τµ (Ν)Τϕ/Φ5 12 Τφ1 0 0 1 144.96 351.33 Τµ (κ)Τϕ/Φ5 12 Τφ1 0 0 1 204.24 380.61 Τµ (κ)Τϕ/Φ5 12 Τφ1 0 0 1 178.08 366.69 Τµ (κ)Τϕ/Φ5 19.996 Τφ1 0 0 1 286.08 244.77 Τµ 19.996 ΤΛ(β)Τϕ/Φ5 19.996 Τφ1 0 0 1 245.04 244.77 Τµ (ζ)Τϕ/Φ5 19.996 Τφ1 0 0 1 218.64 244.77 Τµ (β)Τϕ/Φ5 19.996 Τφ1 0 0 1 143.28 244.77 Τµ (β)Τϕ/Φ5 19.996 Τφ1 0 0 1 286.32 308.37 Τµ (ζ)Τϕ/Φ5 19.996 Τφ1 0 0 1 262.56 292.53 Τµ (β)Τϕ/Φ5 19.996 Τφ1 0 0 1 262.56 321.33 Τµ (β)Τϕ/Φ5 19.996 Τφ1 0 0 1 219.36 308.37 Τµ (ζ)Τϕ/Φ5 19.996 Τφ1 0 0 1 195.84 292.53 Τµ (β)Τϕ/Φ5 19.996 Τφ1 0 0 1 196.8 321.33 Τµ (β)Τϕ/Φ5 19.996 Τφ1 0 0 1 143.28 308.37 Τµ (β)Τϕ/Φ5 19.996 Τφ1 0 0 1 284.64 371.73 Τµ (ζ)Τϕ/Φ5 19.996 Τφ1 0 0 1 269.04 371.73 Τµ (β)Τϕ/Φ5 19.996 Τφ1 0 0 1 233.28 371.73 Τµ (β)Τϕ/Φ5 19.996 Τφ1 0 0 1 186.96 371.73 Τµ (ζ)Τϕ/Φ5 19.996 Τφ1 0 0 1 168.48 371.73 Τµ (β)Τϕ/Φ5 19.996 Τφ1 0 0 1 108.72 371.73 Τµ (ζ)Τϕ/Φ5 19.996 Τφ1 0 0 1 84.48 371.73 Τµ (Η)Τϕ/Φ2 12 Τφ1 0 0 1 178.56 224.13 Τµ 12 ΤΛ(1)Τϕ/Φ2 12 Τφ1 0 0 1 261.12 253.65 Τµ (1)Τϕ/Φ2 12 Τφ1 0 0 1 237.6 239.73 Τµ (1)Τϕ/Φ2 12 Τφ1 0 0 1 234.96 239.73 Τµ (,)Τϕ/Φ2 12 Τφ1 0 0 1 152.64 239.73 Τµ (0)Τϕ/Φ2 12 Τφ1 0 0 1 272.16 287.49 Τµ (0)Τϕ/Φ2 12 Τφ1 0 0 1 272.16 316.29 Τµ (2)Τϕ/Φ2 12 Τφ1 0 0 1 235.44 317.25 Τµ (1)Τϕ/Φ2 12 Τφ1 0 0 1 205.2 287.49 Τµ (0)Τϕ/Φ2 12 Τφ1 0 0 1 205.2 316.29 Τµ (1)Τϕ/Φ2 12 Τφ1 0 0 1 152.64 303.33 Τµ (0)Τϕ/Φ2 12 Τφ1 0 0 1 277.44 366.69 Τµ (1)Τϕ/Φ2 12 Τφ1 0 0 1 242.88 366.69 Τµ (0)Τϕ/Φ2 12 Τφ1 0 0 1 161.04 392.85 Τµ (1)Τϕ/Φ2 12 Τφ1 0 0 1 159.36 351.33 Τµ (0)Τϕ/Φ2 19.996 Τφ1 0 0 1 192.72 244.77 Τµ 19.996 ΤΛ(1)Τϕ/Φ2 19.996 Τφ1 0 0 1 168 308.37 Τµ (1)ΤϕΕΤ0.996 0.608 0.012 ργ0.996 0.608 0.012 ΡΓΒΤ/Φ3 24 Τφ1 0 0 1 −2.64 581.49 Τµ 24 ΤΛ(υ)Τϕ/Φ3 24 Τφ1 0 0 1 −2.64 581.49 Τµ (υ)Τϕ1 γ1 Γ/Φ0 31.922 Τφ1 0 0 1 40.8 581.49 Τµ 31.922 ΤΛ(∆)Τϕ1 0 0 1 63.84 581.49 Τµ (ι)Τϕ1 0 0 1 72.72 581.49 Τµ (ρ)Τϕ1 0 0 1 86.4 581.49 Τµ (ε)Τϕ1 0 0 1 100.56 581.49 Τµ (χ)Τϕ1 0 0 1 114.72 581.49 Τµ (τ)Τϕ1 0 0 1 133.44 581.49 Τµ (φ)Τϕ1 0 0 1 144 581.49 Τµ (ο)Τϕ1 0 0 1 160.08 581.49 Τµ (ρ)Τϕ1 0 0 1 174.24 581.49 Τµ (µ)Τϕ1 0 0 1 208.8 581.49 Τµ (σ)Τϕ1 0 0 1 221.28 581.49 Τµ (τ)Τϕ1 0 0 1 231.84 581.49 Τµ (ρ)Τϕ1 0 0 1 246 581.49 Τµ (υ)Τϕ1 0 0 1 263.76 581.49 Τµ (χ)Τϕ1 0 0 1 277.92 581.49 Τµ (τ)Τϕ1 0 0 1 288.48 581.49 Τµ (υ)Τϕ1 0 0 1 306.24 581.49 Τµ (ρ)Τϕ1 0 0 1 319.92 581.49 Τµ (ε)Τϕ1 0 0 1 342.48 581.49 Τµ (φ)Τϕ1 0 0 1 353.04 581.49 Τµ (ο)Τϕ1 0 0 1 369.12 581.49 Τµ (ρ)Τϕ1 0 0 1 390.72 581.49 Τµ (α)Τϕ1 0 0 1 406.8 581.49 Τµ (ν)Τϕ1 0 0 1 432.48 581.49 Τµ (Φ)Τϕ1 0 0 1 451.92 581.49 Τµ (Ι)Τϕ1 0 0 1 464.4 581.49 Τµ (Ρ)Τϕ1 0 0 1 495.6 581.49 Τµ (φ)Τϕ1 0 0 1 506.16 581.49 Τµ (ι)Τϕ1 0 0 1 515.04 581.49 Τµ (λ)Τϕ1 0 0 1 523.92 581.49 Τµ (τ)Τϕ1 0 0 1 534.48 581.49 Τµ (ε)Τϕ1 0 0 1 548.64 581.49 Τµ (ρ)Τϕ1 0 0 1 562.8 581.49 Τµ (:)Τϕ/Φ0 31.922 Τφ1 0 0 1 40.8 581.49 Τµ (∆)Τϕ1 0 0 1 63.84 581.49 Τµ (ι)Τϕ1 0 0 1 72.72 581.49 Τµ (ρ)Τϕ1 0 0 1 86.4 581.49 Τµ (ε)Τϕ1 0 0 1 100.56 581.49 Τµ (χ)Τϕ1 0 0 1 114.72 581.49 Τµ (τ)Τϕ1 0 0 1 133.44 581.49 Τµ (φ)Τϕ1 0 0 1 144 581.49 Τµ (ο)Τϕ1 0 0 1 160.08 581.49 Τµ (ρ)Τϕ1 0 0 1 174.24 581.49 Τµ (µ)Τϕ1 0 0 1 208.8 581.49 Τµ (σ)Τϕ1 0 0 1 221.28 581.49 Τµ (τ)Τϕ1 0 0 1 231.84 581.49 Τµ (ρ)Τϕ1 0 0 1 246 581.49 Τµ (υ)Τϕ1 0 0 1 263.76 581.49 Τµ (χ)Τϕ1 0 0 1 277.92 581.49 Τµ (τ)Τϕ1 0 0 1 288.48 581.49 Τµ (υ)Τϕ1 0 0 1 306.24 581.49 Τµ (ρ)Τϕ1 0 0 1 319.92 581.49 Τµ (ε)Τϕ1 0 0 1 342.48 581.49 Τµ (φ)Τϕ1 0 0 1 353.04 581.49 Τµ (ο)Τϕ1 0 0 1 369.12 581.49 Τµ (ρ)Τϕ1 0 0 1 390.72 581.49 Τµ (α)Τϕ1 0 0 1 406.8 581.49 Τµ (ν)Τϕ1 0 0 1 432.48 581.49 Τµ (Φ)Τϕ1 0 0 1 451.92 581.49 Τµ (Ι)Τϕ1 0 0 1 464.4 581.49 Τµ (Ρ)Τϕ1 0 0 1 495.6 581.49 Τµ (φ)Τϕ1 0 0 1 506.16 581.49 Τµ (ι)Τϕ1 0 0 1 515.04 581.49 Τµ (λ)Τϕ1 0 0 1 523.92 581.49 Τµ (τ)Τϕ1 0 0 1 534.48 581.49 Τµ (ε)Τϕ1 0 0 1 548.64 581.49 Τµ (ρ)Τϕ1 0 0 1 562.8 581.49 Τµ (:)ΤϕΕΤ296.16 152.61 295.92 539.76 ρε0.016 0.165 0.643 ργφ0.996 0.608 0.012 ργ0.996 0.608 0.012 ΡΓΒΤ/Φ2 12 Τφ1 0 0 1 327.84 155.25 Τµ 12 ΤΛ(∆)Τϕ1 0 0 1 336.48 155.25 Τµ (ρ)Τϕ1 0 0 1 339.84 155.25 Τµ (.)Τϕ1 0 0 1 345.6 155.25 Τµ (Ν)Τϕ1 0 0 1 354.24 155.25 Τµ (α)Τϕ1 0 0 1 359.52 155.25 Τµ (ι)Τϕ1 0 0 1 362.4 155.25 Τµ (µ)Τϕ1 0 0 1 375.36 155.25 Τµ (∆)Τϕ1 0 0 1 384 155.25 Τµ (α)Τϕ1 0 0 1 389.28 155.25 Τµ (η)Τϕ1 0 0 1 395.04 155.25 Τµ (ν)Τϕ1 0 0 1 400.8 155.25 Τµ (ο)Τϕ1 0 0 1 407.04 155.25 Τµ (υ)Τϕ1 0 0 1 413.04 155.25 Τµ (ν)Τϕ1 0 0 1 418.8 155.25 Τµ (,)Τϕ1 0 0 1 425.28 155.25 Τµ (Β)Τϕ1 0 0 1 433.2 155.25 Τµ (ρ)Τϕ1 0 0 1 437.28 155.25 Τµ (ι)Τϕ1 0 0 1 440.16 155.25 Τµ (σ)Τϕ1 0 0 1 444.72 155.25 Τµ (τ)Τϕ1 0 0 1 448.32 155.25 Τµ (ο)Τϕ1 0 0 1 454.56 155.25 Τµ (λ)Τϕ1 0 0 1 461.04 155.25 Τµ (Υ)Τϕ1 0 0 1 469.68 155.25 Τµ (ν)Τϕ1 0 0 1 475.44 155.25 Τµ (ι)Τϕ1 0 0 1 478.32 155.25 Τµ (ϖ)Τϕ1 0 0 1 484.32 155.25 Τµ (ε)Τϕ1 0 0 1 489.6 155.25 Τµ (ρ)Τϕ1 0 0 1 493.68 155.25 Τµ (σ)Τϕ1 0 0 1 498.48 155.25 Τµ (ι)Τϕ1 0 0 1 501.6 155.25 Τµ (τ)Τϕ1 0 0 1 505.2 155.25 Τµ (ψ)Τϕ1 0 0 1 510 155.25 Τµ (,)Τϕ1 0 0 1 520.08 155.25 Τµ (()Τϕ1 0 0 1 524.16 155.25 Τµ (χ)Τϕ1 0 0 1 529.44 155.25 Τµ ())Τϕ1 0 0 1 536.16 155.25 Τµ (Τ)Τϕ1 0 0 1 542.64 155.25 Τµ (ε)Τϕ1 0 0 1 547.92 155.25 Τµ (ξ)Τϕ1 0 0 1 553.68 155.25 Τµ (α)Τϕ1 0 0 1 558.96 155.25 Τµ (σ)Τϕ1 0 0 1 567.12 155.25 Τµ (Ι)Τϕ1 0 0 1 571.2 155.25 Τµ (ν)Τϕ1 0 0 1 576.96 155.25 Τµ (σ)Τϕ1 0 0 1 581.52 155.25 Τµ (τ)Τϕ1 0 0 1 585.12 155.25 Τµ (ρ)Τϕ1 0 0 1 589.2 155.25 Τµ (υ)Τϕ1 0 0 1 595.2 155.25 Τµ (µ)Τϕ1 0 0 1 604.08 155.25 Τµ (ε)Τϕ1 0 0 1 609.36 155.25 Τµ (ν)Τϕ1 0 0 1 615.12 155.25 Τµ (τ)Τϕ1 0 0 1 618.72 155.25 Τµ (σ)Τϕ1 0 0 1 626.88 155.25 Τµ (2)Τϕ1 0 0 1 632.88 155.25 Τµ (0)Τϕ1 0 0 1 638.88 155.25 Τµ (0)Τϕ1 0 0 1 644.88 155.25 Τµ (4)Τϕ/Φ0 36 Τφ1 0 0 1 −7.44 651.33 Τµ 36 ΤΛ(Ρ)Τϕ1 0 0 1 18.48 651.33 Τµ (ε)Τϕ1 0 0 1 34.56 651.33 Τµ (α)Τϕ1 0 0 1 52.56 651.33 Τµ (λ)Τϕ1 0 0 1 62.64 651.33 Τµ (ι)Τϕ1 0 0 1 72.72 651.33 Τµ (σ)Τϕ1 0 0 1 86.64 651.33 Τµ (α)Τϕ1 0 0 1 104.64 651.33 Τµ (τ)Τϕ1 0 0 1 116.64 651.33 Τµ (ι)Τϕ1 0 0 1 126.72 651.33 Τµ (ο)Τϕ1 0 0 1 144.72 651.33 Τµ (ν)Τϕ1 0 0 1 173.28 651.33 Τµ (Σ)Τϕ1 0 0 1 193.2 651.33 Τµ (τ)Τϕ1 0 0 1 205.2 651.33 Τµ (ρ)Τϕ1 0 0 1 221.28 651.33 Τµ (υ)Τϕ1 0 0 1 241.2 651.33 Τµ (χ)Τϕ1 0 0 1 257.28 651.33 Τµ (τ)Τϕ1 0 0 1 269.28 651.33 Τµ (υ)Τϕ1 0 0 1 289.2 651.33 Τµ (ρ)Τϕ1 0 0 1 305.28 651.33 Τµ (ε)Τϕ1 0 0 1 330.24 651.33 Τµ (Σ)Τϕ1 0 0 1 350.16 651.33 Τµ (ε)Τϕ1 0 0 1 366.24 651.33 Τµ (λ)Τϕ1 0 0 1 376.32 651.33 Τµ (ε)Τϕ1 0 0 1 392.4 651.33 Τµ (χ)Τϕ1 0 0 1 408.48 651.33 Τµ (τ)Τϕ1 0 0 1 420.48 651.33 Τµ (ι)Τϕ1 0 0 1 430.56 651.33 Τµ (ο)Τϕ1 0 0 1 448.56 651.33 Τµ (ν)Τϕ/Φ0 36 Τφ1 0 0 1 −7.44 651.33 Τµ (Ρ)Τϕ1 0 0 1 18.48 651.33 Τµ (ε)Τϕ1 0 0 1 34.56 651.33 Τµ (α)Τϕ1 0 0 1 52.56 651.33 Τµ (λ)Τϕ1 0 0 1 62.64 651.33 Τµ (ι)Τϕ1 0 0 1 72.72 651.33 Τµ (σ)Τϕ1 0 0 1 86.64 651.33 Τµ (α)Τϕ1 0 0 1 104.64 651.33 Τµ (τ)Τϕ1 0 0 1 116.64 651.33 Τµ (ι)Τϕ1 0 0 1 126.72 651.33 Τµ (ο)Τϕ1 0 0 1 144.72 651.33 Τµ (ν)Τϕ1 0 0 1 173.28 651.33 Τµ (Σ)Τϕ1 0 0 1 193.2 651.33 Τµ (τ)Τϕ1 0 0 1 205.2 651.33 Τµ (ρ)Τϕ1 0 0 1 221.28 651.33 Τµ (υ)Τϕ1 0 0 1 241.2 651.33 Τµ (χ)Τϕ1 0 0 1 257.28 651.33 Τµ (τ)Τϕ1 0 0 1 269.28 651.33 Τµ (υ)Τϕ1 0 0 1 289.2 651.33 Τµ (ρ)Τϕ1 0 0 1 305.28 651.33 Τµ (ε)Τϕ1 0 0 1 330.24 651.33 Τµ (Σ)Τϕ1 0 0 1 350.16 651.33 Τµ (ε)Τϕ1 0 0 1 366.24 651.33 Τµ (λ)Τϕ1 0 0 1 376.32 651.33 Τµ (ε)Τϕ1 0 0 1 392.4 651.33 Τµ (χ)Τϕ1 0 0 1 408.48 651.33 Τµ (τ)Τϕ1 0 0 1 420.48 651.33 Τµ (ι)Τϕ1 0 0 1 430.56 651.33 Τµ (ο)Τϕ1 0 0 1 448.56 651.33 Τµ (ν)Τϕ/Φ0 36 Τφ1 0 0 1 477.36 651.33 Τµ (−)Τϕ/Φ0 36 Τφ1 0 0 1 477.36 651.33 Τµ (−)Τϕ/Φ0 36 Τφ1 0 0 1 498.24 651.33 Τµ (Σ)Τϕ1 0 0 1 518.16 651.33 Τµ (τ)Τϕ1 0 0 1 530.16 651.33 Τµ (ε)Τϕ1 0 0 1 546.24 651.33 Τµ (π)Τϕ1 0 0 1 575.28 651.33 Τµ (3)Τϕ/Φ0 36 Τφ1 0 0 1 498.24 651.33 Τµ (Σ)Τϕ1 0 0 1 518.16 651.33 Τµ (τ)Τϕ1 0 0 1 530.16 651.33 Τµ (ε)Τϕ1 0 0 1 546.24 651.33 Τµ (π)Τϕ1 0 0 1 575.28 651.33 Τµ (3)ΤϕΕΤ296.16 500.61 65.76 59.76 ρε1 1 1 ργφθ297.16 500.41 64.96 59.44 ρεΩ ν223.68 560.85 µ362.88 560.85 λ362.88 499.65 λ223.68 499.65 λ223.68 560.85 λη224.88 500.37 µ224.16 500.85 λ362.16 500.85 λ361.68 500.37 λ361.68 560.37 λ362.16 559.65 λ224.16 559.65 λ224.88 560.37 λ224.88 500.37 ληφΘ0 γ0 ΓΒΤ/Φ4 31.828 Τφ1 0 0 1 288 519.33 Τµ 31.828 ΤΛ(∑)Τϕ/Φ4 11.941 Τφ1 0 0 1 299.04 545.73 Τµ 11.941 ΤΛ(−)Τϕ/Φ4 11.941 Τφ1 0 0 1 296.16 504.45 Τµ (=)Τϕ/Φ4 11.941 Τφ1 0 0 1 340.8 533.49 Τµ (−)Τϕ/Φ2 11.941 Τφ1 0 0 1 305.28 545.73 Τµ (1)Τϕ/Φ2 11.941 Τφ1 0 0 1 303.6 504.45 Τµ (0)Τϕ/Φ5 11.941 Τφ1 0 0 1 288.96 545.73 Τµ (Ν)Τϕ/Φ5 11.941 Τφ1 0 0 1 348.24 533.49 Τµ (κ)Τϕ/Φ5 11.941 Τφ1 0 0 1 322.32 519.57 Τµ (κ)Τϕ/Φ5 19.891 Τφ1 0 0 1 330.96 524.61 Τµ 19.891 ΤΛ(ζ)Τϕ/Φ5 19.891 Τφ1 0 0 1 312.72 524.61 Τµ (β)Τϕ1 γ1 Γ/Φ0 31.922 Τφ1 0 0 1 40.8 449.49 Τµ 31.922 ΤΛ(Χ)Τϕ1 0 0 1 63.84 449.49 Τµ (α)Τϕ1 0 0 1 79.92 449.49 Τµ (σ)Τϕ1 0 0 1 92.4 449.49 Τµ (χ)Τϕ1 0 0 1 106.56 449.49 Τµ (α)Τϕ1 0 0 1 122.64 449.49 Τµ (δ)Τϕ1 0 0 1 140.4 449.49 Τµ (ε)Τϕ1 0 0 1 162.48 449.49 Τµ (σ)Τϕ1 0 0 1 174.96 449.49 Τµ (τ)Τϕ1 0 0 1 185.52 449.49 Τµ (ρ)Τϕ1 0 0 1 199.68 449.49 Τµ (υ)Τϕ1 0 0 1 217.44 449.49 Τµ (χ)Τϕ1 0 0 1 231.6 449.49 Τµ (τ)Τϕ1 0 0 1 242.16 449.49 Τµ (υ)Τϕ1 0 0 1 259.92 449.49 Τµ (ρ)Τϕ1 0 0 1 273.6 449.49 Τµ (ε)Τϕ1 0 0 1 287.76 449.49 Τµ (σ)Τϕ1 0 0 1 300.24 449.49 Τµ (:)Τϕ/Φ0 31.922 Τφ1 0 0 1 40.8 449.49 Τµ (Χ)Τϕ1 0 0 1 63.84 449.49 Τµ (α)Τϕ1 0 0 1 79.92 449.49 Τµ (σ)Τϕ1 0 0 1 92.4 449.49 Τµ (χ)Τϕ1 0 0 1 106.56 449.49 Τµ (α)Τϕ1 0 0 1 122.64 449.49 Τµ (δ)Τϕ1 0 0 1 140.4 449.49 Τµ (ε)Τϕ1 0 0 1 162.48 449.49 Τµ (σ)Τϕ1 0 0 1 174.96 449.49 Τµ (τ)Τϕ1 0 0 1 185.52 449.49 Τµ (ρ)Τϕ1 0 0 1 199.68 449.49 Τµ (υ)Τϕ1 0 0 1 217.44 449.49 Τµ (χ)Τϕ1 0 0 1 231.6 449.49 Τµ (τ)Τϕ1 0 0 1 242.16 449.49 Τµ (υ)Τϕ1 0 0 1 259.92 449.49 Τµ (ρ)Τϕ1 0 0 1 273.6 449.49 Τµ (ε)Τϕ1 0 0 1 287.76 449.49 Τµ (σ)Τϕ1 0 0 1 300.24 449.49 Τµ (:)ΤϕΕΤ296.16 212.61 215.76 197.76 ρε1 1 1 ργφθ297.16 212.41 214.96 197.44 ρεΩ ν73.68 410.85 µ512.88 410.85 λ512.88 211.65 λ73.68 211.65 λ73.68 410.85 λη74.88 212.37 µ74.16 212.85 λ512.16 212.85 λ511.68 212.37 λ511.68 410.37 λ512.16 409.65 λ74.16 409.65 λ74.88 410.37 λ74.88 212.37 ληφΘ0 γ0 ΓΒΤ/Φ4 16.246 Τφ1 0 0 1 469.44 380.85 Τµ 16.246 ΤΛ(()Τϕ/Φ4 16.246 Τφ1 0 0 1 496.56 380.85 Τµ ())Τϕ0 0 0 ΡΓΕΤ361.68 313.41 µ397.92 313.41 λΣ0 ΓΒΤ/Φ4 16.246 Τφ1 0 0 1 419.52 317.49 Τµ 16.246 ΤΛ(()Τϕ/Φ4 16.246 Τφ1 0 0 1 446.64 317.49 Τµ ())Τϕ/Φ4 33.48 Τφ1 0 0 1 341.52 244.77 Τµ 33.48 ΤΛ()−

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z -1

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z -1

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b 1,2

z -1

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uu Direct form structure for an FIR filter:Direct form structure for an FIR filter:



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ImplementationImplementation

uu Implementation procedure inImplementation procedure infixedfixed--point:point:ww Set up the codec (Set up the codec (\\

ww Transform: toTransform: to

((\\LinksLinks\\FIRFixed.pdfFIRFixed.pdfww Configure timer 1 to generate an interrupt atConfigure timer 1 to generate an interrupt at

8000Hz (8000Hz (\\LinksLinks\\TimerSetup.pdfTimerSetup.pdfww Set the interrupt generator to generate anSet the interrupt generator to generate an

interrupt to invoke the Interrupt Serviceinterrupt to invoke the Interrupt ServiceRoutine (ISR) (Routine (ISR) (\\LinksLinks

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ImplementationImplementation -- Step 5Step 5

Implementation procedure inImplementation procedure in ‘‘CC’’ withwith

\\LinksLinks\\CodecSetup.pdfCodecSetup.pdf).).

Transform: toTransform: to ‘‘CC’’ code.code.

FIRFixed.pdfFIRFixed.pdf))Configure timer 1 to generate an interrupt atConfigure timer 1 to generate an interrupt at

TimerSetup.pdfTimerSetup.pdf).).Set the interrupt generator to generate anSet the interrupt generator to generate aninterrupt to invoke the Interrupt Serviceinterrupt to invoke the Interrupt Service

LinksLinks\\InterruptSetup.pdfInterruptSetup.pdf).).

[ ]−⋅k knxb





uu Implementation procedure inImplementation procedure infloatingfloating--point:point:Same set up as fixedSame set up as fixedww Convert the input signal to floatingConvert the input signal to floating

format.format.ww Convert the coefficients to floatingConvert the coefficients to floating

format.format.ww With floatingWith floating--point multiplications there ispoint multiplications there is

no need for the shift required when usingno need for the shift required when usingQ15 format.Q15 format.

uu SeeSee \\LinksLinks\\FIRFloat.pdfFIRFloat.pdf



Implementation procedure inImplementation procedure in ‘‘CC’’ withwith

Same set up as fixedSame set up as fixed--point plus:point plus:Convert the input signal to floatingConvert the input signal to floating--pointpoint

Convert the coefficients to floatingConvert the coefficients to floating--pointpoint

point multiplications there ispoint multiplications there isno need for the shift required when usingno need for the shift required when using

FIRFloat.pdfFIRFloat.pdf





uu Implementation procedure in assembly:Implementation procedure in assembly:Same set up as fixedSame set up as fixedww is written in assembly.is written in assembly.

((\\LinksLinks\\FIRFixedAsm.pdfFIRFixedAsm.pdf

ww The ISR is now declared as external.The ISR is now declared as external.

[] [ ]∑−

=

−⋅=1

0

N

kk knxbny



Implementation procedure in assembly:Implementation procedure in assembly:Same set up as fixedSame set up as fixed--point, however:point, however:

is written in assembly.is written in assembly.

FIRFixedAsm.pdfFIRFixedAsm.pdf))

The ISR is now declared as external.The ISR is now declared as external.





uu Implementation procedure in assembly:Implementation procedure in assembly:The filter implementation in assembly isThe filter implementation in assembly isnow usingnow using circular addressingcircular addressingtherefore:therefore:ww The circular pointers and block size registerThe circular pointers and block size register

are selected and initialised by setting theare selected and initialised by setting theappropriate values of the AMR bit fields.appropriate values of the AMR bit fields.

ww The data is now aligned using:The data is now aligned using:

ww Set the initial value of the circular pointers,Set the initial value of the circular pointers,seesee \\LinksLinks\\FIRFixedAsm.pdfFIRFixedAsm.pdf

#pragma DATA_ALIGN (symbol, constant (bytes))#pragma DATA_ALIGN (symbol, constant (bytes))



Implementation procedure in assembly:Implementation procedure in assembly:The filter implementation in assembly isThe filter implementation in assembly is

circular addressingcircular addressing andand

The circular pointers and block size registerThe circular pointers and block size registerare selected and initialised by setting theare selected and initialised by setting theappropriate values of the AMR bit fields.appropriate values of the AMR bit fields.The data is now aligned using:The data is now aligned using:

Set the initial value of the circular pointers,Set the initial value of the circular pointers,FIRFixedAsm.pdfFIRFixedAsm.pdf..

#pragma DATA_ALIGN (symbol, constant (bytes))#pragma DATA_ALIGN (symbol, constant (bytes))





y0 = b0*x0y0 = b0*x0 + b1*x1+ b1*x1

Circular addressing link slide.Circular addressing link slide.

y[n]y[n]

00 11 22

bb00bb11bb22bb33



+ b1*x1+ b1*x1 + b2*x2+ b2*x2 + b3*x3+ b3*x3


timetime22

xx00xx11xx22xx33





y0 = b0*x0y0 = b0*x0 + b1*x1+ b1*x1y1 = b0*x4y1 = b0*x4 + b1*x1+ b1*x1


y[n]y[n]

00 11 22

bb00bb11bb22bb33


xx44xx11xx22xx33


+ b1*x1+ b1*x1 + b2*x2+ b2*x2 + b3*x3+ b3*x3+ b1*x1+ b1*x1 + b3*x3+ b3*x3+ b2*x2+ b2*x2


timetime22





y0 = b0*x0y0 = b0*x0 + b1*x1+ b1*x1y1 = b0*x4y1 = b0*x4 + b1*x1+ b1*x1y2 = b0*x4y2 = b0*x4 + b1*x5+ b1*x5


y[n]y[n]

00 11 22

bb00bb11bb22bb33


xx44xx55xx22xx33


+ b1*x1+ b1*x1 + b2*x2+ b2*x2 + b3*x3+ b3*x3+ b1*x1+ b1*x1 + b3*x3+ b3*x3+ b2*x2+ b2*x2+ b1*x5+ b1*x5 + b3*x3+ b3*x3+ b2*x2+ b2*x2


timetime22




FIR CodeFIR Code

uu Code location:Code location:ww CodeCode\\Chapter 14Chapter 14 -- Finite Impulse Response FiltersFinite Impulse Response Filters

uu Projects:Projects:ww Fixed Point in C:Fixed Point in C:ww Floating Point in C:Floating Point in C:ww Fixed Point in Assembly:Fixed Point in Assembly:ww Floating Point in Assembly:Floating Point in Assembly:


FIR CodeFIR Code

Finite Impulse Response FiltersFinite Impulse Response Filters

\\FIR_C_FixedFIR_C_Fixed\\\\FIR_C_FloatFIR_C_Float\\

Fixed Point in Assembly:Fixed Point in Assembly: \\FIR_Asm_FixedFIR_Asm_Fixed\\Floating Point in Assembly:Floating Point in Assembly: \\FIR_Asm_FloatFIR_Asm_Float\\




-- EndEnd


EndEnd --



Chapter 14 Finite Impulse Response (FIR) Filters ... - EE 301

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