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$: H z) = H z - lttm.dei.unipd.it · fg_acafw «fgcanv\ Z]\ cv\ na_a\ JVYSaSafh\kcafA "QJRW F afA JVYRWRyfh{yS~VY_UcaQ"_ Z]f fh_ \ " "V^RWcaV "V^b j \ ZyZ]V F aQTfHXYR]\ caf(caf(bhfh$
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s → z��fh_�Sv\^R}Z]fg�"VYbh\^RWQd_aQ"bUSaRWVYX^QJZWZWV¦cafotvbgZWRWfU{yQ"bgQJZWZWfg�of�©

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zS�Q"R1fhbASURWVYXYQ[ZWZWV,cafAtvbwZ]RWf/µ�Sv\^{W{W\£¶³ZWnUZWZWV�·

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H1(z)RWQJ{WS�VY_a{W\^�afhbgQ

cUQ"bhbh\¦\^SaSURWVY{y{Wfhe \^pJfhVY_aQ>caQ"bae*VocanabhVQOcafana_a\6jrqsc�q½Z�qH2(z)

Sv\�{W{]\£Z]nUZyZ]V ¸H2(z) =

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H(z) = H1(z)H2(z).¸�À q ¨ ¹

Á f2�v\�monUfh_acaf2S�Q"R=fgb�e VocanabgV|H(ejθ)| = |H1(e

jθ)|¸�À q ª ¹

e*Q"_�ZWRWQ»S~QJR=bi\Tj�\^{yQ1Q»fgb RWfwZF\^RycaV`caf�XYRynaSaS�Vϕ(θ) = ϕ1(θ) + ϕ2(θ)τ(θ) = τ1(θ) + τ2(θ)

¸�À q ¬ ¹

ª À ¬

� �� # $&%('*)�' )�$ODJD[$�� 3»-�-;$s −→ z

µr�¦z��x��¦x·Á �� x6�x

Ha(ω) ' Da(ω)

Da(ω)D(ejθ)

H(ejθ) ' D(ejθ)

z −→ s

z ←− s

Á f(fh_acUfh�F�af��JVY_D(ejθ)

bi\�RWfg{WS�VY{yZ]\*fg_�jlRWQ"m�naQJ_ap�\*caQ"{yfhcaQJR]\�Z]\*Q1bi\�{yf�Z]R]\�{yjlVYRye f�_UQ"bw¶bi\�RWfh{yS~V^{yZF\�fh_ jlRyQ"m�naQ"_Up�\�caf>na_�{yfh{yZWQ"e \�Z]QJe S�V��JVY_�Z]fh_�naV�µr\^_v\^bgVYXYfg�"V�·�ZWR]\^e*fgZWQ nU_v\Z]RW\^{yjlV^RWe \^p"fgVY_aQ1cv\�Safh\^_aV

z\ Safi\^_UV

s � z → s�~V^ZWZWQ"_aQJ_acaV�bi\`jlna_apJfhVY_aQ

Da(ω)q��Q"b ¶

b�� µ}\�RW�F�afw�«fgV caQJbhbgQ�{WVYbgnap"fgVY_af�\^_v\^bgVYXYfh�F�UQ[·�{Wf¦�JQ"Ry��\�bi\ {ynv\ e fgXYbhfgVYRWQ�\^SaSaRyVY{W{yfhe \^p"fgVY_aQ_aQJbhb��sfh_U{WfhQJe Q1caQJbhbhQG{WVYbgnap"fgVY_af7�"VY_aV^{W�"fgnUZ]Q�q��¦naQJ{yZ]\�\�SaSaRWV^{W{Wfge�\�p"fhV^_aQ

Ha(ω) ' Da(ω)�ofhQJ_aQ»S~VYf2RyfhS�VYRyZ]\�ZF\��JVY_�bi\�Z]RW\^{yjlVYRye�\�p"fhV^_aQ¦fh_��^Q"RW{W\s −→ z

fg_monUQ"bhbh\�caf(na_�{yfh{}Z]Q"e \\�Z]QJe S�V�cafh{y�"RyQJZ]V

H(ejθ)�aSaRyQ"{WQJRy��\^_acUVY_aQ1bhQ¦SaRyVYSaRyfhQJZ��Q"{W{yQ"_apJfi\^bgf � H(ejθ) ' D(ejθ)

q��bgbi\�e \^SaSv\

s −→ z{Wf2RWfg�F�afhQJcaQ6�F�aQ

¨ q=e�n«Z]f�na_a\`jrqsc�q½Z�qAR]\�p"fhV^_v\^bhQ�fh_sfg_�nU_v\`jrqsc�q½Z�qARW\^p"fgVY_v\^bgQ¦fh_

z©

ª q=SaRyQ"{yQ"Ry�of(bi\�{}ZF\^�afgbhfwZ�� Ha(s){yZF\��afhbgQ −→ H(z)

{}ZF\^�afgbhQ�©¬ qOZWR]\^{}jlVYRWe*f b�� \^{y{WQ»fhe*e \^XYfh_a\^RWfgV

s = jω_UQ"b2�"QJRW�F�afgV`caf2R]\^X^XYfhV�na_UfgZF\�RWfhV

z = ejθ qx�_ bhfg_aQ�\ caf¦SaRyfh_a�JfhSafgV��;na_v\ e \^SaSa\ S~V�Z]Q"_Up"fi\�bhQ,m�nv\�bh{Wfh\^{WfSaRWVo�"QJcanaRW\ caf¦cafg{W�"RyQJZWfhp[¶p�\�p"fhV^_aQ�caf7na_v\�QJmona\^p"fgVY_aQ�caf�� QJRWQ"_Up"fi\�bhQ1bhfh_UQ�\^RyQ�\��JV«Q�� JfhQJ_�ZWf/�JVY{yZ]\^_�Z]f �F�aQ�S�VYR}Z]f7\^cna_v\`Q"m�nv\�p"fhV^_aQ»bhfh_UQ�\^RyQ6\^bhbgQ¦caf�� QJRWQ"_Up"Q6tv_afwZ]Q»\��"VoQ �*�"fgQ"_�Z]f(�JVY{yZ]\^_�Z]f�q�Oc�Q"{WQJe SUfhVa�v\�bhbi\G{WVY{}Z]fgZWnap"fgVY_aQ6caQJbhbh\�caQ"Ryfg��\�ZF\��JVY_�fgb RW\^SaS�VYRyZWV*cUf2fh_a�JRWQ"e*Q"_�ZWf tv_afwZ]f

y′(t) −→ y(tn)− y(tn−1)

tn − tn−1tn = nT

¸�À q"!�¹�"V^RWRWfg{WS�VY_acaQ6bh\�e�\^SUSv\

s = (1− z−1)/T�U�F�aQG{yV«cUcafh{}j�\`{yVYbhV�bh\��"VY_acUfhp"fgVY_aQ ¸ ¨ ¹[q

x�_�SaRW\�Z]fg��\U�£XYbgfonU_afh�Jfoe*QJZWV«cUfo�F�UQ&{Wfo{WVY_UV¦RWfw�YQ"bh\�Z]f�nUZWfhbgf�{WVY_aVfhb�e*QJZ]VocaVcaQ"bgbi\=fh_��\^RWfh\^_ap�\caQJbhbi\�Ryfh{yS~VY{}ZF\�fhe*Sanabg{Wfw�^\1Q»fhb ¤��[²�¡"¥o¡ ¥o�[�®�w¯ ²�±y¯£�l°�¡£±[¤*¯$#"��¡&% �('"�®� �)% �W¯^±F�[q

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z\

s

s =αz + β

γz + δ

¸�À q54Y¹

ª À !

�JVY_s, z, α, β, γ, δ ∈ C

Qαδ − βγ 6= 0

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α, β, γ, δ¹[q

x�_ Sv\^R}Z]fg�"VYbh\^RWQ��GfgbT�"QJRW�F�afgV caf�R]\�XYXYfhV na_afgZ]\^RWfgVz = ejθ �ofhQJ_aQ e�\�SaSv\�ZWV _aQ"bgb�� \^{W{yQfge e \^XYfg_v\^RWfgV

s = jωcv\^bhbh\

�s = α

z − 1

z + 1

¸�À q À ¹�JVY_

αRyQ�\^bgQ^q|PHVY_aQ"_UcaV

α = 1{yf2V^ZWZWfhQJ_aQ

�

s =z − 1

z + 1←→ z =

1 + s

1− s

¸�À q��^¹x�_�Sv\^R}Z]fh�JVYbi\�RWQ¦{yf2�v\�bi\��"VYRyRWfg{WS�VY_acaQJ_ap�\ �

s = 0 ←→ z = 1s =∞ ←→ z = −1s = −1 ←→ z = 0

¸�À q��Y¹

�(\�SaRWfge�\*caQ"bgbhQ ¸�À q��¹�\^{y{Wfh�JnaR]\��F�aQ�fgb/�JVYe S�VYR}ZF\ê*Q"_�Z]Vk\�jlRWQ"m�naQJ_ap�\p"QJRWVfh_s�YQJ_aX�\

e \^SaSv\£Z]V _UQ"b��"VYe*S~V^RyZF\�e QJ_�ZWV \�jlRWQJmonUQ"_ap"\ p"Q"RyV fh_z¸s = j0 ↔ z = 1 = ej0 ¹Q`bi\�{yQ"�"V^_acv\�F�aQ fgb �"Vê S�VYRyZ]\ê*Q"_�Z]V\kjlRWQJm�naQ"_ap"\�fg_Utv_afwZF\kfh_

s�YQ"_UX�\�e \^SaSv\£Z]V�_aQJb

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R[s] < 0 ←→ |z| < 1.¸�À q��¹

¦S�Q"R]\£Z]fg��\ê*Q"_�ZWQ^�Uca\�ZF\�bh\Tjrq c�q½Z�qARW\^p"fgVY_v\^bgQHa(s)

�a{Wf�V�ZWZ]fgQ"_aQ�bi\H(z)

�JVYRWRyfh{WS�VY_acUQ"_�Z]Q�JVY_�fgb��\�e��afhV�cUf��^\�RWfi\��afhbgQ

s = (z − 1)/(z + 1)

H(z) = Ha

(

z − 1

z + 1

) ¸�À q ¨ ¹�F�aQ1e�nUZ]\�fhb�caVê fg_afhVGe�\�bi\�{W�"fh\�fh_v\�bgZ]QJR]\�ZWV�fhb �"VocaVYe*fh_afgVaq

�|Q"Ryfgtv�F�afh\ê*V1�F�UQ�fhbv�JQ"Ry�F�afhVGcaf~RW\^XYXYfgV»na_afgZ]\^RWfgV»¿e \^SaSv\£Z]V1_aQ"bgb�� \^{W{yQfge e \^XYfg_v\^RWfgVz = ejθ ←→ s = jω.

¸�À q Ÿ¨ ¹Á V^{yZ]fwZ]naQJ_acaV

z = ejθ _aQ"bgbi\ ¸�À q��^¹>{Wf2V^ZyZ]fhQJ_aQ6fh_«j�\�ZWZWf

s =ejθ − 1

ejθ − 1=

ej θ2 − e−j θ

2

ej θ2 + e−j θ

2

= j tanθ

2= jω

¸�À q ¨¢ª ¹�¦nafh_acUf(\^bhbh\�ZWR]\^{}jlVYRWe \^pJfhVY_aQ»�afgbhfg_aQ�\^RyQ¦Z]RW\ f(Safi\�_af

zQ"c

s�"VYRyRWfg{WS�VY_acaQ1bi\�Z]RW\^{yjlVYRr¶

e \^p"fgVY_aQ6_aV^_�bgfh_aQ"\^RWQZ]R]\θQ

ω

ω = tanθ

2←→ θ = 2 arctanω

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ª À 4

−18 −12 −6 0 6 12 18

ω (rad/sec)

θ

−π

π

0

�� ! "��#%$'&( *)* *#%$+��,$.-/�,$0)1��! "��#%$32��θ$

ω

�A{W{W\ fh_�ZWRWVocana�"Q6nU_v\*cUfh{yZWVYRW{yfhVY_UQ¦caQ"bgb�� \^{W{yQ»caQ"bgbhQ�jlRyQ"m�naQ"_Up"Q �Ha(ω) −→ H(ejθ) = Ha(ω)|ω=tan θ/2

¸�À q ¨ !�¹Q6m�nafh_Ucaf2fhb~e Q[Z]VocaV�¿6nUZ]fgbhQ¦m�nv\�_acaV�ZF\^bgQ¦cafg{yZ]V^RW{WfgVY_aQ6SanU§*QJ{W{yQ"RWQ»Z]V^bhbhQJR]\�Z]\GV`�"V^e S�Q"_«¶{]\£ZF\Uq �6nUQ"{yZ � nUbgZ]fge V��\^{yV1{yfaSaRyQ"{WQJ_�ZF\»_aQ"baSURWVYXYQ[ZWZWV6cafUtvbwZ]RWfo{WQJbhQ[ZWZ]fw�of��Y�JfhV �>�"V^_*e*VocanabhVcaQJbhbi\�Ryfh{yS~VY{}ZF\�fh_�jlRyQ"m�naQ"_Up�\Tµr�"V^{yZF\�_�ZWQ1\`ZWR]\�ZyZ]fw·]q

�(\ Z]R]\�{yjlVYRye�\^pJfhVY_UQ3�;nUZWfhbgQ^�Ofh_UVYbgZWRWQ^�Om�nv\�_acaV {Wf�«nUVYbhQ�Ryfh�JVY_acanaRyRWQ�SURWVYSaRyfhQ[Z�� caf{Wfge e*QJZWRWfh\�RWfg{WS�QJZyZ]V�\^b��"Q"Ry�F�afhV |z| = 1

\�SaRyVYSaRWfgQJZ��caf>{yfhe*e Q[Z]RWfh\�Ryfh{WS�QJZyZ]V�\^bgb�� \^{W{yQs = jω

�aQ6�«fg�"Q[�YQ"Ry{]\Uq

s z

p1

p∗1

1

p1

1

p∗1−s∗

1

−s1s∗1

s1

Á f¾�"VY_U{WfhcUQ"RWf¾VYRW\1b�� Q�� Q[ZWZWV1cUQ"bhbh\¦Z]R]\�{yjlVYRye�\^pJfhVY_UQO�afhbgfh_aQ"\^RWQO\�SaSabhfg��\�Z]\»\^c nU_v\GRyfh{WS�V�¶{yZ]\*fg_kjlRWQJmonUQ"_ap"\

Ha(ω)¸ tvXaq À q ª ¹[�v\�c�QJ{WQJe SafgVa�aSv\�{W{]\*�v\^{y{WV

ª À^À

q54¨

θp

θs

π

θp

θs

π

ωpωs 6

ωpωs 6

q54

¨|H(ejθ)|

|Ha(j

ω)|

(rad/sec)

�� ()* ��+��! "��#%$($0)*)1� 2��! "��#%$'&( *)* *#%$+��,$

ª À �

��

0 ≤ |ω| ≤ ωp = tan θp

2

1− δp ≤ |Ha(ω)| ≤ 1

=⇒0 ≤ |θ| ≤ θp

1− δp ≤ |H(ejθ)| ≤ 1

��

|ω| > ωs

|Ha(ω)| ≤ δs

=⇒θs ≤ |θ| ≤ π

|H(ejθ)| ≤ δs

� � QJRWRyVYRWQ»caf(\^SUSaRWVY{y{Wfge�\^pJfhVY_UQ6{Wnab2e*V«cUnabhV��JVY_a{WQJRy��\ bgV`{yZ]QJ{W{yVk\^_aca\ê QJ_�Z]V ¸ \`Sv\^R}Z]Q»fhb��\�e��afi\�e QJ_�ZWV,caf�{W�"\^bi\

ω −→ θ¹ � \^c Q"{yQ"e*SafhVU�A{WQ |Ha(ω)| VY{y�"fgbhbi\�� ^VYbgZWQ�fh_ na_v\�v\^_Ucv\»ZWR]\6na_*e fg_afhe*V¦Q=e \^{W{yfhe*Va��\^bgbhVYRW\¦\^_a�F�aQ |H(ejθ)| VY{y�"fgbhbi\�� ^VYbgZWQ=_aQ"bgbi\¦�v\^_Ucv\�"V^RWRWfg{WS�VY_acaQJ_�Z]Q6Z]RW\`X^bhf�{}Z]Q"{y{Wf2��\^bhV^RWf e*fh_Ufhe*VTQ»e�\�{W{Wfge VUq

� � \�_acv\ê*Q"_�Z]V�caQ"bgbi\Oj�\^{WQ&fg_��YQ"�JQO��\êT�afi\ � {yQ^��\^cTQ"{WQJe SUfhVa� Ha(ω)�v\�j�\^{WQ�mona\^{Wf«bhfh_UQ�\^RyQ

fh_ �v\^_Ucv\�Sa\^{W{W\^_�Z]Q^�∠Ha(ω) ' −α ω

�/\�bhbhV^R]\kRWfg{WnabwZF\∠H(ejθ) = ∠Ha(tan θ/2) '

−α tan(θ/2)�¾QGmonUfh_acafHbi\`bhfg_aQ�\�RWfgZ��{WfH�"VY_a{yQ"R}�^\ {WVYbgV�S�Q"R�Safh�J�"VYbgf(��\^bgVYRWf2cUf

θ�~m�nafg_acaf

cafθpq

�¦\�Z]\*bh\Tjrq c�q½Z�qHa(s)

Ha(s) = b0

∏Mk=1(s− zak)

∏Nk=1(s− pak)

(M Q N)¸�À q ¨ 4Y¹

�"V^_�bh\TZ]R]\�{yjlVYRye�\^pJfhVY_UQ¦�afhbgfh_aQ"\^RWQ�{Wf2V^ZyZ]fhQJ_aQ6bi\Tjrqsc�q½Z�qH(z)

H(z) = Ha

(

z − 1

z + 1

)

= b0

∏Mk=1

(

1−z−1

1+z−1 − zak

)

∏Nk=1

(

1−z−1

1+z−1 − pak

) = b′0

∏Mk=1(1− zkz

−1)∏N

k=1(1− pkz−1)(1+z−1)N−M

¸�À q ¨ À ¹caV¢�YQ

zk =1 + zak

1− zakpk =

1 + pak

1− pak

¸�À q ¨ �Y¹{WV^_aVTbgQfge e \^XYfg_af«_UQ"b�Safi\^_UV

zcUQ"XYbgfvp"Q"Ryf

zak

Q�caQJf�S�VYbgfpak

_aQJb~Safh\^_aVsQfgb¾S~VYbgV ��pJQ"RWV

fh_z = −1

¿6bi\�fhe*e�\�XYfh_aQ�caQJb2S~VYbgV ��pJQ"RWV�fg_s =∞ q

!|�¢�[¤��¡#"%$'& $)(�* Á f2�"VY_a{yfhcaQJRWf�b�� fg_�Z]Q"X^R]\�ZWVYRWQ6\�_v\^bhV^XYfh�JV

y(t) =∫ t

0x(τ)dτ −→ Ha(s) =

1

s

¸�À q ¨ ��¹�(\ jlna_apJfhVY_UQ�caf7ZWR]\^{}jlQ"Ryfhe*Q"_�Z]V

Ha(s) = 1/s�ofhQJ_aQTZ]RW\^caV^ZyZF\�fg_�m�naQ"bgbi\*caQ"b7{Wfh{}Z]QJe�\k\

Z]QJe S�V`cafh{y�"RyQJZ]VH(z) = Ha

(

z − 1

z + 1

)

=1 + z−1

1− z−1

ª À �

�(\H(z)

�v\�na_S�VYbhV�fg_z = 1

¸ fhe*e \^XYfh_UQ�cUf�m�naQ"bgbhV�fh_s = 0

¹�Q6na_aV`p"QJRWV`fh_z = −1¸ fhe*e�\^X^fh_aQcaf�m�naQ"bgbhV`fh_

s =∞ ¹>Q»�"VYRyRWfg{WS�VY_acaQ»\^bgbi\TQJm�nv\^p"fgVY_aQ»x � y(n) = y(n− 1) + x(n) + x(n− 1)

¸�À q ¨ ��¹�F�aQ ��bh\Q"m�nv\�p"fhV^_aQ�cUQ"bhb�� \��"�"nUe�nabi\£Z]VYRyQ µ��"VYRyRWQ[ZWZF\L·d\^XYXYfgna_aXYQJ_acaV�fhb��"V^_�ZWRWfg�anUZ]V

x(n−1)q

P7QJR�bh\�RWfg{WS�VY{yZ]\`fg_�jlRWQ"m�naQJ_ap�\*{Wf2�v\

Ha(jω) =1

jωω = tan

θ

2' θ

2

S�Q"Rθ ' 0

H(ejθ) =1 + e−jθ

1− e−jθ' 2

jθθ ' 0

¸�À q ª ¹

�

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f/Fc

|H(f

)|

a=0.7265 a=−0.7265

�� $0� )�� $!��$0� �( "��

!|�¢�[¤��¡ "%$'& $��* Á Q6\�SaSabhfg�F�afi\�e V1bi\»Z]R]\�{yjlVYRye�\^pJfhVY_UQ�afgbhfg_aQ�\^RyQ�\�bhbi\»jrq c qsZ"q Sv\^{y{]\��v\^{y{WV\^_a\^bhVYX^fh��\ ¸ {yZ]\^�afgbhQ¢¹

Ha(s) =1

1 + s/ω0

¸�À q ª«¨ ¹caV¢�^Q

ω0 > 0R]\�SaSaRWQJ{WQJ_�Z]\ bi\�Sanabg{]\^pJfhVY_UQ�cUf&ZF\^XYbgfhV�\^f

3cv¼

(|Ha(jω0)| = 1/√

2)�

V^ZyZ]QJ_afi\^e*V�bi\Tjrqsc�q½Z�q

H(z) = Ha(z − 1

z + 1) =

1− a

2· 1 + z−1

1− az−1

¸�À q ªYª ¹

ª À �

a =1− ω0

1 + ω0(−1 < a < 1)

�"V^_�nU_aV�p"Q"RyVufg_z = −1

�dfhe*e \^XYfh_UQ*cUQ"bhbgV;p"QJRWV�fh_s = ∞ �|Q�na_�S�VYbhV;fg_ z = a

�fhe*e \^XYfh_UQGcaQJb|S~VYbgV�RWQ"\^bhQ�_aQJX�\�ZWfg�YVfg_

s = −ω0

q Á fAVY{y{WQ"R}�ofd�F�UQ S�Q"Rω0 = 1

RW\^c ��{J�a = 0

Q1bh\H(z)

RWfh{ynabgZ]\��/xrz»q�(\ RWfg{WS�VY{yZ]\ fh_�jlRWQJmonUQ"_ap"\

H(ejθ)QJRWQ"cUfgZF\ufhb=�"VYe*S�VYRyZ]\ê QJ_�Z]V cUf

Ha(jω)Q�m�nafh_acUf

�v\�e*V«canUbhVke*VY_aV^ZWVY_aVkcUQ"�"RyQ"{y�"Q"_�ZWQ�cv\^b7�^\�bhVYRyQ�e�\�{W{Wfge V�Sv\^RWf \^c�na_aVS~QJRθ = 0

�(\�be*fh_afge VGSv\^RWf(\�pJQ"RWV*S�Q"R

θ = πq

� fh_�Z]QJRWQJ{W{]\�_�ZWQ�VY{y{WQ"R}��\^RWQ��F�UQ�bi\*�v\^_acv\�\^f3cv¼ cUf

H(ejθ)¸ |H(ejθ3dB)| = 1/

√2)¹ �

b��sfhe*e \^XYfh_UQ�cUfω0�vmonUfh_acaf

ω0 = tan(θ3dB/2)Q»fhb��JV«Q��*�"fhQJ_�Z]QaRyfh{WnUbgZF\�cUfhS�Q"_acaQJRWQ»cv\^bgbi\T�a\^_acv\*\^f

3cv¼ Z]RW\ê fwZ]Q¦bh\

a =1− ω0

1 + ω0=

1− tan(θ3dB/2)

1 + tan(θ3dB/2)

¸�À q ª^¬ ¹��\^bhfgcv\¦S�Q"RdVYXY_afa��\^bgVYRWQOcaf

θ3dB

q Á f¾VY{y{WQ"R}��\G�F�aQ�S~QJR�SUfh�"�JVYbhf«�^\�bhVYRyfacafθ3dB

��Ryfh{WnUbgZF\a '

1− θ3dB

¸ m�nafg_acafθ3dB ' 1− a

¹�Q^�«�"VYe*Q�{ynaXYXYQJRWfgZWVTcv\�bhbi\1fh_�Z]QJRWSaRyQJZ]\^p"fgVY_aQ�X^Q"VYe*QJZWRWfh�"\caQJbhbi\GRWfh{yS~V^{yZF\�fg_kjlRWQJmonUQ"_ap"\U�¾b��sQ �~QJZWZWV�caQ"bgbhV�pJQ"RyV fg_ −1

Ryfh{WnUbgZF\Gfh_afg_��vnaQJ_�ZWQ6{Wnab~�^\�bhVYRyQcaf

θ3dBq

x�_��/fhXaq À q ¬ �|RWfhS�VYR}ZF\�ZWVOfhbYe*V«cUnabhVOcaQJbhbh\&RWfh{yS~V^{yZF\Ofg_1jlRyQ"m�naQ"_Up�\ |H(ej2πfT )| caf�canaQ|tvbgZWRWf\��YQJ_�ZWf��¾RWfg{WS�QJZyZ]fg��\ê*Q"_�ZWQ^�θ3dB = π/10

¸ bgfh_aQ"\�\`ZWR]\�ZyZ]V`�"VY_�ZWfh_�naV�¹&Qθ3dB = π − π/10¸ bgfh_aQ"\�Sana_�Z]QJXYXYfh\�ZF\�¹8�a�"nUf2�"VYRyRWfh{yS~V^_acaVY_aV��\^bgVYRWf~VYSaS�VY{yZWf�S~QJR=fhb�S�VYbhV ¸a ' ±0.72

¹8qÁ fH\^SaSabgfh�F�af2bh\�Z]R]\�{yjlVYRye�\^pJfhVY_UQ»\^bhbh\Tjrq c�q½Z�q>Sv\^{W{W\�\^bwZ]V

Ga(s) = (s/ω0)/(1 + s/ω0)¸ SUna§ QJ{W{yQ"RWQ1nUZWfhbhQ�VY{W{yQ"Ry��\^RyQG�F�UQHa(s) + Ga(s) = 1

¹[q�

π

4

π

23π

4π

1

4

1

2

√2

2

1

θ

|H(e

jθ)|

0 5 10 15 20 25 30−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

n

h(n

)

�� $0� )�� $!�,$0� �( "�� - � � �()"� ($0)*)1� �� "� � �.��2�� *# � �,$ ��%$0#(� � $ �� "� � �.��2�� *� ��()"�� .��

!|�¢�[¤��¡#"%$'& $'& * Á f2�"VY_a{yfhcaQJRWf�fgb tvbwZ]RWV`Sv\�{W{]\L¶��v\^{y{WV*caf(¼>nUZyZ]QJR��VYRyZW��caQJb�Z]QJRWp"V*VYRycafh_aQ

Ha(s) =ω3

0

(s + ω0)(s2 + sω0 + ω20)

ª �

caV¢�^Qω0

S~V^RWXYQ�bi\=�v\^_acv\�\^f3cv¼ ¸ |Ha(jω0)| = 1/

√2¹[q �|VYXYbhfh\ê*V��JVY{yZWRWnafgRW_aQ��ZWR]\ê*fgZWQ

bh\uZ]RW\^{yjlVYRye�\�p"fhV^_aQ�afhbgfh_aQ"\^RWQ��&b��sQ"m�nafw�^\�bhQ"_�ZWQ�\uZWQ"e*S~Vucafh{y�"RWQ[Z]V �JVY_ �v\^_Ucv\ \^f3cv¼

Sv\�RWfA\^c;nU_ m�nv\�RyZ]V�cUQ"bhbh\ jlRWQJm�naQ"_ap"\�cUf ��\ê*SafgVY_v\ê*Q"_�Z]VUq(�6nUfh_acaf�� fg_acafh�"\^_acaV��"VY_θ0b�� fge e \^XYfg_aQcUf

ω0�acaQJ�^QTQJ{W{WQJRWQ

θ0 = 2π/4 = π/2¸ |H(ejθ0)| = 1/

√2¹8�aQ

ω0 = tanθ0

2= tan

π

4= 1

¸ R]\^c��{F¹S�Q"R=�"nUf

Ha(s) =1

(s + 1)(s2 + s + 1)�SaSabhfg��\^_UcaV`bi\TZ]RW\^{yjlV^RWe \^p"fgVY_aQ¦�afgbhfg_aQ�\^RyQ�{yfHV�ZWZ]fgQ"_aQ

H(z) = Ha

(

z − 1

z + 1

)

=(1 + z−1)3

6(1 + 0.3z−2)

¸�À q ª !�¹

caV¢�^QH(ejθ) = Ha(jω)|ω=tan(θ/2)

q � � \�_acv\ê*Q"_�Z]V`caQJb2e VocanabgV |H(ejθ)| Q»caQJbhbi\�Ryfh{yS~V�¶{}ZF\`fhe*Sanabh{yfg��\h(n)

{WV^_aV RyfhS�VYRyZ]\�Z]f~fh_ �/fgXaq À q !aq�

� �� 9--�./'G:(:2+ � 3»N&+['TI�+�3GI3»DJ'10G+J<A@�$ )$OD5?�D8%2./' � -�31:(:(3/�31:(:2'��

��Q"bd{WQ"X^nafgZWV�Ryfh�F�afh\ê fh\ê*VkbhQ�jrqsc�qsZ"q\^_v\^bgVYXYfg�F�aQ�fhbAe*V«cUnabhVcaQJbhbi\��Jnaf�Ryfh{yS~VY{}ZF\fh_�jlRWQ8¶m�naQJ_ap�\�\^SaSURWVY{y{Wfhe \ ¸ {WQJ�"VY_acUV;na_ m�nv\^bh�F�UQk�"RyfgZ]QJRWfgV�caf�V^ZWZWfhe \^bhfwZ��¹�b�� \^_acv\ê*Q"_�ZWV,caQJbtvbwZ]RyV*Sa\^{W{W\*�v\�{W{WV*fhcUQ�\^bgQ^q�>fARyfgjlQ"RyfhRyQ"e*Vk{WVYbgV�\^bhbgQ�\^SaSaRyVY{W{yfhe \^p"fgVY_af7caQ"bASv\�{W{]\�v\^{y{WV�cv\�ZWV��F�aQ��7�JVYe*Q��YQ"cURWQ"e*V_aQJb({WQ"X^nafgZWVa�¾XYbhf2\�_acv\ê*Q"_�Z]f(Sv\�{W{]\ \^bgZWVa�vSv\^{y{]\*�v\^_acv\`Q"c�Q"bgfhe*fh_v\��v\�_acv\`S~VY{y{WVY_UV�Q"{r¶{yQ"RWQ1RWfg��\��\�Z]f cv\�monUQ"bhbgV`Sv\^{W{W\*�a\^{W{yV �JVYb2e Q[Z]VocaV�caQJbhbhQ¦ZWR]\^{}jlVYRWe \^pJfhVY_af�caf�jlRyQ"m�naQ"_Up�\Uq

�(\�e�\^{y�F�aQ"RW\��"VY_�"nUf�{WVY_UV*ca\�Z]Q¦bgQ¦{WS�Q"�Jfgtv�F�UQ1cUf�na_�tvbgZWRWV�Sv\�{W{]\��v\�{W{WV*\^_a\^bhVYX^fh�"V¿»RWfgS~V^RyZF\£ZF\�fh_ �/fgXaq À q 4Y¹

�{yf2�v\`na_�fg_�Z]Q"R}��\^bhbgV

[0, ωp]�vcUQJZWZWV3'F¯ %�¥�¯1�U¯£�F�"¯ %v²r�»fg_�"naf2{yf

�onaVYbgQ |Ha(ω)| ' 1Saf��SaRyQ"�"fg{]\ê*Q"_�ZWQ

1− δp ≤ |Ha(ω)| ≤ 1Q»na_fh_�Z]QJRy��\^bhbgV

[ωs,∞]�

caQ[ZWZWV 'F¯ %�¥�¯G¯^²�²��%v�U¯^²³¯^��fg_��"nafU{Wf«�onaV^bhQ |Ha(ω)| ' 0�YSaf��SURWQ"�Jfh{W\ê QJ_�Z]Q |Ha(ω)| ≤ δs

q�>VYe*Q e*fh{ynaR]\�caQJXYbhfAQJRWRyVYRWfAcafd\^SUSaRWVY{y{Wfge�\^pJfhVY_UQ�¿ �JVY_a{WnUQJZ]nUcafh_aQ*na{W\^RWQ fhb&±8� ��h�*�)%

'F¯ %�¥�¯»�U¯£�F�"¯ %v²��Rp

caQ[tv_afgZWV �JVYe*Q

Rp , 20 log10

maxB.P. |Ha(ω)|minB.P. |Ha(ω)| = −20 log10(1− δp)

¸�À q ª 4^¹

Q»bi\�¯^²�²r��%v�U¯$#"��¡&% � �³¤��)%v�®¤*¯��)% 'F¯ %�¥�¯�¯^²�²r��%v�U¯^²³¯�

Rs = AdB , −20 log10 maxB.A.|Ha(ω)| = −20 log10 δs

¸�À q ª À ¹Á f��v\�_a_aV*S~VYf canaQ»XYR]\�_acaf�j�\ê*fhXYbgfhQ�caf tvbwZ]Ryf2\^_v\^bgVYXYfg�"f Sa\^{W{W\*�v\�{W{WV �

ª � ¨

11− δp

ωp ωs ω

δs

�� ()"$ *# & ��# %� � ��,� ��# 2,$'$'��2�2,$0#� ��! "��#%$ *# & ��# %� ��2�2,$0# ��2��

�� v\^_a_aV*bi\TjlnU_ap"fgVY_aQ»caf ZWR]\^{}jlQ"RWfge QJ_�Z]V`caQ"b�ZWfhS�V

Ha(s) =1

Da(s)

caV¢�^QDa(0) =

{

1' 1

¸�À q ª �Y¹�JVY_

Da(s)S~V^bhfh_UVYe fgVkfh_

sQ�m�nafg_acafAZ]n«ZWZ]f|XYbgfAp"QJRWf|caf

Ha(s){WV^_aV�\^bgb��sfh_Uta_afgZWVaq

�¦naQJ{yZ]V*j�\�ZWZWV*\^{W{yfh�"nUR]\limω→∞ |Ha(ω)| = 0

q��SaSv\�RyZ]QJ_aXYVY_UV�\`m�naQ"{}ZF\ �Jbi\^{y{WQ1ftvbwZ]Ryf�caf2¼>n«ZWZ]QJR��VYRyZW�,Q»caf �>�UQ"��o{W�aQ[�,caQJb2SaRWfge V�ZWfhS�Vaq

�� v\^_a_UV*bh\�jlna_ap"fgVY_aQ6caf�ZWR]\^{}jlQ"Ryfhe*Q"_�Z]V�caQJb2Z]fhS�V

Ha(s) =Na(s)

Da(s)

¸�À q ª ��¹�JVY_

Na(s)QDa(s)

S�VYbhfg_aVYe*fYfo�"naf«XYR]\^cUf«{yV«cacUfh{yj�\�_aV¦bi\deg Na(s) ≤ deg Da(s) �{yVY_aV�m�nafg_acaf2tvbwZ]RWf(�JVY_,pJQ"RWf7\^b2tv_afwZ]Vaq>��SaSv\�RyZ]QJ_aXYVY_UVk\ m�naQJ{yZ]\��"bh\^{W{yQTf2tvbgZWRWf2caf

�&\^nUQ"R ¸ V`Q"bgbhfwZWZ]fg�"fi¹|Q»caf �>�UQ"��o{W�aQ[�,caQJb2{WQJ�"VY_acUV Z]fgS~VUq

� �� +[D[%2.7+ )�+�� Ck%2%($&.��'».7%H@x�tabgZ]Ryf2cafH¼>nUZyZ]Q"R �>V^RyZ]��{WVY_UV �"\^R]\�ZyZ]QJRWfhpJp�\�ZWf2cv\^b�j�\�ZyZ]V*�F�aQ1�v\^_a_aV e VocanabgVGµre \^{W{yfhe \£¶e*Q"_�Z]Q»Safh\�ZWZWV�·¦_UQ"bhb�� fg_�Z]VYRy_aV�caQJbhb��sVYRyfhXYfg_aQ�Q6caQJbhb�� fg_Utv_afwZ]Vaq�>VY_a{yfhcaQJRWfh\ê V�bh\Tjlna_apJfhVY_aQ�� ! ��"#� \^{W{yV«�Jfi\�Z]\*\�bhbi\ ¸�À q ª �Y¹

Ma(ω) , |Ha(ω)|2 =1

PN(ω2)

¸�À q ª ��¹�"V^_

PN(ω2) ≥ 0S~V^bhfh_UVYe fgV�cafdXYR]\^cUV

Nfh_

ω2 q�x�e S�VY_afh\ê V Ma(0) = 1Q�m�nafh_Ucaf

PN(0) = p0 = 1q �|VYXYbgfi\ê*V��F�aQ

Ma(ω)\^�a�afh\ue*fh_afge V�{y�"VY{}ZF\ê*Q"_�Z]V cv\^b=�^\�bhVYRyQ

na_afwZF\^RyfhV_aQ"bgb��sfh_�Z]VYRy_aV¦caQJbhb�� VYRyfhXYfg_aQ^q(��Q&�ofhQ"_UQO�F�aQPN(ω2)−1

cUQJ�YQ=QJ{W{WQJRWQ�fh_Uta_afgZWQ"{Wfge Vcaf2VYRycafh_UQ6e�\^{y{Wfge V ¸ �"VYe*Sv\�ZWfh�afgbhe*Q"_�Z]Q�"VYb2X^R]\^caV�¹�_aQ"bgb��sfh_�Z]V^RW_aV�caQJbhb�� VYRyfhXYfg_aQ^��m�nafg_acaf

PN(ω2) = 1 + pNω2N¸�À q ¬ ¹

ª � ª

x�_;�JVYRWRyfh{WS�VY_acUQ"_ap"\k{Wf V^ZyZ]fgQ"_aQGmonUfh_acaf7bi\`jlna_ap"fgVY_aQ�e*VY_aV�Z]VY_v\*caQ"�JRWQJ{W�"QJ_�Z]Q*cv\^b7��\^bgVYRWQna_UV�\^b~�^\�bhVYRyQ6p"Q"RyV

Ma(ω) = |Ha(ω)|2 =1

1 +(

ωω0

)2N

¸�À q ¬U¨ ¹

caV¢�^Qω0�v\�fhb�{yfhXY_afwtv��\£Z]V�caf2SanUbh{]\�p"fhV^_aQ»\^f ¬ cv¼��v�JfhVo¿

Ma(ω0) = |Ha(ω0)|2 =1

2∀N ¸�À q ¬Yª ¹

� m�naQ"{}Z]V�SUna_�Z]V*_aV^ZF\�bh\Tjlna_ap"fgVY_aQ»e*V«canUbhV�m�nv\^cURWVa�Ma(ω)

�a{yf2caQJ�^QGcaQ[Z]QJRWe*fh_v\^RyQbh\�jrqsc�q½Z�q

Ha(s)R]\�p"fhV^_v\^bhQ�Q1{}ZF\^�UfhbhQZF\^bgQ6�F�aQ |Ha(ω)|2 = Ma(ω)

q�bhbhV�{y�"VYS�V*{Wf2Sana§`{y�"RWfw�YQJRWQ

Ma(ω) = Ha(ω) ·H∗a(ω) = Ha(ω) ·Ha(−ω)

¸�À q ¬^¬ ¹Q��JVY_;bi\ S~VY{yfhpJfhVY_aQ

s = jω{Wf QJ{yZ]QJ_acaQ`\ ZWnUZWZWV�fhb(Safh\^_aVkbh\*jlna_apJfhVY_UQ�e VocanabgV m�nv\�caRWV

V^ZyZ]QJ_aQ"_acUVMa(s) = Ha(s)Ha(−s) =

1

1 +(

sjω0

)2N

¸�À q ¬ !�¹

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

N = 1N = 3N = 5N=20

ωω0

|Ha(ω

)|

�� "� �� *)*2��,� �� (2�2,$0�� 2��- ��# %�� $0# 2,� |Ha(ω)| ��) �.�� 1��,$ N�

x�S�VYbgf�cafMa(s)

{yVY_aV`bhQ6{yVYbhnapJfhVY_Uf�caQ"bgbi\�QJmona\^p"fgVY_aQ(

s

jω0

)2N

= −1 = ej(−π+2kπ) ¸�À q ¬ 4^¹

ª � ¬

Q»m�nafh_acUfsk = ω0e

jπ( 1

2+ 2k−1

2N ) k = 1, 2, . . . , 2N¸�À q ¬ À ¹

xkS�VYbhf�{WVY_aV {Wfge e*QJZWRWfh�Jf�Ryfh{WS�QJZyZ]V \^bhb�� \�{W{WQ�RyQ�\^bgQ Q�fge e \^XYfg_v\^RWfgV Q ¥^�h��v¡¢�[²��1�[� � %�¯�[�®±]�F¡&%�°��[±]��% #¢¯�¥^�A±y¯¢´¢´^��¡

ω0

qx>S�VYbgf caf

Ha(s)�"V^RWRWfg{WS�VY_acaV^_aV`\^XYbgf

sk

�JVY_�Sv\^RyZWQ6RWQ"\^bhQ6_aQJX�\�ZWfg��\

sk = ω0

[

− sin

(

2k − 1

2Nπ

)

+ j cos

(

2k − 1

2Nπ

)]

k = 1, 2, . . . , N¸�À q ¬ �Y¹

Q»bi\Tjlna_Up"fhV^_aQ»caf Z]RW\^{yjlQJRWfge QJ_�ZWV`{Wf2Sana§*RWfg{W�JRWfg�^Q"RyQ6�"VYe*Q

Ha(s) =

N∏

k=1

(−sk)

N∏

k=1

(s− sk)

=ωN

0

N∏

k=1

(s− sk)

¸�À q ¬ ��¹

��SaSUbhfh�"\^_acaV`bh\TZ]R]\�{yjlVYRye�\^pJfhVY_UQ¦�afhbgfh_aQ"\^RWQ�\^bhbh\ ¸�À q ¬ ��¹d{Wf(\^RyRWfg��\`\^bgbi\Gjrqsc�q½Z�q

H(z) = A(1 + z−1)N

N∏

k=1

(1− pkz−1)

¸�À q ¬ ��¹

caV¢�YQA

�»na_�XYnv\^cv\�XY_aV��F�aQG\^{W{yfh�JnaR]\�bi\��JVY_acafgp"fgVY_aQH(1) = 1

Q6bgV�p"QJRWV*caf2VYRWcUfh_aQN�»b��sfhe*e�\�XYfh_aQ=caQJbhbgV�p"Q"RyV cUf2VYRWcafg_aQ

N\

s =∞ qdxdS�VYbhf pk{yVY_aV`cv\�ZWf2cv\^bhbh\

pk =1 + sk

1− sk

k = 1, 2, . . . , N¸�À q ! ¹

Q^��RWfg�"VYRycv\^_acUV bh\`SURWVYSaRyfhQ[Z��jlVY_acv\^e*Q"_�Z]\^bhQ»caQJbhbi\�ZWR]\^{}jlVYRWe \^p"fgVY_aQ»�afgbhfh_UQ�\^RyQ^�a{yVY_aV�cUfh{}¶S�VY{yZWf2_aQ"b2Safh\^_aV

z{Wn�na_��JQ"Ry�F�afhV`�F�aQ(�»b��sfhe*e�\^X^fh_aQ=caQJb2�"Q"Ry�F�afhV |s| = ω0

q!|�¢�[¤��¡ "%$��$)(�* Á fA�onaVYbhQ*�"V^{yZ]RynafhRyQ`bi\�jrq c�q½Z�q�caf|na_utvbwZ]RyV�cafd¼>n«ZWZ]QJR��VYRyZW� \Z]Q"e*S�Vcafg{W�"RyQJZWV��"VY_ {WS�Q"�"fwtv�F�aQ � jlRyQ"m�naQJ_ap�\�cafd��\�e SafgVY_v\^e*Q"_�ZWV Fc = 8 � �p�� jlRWQ"m�naQJ_ap�\�cUfZF\�XYbhfgV�\^f ¬ cv¼

f0 = 1 � ��pGQ6jlRyQ"m�naQ"_Up�\ fscaf2fh_afgp"fgV�caQ"bgbi\��v\^_Ucv\`\�ZWZWQ"_�nv\�Z]\*Sv\�RWf(\ ª

� �p��JVY_na_v\*\�ZWZWQ"_�nv\^pJfhVY_UQ AdB ≥ 40cv¼q

�¦\^f2ca\�Z]f ¸ fh_z¹&{Wf2V�ZWZ]fgQ"_aQ

θ0 = 2πf0

Fc

=π

4θs = 2π

fs

Fc

=π

2Q»m�nafh_acUf2S~QJROfhb tvbwZ]RyV`\^_v\^bgVYXYfh�JV�{Wf2�a\

ω0 = tan

(

θ0

2

)

' 0.383

( RW\^c{

)

ωs = tan

(

θs

2

)

= 1

( R]\�c{

)

�¦\^bgbi\�{yS~QJ�"fgta��\`{Wnabgb�� \�ZWZWQ"_�nv\^pJfhVY_aQ�fh_�v\^_aca\ \�ZyZ]QJ_ona\�ZF\

|H(ωs)| =1

√

1 +(

ωs

ω0

)2N≤ 1

100

ª �&!

{yf2V^ZWZWfhQ"_UQ6b��sVYRWcafg_aQ6caQJb�tvbgZWRWV

N ≥ 4.8 −→ N = 5

m�nafg_acafsk = 0.383ejπ( 1

2+ 2k−1

10) k = 1, . . . , 5.x�S~VYbgf caf

H(z){yf�Z]RWV¢��\^_aV*\^SUSabhfg��\^_acUV�bi\ ¸�À q"! ¹[q

�

� �y� ��+[D[%2.7+ )+��@�$ /��:(@�$��xdtvbgZWRWf cUf��>�aQ"��o{W�aQ[� ¸ caQJb2SaRWfge V�ZWfhS�V�¹�{WV^_aV�tvbgZWRWf�\Gµ�{WVYbgf�S~V^bhfw·��"\^R]\£ZWZ]QJRWfgp"p�\£Z]f�cv\�bhb�� \£¶�^Q"RWQ»fgb~e*fh_afge V1{W�JVY{yZ]\ê*Q"_�Z]V�cv\�b~��\^bhV^RWQ�nU_afgZ]\^RWfgVTfh_ Z]nUZyZF\�bi\G�v\^_acv\�Sa\^{W{W\^_�Z]Q

[0, ωp]�

\^_Up"fh�F� �^�v�"Vê Q»f tvbwZ]Ryf2¼>nUZyZ]Q"R �>V^RyZ]� ��{yVYbhV�_UQ"bhb�� fg_�Z]VYRy_aV�caQ"bgb��sVYRWfgXYfh_UQ^qPHVY{yZWV

Ma(ω) =1

1 + PN(ω2)

�"VY_PN(ω2) ≥ 0

¸�À q"! ¨ ¹{yf��v\�fhb|e fg_afhe*V�{W�JVY{yZ]\ê*Q"_�Z]V�ca\^bd��\^bgVYRWQ`na_UfgZF\�RWfhV�fh_

[0, ωp]fh_ �JVYRWRyfh{yS~VY_UcaQ"_ap"\,caQJb

S�VYbhfg_aVYe*fhVPN (ω2)

\�e*fh_Ufhe*V*{y�"VY{}ZF\ê*Q"_�ZWV�cv\^b(�^\�bhVYRyQGpJQ"RWV�fg_[0, ωp]

q �(\�{yVYbhnapJfhVY_UQ¿»cv\�Z]\�cv\

PN(ω2) = ε2 T 2N

(

ω

ωp

) ¸�À q"! ª ¹�JVY_

ε2 j�\�ZyZ]VYRyQ=caf~{y��\^bh\1Q TN(x)S�VYbhfg_aVYe*fhV¦cUf �>�aQJ��«{y�aQJ��caf¾VYRWcUfh_aQ� cUQJtv_afwZ]VG�"VYe*Q

TN (x) =

{

cos(N arccos x) |x| ≤ 1cosh(N

\^Ry�"�"V^{W�x) |x| > 1

¸�À q"! ¬ ¹PHVY{yZWV

x = cos ϕ�v{Wf2�a\

TN (cos ϕ) = cos(Nϕ)¸�À q"! !�¹

cv\`�"nUf��aS�VY_aQJ_acaVϕ = arccos x

_aQJbhbh\

cos(N + 1)ϕ = 2 cosϕ cos(Nϕ)− cos(N − 1)ϕ¸�À q ! 4Y¹

{yf2RWfh�"\��^\�bh\`QJm�nv\^p"fgVY_aQ»RWfg�"VYRy{Wfw�^\

TN+1(x) = 2xTN(x)− TN−1(x)¸�À q"! À ¹

�F�aQ��"VY_ bgQ �JVY_acafgp"fhV^_af|fh_afgp"fi\�bhfT0(x) = 1

Q��F�UQT1(x) = x

�"VY_a{yQ"_�Z]Q�fgb|��\^bg�"VYbgV�caQJfS�VYbhfg_aVYe*f

TN(x)caf2m�nv\^bg{Wfh\^{Wf�V^RWcafg_aQ

N ≥ 2q

x�b�S�VYbhfg_aVYe*fhVTcUf��>�aQ"��o{W�aQ[�TN(x)

�v\�bhQ6{yQ"XYnUQ"_�Z]f(SaRyVYSaRWfgQJZ�� • TN (x)

�na_�S~VYbgfh_aVê fgV�Sv\�RWf ¸ �v\ {yVYbhV;S�V^ZWQ"_apJQ�Sa\^RWfOcafx¹�S�Q"R

NSv\^Ryf��AQna_

S�VYbgfh_aVYe*fhVTcafh{ySv\^RWf(S�Q"RNcafh{ySv\^RWf�q&x�_aVYbwZ]RyQ

TN(1) = 1¸ monUfh_acaf

TN(−1) = ±1\;{yQ"�"V^_acv\��F�aQN{Wfh\�Sv\^RWf>V;cafh{ySv\^RWfi¹GQ�fhb��"VoQ �*�"fgQ"_�Z]QcaQJb>Z]QJRWe*fh_aQ�caf&XYRW\^caV

e \^{y{Wfhe*VxN �

2N−1 ©

ª � 4

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−10

−8

−6

−4

−2

0

2

4

6

8

10

N = 2N = 3N = 4N = 5

x

TN

(x)

�� )* *#%�� %$0&�� %$ � ��) �� 1��,$ ($0)*)�� *#%$N

• X^bhf N pJQ"RWf/cafTN(x)

{yVY_aVRWQ"\^bhf7Q*cUfh{yZWfh_�Z]f7Q��\^cUVY_aVfh_(−1, 1)

Q�Z]RW\�canaQ�p"QJRWf�JVY_a{WQJ�"nUZWfg�of

TN(x)\�{W{WnUe Q»fhb~�^\�bhVYRyQ

+1V −1

¸ {yf2RWfh�JVYRWcUf�bi\ ¸�À q"! !�¹W¹8©• jlRW\ZWnUZWZWf|fdS�VYbgfh_aVYe*f PN(x)

�"VY_u�"VoQ �*�"fgQ"_�Z]Q caf|XYR]\^cUV�e \^{W{yfhe*VSv\^RWf|\2N−1 �

TN(x)¿»fhb�S�VYbhfg_aVYe*fhV1�F�aQ1RWfh{yVYbg�^Q6fhb~SaRWVY�UbhQ"e \

minPN

max|x|≤1|PN(x)| ¸�À q"! �^¹

�JfhV �¿�fhb¾S�VYbhfg_aVYe*fhV6\1e fg_afhe*V»{W�"V^{yZF\�e QJ_�ZWVTcv\�b��\^bgVYRWQ=p"QJRWV�fg_[−1, 1]

ZWR]\1ZWnUZWZWff¦S�VYbgfh_aVYe*f�\��YQ"_�ZWf6na_aV�{yZWQ"{W{yV��\^bgVYRWQ,caQ"b¦�JV«Q��*�"fhQJ_�Z]Q�caQ"bZ]Q"Rye fg_aQ�caf�XYR]\^cUVe \^{W{yfhe*V ¸ {WQ�_UVY_�{Wf7fhe*S~V^_aQ"{y{WQTm�naQJ{yZ]\k�"VY_acUfhp"fgVY_aQ1bi\*{WVYbgnap"fgVY_aQT{W\^RWQJ�a��Q�cv\�Z]\cv\�b�S~V^bhfh_UVYe fgVTfgcaQ"_�Z]fg��\^e*Q"_�ZWQ6_onUbhbhV�¹�Q»RWfh{ynabgZ]\

max|x|≤1|TN(x)| = max

ϕ∈[0,π]| cos(Nϕ)| = 1

• jlRW\ Z]nUZyZ]f¦f�S~VYbgfh_aV^e f Qi(x)�JVY_ XYR]\^cUV

i ≤ NQ�ZF\^bgf��F�aQ |Qi(x)| ≤ 1

S�Q"Rx ∈ [−1, 1]

�TN(x)

{WVocacafg{yj�\�bi\

|TN(x)| ≥ |Qi(x)|, |x| > 1¸�À q"! ��¹

�¦\�Z]\U�vm�nafh_Ucaf��«bi\Ma(ω) = |Ha(ω)|2 =

1

1 + ε2T 2N

(

ωωp

)

¸�À q ! ��¹

ª � À

bgQ1SURWVYSaRyfhQ[Z��`{WVYSaRW\ �JfgZF\£Z]Q»{Wf2Z]RW\^cana�JVY_aV*_aQ"bgbi\�SaRyVYSaRWfgQJZ��*�F�aQG_aQ"{y{Wna_�\^bgZWRWV�tvbgZWRWV*\^_v\L¶bgVYXYfh�JV \ µ�{WVYbgf�S�VYbgfw·caf�VYRWcafg_aQ

N�v\ bhQ��\^RW\�ZWZWQ"Ryfh{yZWfh�F�aQ,�F�aQ�\^SaSaRyVY{W{yfhe \^_aVue QJXYbhfgV

b�� \^_Ucv\ê*Q"_�Z]VGcaQ"b~Sv\^{W{W\��v\^{y{WV�fhcUQ�\^bgQ`�F¡&%v²��[¤��v¡£±y¯ % �[¤��%v²r�fh_�¼�q P7qUQfh_�¼q �Gq ¸ �"fhV ��{yQna_�tabgZ]RyV�\{WVYbgfoS�VYbgfo�a\�caQJ�ofh\^p"fgVY_aQdfh_�¼q PHq ≤ (1+ε2)−1 ��Q"{y{WV6�v\�e*fh_UVYRWQd\�ZyZ]Q"_�nv\�p"fhV^_aQfg_�¼q �Gq ¹8q

�1 ωs/ωp 3

�δs

δp = 1√1+ε2

1

ω/ωp

|Ha(ω

)|

�� ()"� ($0)*)1�� "� � �.��2�� *# ��,$ ��%$0#(� � (#��%)*2��,� � ��,�� & ��,�,� � �%$0&�� %$ � �� *#%$ �

x�b�JVYRWRyfh{yS~VY_UcaQ"_�Z]Q,e VocanabgVucaQJbhbh\uRWfg{WS�VY{yZ]\ fh_�jlRWm�naQJ_ap�\ |Ha(ω)| �v\ \^_aca\ê QJ_�Z]VQJmonUfhVY{y�"fhbgbi\�ZWVYRWfgV ¸ µrQJm�nafhRyfhSaSabgQ[·r¹(fg_��v\^_aca\¦Sv\^{W{W\^_�Z]Q��oe*Q"_�ZWRWQ&�v\�\^_acv\ê*Q"_�ZWV6e VY_UV^Z]VY_UVcaQJ�"RyQ"{W�JQ"_�Z]QTfh_�v\^_aca\*\£ZWZ]QJ_�nv\�ZF\«q�¦\�bhbi\ �/fgXaq À q��`{Wf(�YQJcaQT�F�UQTfh_�v\�_acv\ Sa\^{W{W\^_�Z]QTf2��\^bgVYRWf�e \^{y{Wfhe*f�Q1e fg_afhe*f~ZWR]\`�"naf({yfVY{y�"fgbhbi\G{WVY_aV

maxB.P. |Ha(ω)| = 1 TN (x) = 0 cos(Nϕ) = 0

minB.P. |Ha(ω)| = 1√1+ε2

TN (x) = ±1 cos(Nϕ) = ±1

¸�À q54 ¹

e*Q"_�Z]RyQ»S~QJR�bh\��v\^_aca\ \�ZyZ]QJ_ona\�ZF\�{Wf2�a\

ω −→∞ |Ha(ω)| ' 1

ε2N−1(

ωωp

)N

¸�À q54 ¨ ¹

PHQ"R��\^bg�"VYbh\^RWQ¦f S~VYbgf�cafMa(s)

{yf2caQJ�^QT\�_a_�nabhbh\^RWQ¦fgb�caQJ_aVYe*fh_v\�ZWVYRWQ�caQ"bgbi\ ¸�À q"! �Y¹

1 + ε2 T 2N

(

s

jωp

)

= 0¸�À q54 ª ¹

ª � �

Qfh_�Z]RyV«cUna�"QJ_acaV bh\,��\^RWfh\^�afhbgQs , s/ωp

QJc nU{]\^_acUV bi\;caQ[tv_afgp"fhV^_aQcafOS~V^bhfh_UVYe fgV,caf�>�aQJ��o{y�aQJ��{Wf2�v\

1 + ε2 cos2(N arccoss

j) = 0

¸�À q54 ¬ ¹P7V^_aQ"_acUV

ξ , arccos(

sj

) {Wf(\^RyRWfw�^\�\�bhbi\GQ"m�nv\^pJfhVY_aQ

1 + ε2 cos2(Nξ) = 0¸�À q54&!�¹

cv\*�"nafH{WfHV^ZWZWfhQJ_aQG�F�UQ�f(S~VYbgf(XYfi\^�J�"fgVY_aV`{Wn�nU_v\�Q"bgbhfg{W{WQ»fH�"nafHSv\^RW\^e*QJZ]Ryf2cafhS�Q"_UcaVY_aV cv\N, ε, ωp

�a�"VY_�{WQJe fh\^{W{yf

B =ωp

2

{

[√

1 + ε−2 + ε−1]1/N + [√

1 + ε−2 + ε−1]−1/N} ¸�À q 4 4Y¹

A =ωp

2

{

[√

1 + ε−2 + ε−1]1/N − [√

1 + ε−2 + ε−1]−1/N} ¸�À q 4 À ¹

x>S�VYbgf cafHa(s)

{WV^_aV`cv\�Z]f2ca\

sk = −A sin

[(

2k − 1

2N

)

π

]

+ jB cos

[(

2k − 1

2N

)

π

]

k = 1, 2, . . . , N.¸�À q 4 �Y¹�¦\^bgbi\ ¸�À q 4 �Y¹8�&RWfg�"VYRycv\^_acUV bi\ Q"{ySaRWQJ{W{WfgVY_aQ�cUQ"f�S~VYbgf=cafnU_ tvbgZWRWVucaf�¼�nUZWZWQ"R��VYR}Z]��={WfcaQJcana�"Q1�F�aQ1bi\�Sv\^R}Z]Q»RWQ"\^bhQ6cUQ"f2S�VYbhf�cUf2na_�tabgZ]RyV�caf��>�aQJ��o{y�aQJ��"VYfg_a�"fgcaQ6�"V^_�bh\`Sa\^RyZWQRWQ"\^bhQ�caQJf~S�VYbhf¾caf¾na_ tvbgZWRWVTcUf~¼�nUZWZWQ"R��VYR}Z]��caf�Sv\^Ryf�V^RWcafg_aQQ�v\^_aca\

ω0 = AQbi\»Sv\^RyZWQ

fhe*e \^XYfh_a\^RWfh\=�JVYfh_U�"fhcUQ&�"VY_*bi\¦Sa\^RyZWQOfhe*e�\^X^fh_v\^Ryfi\OcafanU_*tvbwZ]RyV6cafv¼>nUZyZ]QJR��VYRyZW� cafUSv\^RWfVYRycafh_aQ6Q»�v\�_acv\

ω0 = Bq

P7QJR7caQ[Z]QJRWe*fh_v\^RyQAb��sVYRWcUfh_aQAcaQJbYtvbgZWRWVN\OSv\^RyZWfhRyQAcv\^f�Sv\^RW\^e*QJZ]Ryf�{Wnab�RyfhSaSUbhQAfh_»�v\�_acv\

Sv\^{y{]\^_�ZWQ^�Ub�� \£ZWZ]QJ_�nv\^p"fgVY_aQOfg_ �v\�_acv\1\�ZWZWQ"_�nv\�Z]\1Q�bhQ=Sanabg{]\^pJfhVY_UfωpQ

ωs{Wf¾fh_�Z]RyV«cUna�"VY_UV

bhQ6m�nv\�_�ZWfgZ��η−1 ,

√

δ−2s − 1

ε2' 1

εδs

¸�À q 4 ��¹

κ−1 ,ωs

ωp

¸�À q 4 ��¹Q»{Wf2�v\

N =log10(η

−1 +√

η−2 − 1)

log10(κ−1 +

√κ−2 − 1)

¸�À q À ¹Q^�v�JVYe*Q6�"f2{Wf2SUn��\�{WS�QJZWZ]\^RWQ��vb��sVYRWcUfh_aQ6� cafw�YQJRWXYQ»{WQ

η−1 −→∞ Q��Vκ −→ 1

q� �� Á f¾�«nUVYbhQ��\^bg�"VYbh\^RWQ=b��sVYRycafh_aQ=caf�na_�tabgZ]RyVTSv\�{W{]\G�v\^{W{yV�caf �>�aQ"��o{W�UQJ�\��YQJ_�Z]Qna_;RyfhSaSabgQTfg_��v\^_aca\kSv\^{W{W\^_�Z]Q�Sv\�RWf7\ ¨ cv¼ �"VY_

ωs = 1.28ωp

Q�na_v\�\�ZyZ]QJ_ona\^p"fgVY_aQ�fg_�v\^_Ucv\ \£ZWZ]QJ_�nv\�ZF\

AdB ≥ 30cv¼�q �{]\^_UcaV bgQ¦jlVYRWeTnabhQ¦SURWQ"�JQ"caQJ_�Z]f

κ−1 =ωs

ωp' 1.28 δp ≡ 20 log10

√1 + ε2 = 1

¸ cv¼O¹ −→ ε ' 0.509

�¦\^bgbi\�{yS~QJ�"fgta��\`fh_��v\^_Ucv\ \£ZWZ]QJ_�nv\�ZF\`{Wf2V�ZWZ]fgQ"_aQδs ≤ 1/31

�«S~QJR��Jnafη−1 ' 62.1 −→ N ' 6.59 −→ N = 7

ª � �

� �� +[D[%2.7+ )+��31C$&.�' $ODJD[+8%2%(+"<>+xdtvbgZWRWf�caf �&\^naQ"R ¸ V`QJbhbhfwZWZWfh�"fi¹ {WVY_aV*��\�R]\�ZyZ]Q"RyfhpJp�\�ZWf�cv\^bgb�� \¢�^Q"RyQ6na_v\�jlna_ap"fgVY_aQ6caf�ZWR]\^{}jlQ"Rr¶fge QJ_�ZWV`R]\^pJfhVY_v\�bhQ��JVY_�p"Q"Ryf(Q6S~VYbgf

�

Ha(s) =

∑Mk=0 bks

k

∑Nk=0 aksk

M ≤ N¸�À q À ¨ ¹

Q»�ofH{yVY_aVa�¾m�nafg_acaf�� 1�p"QJRWfH\^b2tv_afwZ]Va�¾\^b2�JVY_�Z]RW\^RWfgV*cUQ"f2tvbgZWRWf(¼>n«ZWZ]QJR��VYRyZW��Q»caf �>�aQ"��¶{y�aQJ�caQJb2x�ZWfhS�V��F�aQG{yVY_aV*��\^RW\�ZWZWQ"Ryfhp"p"\�Z]f cv\^bhb�� \��^Q"RWQ¦ZWnUZWZWf�XYbgf�p"QJRWf(\^bgb��sfh_Utv_UfgZ]VUq�>VY_�f>tabgZ]Ryf>caf �>�aQ"��o{W�aQ[��caQ"b&x»Z]fgS~VU�|\^c�Q"{WQJe SUfhVa�A{yfORWfhQJ{W�JQ\�RWQ"_UcaQ"RyQ�b�� QJRWRyVYRWQkcUf\^SUSaRWVY{y{Wfge�\^pJfhVY_UQ na_UfgjlVYRye Q`fh_ ¼�q PHq7e*Q"_�ZWRWQ�b�� \�_acv\ê*Q"_�Z]V�fh_ ¼�q �Gq.� e VY_UV^Z]VY_UV�caQ8¶�JRWQ"{y�"QJ_�ZWQ^�ofh_`\^bwZ]RyQ>Sv\^RyVYbhQ��Yfh_`¼q �Gq^b�� \^SaSURWVY{y{Wfhe \^pJfhVY_aQdcaQJbU��\^bhVYRyQ>pJQ"RWV �ObgQ"X�\�Z]\na_Ufh��\L¶e*Q"_�Z]Q»\�b�j�\�ZyZ]V��F�aQ¦�of �¦na_Q"�J�"Q"{y{WV*caf S�VYbgf~Ryfh{WS�QJZyZ]V�\�XYbhf�p"QJRWf �F�UQ¦j�\�{��7�F�UQ6fhb e*VocanabhVcaf

Ha(ω)caQ"�JRWQ"{y��\k\�{Wfh_�ZWV^Z]fg��\ê*Q"_�Z]Q��"VYe*Q

K/ωN qÁ Q6{yf��v\^_U_aV*\�cafh{yS~VY{yfhpJfhVY_aQcaQ"X^bhf pJQ"Ryf2\^b~ta_afgZWV ¸ �JVYe Q¦\¢��ofhQJ_aQ»_aQ"f~tvbgZWRWf�caf �&\^naQJRF¹8� �RW\^XYfhV^_aQJ�^VYbhQ�S~QJ_a{]\^RyQ&�F�aQOQJ{W{WfU\^_acaRW\^_a_aV6cUfh{WS�VY{}Z]fo{Wnabgb�� \^{W{yQ&fhe*e�\�XYfh_v\�RWfhV>fh_��JVYRWRyfh{WS�VY_«¶caQJ_ap�\*caQ"bgbi\�¼�qs�Tqvfg_�e VocaV�Z]\^bhQ6cv\*\^{y{Wfh�JnaR]\�RWQ»\^bwZF\�\�ZyZ]Q"_�nv\�p"fhV^_aQ»fh_�ZF\^bgQ¦RWQJXYfhV^_aQ^qxtvbwZ]Ryf caf �&\^naQJRG{yVY_aV��\^RW\�ZWZWQ"Ryfhp"p"\�Z]f/cv\^b j�\�ZyZ]V��F�aQ bh\�jlna_apJfhVY_aQè*V«cUnabhV�m�nv\^cURWV

�JVYRWRyfh{WS�VY_acUQ"_�Z]Q |Ha(ω)|2 �dbi\�jlnU_ap"fgVY_aQdR]\^pJfhVY_v\�bhQ|�"VY_TS~VYbgf�Q>pJQ"RWf«�F�aQ>e*Q"X^bhfhV�\^SUSaRWVY{y{Wf ¶e \T�"V^_�ZWQ"e*S~V^R]\^_aQ"\ê*Q"_�Z]Qfhb¾�^\�bhVYRyQna_UfgZF\�RWfhVGfh_�¼�q PH�UQJc�fhb¾�^\�bhVYRyQ_�nabgbhVTfg_�¼�qs�Tq ¸ �JfhVo¿QJ{W{Wf(e*fh_afge fgp"p"\^_aVTb�� QJRWRyVYRWQ6e \^{W{yfhe*V�S~QJ{]\�ZWV�¹[q�(\�jlna_ap"fgVY_aQ6e*V«cUnabhV�m�nv\�caRWV \^{W{ynae*Q1bh\�jlVYRWe \ �

Ma(ω) = |Ha(ω)|2 =1

1 + ε2 R2N

(

ωωp

; L)

¸�À q À ª ¹

caV¢�^QRN(ω/ωp; L)

��caQ[ZWZF\6jlna_apJfhVY_UQ�RW\^p"fgVY_v\^bgQ&caf �>�aQJ��o{y�aQJ� ¸ V»jlna_apJfhVY_aQ=QJbhbhfwZWZWfh��\caf�Y\^�JVY�afi¹>caf2VYRWcUfh_aQ

NQ»caf�Sa\^R]\ê*QJZWRWV

Lq

PHVY{yZWVx = ω/ωp

bh\ jlna_apJfhVY_UQ�RW\^p"fgVY_v\^bgQRN(x; L)

Q"{}Z]QJ_acaQ�\^b��\^{yV caQJbhbhQjlna_apJfhVY_UfRW\^p"fgVY_v\^bgf bhQ6SURWVYSaRyfhQ[Z��caQJfHS�VYbgfh_aVYe*f¾caf��>�aQJ��o{y�aQJ� ¸ {yf��YQ"ca\*Sv\�R"q|SaRWQJ�"QJcaQ"_�Z]Q�¹[q�2Q»SaRWV^SaRWfgQJZ��caf�Z]\^bhQjlna_ap"fgVY_aQ»{WV^_aV`fh_Uj�\�ZyZ]f�bgQ¦{WQJXYnaQJ_�ZWf �¨ q

RN (x; L)¿6Sa\^RWf2S�Q"R

NSv\^Ryf��UcUfh{WSa\^RWf�S�Q"R

Ncafg{WSv\�RWf�©

ª qRN (x; L)

V^{W�"fgbhbh\GjlR]\�f ��\^bgVYRWf�� ¨ Q�¶ ¨ S�Q"Rx ∈ [−1, 1]

\^_a_�nabgbi\^_acUVY{Wf � �^VYbgZWQ^©¬ q

RN (1; L) = 1¸ �JVY_acafgp"fgVY_aQ6caf�_UVYRWe \^bhfgp"p"\^p"fgVY_aQ¢¹8©

!aqObh\ jlna_apJfhVY_UQ�RWQJ�"fhSURWVo��\1/RN(x; L)

VY{y�"fhbgbi\`jlR]\kf7��\^bgVYRWf1/L

Q −1/LS�Q"R |x| ≥

xL > 1�JVY_

xL

naX^nv\^bhQT\^bhbh\*SUf��,SUfh�"�JVYbi\*{WVYbgnap"fgVY_aQGcaQ"bgb��sQ"m�nv\^pJfhVY_aQRN (x) = L¸

L �¨ ¹8q �6nUfh_acaf

RN(x; L)V^{W�"fgbhbh\ \�ZyZ]VYRy_aV \^bgb��sfh_Utv_UfgZ]V�Q |RN(x; L)| ≥ L

�|x| ≥ xL

q�(\�jlna_apJfhVY_aQ

RN(x; L)¿�bi\,jlna_apJfhVY_UQ�R]\^pJfhVY_v\�bhQ��JVY_�XYR]\�caV;caQ"bOcUQ"_aVYe*fh_a\�Z]VYRyQ

NQ

XYRW\^caV`caQJb2_onUe QJR]\�ZWVYRWQ»_aVY_{ynaS~QJRWfgVYRWQ1\^cN�F�aQ �

\Y¹�¿»bh\�e fgXYbhfgVYRWQ\^SaSaRyVY{W{yfhe \^p"fgVY_aQ¦cUQ"bhbgV�p"QJRWV*S~QJRx ∈ [−1, 1]

©

ª � �

�v¹¿»bi\�e*fhX^bhfhV^RWQ\^SaSaRyVY{W{yfhe \^p"fgVY_aQ¦caQJbhb�� fg_Utv_afwZ]V�S�Q"R |x| ≥ xL > 1q

�(\>jlVYRWeTnabi\^pJfhVY_UQ/caQJbYSaRWV^�abhQJe�\>cUf�\^SaSaRyVY{W{yfhe \^p"fgVY_aQA\Y¹~QA�¾¹ {Wf�Z]R]\�cana�"QA_UQ"bhbh\>�"V^{yZ]Rynap"fgVY_aQcafona_v\¦QJmona\^p"fgVY_aQ>cUf �~Q"RWQJ_ap"fh\^bhQdbi\��"naf«{WVYbgnap"fgVY_aQd¿

RN(x; L)Q&�F�aQ&�JVYe S�VYR}ZF\fgb��\^bg�"VYbgV

caf2na_�fh_�ZWQ"XYRW\^bhQ¦QJbhbgfgZWZWfh�JV ¸ ca\`�Jnaf(fhb~_aVYe*Q¢¹[q�QJbG�"\^{WV b��sVYRWcafg_aQ

N{Wfi\�Sv\^Ryf��¦bh\ jlna_ap"fgVY_aQ R]\^pJfhVY_a\^bhQ;caf �>�aQJ��«{y�aQJ� {Wf1bi\^{y�"fh\

{W�JRWfw�YQ"RyQG_UQ"bhbh\TjlVYRWe \�

RN (x; L) = c

N2

∏

i=1

x2 − (xL/xi)2

x2 − x2i

¸�À q À ¬ ¹fh_*�"nUfvp"Q"Ryf¾QS�VYbhfU{WVY_aV1{Wfge e*QJZWRWfg�"faRyfh{WS�QJZyZ]V1\^bgb��sVYRWfgXYfh_aQ�q(�QJb��"\^{WVGfh_`�"nUf

N{yfi\6cafh{ySv\^Ryf

�v\^{}ZF\*\^XYXYfgna_aXYQJRWQ6\�bhbi\ ¸�À q À ¬ ¹>fgb j�\�ZWZWVYRWQxq

PARWVYSURWfhQ[Z��\^RW\�ZWZWQ"RWfgp"p"\^_�Z]QdcaQ"bgbi\ ¸�À q À ¬ ¹2¿>�F�aQ>\^c�VYX^_af�S~VYbgVxi (|xi| > xL)

�JVYRWRyfh{WS�VY_acUQna_aV p"QJRWV�fg_

xL/xi ∈ (−1, 1)q�� \^_aca\ê QJ_�Z]V�cUf

R2N(x; L)

S~QJRN = 5

¿1RWfgS~VYR}ZF\�ZWV`fh_�/fgXaq À q��Uq

�� # %�� $0# 2,� R2

N (x;L)� $0�

N = 5�

�¦\�ZWV �F�UQx = ω/ωp

�|b��sfh_�Z]QJRy��\^bgbhV[0, 1]

�JVYRWRyfh{WS�VY_acUQ�\[0, ωp]

VY{W{yfi\;\^bgbi\�¼�q P7q/Qb��sfh_�ZWQ"Ry��\^bgbhV

[xL,∞)�"V^RWRWfg{WS�VY_acaQ»\

[ωs,∞)m�nafg_acaf2\�bhbi\�¼q �Gqgq

z=fh�"V^RWcv\^_UcaV�bi\ ¸�À q À ª ¹>{Wf2�v\��F�aQ �¨ q

Ma(ω)VY{y�"fhbgbi\

N + 1�^VYbgZWQ¦fh_k¼�q PHqUZ]RW\Tfhb��\^bhVYRyQ�e*fh_afge V

1/(1 + ε2)Q¦fgb��\^bgVYRWQ

e \^{W{yfhe*V ¨ ©ª q

Ma(ω)V^{W�"fgbhbh\

N +1�^VYbgZWQfg_k¼�qs�Tq�ZWR]\»fhbv��\^bhVYRyQOe fg_afhe*V Q�fhba��\^bgVYRWQ=e \^{W{yfhe*V

1/(1 + ε2L2)q

xOSana_�ZWf7cafHe \^{W{yfhe*V QGe fg_afhe*V�cafMa(ω)

{WVY_UVN + 1

{Wfh\ fg_�¼�qs�Tq ¸ �JVYe*QTS�Q"R�fHtvbwZ]Ryfcaf �>�aQ"��o{=�aQJ��caQ"b�SURWfhe*VTV^RWcafg_aQ¢¹|�F�aQ6fh_¼�q P7q«Q¦fhb�bhV^RWV�_�nae*Q"RyV�¿¦�"VYe*SabgQ"{W{yfg��\ê*Q"_�Z]Q2N +2

m�nafh_UcafU{ynaS~QJRWfgVYRWQ�cafUna_a\�na_afwZ��¦\^bo_�nae*Q"RWV�cafUXYRW\^cafocafUbgfh��Q"R}Z��=cafMa(s)

q.�>VYe*Q{Wf��YQJcaR��S�Q"ROf~tabgZ]Ryf �7xrz µ�Q"m�nafhRyfhSaSUbhQ[· ¸ Z]QJVYRWQJe�\TcUQ"bhb�� \�bgZ]QJRW_v\�_ap�\�¹8�oZ]\^bhQ_�nae QJRWV�_aV^_�¿��\�{Wnv\^bgQ^q �(\ �/fhXaq À q ¨� e VY{}Z]RW\`b�� \�_acv\ê*Q"_�Z]V�cUf

Ma(ω)S�Q"R

N = 4Q

N = 5q

�VY_ Q �6fg_aVYbwZ]RWQ caf �*�"fgbhQ cUfhe*VY{yZWR]\^RyQ ¸ �v\�{yZF\ \^_a_�nabgbi\^RyQdMa(ω)/dω

na{]\�_acaV bi\¸�À q À ª ¹W¹ �F�aQ�f��^\�bhVYRyf�caf

ωfg_ �"naf

Ma(ω)\^{y{Wnae*Q;f��\^bgVYRWf�e�\�{W{Wfge f�Q;e*fh_afge f % ¡&%

ª �

�� # %�� $0# 2,� Ma(ω)

� $0�N = 5

$N = 4 � As = 1/(1 + ε2L2) � �

cafgS~QJ_acaVY_UV cv\ε�ve�\�cafgS~QJ_acaVY_UV {yVYbhV`cv\^bgbi\Tjlna_Up"fhV^_aQ

R2N

�vm�nafg_acaf2cv\ωs/ωp

QGca\Nq

��Q`�ofgQ"_aQ��F�aQ��2tv{W{W\�Z]fNQ

ωs

Qωp

�(\^b/��\^Ryfi\^RyQ�cafε{Wf V�ZWZ]fgQ"_aQ�nU_v\kj�\^e fgXYbhfh\�caf/tvbwZ]RWf

\��YQJ_�Z]f7ZWnUZWZWf(bi\`{yZ]QJ{W{W\�¼�q PHq¾Q�¼�qs�Tqw�~e \`cafg�^Q"RW{yf7QJRWRyVYRWf(cafH\^SaSURWVY{y{Wfhe \^pJfhVY_aQ»_aQJbhbgQ1cUnaQ�v\�_acaQ^q��HnUZWZWf2f�tvbgZWRWf��JVYRWRyfh{yS~VY_UcaQ"_�Z]f2SURWQ"{yQ"_�ZF\�_aV X^bhf�{}Z]Q"{y{Wf(Sana_�ZWfHQJ{yZWRWQ"e \^bgf�q�V^Z]\

Ma(ω)�afg{WVYX^_v\�caQJZWQ"RWe*fh_a\^RWQ�bh\�jrq c�q½Z�q

Ha(s){yZ]\^�afgbhQ��F�aQ*\^e*e*QJZWZWQ

Ma(s)�JVYe Q6jlna_Up"fhV^_aQ6e VocanabgV�m�nv\^caRyV �Ma(s) = Ha(s)Ha(−s)

¸�À q À !�¹�(\

Ha(s)�v\*bgQ6{WQJXYnaQ"_�ZWfH�"\^R]\£ZWZ]QJRWfg{yZ]fg�F�aQ �

• XYbhf�pJQ"Ryf2caf Ha(s)¸ S�VYbhf�cUf

RN(s/(jωp); L)¹>{WVY_UV fge e \^XYfg_v\^RWf¾SanURWf

zi = jωi = jωpxi

¸�À q À 4^¹Q»cafgS~QJ_acaVY_UV*ca\

ωs/ωp

Q»cv\N�ve�\ % ¡&%*cv\

ε©

• f2S~VYbgf caf Ha(s){WV^_aV`bhQ6R]\�cafh�Jf2\*Sa\^RyZWQ6RWQ�\�bhQ6_aQJX�\�Z]fw��\`cUf

1 + ε2R2N (

s

jωp) = 0

¸�À q À^À ¹Q»cafgS~QJ_acaVY_UV*ca\

ωs/ωp

�NQ

ε©

Á QN¿�Sa\^RWfO\�bhbhV^R]\

M = N¸ _�nae*Q"RyV;caQ"X^bhf&pJQ"RWf>naXYnv\^bgQ�\^b&_�nae*Q"RyV;caQ"fOS�VYbgf®¹ � XYbgfpJQ"RWf/{WVY_aV{ynabhb�� \^{y{WQ�fge e \^XYfg_v\^RyfhV�Q"c \��JVYSaSafgQ��"VYe*SabgQ"{W{yQ��"V^_afhnaXY\�Z]Q�q1x�_UVYbgZWRWQ

Ma(ω)SaRyQ"{yQ"_�ZF\*na_�e*fh_Ufhe*VTS�Q"Rω = 0

qÁ Q

N¿»cafh{ySv\^Ryf(\^bhbgVYR]\

M = N − 1Q»na_aV*p"Q"RyV {yf2Z]RWV¢��\`\^bhb�� fg_Utv_afwZ]Va�oQ»fh_aV^bgZ]RyQ

Ma(ω)SaRyQ"{yQ"_�ZF\*na_�e \^{y{Wfhe*V�fh_ω = 0

Sv\�RWf(\�na_aV*caV¢�onUZ]V \^bhbh\TSaRyQ"{WQJ_ap�\*caf2na_�S~V^bhV�RWQ"\^bhQ�q��Sv\^RyfgZ��,caf

ωs/ωp

QN��\^bd�^\�RWfi\�RWQ�cUf

ε2 ��\^Ryfi\,bh\,S�VY{Wfgp"fhV^_aQkcaQJf>S�VYbhf�Qkm�nafh_acUfXYbgf/QJRWRWV^RWf caf \^SaSaRyVY{W{yfhe \^p"fgVY_aQδp

Qδs

Q^�(\�c QJ{WQJe SafgVa�2\^b/��\^bh\^RWQ�cUfε2 {WfARWfgcana�JQ δp

Q�JRWQ"{y�"Q

δs

q��Z]\^bhQ1SaRWV^S~VY{yfgZWVa�¾{WfHSana§kVY{y{WQJRy��\^RWQT�F�aQ^�~{WQε2 � 1

Qε2L2 � 1

�¾��\^bhXYV^_aVbgQ»\^SaSaRyVY{W{yfhe \^p"fgVY_af

δp = 1− 1√1+ε2

δs = 1√1+ε2L2

' 12ε2 ' 1

εL

¸�À q À �^¹

ª � ¨

Q»m�nafh_acUfδp

Qδs

{WV^_aV bgQ"X�\£Z]f ca\^bhbh\�bhQ"X^XYQ√

δp · δs∼= cost = f

(

N ;ωs

ωp

) ¸�À q À ��¹� Sv\^RWfwZ��»caf

NQ

ωs/ωp

S�VY{y{Wfi\�e Va��\�c�Q"{yQ"e*SafhVU�om�nv\�caRWnaSUbhfh�"\^RWQδp

V^ZWZWQ"_aQJ_acaV�fg_ �"\ê`¶�afgV*nU_�cUfhe*Q"p"p"\ê*Q"_�Z]V`caf

δs

qP7QJRfgb SaRyVYXYQJZyZ]Va�vca\�Z]f

ωp

Qωs

{Wf2��\�bh�"V^bi\�

k ,ωp

ωs

η ,1

L∼= εδs −→ N

¸�À q À ��¹Q&{yfURWfg{]\^bgQ>\�bhb��sVYRycafh_aQd��Z]R]\�e fwZ]Q|ZF\^��Q"bgbhQ|V6fh_�1,\�Z]bh\^�TZWR]\ê*fgZWQ��Q �� ¸ �YQ"cUfcaVYS�V�¹8qÁ Q�{Wf/nUZ]fgbhfhpJp�\*bi\ ZWR]\^{}jlVYRWe \^p"fgVY_aQ1�afhbgfh_aQ"\^RWQ

s = (z − 1)/(z + 1)S�Q"R¦Sv\�{W{]\�RWQ�cv\^b

Safh\^_aV*µs·G\^b/SUfi\^_aV�µ

z·F� bhQTSURWVYSaRyfhQ[Z�� cUQ"b/tvbwZ]RyV�Q"bgbhfwZWZ]fg�"V*\^_v\�bhVYXYfg�"V

Ha(s)cafg�^Q"_aX^VY_aV

\^bwZ]RWQ[ZWZ]\^_�Z]Q6SaRyVYSaRWfgQJZ��`cUQ"b�tvbwZ]RWV�QJbhbgfgZWZWfh�JVT_�nae*Q"RWfg�"V �H(z) = Ha

(

z − 1

z + 1

) ¸�À q�� ¹x�_�Sv\^RyZWfh�JVYbi\^RyQ^�«bhQ6RWfg{WS�VY{yZWQ6fh_�jlRWQJm�naQ"_ap"\ Ryfh{WnUbgZF\�_aV�bhQJX�\�ZWQ¦Z]R]\�e fwZ]Q¦bgQ �

ω = tanθ

2θ = 2 arctan ω

¸�À q�� ¨ ¹caV¢�YQ

θ = 2πfT = 2πf/Fc

�JVY_TQ

Fc

S�Q"RWfgV«cUV*Q6jlRyQ"m�naQ"_Up�\�caf2�"\ê*SafhVY_a\ê QJ_�Z]V`caQ"btvbwZ]RWV�_onUe QJRWfh�JVaq Á f �v\km�nafg_acaf nU_v\kcafh{}Z]VYRy{WfgVY_aQ�caQJbhb�� \^{y{WQ�cUQ"bhbgQGjlRyQ"m�naQ"_Up"Q^�(e*Q"_�Z]RyQ�fhb�"Vê S�VYRyZ]\ê*Q"_�Z]V�caQJb2e VocanabgV�caf

H(ejθ)Ryfhe \^_aQ¦QJm�nafhRyfhSaSabgQ^q

Á f6VY{y{WQJRy��\ �F�aQ��6cv\�Z]\�bh\��JVYRWRyfh{yS~VY_UcaQ"_ap"\s = ∞ ←→ z = −1

�¦fgb¦XYRW\^caV�caQ"b_�nae*Q"R]\£Z]VYRyQ1cUf

H(z)Ryfh{WnUbgZF\�{yQ"e*SaRWQ1naXYnv\�bhQ1\*m�naQJbhbgV*cUQ"b(caQ"_UVYe fg_v\�ZWVYRWQ�q�x�_aV^bgZ]RyQ6XYbhf

p"QJRWf�{WnUbhb�� \^{y{WQdfhe*e�\^X^fh_v\^RyfhV�caQ"b�Safh\^_aVs�YQJ_aXYVY_aV�e�\^SUSv\�Z]f^fh_1p"Q"Ryf«{ynab��"QJRW�F�afgV�caf�R]\�XYXYfhV

na_afwZF\^RyfhV � |z| = 1caQJb Safh\^_aVkcUf

zq �>VY_a{yQ"XYnUQ"_�Z]QJe QJ_�ZWQ^�(fhbHS�VYbhfg_aVYe*fhV`_�nae QJR]\�ZWVYRWQTcaf

H(z)RWfg{WnabwZF\&RWQJ�"fgSaRWVo�"Va�£Q|fhbY�JVY_�Z]Ryfh�anUZWV=caQ"XYbgf�p"QJRWf�\^bgbi\>j�\^{yQdcUQ"bhbh\ORWfh{yS~V^{yZF\Ofg_»jlRyQ"m�naQJ_ap�\

H(ejθ)RWfg{WnabwZF\`bhfh_UQ�\^RyQG�JVY_�S�Q"_acaQJ_ap�\ −N/2

_aQ"bgbi\*¼�q P7qw� S�Q"R��JnafHbi\ _UVY_�bhfg_aQ�\�RWfgZ��cafj�\^{yQ1Ryfh{ynabgZ]\TZ]nUZyZF\�fhe*SanUZ]\^�afhbgQ¦\^f S~VYbgf�qx�b�j�\�ZWZWV`�F�aQ»XYbhf�pJQ"Ryf2{Wfi\�_aV`{Wnab2�JQ"RW�F�UfhV*caf2R]\^X^XYfhV�nU_afgZ]\^RWfgV��v\U�afg_aVYbgZWRWQ��Ufhe*S�VYRyZ]\^_�Z]f

�"V^_a{WQJXYnaQ"_Up"Q^�~{WQ1fhb�tvbwZ]RyV`�ofgQ"_aQ1RWQ"\^bhfgp"p�\£Z]V`�"VYe*Q»��\^{y��\�Z]\ caf(�JQ"bhbgQ6caQ"b({yQ"�"V^_acaV ¸ Q1SURWfw¶e*V�¹�VYRycafh_aQ�qx�_Uj�\�ZyZ]f��~f(j�\�ZWZWVYRWfHm�nv\^caRW\�Z]fg�"fHfh_��Jnaf/SUna§kQ"{y{WQJRWQ�j�\�ZyZ]VYRyfhp"p"\�Z]V fhbH_�nae QJR]\�ZWVYRWQGRWfg{WnabwZF\^_aVcaQJb=ZWfhS�V

1 + b1kz−1 + z−2 ��Q"{y{WQ"_UcaV b2k = zkz

∗k = 1

q �>fh§ �"VYe*S~V^RyZF\;cv\�na_ bi\�ZWVna_v\1RWfgcanap"fgVY_aQO_aQJbhbh\»�JVYe*SabhQJ{W{WfwZ��»caf¾��\^bg�"VYbgV»\^{W{yV«�Jfi\�Z]\G\�c VYX^_onU_aVTcafvZ]\^bhfUj�\£ZWZ]V^RWf ¸ nU_v\e*VYbgZWfhSabgfh��\�p"fhV^_aQTfg_ue*Q"_aV�¹[�/Q cv\^bgb�� \^bgZWRWV��F�aQkfg_unU_ \ê��afgQ"_�Z]Q*�"V^_ \^RWfwZ]e*QJZWfh��\\�SaRWQ8¶�"fg{WfgVY_aQ»tv_afgZ]\U�¾fh_�Sv\^R}Z]fh�JVYbi\�RWQ»�"VY_�\^RWfwZ]e*QJZ]fg��\�fh_�ofhRyXYVYbi\�tv{y{]\U��XYbhf(pJQ"RWfHRWfge�\�_aX�\^_aV`cafe*V«canUbhV�na_afwZF\^RyfhV�fg_acafgS~QJ_acaQ"_�ZWQ"e*Q"_�Z]Q1cv\^bgbi\TSaRyQ"�Jfh{WfgVY_aQ»�"V^_��Jnaf2{WVY_UV RW\^SaSaRyQ"{WQJ_�ZF\�ZWfHf�"VoQ �*�JfhQ"_�ZWf

b1k = −(zk + z∗k) = −2Re(zk) = −2 cos θk� Q S~V^{W{]\�_aV�{WV^bhV�{WS�VY{yZ]\^RW{yf

\^_aX^VYbi\^Rye QJ_�Z]Q^q

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� �� x�baSaRWV^XYQJZyZ]V»cafUtvbwZ]RyfUQ"bgbhfwZWZ]fg�"f�fh_`\�e��afhQJ_�Z]Q�� ofhQJ_aQ�Q�� Q[ZWZ]na\�Z]VTZWR]\ê*fgZWQ=bhQ=jlna_ap"fgVY_afUµrQJbhbgfhS�VYRWc«·]�Y�F�aQ¦�"\^bh�JVYbi\»b��sVYRycafh_aQ=caQJbhbh\jrqsc�q½Z�qw�vQ"ckµrQJbhbhfgS«·F��F�aQ1�"V^{yZ]Rynafh{y�"Q»bi\�jrqsc�q½Z�qA�"V^RWRWfg{WS�VY_acaQJ_�Z]Q^q �_�Q"{yQ"e*SafhV�cUf~SaRyVYXYQ[ZWZ]VTcaf~tvbwZ]RyVTSv\^{y{]\£¶³�v\^{y{WV�fhbgbhna{}Z]RW\1fgb~SaRyV«�JQ"cafge QJ_�Z]Va�oe Q[ZWZWQfg_bhna�JQ6na_�SURWVY�abgQ"e \�caf~µrQJbhbgfhS«·Q6e*VY{yZWR]\��"VYe*Q6{WnUS~QJR]\^RybhVaq� _��

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ncUf�na_ktvbwZ]RWVT�"VY_��v\�_acv\�Sv\^{y{]\^_�ZWQ

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q Á f2�v\�fh_UVYbw¶ZWRWQ

wn = wp

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EllipordOttimo in B.A.Ottimo in B.P.

fp fs

Fc

2

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fp

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0.25ca¼�q

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N = 5q x�b=Ryfh{WnUbgZF\£Z]V �,RW\^SaSaRyQ"{WQJ_�ZF\�ZWV fh_ �/fhXUq

À q ¨¢¬ � \^c Q"{WQJe SUfhVa�=�"V^_ fp = 1.5 � ��p^� fs = 2.5 � ��p�Q Rp = 0.1 < 2.5cv¼�{Wf

V^ZyZ]fgQ"_aQRs = 58 > 45

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ª � 4

0.05 0.1 0.15 0.2 0.25

46

48

50

52

54

56

58

60

62

Rp

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Rs

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Rs

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ª � À

Ellittico NE = 5

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= 8Butterworth N

B = 13

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Q�monUfh_acaf=cv\�bSanU_�ZWV caf=�ofh{}ZF\u�JVYe*SanUZF\�p"fhV^_v\^bhQ ¸ _�nae QJRWVucaf{yVYe*e QQ,SURWVocaV^ZWZWfOS~QJR�"\^e SUfhVY_aQ6cUf2na{W�JfgZF\Y¹>fhb~tabgZ]RyV`Saf��Q"�JVY_aVYe*fh�JV�¿»monUQ"bhbgV`Q"bhbgfgZyZ]fh�JVaq�¦\^bgbi\ �/fhXUq À q ¨ 4u{Wf=�YQJcaQ��"V^e Q�b�� \�ZWZWQ"_�nv\^pJfhVY_aQ�fh_ �v\^_aca\ \�ZWZWQ"_�nv\�Z]\ _aQJf�tvbwZ]Ryf�cUf

�>�aQJ��«{y�aQJ��Q6¼�nUZWZWQ"R��VYR}Z]��"RyQ"{W�JQ1\^bhbgVY_�ZF\^_v\�_acaVY{yf~cv\fs

Q�ZWQ"_acUQ»\^bhb�� fg_Utv_afwZ]Va��e*Q"_�Z]RyQ_aQJbHtabgZ]RyV cUf.�&\�naQ"Rbi\*\^SaSaRyVY{W{yfhe \^p"fgVY_aQ6caQJbH��\^bgVYRWQ»p"QJRWV �Tna_afwjlVYRWe*Q"e*Q"_�ZWQ1�UnaVY_v\*{WnZWnUZWZ]\*bh\��v\^_acv\«qx�_ �/fhXUq À q ¨ À �ofgQ"_aQ��2fh_Utv_UQ^��RyfhS�VYRyZ]\�Z]Vkfgb7RyfgZ]\^RWcaV�cUf/XYRynaSaS�V�fg_��v\^_Ucv\�Sv\^{W{W\^_�Z]Q`caQJfZWRWQ6tvbgZWRWf � τ(θ) = −dϕH(θ)/dθ

�"VY_ϕH(θ)

�"\^R]\£ZWZ]QJRWfg{yZ]fg��\�caf j�\�{WQ»cafH(ejθ)

qÁ f2V^{W{WQJRy��\ �F�UQGfgb tvbgZWRWV�QJbhbhfwZWZWfh�"VU�«�F�aQ»�v\`VYRycafh_aQ6fg_UjlQ"RyfhVYRyQ^�afg_�Z]RWVocana�JQ6na_�RWfgZ]\^RWcUV*fg_UjlQ[¶RyfhVYRyQGRyfh{WS�QJZyZ]V�\�XYbhfH\^bwZ]RWf2cUnaQ^��e*Q"_�ZWRWQTfgbH��\^bgVYRWQ

max(τ(θ) − τ(0))¸ �F�aQ�_aV^_�cafhS�Q"_UcaQ

cv\�bA�"VY_�Z]Ryfh�an«Z]V��"V^{yZF\�_�ZWQ�\�b RWfwZF\^RycaVkcafAXYRynaSaS�VcaQ"f _�nae*Q"RW\�Z]V^RWf caQJf Z]RyQ`tabgZ]Ryf®¹8�~�ofhQJ_aQ\^{y{Wna_�ZWV�fh_SaRyVY{W{yfhe*fgZ��Tcaf

θp

QJc�¿6�JfhRy��\�bhV`{yZWQ"{y{WV�S�Q"ROf�ZWRWQ6tvbwZ]RWf�q

� �� . 31:2K�'». � 3»N&+['TI�+ )�+ K£.7$=M*C$=I�NO3x�_�SaRWQJ�"Q"cUQ"_ap"\k{Wf(¿G�JVY_a{yfhcaQJR]\�ZWV�fhb SaRWVYX^QJZWZWV�{yVYbhV`caf2tvbwZ]RWf2Sa\^{W{W\ �v\^{y{WVa��fh_�m�nv\^_�Z]V na_m�nv\^bg{Wfh\^{Wf&Sa\^{W{W\ \�bgZ]V,V Sa\^{W{W\��a\^_acv\;V�Q"bgfhe*fh_v\��v\^_acv\;SanU§ QJ{W{yQ"RWQV^ZyZ]QJ_on«Z]V�cv\;na_

ª � �

Ellittico NE = 5

Chebyshev NC

= 8Butterworth N

B = 13

fp fs Fc

Rs

0

f

|H(f

)|

� ��

�� # ��,��# 2,� ($0 �2��,$ �%)*2�� (2�2,$0�� 2�� %$0& � ��%$ � $ � � %$0�

0

5

10

15

20

25

30Ellittico N

E = 5

Chebyshev NC

= 8Butterworth N

B = 13

fp

f

Rit

ard

odigru

ppo

�� # ��,��# 2,� ($0 �2��,$ �%)*2�� (2�2,$0�� 2�� %$0& � ��%$ � $ � � %$0�

ª � �

Sv\�{W{]\`�v\^{y{WV e QJcafi\^_�ZWQ6na_v\�VYSaS�VYRyZWna_v\�Z]RW\^{yjlV^RWe \^p"fgVY_aQ¦caf�jlRyQ"m�naQJ_ap�\Uq�bhbhV {y�"VYS�V�fh_acafg�F�afi\�e V �JVY_

H`(z)na_a\�jlna_Up"fhV^_aQ;caf¦Z]RW\^{yjlQJRWfge QJ_�ZWV�caf¦Z]fgS~V�Sv\^{y{]\

�v\�{W{WV`Q6�JVYe fg_a�"fh\ê*VGVY{y{WQJRy��\^_acaV`�F�aQ1\^bgbi\»Z]R]\�{yjlVYRye�\^pJfhVY_UQz −→ −z

�v�JVYRWRyfh{WS�VY_acUQ¦fh_jlRyQ"m�naQ"_Up�\ nU_v\`ZWR]\^{ybi\^pJfhVY_UQ¦Sv\^Ryf2\

Fc/2 �

H(z) = H`(−z)H(ejθ) = H`(e

j(θ+π))

¸�À q�� ª ¹QJc fh_ ZF\^b|e*V«caVU�7S�VYfh�F� ��S�Q"R1fhS�V^Z]QJ{Wf

H`(z)¿ cafdZWfhS�V�Sa\^{W{W\,�v\^{y{WVa�

H(z)RWfg{WnabwZF\caf

ZWfhS�V*Sa\^{W{W\ \^bwZ]Vaq��bgbi\uZWR]\^{}jlVYRWe \^p"fgVY_aQ

z −→ ±z2 �JVYRWRyfh{yS~VY_UcaQ;bi\ H(z) = H`(±z2)m�nafg_acaf6bh\

H(ejθ) = H`(−ej2θ)q Á f��YQ"Ryfgtv�"\Tj�\^�"fgbhe*Q"_�Z]Q��F�aQ6bh\1Z]RW\^{yjlV^RWe \^p"fgVY_aQ

z −→ −z2 e�nUZ]\fgb��JVYe*S~VYR}ZF\ê*Q"_�ZWV1Sa\^{W{W\T�v\^{y{WV�cafH`(z)

fh_�m�naQJbhbgVGSv\�{W{]\G�v\^_acv\1cafH(z) = H`(−z2)Q�bi\;Z]R]\�{yjlVYRye�\^pJfhVY_UQ

z −→ z2 e�nUZ]\ufhbO�JVYe*S~VYR}ZF\ê*Q"_�ZWVuSv\�{W{]\ �v\^{y{WV caf H`(z)fh_

m�naQJbhbhV`QJbhfhe*fh_a\G�v\^_Ucv\*cUfH(z) = H`(z

2)q

Á foVY{y{WQ"R}��\¦�F�aQO\^bgbi\&Z]RW\^{yjlVYRye�\�p"fhV^_aQz −→ ±z2 �JVYRWRyfh{WS�VY_acUQ�na_�RW\^cacaV^SaSafhV�caQJbhb��sVYRycafh_aQcaQJbhbh\Tjlna_ap"fgVY_aQ»caf ZWR]\^{}jlQ"Ryfhe*Q"_�Z]Vaq

_�\^bgZWRWV1SaRWVo�"QJcafhe*Q"_�Z]V1�F�aQ¦�JVY_a{yQ"_�Z]Q¦cUf�V�ZWZ]QJ_aQ"RyQ�na_a\Gjlna_apJfhVY_aQ�cafvZ]RW\^{yjlQJRWfge QJ_�ZWVcaf�Z]fhS�V�Sv\�{W{]\;�v\�_acv\�cv\�na_�Sv\�{W{]\��a\^{W{yV �JVY_a{yfh{yZWQ�_aQ"bgb��sVYS�Q"R]\�RWQ�nU_v\,Z]RW\^{Wbh\^p"fgVY_aQ�fg_jlRyQ"m�naQ"_Up�\U�

H`(ejθ)⇒ 1

2

(

H`

(

ej(θ−θ0))

+ H`

(

ej(θ+θ0)))

QJmonUfg��\^bhQJ_�Z]Q»_aQ"b�ZWQ"e*S~V*\^bgbi\h(n) −→ 2h(n) cos(θ0n)

qP7QJRdV�ZWZ]QJ_aQ"RyQOna_`tvbgZWRWV6Sv\^{y{]\»�v\^_aca\»S�VY{y{Wfi\�e V6\�_a�F�aQ=e Q[ZWZWQ"RWQOfg_`��\^{y��\�Z]\»nU_*tvbwZ]RyV

Sv\�{W{]\`�v\^{y{WV Q6na_�tabgZ]RyV`Sv\^{W{W\ \�bgZ]V �H1(z)

S�q|�v\^{y{WV*VYRWcUfh_aQN1

H2(z)S�qd\^bwZ]V�VYRycafh_aQ

N2

}

=⇒ H(z) = H1(z) ·H2(z)¸�À q�� ¬ ¹

Q�b��sVYRWcafg_aQ�cafw�YQ"_�Z]\N = N1 + N2

¸ \�_a�F�aQ,fg_ m�naQJ{yZWV ��\^{yVa�O{WQN1 = N2

�&b��sVYRycafh_aQRW\^cacaVYSUSafi\�¹8qÁ f�VY{y{WQ"R}��\��F�aQ¦�"V^_kbi\»Z]R]\�{yjlVYRye�\^pJfhVY_UQ

z → −z2 �oV^ZWZWQ"_afh\ê V1na_v\1jlna_ap"fgVY_aQcaf�ZWR]\^{}jlQ"Rr¶fge QJ_�ZWVfhbA�"nUfdQJRWRyVYRWQ`cafd\^SaSURWVY{y{Wfhe \^pJfhVY_aQ�fg_ ¼�q PHq(¿δp

�He*Q"_�ZWRWQ*XYbhf QJRWRyVYRWfA_aQJbhbgQ*cUnaQ¼�qs�Tqo{WV^_aVTQJ_�ZWR]\êT�af

δs

�«m�nafg_acaf�{WQ�{yQ"R}�«fg{W{yQ"RWVTcanaQ�QJRWRyVYRWf¾cafw�YQ"Ry{Wf�fh_*m�naQ"{}Z]Q¦canUQ��v\^_UcaQ{yfdcUVL�oRyQ"�a��Q�{yV«cacUfh{yj�\�RWQ*\^b|Saf�� Safg�"�"V^bhVcaQ"f|canUQ^q �>VY_ubi\�{WV^bhnapJfhVY_aQ`�"\^{W�"\�ZF\U�Hfh_��^Q"�"Q��S�V^Z]RyQ"e*e VV^ZyZ]QJ_aQ"RyQ canUQ QJRWRWV^RWfAcafd\^SaSURWVY{y{Wfhe \^pJfhVY_aQ�cUfg�YQJRW{yfd_UQ"bhbgQ`canaQ ¼�q �Gqg�He \fg_m�naQJ{yZ]V*��\�{WV`caV¢�oRWQ"e*e*V�Z]Q"_UQ"RWQ6�JVY_�Z]V`�"Vê�na_am�naQ»�F�aQ6XYbgf QJRWRWV^RWf fg_¼�q P7qa{yf�{WVYe*e \^_aVaq�QJb��\^{yV cUf tvbgZWRWf�eTnabgZWfh�v\^_Ucv\T�"V^_�naX^nv\^bhf�Z]VYbgbhQJR]\^_apJQ¦_aQJbhbhQ¦¼�q �GqvQ»¼�q P7qv�"f2{yf�Sana§U�

fg_*X^Q"_aQJR]\^bgQ^��RWfg�"VY_UcanaRWRyQ=\^bvSaRyVYXYQJZyZ]V»cafanU_*tvbwZ]RyV6Sv\^{W{W\£¶��a\^{W{yVTQ"m�nafw��\^bhQJ_�ZWQO�"VYbve*QJZWV«cUVcaQJbhbgQ�Z]RW\^{yjlVYRye�\�p"fhV^_af~fg_�jlRWQJm�naQ"_ap"\U�¾�"naf�\^SaSv\^R}Z]QJ_aXYVY_aV�bgQ�Z]RW\^{yjlV^RWe \^p"fgVY_af

z −→ −zQ

z −→ ±z2 �JVY_a{WfgcaQ"RW\�Z]Q�q�¦\�Z]\una_v\ jrq c�q½Z�q SaRyV^Z]V�Z]fhS�V

H`(w)cUf�ZWfhS�VuSv\�{W{]\L¶��v\^{y{WVa�O{yf�JQ"Ry��\ na_v\�Z]RW\^{yjlVYRr¶

e \^p"fgVY_aQ ¸ ��\ê��UfhV�caf��\^Ryfi\^�afgbhQ�¹w = F (z)

ZF\^bgQ¦cv\`V^ZWZWQ"_aQJRWQ»bi\Tjrqsc�q½Z�q|caQJ{WfhcUQ"R]\£ZF\ �H(z) = Hl(w)|w=F (z).

¸�À q�� !�¹

ª � �

�(\�Z]R]\�{yjlVYRye�\^pJfhVY_UQw = F (z)

cUQJ�YQ�

¨ q=QJ{W{WQJRWQ1R]\^pJfhVY_a\^bhQ�©ª q=e \^SaSv\�RWQAfhb^�"Q"Ry�F�afhVcaf�R]\�XYXYfhVOnU_afgZ]\^RWfgVOfh_»{WQd{yZWQ"{W{yVa�Y�JfhV � |F (ejθ)| = 1

�Lfh_1mona\^_�Z]V{yf�caQJ{WfgcaQ"RW\��F�aQ�Z]\^bhQ�jlna_Up"fhV^_aQ RW\^SaSaRyQ"{yQ"_�Z]f�na_ ��\�e��afhV�caf|{W�"\^bi\ ¸ _aV^_ bhfg_aQ�\^RyQ¢¹caQJbhb�� \^{y{WQ6caQJbhbgQ¦jlRWQ"m�naQJ_ap"Q�©

¬ q=SaRyQ"{yQ"Ry��\^RyQGbh\�{yZ]\^�afhbgfgZ��Uqx�_�caQJta_afgZWfg��\�bi\

F (z)cUQJ�YQ1Q"{y{WQJRWQTRW\^p"fgVY_v\^bgQ¦Q6{WVocacafg{yj�\^RyQ»bhQ6�"V^_acafhpJfhVY_Uf

|F (z)|

< 1{yQ |z| < 1

= 1{yQ |z| = 1

> 1{yQ |z| > 1

¸�À q�� 4Y¹

cv\ �Jnaf�Ryfh{WnUbgZF\u�F�aQF (z)

cUQJ�YQ;QJ{W{yQ"RWQ�na_a\ Z]R]\�{yjlVYRye�\^pJfhVY_UQ�Sv\�{W{]\L¶³Z]n«ZWZ]Vuµ�fh_a{}ZF\^�UfhbhQ8·¸ SURWVocaV^ZWZWV`caf2¼>bh\^{W�F� � Q¢¹ �

w = F (z) = ±N∏

k=1

(

1− αkz−1

−α∗k + z−1

)

|αk| < 1¸�À q�� À ¹

�F�aQ»e \^SaSv\�fgb �"QJRW�F�afgV`caf�R]\�XYXYfhVTna_afgZ]\^RWfgVN�^VYbgZWQ6{Wn{WQ»{yZWQ"{y{WV ¸ Z]QJVYRWQJe�\�cUQ"bhb�� fg_acafg��\£¶

Z]V^RWQ6bhV^X�\^RWfwZ]e*fh�JV � bh\ F (z)�v\

Np"QJRWf(fh_�ZWQ"RW_Uf�Q

NS�VYbgf Q"{}Z]QJRW_afH\^b2�JQ"RW�F�UfhV � p � � ¨ ¹8q�(\*ZWR]\^{}jlVYRWe \^p"fgVY_aQ»¿1Q"m�nafg��\^bgQ"_�Z]QT\ {yVY{yZWfgZ]nUfhRWQ»fgbHRyfgZ]\^RWcaV*Q"bgQ"e*Q"_�ZF\�RWQ � w−1 fg_ H`(w)�"V^_�bh\Tjrq c�q½Z�q

1/F (z) �

w−1 −→ ±N∏

k=1

(−α∗k + z−1

1− αkz−1

) ¸�À q�� Y¹x�b SaRyVY�abgQ"e \k�F�aQ`{WfARWfg{W�JVY_�Z]R]\kfg_�monUQ"{yZWV�e*VocaVkcafASaRyV«�JQ"caQJRWQ`¿��F�aQ��7{yQ�{WfAQ �~QJZyZ]nv\�bh\Z]RW\^{yjlV^RWe \^p"fgVY_aQ6{Wn�na_v\*RWQ�\�bhfhpJp�\^pJfhVY_UQ6�"VYe*SanUZ]\^�afhbgQ6caf

H`(w)¸ {yf2�YQ"ca\�fhb��&\^S q@��¹[�¾{yf

\^RyRWfg��\*\^c�na_a\ RWQ"\^bhfgp"p"\^p"fgVY_aQ�µr_aV^_,�"V^e San«ZF\^�afgbhQ8·F�vQ"�J�"Q[ZWZ]V�fgb2��\^{yVkfh_��"naf7\^bge QJ_aV nU_aVcaQJXYbhf

αk

{Wfh\�_onUbhbhV ¸ \^c�Q"{WQJe SUfhVa�w−1 = ±z−2 ¹[q

�� −→ �� P7V^_afi\^e*V

w−1 :RWfwZF\^RycaV�Q"bgQ"e*Q"_�ZF\�RWQ»fh_

Hl(w)z−1 :

RWfwZF\^RycaV�Q"bgQ"e*Q"_�ZF\�RWQ»fh_H(z)

�(\�Z]R]\�{yjlVYRye�\^pJfhVY_UQ¦�"QJRW��\£ZF\`¿G\�bhbhV^R]\ �w−1 =

−α + z−1

1− αz−1|α| < 1

¸�À q�� Y¹Q`{Wf V^{W{WQJRy�of|�"VYe*Q�fhb7_�nae QJR]\�ZWVYRWQ`Q�fgb/caQJ_aVYe*fh_v\�ZWVYRWQT{Wfi\�_aVb��sna_aV�fhb/RWQ"�JfhSaRyV«�"\�Z]V�cUQ"bw¶b�� \^bwZ]RWVUq� j�\^�"fgbhQdcafhe*VY{yZWR]\^RyQ��F�UQ&m�naQ"{}ZF\¦Z]RW\^{yjlVYRye�\�p"fhV^_aQd�^Q"Ryfgtv�"\¦bhQ>SURWVYSaRyfhQ[Z�� ¨ � ª Q ¬ �^fh_aVYbwZ]RyQ �

z = α←→ w = 0 z =1

α←→ w =∞

¸�À q�� ¹

ª �

PHQ"R�bi\��"V^RWRWfg{WS�VY_acaQJ_ap�\�Z]RW\wQ

z{ynab2�"QJRW�F�afgV caf2RW\^XYXYfgV�na_afwZF\^RyfhV

�

w = ejλ ←→ z = ejθ¸�À q�� ¹

{yf�Z]RWV¢��\�

θ = arctan

[

(1− α2) sin λ

2α + (1 + α2) cos λ

] ¸�À q�� ¨ ¹�F�aQ {WVY{}ZF\^_Up"fi\�bhe*Q"_�Z]Q��JVY{yZWfgZ]nUfh{W�JQ`na_u�"\ê��afgVcafA{W�"\^bi\_aVY_ bhfg_aQ�\^RyQ ¸ �JVY_�Z]RW\^p"fgVY_aQ`S~QJR1 > α > 0

Q1cafgbi\�Z]\^p"fgVY_aQ6S�Q"R −1 < α < 0¹8q �(\ ¸�À q�� ¨ ¹&¿GRyfhS�VYRyZ]\�ZF\�fh_ �/fgXaq À q ¨ ��S~QJR

α = −0.5, 0, 0.5q

α = 0.5α = 0α=−0.5

� π8

π4

3π8

π2

5π8

3π4

7π8

π

π8

π4

3π8

π2

5π8

3π4

7π8

π

θ

λ�� ! "��#%$ � ��,�� & ��,� −→ � ��,�� & ��,�(- ��

λ −→ θ

λp λs π 0 θp θs π�� $0� )1� 2��! "��#%$ � ��,�� & ��,� −→ � ��,�� & ��,�

��fh{yS~V^_aQ"_acUV�cUQ"bY{yVYbhVOXYRW\^caVOcaf^bhfh��Q"R}Z��dcv\�Z]V=cv\α�L{WfYS�V^Z]R �&{W�JQ"XYbgfhQ"RyQ ZF\^bgQ/Sv\�R]\ê*QJZWRWV

fg_�e*VocaV��F�aQλp −→ θp

Vλs −→ θs

qÁ fU{WnaSaS�VY_aXY\¦cafUcafg{WS�VYRWRyQ&caf«na_�tvbgZWRWV¦xyxrz cafUVYRWcafg_aQ&�1�^caf«Z]fhS�V�Sa\^{W{W\»�a\^{W{yV1�JVY_�Z]V^bhbhQJR}¶\^_Up"Q

δp

Qδs

Q�cUfA\^SaSabgfh�"\^RWQGbi\�Z]RW\^{yjlV^RWe \^p"fgVY_aQ ¸�À q�� ¹�j�\^�JQ"_acUV��\^Ryfi\^RyQα ∈] − 1, 1[

qx�_�ZF\�b e*V«caVTV^ZWZWQ"_afh\ê V1Z]nUZyZ]f~f~tvbwZ]RWf~xWxrz caf VYRycafh_UQ6� Q"m�nafw��\^bhQJ_�ZWf cv\�b�Sana_�Z]V�cUf �ofh{yZ]\caQJbhbgQ>{WS�Q"�Jfgtv�F�UQ

δp

Qδs

¸ XYbgf«QJRWRyVYRWf«cafa\^SUSaRWVY{y{Wfge�\^pJfhVY_UQ>RWfge�\�_aXYVY_aV�fh_a\^bgZWQ"R]\£Z]f�fh_�m�nv\�_«¶ZWV�bi\»Z]R]\�{yjlVYRye�\^pJfhVY_UQ�afgbhfg_aQ�\^RyQ=e Vocafgta��\»bhQ�{yVYbhQ�\^{W�Jfh{y{WQ¢¹8q _�QJ{WQ"e*SafgVTS�Q"R

α = −0.5Qα = 0.5

¿»RWfgS~V^RyZF\£Z]V�fh_ �7fhXaq À q ¨ �«q

ª � ¨

α = 0.5α = 0α=−0.5

0 fp/Fc 0.5f/Fc

δs

1− δp

1

|H(f

)|

�� /�,$0� �( 2��! "��#%$ � � & � → � � & �

�� −→ �� x�_�m�naQ"{}Z]V*��\^{yV�bi\GZWR]\^{}jlVYRWe \^pJfhVY_aQcafg�ofgQ"_aQbi\ ¸�À q�� ¹[�U\^SaS�Q"_v\T�ofh{}ZF\U�Ue VYbwZ]fgSabhfg��\�Z]\¦S~QJR−1

w−1 = −−α + z−1

1− αz−1

¸�À q�� ª ¹S�Q"R=�"naf �

H`(w) −→ H(z)Sv\^{W{W\£¶r\�bgZ]V ¸�À q�� ¬ ¹

�(\*e \^SaSv\w = ejλ → z = ejθ ¿»RWfgS~VYR}ZF\�Z]\Tfh_ �/fgXaq À q ¨ ��S�Q"R α = −0.5, 0, 0.5

q�(\��"VYRyRWfh{yS~V^_acaQ"_Up�\�Z]R]\��v\�_acaQ»¿»{W�F�aQJe�\�ZWfhpJp�\�Z]\`fg_ �/fhXaq À q ªo¨ qx�_�Sa\^RyZWfh�"V^bi\^RyQ¦S~QJR

α = 0{Wf2RyfgZ]RyV¢�^\�bh\

w−1 = −z−1

θ = λ + π

¸�À q�� !�¹Q^�«fh_k�JVYRWRyfh{yS~VY_UcaQ"_ap"\U�Ubh\�¼�q P7q

[0, λp]cafw�«fgQ"_aQ�bi\

[θp, π] = [π− λp, π]Qbi\�¼q �Gq

[λs, π]cafw�«fgQ"_aQ6bh\[0, θs] = [0, π − λs]

q�� −→ �� (\�Z]R]\�{yjlVYRye�\^pJfhVY_UQ¦cv\*\^SaSabgfh��\�RWQ¦¿»caQJb2{WQ"�JVY_acaV*VYRycafh_aQ

w−1 =a2 + a1z

−1 + z−2

1 + a1z−1 + a2z−2

¸�À q�� 4Y¹

ª � ª

α = 0.5α = 0α=−0.5

−π −π2 0

π2 π0

π2

π

3π2

2π

λ

θ

�� ! "��#%$ � ��,� � & ��,�,� −→ � ��,� � ��)*2,�(- �� λ −→ θ

Q¦{yf S�VY{W{yVY_aV�{y�"Q"X^bhfhQJRWQa1Q

a2fg_ke*V«caVGcv\Tfg_a{yZ]\^naR]\�RWQ�bh\G�JVYRWRyfh{WS�VY_acUQ"_ap"\�{W�F�aQJe�\�ZWfhp8¶

p"\�ZF\`fh_ �/fgXaq À q ªYªÁ f�VY{y{WQJRy��\ �F�aQ`�®��´^±y¯�¥o¡ ¥o�[�®�w¯�°8� % #"��¡&% �G¥^�A²�±y¯£�l°��[±[�®¤��%v²�¡`±y¯�¥�¥o¡F��¯ �

Hl(w)caf2XYRW\^caV*� −→ H(z)

caf�X^R]\^caV ª �Á Q fh_ Sv\^R}Z]fh�JVYbi\�RWQ�{yf�S�VY_aQ`bi\�v\^_Ucv\�Sv\�{W{]\�_�ZWQ�caQJb|Sv\^{W{W\��v\�{W{WV,naXYnv\^bgQ*\�m�naQJbhbi\caQJbSv\�{W{]\`�v\^_Ucv\ �

λp , θp2− θp1

¸�À q�� À ¹bh\�ZWR]\^{}jlVYRWe \^pJfhVY_aQ ¸�À q�� 4Y¹>\�{W{WnUe Q»bi\�jlVYRye�\ �

w−1 = −z−1−α + z−1

1− αz−1

¸�À q�� ^¹

−λp 0 λp θp π π + θp

−π −λs 0 θs

λs π π + θs 2π

¼�q �Gq

¼�q P7q

�� ! "��#%$� ��,�� & ��,�,� � �� )*2,�

ª � ¬

0 λp θp1θp2

λs π 0 θs1θs2

π

¼�q P7q ¼�q P7q

¼q �Gq ¼�q �Gq ¼�qs�Tq

−π −λs −λp 0 λp λs π −π −θs2−θp2

−θp1−θs1

θs1θp1

θp2θs2

π

�� ! "��#%$� ��,�� & ��,�,� → � �� & ��# %�

�"V^_

α =cos

(

θp1+θp2

2

)

cos(

θp1−θp2

2

)

¸�À q�� ¹

�(\�fhe*e \^XYfh_UQθ0

cafλ = 0

¸ m�nafh_acUfocUfw = ej0 = 1

¹H{yf«V�ZWZ]fgQ"_aQ�RWfg{WVYbw�YQ"_UcaV�bh\Q"m�nv\�p"fhV^_aQ

1 = −e−jθ0−α + e−jθ0

1− αe−jθ0

Q»{Wf�ZWRWV¢��\cos θ0 = α.

¸�À q�� Y¹�>VY_

α = 0{WfaRyfhV^ZyZ]fhQJ_aQ>bh\

w−1 = −z−2 Q&fh_��JVYRWRyfh{WS�VY_acUQ"_ap"\¦RWfh{ynabgZ]\ θ0 = π/2�Ym�nafh_Ucaf

fhb �"Q"_�ZWRWV*�v\^_acv\��ofhQJ_aQ»S~VYR}ZF\�ZWV`\^cna_�mona\^RyZWV caQJbhbh\TjlRWQ"m�naQJ_ap�\*caf��"\^e SUfhVY_v\�e QJ_�ZWVaqÁ f2VY{y{WQJRy�of7�F�UQGbh\�ZWR]\^{}jlVYRWe \^pJfhVY_aQ ¸�À q�� 4Y¹>Sana§*Q"{y{WQJRWQ�\�SaSabhfg��\�Z]\�\^bhbh\�RWQ"\^bhfgp"p�\�p"fhV^_aQ6caQ"bSv\^{y{]\��v\�{W{WV`{yQ"_ap"\�fh_�Z]RyV«canURWRWQ�SaRWVY�UbhQ"e*f¾caf �JVYe*SanUZF\��afhbgfgZ��«q �¦naQJ{yZF\�S�VY{y{Wfh�UfhbhfwZ��6¿¦fge`¶S�VYRyZ]\^_�Z]QT{yQT{yfH�onaVYbgQG��\^Ryfi\^RyQGbh\*�v\�_acv\�Sv\�{W{]\�_�ZWQ�caQJbHtvbwZ]RyV�Sv\^{y{]\��a\^_acv\`µr\^b2�YVYbgV�· � ¿{Wn � �JfhQJ_�ZWQT\�bhbhVT{W�"V^S~V`��\^RWfh\^RWQ

αq

x�b¾SaRWVY�UbhQ"e \»�F�aQ�{yf��a\Gna_v\»�YVYbwZF\»fh_a{}ZF\^nUR]\�Z]\1bh\1�JVYRWRyfh{WS�VY_acUQ"_ap"\1Z]RW\1bgQ�a\^_acaQ�Sv\^{r¶{]\�_�ZWf7¿G�F�aQ�S�Q"R{WVocacafh{}j�\^RWQ1bhQG{WS�Q"�Jfgtv�F�aQT�afh{yVYXY_v\*SaRWVYX^QJZWZ]\^RWQ1fhb2tvbwZ]RyV Sv\�{W{]\L¶��v\^{y{WV�caf{WS�Q"�Jfgtv�F�UQGSaf �RWQ"{}Z]RyfgZWZWfg�^Q^�a�JfhV �6�"VY_�e�\�XYXYfhV^RWQ

λp/λs

qx�_k\�bgZ]RyQOSv\^RyVYbhQ��ta{W{]\£ZF\Gbh\¦Z]R]\�{yjlVYRye�\^pJfhVY_UQ^��S�VY{W{yfi\^e*V»SaRWVYX^QJZWZ]\^RWQ=na_ Sv\^{y{]\£¶³�v\^{W{yVT�JVY_na_v\�¼�qs�Tq2naX^nv\^bhQ`\

λ′s

Vλ′′

s

¸ fge e \^XYfg_afHcafθs1

Qθs2

¹8�7\�bhbhQTmona\^bhf �JVYRWRyfh{yS~VY_UcaQ"RW\^_a_aVa�fh_XYQJ_aQ"RW\^bhQ��aVYRycafh_af cafg�^Q"RW{yf�q Á f2{y�"Q"X^bhfhQJR��U�«Z]R]\�bgQ6canaQ1{WVYbgnap"fgVY_aQ^�UmonUQ"bhbh\�\��YQ"_�ZWQTVYRycafh_aQ{WnUS~QJRWfhV^RWQ ¸ Saf��RWQ"{}Z]RyfgZWZWfg��\�¹Obi\�m�nv\�bhQT�JVY_acanU�"Q`\^c;na_�tabgZ]RyV�Sv\�{W{]\L¶��v\^_Ucv\k�F�aQ��v\�nU_v\

ª � !

caQJbhbgQ�canaQ�¼�qs�Tq Saf ��Q"{yZWQ"{W\ucaQJb&_aQ"�JQ"{y{]\^RyfhVaq Á f&V�ZWZ]fgQ"_aQ��fg_ ZF\^b>e VocaVa�|na_v\;{yVYbhnapJfhV�¶_aQ�{Wna�«¶³V^ZWZWfhe \U�Hfh_ m�nv\�_�ZWV;{WVocacafg{yj�\,\�{WS�Q"�"fwtv�F�aQkSUf�� RyQ"{yZWRWfwZWZ]fw�YQ caf�m�naQJbhbgQ�RWfg�F�afhQJ{yZWQ¸ monUQ"{yZWV¿�na_�bgfhe*fgZ]Q»caQJb/SaRyVYXYQ[ZWZ]V�caf7tvbwZ]RyfHeTnabgZWfh�v\^_Ucv\ e*Q"cafh\^_�Z]Q»Z]RW\^{yjlVYRye�\�p"fhV^_aQGfg_jlRyQ"m�naQ"_Up�\�¹8q�_v\^bhV^X�\ê*Q"_�Z]Q6{yQGbgQ6{WS�Q"�"fwtv�F�aQ1caf(\�ZyZ]Q"_�nv\�p"fhV^_aQ»_aQ"bgbhQ6canaQG¼�qs�Tqv{yVY_aV*cafg�^Q"RW{yQ

� δs16=

δs2

�~¿�_aQ"�JQ"{y{]\^RyfhV�{WVocacafg{yj�\^RyQ�\^bgbi\`Saf��,RWQ"{}Z]RyfgZWZWfg��\ cUQ"bhbgQGcUnaQ�Q"{yfhXYQJ_ap"Q ¸ m�naQ"bgbi\�RyQ"bh\�Z]fw�^\\^b

δs

Saf ��SUfh�"�JVYbhV�¹[�Ae \ \�_a�F�aQfh_�m�naQ"{}Z]Vu��\^{yVubh\�{WVYbgnap"fgVY_aQkV^ZyZ]QJ_on«ZF\�_aVY_�¿�m�naQJbhbh\V^ZyZ]fge�\Uqx�_�\^bgZWQ"Ry_v\�Z]fw��\T{Wf2SanU§*Ryfh�JVYRWRyQ"RWQ»\�bhbi\G��\^{y��\�Z]\*caf na_�Sa\^{W{W\£¶��v\�{W{WV*Q»caf�na_�Sv\^{y{]\£¶�\^bgZWV�VRyfh�"V^RWRWQJRWQ»\^c�na_�e Q[Z]VocaV�caf2SaRyVYXYQ[ZWZ]V`cafgRWQ[ZWZ]V`fg_�p�q!|�¢�[¤��¡#"%$��$)(�* �>VY_a{WfgcaQ"Ryfi\ê*V�bi\TjlnU_ap"fgVY_aQ»caf ZWR]\^{}jlQ"RWfge QJ_�Z]V`Sv\^{y{]\`�v\^{W{yV ¸�À q ªYª ¹

H`(w) =1− a

2

1 + w−1

1− aw−1

¸�À q�� ¹�JVY_��a\^_acv\

λ3dB

bhQ"XY\�ZF\`\^b ��\^bgVYRWQ�cafacv\�bhbi\

a =1− tan(λ3dB/2)

1 + tan(λ3dB/2).

¸�À q�� ¨ ¹Á Q \^bgbi\ ¸�À q�� ¹¦\^SaSabgfh�F�afh\ê*Vkbi\�ZWR]\^{}jlVYRWe \^p"fgVY_aQ ¸�À q�� Y¹[�(bh\�jrq c�q½Z�q

H`(w)Sa\^{W{W\��v\^{y{WV

cafw�ofhQ"_UQ1bh\Tjrq c qsZ"qH(z)

Sa\^{W{W\*�v\�_acv\`caf2VYRWcUfh_aQ2�"V^_��v\�_acv\`\^f

3cv¼

∆θ3dB = θp2− θp1

= λ3dB

¸�À q�� ª ¹x�_aV^bgZ]RyQ&cv\^bgbi\ ¸�À q�� ¹7Ryfh{ynabgZ]\��F�aQ=b��sfhe*e�\�XYfh_aQAcUf

λ = 0cUfg�ofhQJ_aQ

θ = θ0

¸θp1

< θ0 < θp2

¹�JVY_

α = cos θ0

q �¦nafh_UcafH(ejθ0) = H`(e

j0) = 1Q

f0/Fc = θ0/2πcafw�«fgQ"_aQ;bh\

jlRyQ"m�naQ"_Up�\ cUf2RWfh{yVY_v\^_Up�\Uq� �"V^_�ZWf�j�\�ZWZWf�{Wf�ZWRWV¢��\�bi\�jrq c qsZ"qARWfh{yVY_v\^_�ZWQ

H(z) =b0(1− z−2)

1 + a1z−1 + a2z−2

¸�À q�� ¬ ¹caV¢�^Q

b0 =1− a

2

¸�À q�� !�¹a1 = −(1− a) α = −(1− a) cos θ0

¸�À q�� 4^¹

a2 = a =1− tan(∆θ3dB/2)

1 + tan(∆θ3dB/2)

¸�À q�� À ¹�Jnaf|�"VYRyRWfh{yS~V^_acaQ*bi\��"VYSUSafi\caf|p"QJRWf

z = ±1¸ fge e \^XYfg_aQ�caf

w = −1¹»Q*bi\��JVYSaSafh\�caf

S�VYbhfp1,2 = re±jϕ0

�JVY_r =√

a2 =√

aq

PHQ"ROSafg�"�"V^bhf¾�^\�bhVYRyf~caf∆θ3dB

{Wf~�v\a ' 1−∆θ3dB

Q¦{yf~ZWRWV¢��\r =√

a ' 1−∆θ3dB/2q

�¦nafg_acaf�{yf2RWfgZWRWV¢��\�bi\∆θ3dB ' 2(1− r)

�ab�� \^_UVYe�\�bhfi\1caQ"f2S�VYbgf {WVocacafg{yj�\�bi\ϕ′

0 ' θ0Q6fhb

XYna\^cv\^XY_UV*Ryfh{ynabgZ]\b0 = (1− r2)/2 ' 1− r

qÁ f �^Q"RWfwtv�F�af2�F�aQ��¾\^SaSabgfh�"\^_acaV�bh\TZ]RW\^{yjlVYRye�\�p"fhV^_aQ��afgbhfg_aQ�\^RyQ¦\^bgbi\Gjrqsc�q½Z�qASv\�{W{]\��v\�_acv\

\^_a\^bhVYX^fh��\Ha(s) = 1/(1 + Qs/ω0 + ω0Q/s)

�U{Wf2RWfgV^ZWZWfhQJ_aQ¦bi\ ¸�À q�� ¬ ¹[q�

ª � 4

�� −→ � ��

Á fdRWfg{WVYbw�YQ`fh_ue VocaV�Q"m�nafg��\^bgQ"_�Z]Q�\^b|��\^{yV,caf|Z]RW\^{yjlV^RWe \^p"fgVY_aQ`Sv\^{y{]\,�v\�{W{WV −→ Sv\^{y{]\�v\^_Ucv\`�ofg{yZ]V`_aQJb2Sv\^RW\^XYR]\£jlV ¸�À q��Uq ¬ ¹[qÁ Q»{Wf2fge S�VY_aQ�bi\��"VY_UcafhpJfhVY_aQ ¸�À q�� À ¹8�abi\�Z]RW\^{yjlV^RWe \^p"fgVY_aQ�{yf2{WQJe Sabgfgtv�"\�_aQ"bgbi\

w−1 = z−1−α + z−1

1− αz−1

¸�À q�� Y¹

�"V^_α\^_U�"VYRW\`ca\�Z]V`cv\^bgbi\ ¸�À q�� ¹[q

� �"!� # $&%(' )�+�)�+8./$&%2%(+�+"I Nx*e Q[Z]Vocaf¦cafgRWQJZyZ]ffh_ p;RWfg�"VY_acUna�"VY_UV�fgb¦SaRyVY�abhQJe�\ucaf6SaRWV^XYQJZyZ]V�\^c na_ SaRWV^�abhQJe�\ cafV^ZyZ]fhe*fhpJp�\^pJfhVY_UQ^q�QJb��"\^{WV SUf�� {yQ"e*Sabhfg�"Q,{yf�{ynaSaS�VY_aQ�_aV�Z]V b�� \^_Ucv\ê*Q"_�Z]VucaQ"{yfhcaQJR]\�ZWV S~QJRkbi\ RWfg{WS�VY{yZ]\fhe*Sanabg{Wfw�^\ �

hd(n) 0 ≤ n ≤ P − 1¸�À q�� ¹

�F�aQ�{yf/�onaVYbhQ�\^SaSaRyVY{W{yfhe \^RWQG�"VY_;bh\�RWfg{WS�VY{yZ]\kfhe*Sanabh{yfg��\h(n)

\^{W{yV«�Jfi\�Z]\�\^bhbh\`jlna_Up"fhV^_aQcaf�ZWR]\^{}jlQ"RWfge QJ_�Z]V��\�na{]\^bgQ»{yZF\��afhbgQ

H(z) �

H(z) =

∑Mk=0 bkz

−k

∑Nk=0 akz−k

=

∞∑

n=0

h(n)z−n |z| > R−¸�À q�� ¹

�>VYe*Q6e fg{WnaRW\TcaQJbhb��sQ"RyRWVYRyQ¦{Wf �"VY_a{yfhcaQJR]\�b��sQ"RyRWVYRyQ¦m�nv\^caRW\�Z]fg�"V�jlRW\�bi\�RWfg{WS�VY{yZ]\�fhe*Sanabg{Wfg��\caQJ{WfhcUQ"R]\£ZF\�Q*m�naQ"bgbi\�\^SaSaRyVY{W{yfhe \^_�Z]Q��/fh_�ZWRWVocana�"QJ_acaV,\^_a�F�aQ na_v\�jlnU_ap"fgVY_aQ*S~QJ{WV

w(n)¸ S�Q"R=cv\�RWQ^�a\^c�QJ{WQJe SafgVa�ae \^XYXYfgVYRWQS~QJ{WV \^f�SaRyfhe*f��\�e SafgVY_af~RWfh{yS~Q[ZWZWV \�XYbhf~nabgZWfhe*f®¹ �

e =P−1∑

n=0

[ha(n)− h(n)]2w(n) w(n) ≥ 0¸�À q ¨� ¹

� m�naQ"{}Z]V1Sana_�Z]V1{Wfve*fh_afge fgp"p"\bi\ ¸�À q ¨ ¹/Ryfh{WS�QJZyZ]V1\^fa�JV«Q��*�"fhQJ_�Z]fak

Qbk

q x�bvSaRyVY�abhQJe�\RWfg{WnabwZF\^_�ZWQ6¿1_UVY_bhfh_UQ�\^RyQ¦_aQ"f(Sv\�R]\ê*QJZWRWf

ak

Qbk

Q1cUf2caf �*�"fgbhQ¦{yVYbhnUp"fhV^_aQ^qÁ Q

P = M + N + 1¸ _�nae*Q"RWVTcaf XYRW\^caf�caf bgfh��Q"R}Z��»caQ"bgbi\1jlnU_ap"fgVY_aQ�RW\^p"fgVY_v\^bgQ=fh_kQJ{]\ê*Q¢¹

bi\�{WV^bhnapJfhVY_aQ�F�aQ�e fg_afhe*fhpJp�\,bh\ ¸�À q ¨ ¹�{Wf=Ryfhcana�JQ,\�bhbi\�{yVYbhnUp"fhV^_aQcafnU_ {Wfg{yZWQ"e \ cafQ"m�nv\�p"fhV^_af2bhfg_aQ�\^Ryf �

h(n) , hd(n) n = 0, 1, . . . , M + N¸�À q Ü¨ ¹

Q"cuQ"m�nafw�^\�bhQ*\^c fhe*S~VYRyRWQ�bh\�nUXYnv\^XYbgfi\^_Up�\kcaQJfdSURWfhe*fP = M + N + 1

Z]QJRWe*fh_af/caQ"bgbhV{y�ofgbhnaSaS�V�fh_�{WQJRWfgQ�cUf �H\�naRWQJ_�Z*caQJbhbh\

Hd(z)Q�cUQ"bhbh\

H(z)¸ ¤��[²�¡"¥o¡ ¥^��&¯�¥ �¢¹[q �/\^bgQ

e*QJZ]VocaV»¿=Q"m�nafw�^\�bhQ"_�ZWQ\�c fge S�VYRWRyQ&�F�aQbhQOSaRyfhe*Q=P�µ}\^Rye V^_afh�F�aQ8·F��\�{W{WVo�"fh\�Z]Q�\^bgbi\¦Ryfh{WS�V�¶{yZ]\*fg_kjlRWQJmonUQ"_ap"\ caQJ{WfhcUQ"R]\£ZF\`QG\�m�naQJbhbi\�\�SaSaRWV^{W{Wfge�\�_�ZWQ^�v{yfi\^_aV�nUXYnv\^bgf�q

ª � À

� �"!1! # $&%('*)' )�+�� $=<�N��QJ�"p � ��VY{W{yQ"R}�Y§��F�aQ¦bh\1RW\^SaSaRyQ"{yQ"_�ZF\^pJfhVY_UQ¦fh_ ��\^{y��\�Z]\GcaQJbhbi\6jlna_apJfhVY_UQ�cafvZ]RW\^{yjlQJRWfge QJ_�ZWVSanU§T\��YQJRWQQ �~QJZWZWfv�~QJ_aQJta�"f�\^_a�F�aQ�cv\^bvSUna_�Z]VTcUfa�«fg{yZ]\»cUQ"b¾SaRWV^XYQJZyZ]V»caQ"bvtvbwZ]RyV1fg_`mona\^_�Z]V�JVY_a{WQJ_�Z]Q`caf �JVY_�Z]RWV^bhbi\�RWQG\^XYQJ�^VYbhe*Q"_�ZWQ�bi\�cUfh{WS�VY{yfhp"fgVY_aQGcaQ"f S�VYbgf7Q�cUQ"XYbgf/p"QJRWf�q Á f/Sana§U�\^c Q"{yQ"e*SafhVa�>fhe*S~VYRyRWQ��F�aQ,f�S�VYbgfOQ�XYbgfOp"QJRWf�{Wfi\�_aVuRyQ�\^bgf=V �F�aQ�XYbhfOpJQ"RWf�{yZWfi\^_UVu{ynabhbh\�JfhRW�JVY_UjlQJRWQ"_Up�\=cafYR]\�XYXYfhV>na_afgZ]\^RWfgV ¸ |z| = 1

¹ V=�F�aQ|{Wfi\�_aVOfh_»S�VY{Wfgp"fhV^_aQ RWQJ�"fgSaRWVo��\&Ryfh{WS�QJZyZ]V\^f2S�VYbgf�q

�¦\�ZWVb�� \^_Ucv\ê*Q"_�Z]V�caQ"{yfhcaQJR]\�ZWV�S�Q"RHfhb�e*V«cUnabhV |D(ejθ)| Qdbi\=R]\^SaSURWQ"{yQ"_�ZF\�p"fhV^_aQ>caQJbhbh\jrqsc�q½Z�qd\^SaSaRyVY{W{yfhe \^_�Z]Q

H(z) = AL

∏

k=1

1 + b1kz−1 + b2kz

−2

1 + a1kz−1 + a2kz−2

¸�À q ¨�ª ¹

{yfA�"VY_a{yfhcaQJR]\U�(�JVYe Q�fg_acafg�"Q�cafA{y�"VY{}ZF\ê*Q"_�ZWVa�(fhb/��\^bgVYR¦e*Q"cafgVkcaQ"b ��\^bgVYRWQ`\�{W{WV^bhnUZWV�caQ"b ¶b�� QJRWRWV^RWQ�fh_`jlRWQ"m�naQJ_ap�\�QJbhQ[�^\£Z]VG\^c�na_a\TS~V�Z]Q"_Up�\

pQ�S�Q"{W\�Z]VG{WQJ�"VY_acUV�na_v\»jlna_ap"fgVY_aQ�_aVY_

_aQJX�\�ZWfg��\W (ejθ) �

||EM ||pp =

∫ 2π

0

∣

∣|D(ejθ)| − |H(ejθ)|∣

∣

pWM(ejθ)

dθ

2π(WM ≥ 0)¸�À q Ÿ¬ ¹

�d\^bgQ6bi\�RyQ"bi\�p"fhV^_aQ||EM ||1 ≤ ||EM ||2 ≤ · · · ≤ ||EM ||∞

¸�À q ¨ !�¹�JVY_

limp−→∞

||EM ||p = ||EM ||∞ = maxθ{||D(ejθ)| − |H(ejθ)||WM(ejθ)}

¸�À q ¨ 4^¹��fg{W�JRWQJZWfhpJp�\^_acUV bh\ ¸�À q Ÿ¬ ¹>{yf2V^ZWZWfhQJ_aQ �

||EM ||pp ' E ′M ≡

L∑

k=0

||D(ejθk)| − |H(ejθk)||p WM(ejθk)¸�À q ¨ À ¹

�(\&jlna_apJfhVY_aQ/Q"RWRyVYRWQE ′

M(a1k, a2k, b1k, b1k, A)¿Ana_v\&jlna_apJfhVY_UQ _aVY_»bhfg_aQ�\�RWQ caQJfYSv\^RW\ê*QJZ]Ryf��

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E ′M(1, θk) =∞ fhbv�F�aQ�e*VY{yZWR]\

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M

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E ′M

� ¼

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M

*# � (#(�! "��#%$ rk

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Sa\^{W{W\£¶Z]n«ZWZ]V

H(z) =A(z)

A(z)=

∏

k

a2k + a1kz−1 + 1

1 + a1kz−1 + a2kz−1

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∫ 2π

0

∣

∣τD(ejθ)− τ(ejθ; a1k, a2k)∣

∣

pWτ (e

jθ)dθ

2π.

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min{α||EM ||p + (1− α)||Eτ ||p} 0 ≤ α ≤ 1¸�À q ¨� ��¹

caV¢�YQα�JVY_�Z]RyVYbhbh\�b��sfhe*S�VYRyZ]\^_ap�\caQJbhb�� \^SaSURWVY{y{Wfhe \^pJfhVY_aQ*{WnUb&e VocanabgV,Q�{ynab&RWfwZF\^RycaV�caf

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α�RyfhS�VYRyZ]\�Z]V

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p�ve \�{Wf2cafge VY{}Z]RW\

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p = 10÷ 20¹8q

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D , |D(ejθ)|2 ¸�À q ¨ ��¹

ª � �

log ||Eτ ||p

log ||EM ||pα = 0

¸ \^SaSaRyVY{W{yfhe \^p"fgVY_aQ¦{ynabhbh\�{WVYbh\�j�\^{WQ�¹

α = 1¸ \^SaSURWVY{y{Wfhe \^pJfhVY_aQ�{Wnab({WV^bhV�e*V«canUbhV�¹

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α

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M(ejθ) = H(ejθ)H∗(ejθ) =P (ejθ)

Q(ejθ)≥ 0

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pk

Qqk

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∑

k pk cos kθQ

Q =∑

k qk cos kθ��fg_;e VocaV

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maxθ

{∣

∣

∣

∣

D − P

Q

∣

∣

∣

∣

W

} ¸�À q ŸŸ¨ ¹�JVY_ fhb|�ofh_a�JVYbhV

P/Q ≥ 0QkcaV¢�YQ

W ≥ 0��na_v\,jlna_apJfhVY_UQ�S�Q"{WVUq;�Oc�QJ{WQJe SafgVa�A\^bgbi\

{y�"Q"bwZF\W = 1/D

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¸ na_aV \^bhb�� fg_�Z]Q"Ry_aV�Qna_UV�\^bgb��sQ"{}Z]Q"Ry_aV�caQJbH�JQ"Ry�F�afhV`caf2RW\^XYXYfgV�na_afgZ]\^RWfgV�¹[qÁ QT{yf7fg_acafg��\`VYR]\*�"VY_

δb�� QJRWRWV^RWQG\^{W{yVYbhnUZWV�e \^{W{yfhe*V�\�{W{WVo�"fh\�Z]V \^c�nU_v\ �JVYSaSafh\

PQ

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W

∣

∣

∣

∣

DQ− P

Q

∣

∣

∣

∣

≤ δ

P ≥ 0Q > 0minp,q,δ

δ

=⇒

W |DQ− P | ≤ δ ·QP ≥ 0Q > 0minp,q,δ

δ

¸�À q Ÿ¨Lª ¹

x�bHSaRWV^�abhQJe�\*Q"m�nafg��\^bgQ"_�Z]Q1�"V^{yZ]RynafgZWV�¿�nU_�SaRWV^�abhQJe�\`cafHSaRWVYX^R]\ê*e�\�p"fhV^_aQ6_aVY_�bgfh_aQ"\^RWQ¸ \�{WQ"�JVY_acaV e*Q"e��aRyV�caQJbhbi\ ¸�À q Ÿ¨Lª ¹»�"VYe*Sv\^fgVY_aV�f&SaRWVocaV^ZyZ]f

δ qk

¹[�A�F�aQS�Q"R �;cUfg�YQJ_�ZF\bgfh_aQ"\^RWQ¦{yQ

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δ0

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Á Qδ0 < δ∗

�"VY_δ∗ = min

p,qmax

θ

∣

∣

∣

∣

D − P

Q

∣

∣

∣

∣

W¸�À q Ÿ¨¢¬ ¹

ª � �

fhboSaRyVY�abhQJe�\_aVY_��v\¦{yVYbhnapJfhVY_UQ>Q&{yfaSana§6�JVY{yZWRWnafgRWQ&na_�_�naV¢�^V1SURWVY�abgQ"e \��"V^_�na_δ1 > δ0

qÁ Q��~fg_��^Q"�JQ^�

δ0 > δ∗fhb2SaRyVY�abgQ"e \`�JVYRWRyfh{yS~VY_UcaQ"_�Z]QG�v\`{WVYbgnap"fgVY_aQ1QG{yfHSUna§��JVY{yZWRWnafgRWQ1na_

_�naV¢�YVkSURWVY�abgQ"e \*�JVY_δ1 < δ0

q Á f(RWfhQJ{W�JQT�"V^{��d\ �"V^{yZ]RynafhRyQGnU_v\ {yQ"m�naQ"_Up�\�caf(SaRWV^�abhQJe fcaf2SaRyVYXYRW\ê e \^pJfhVY_aQbhfg_aQ�\^RyQ¦bi\��Jnaf2{WV^bhnapJfhVY_aQ6�JVY_��YQJRWXYQG\^bhbh\�{WVYbgnap"fgVY_aQ¦cUf ¸�À q Ÿ¨Lª ¹8q

�� Á Q�{WnaSaS�VY_afh\ê*V �F�UQ,fhb=caQJ_aVYe*fh_v\£Z]VYRyQ�V^ZWZWfhe*V

Q(eθ){Wfh\ �JfhRy��\u�JVY{yZ]\^_�Z]Q��OS~VY{y{Wfh\ê V

Z]RW\^{yjlV^RWe \^RWQ ¸�À q Ÿ¨Lª ¹>_aQJb2SaRWV^�abhQJe�\�bgfh_aQ"\^RWQ

W |DQ− P | ≤ γP ≥ 0Q > 0minp,q,γ

γ

¸�À q Ÿ¨ !�¹

bi\��Jnaf2{WV^bhnapJfhVY_aQ6S�VYRyXYQ6na_v\`{WVYbgnap"fgVY_aQ»\^SUSaRWVY{y{Wfge�\�Z]\�caQ"b2SURWVY�abgQ"e \�VYRWfgXYfh_a\^bhQ�q

�� !� � � � � � �#"%$�&� ��'��

� � \�bhXYVYRyfgZWe VGcaf2�"VYRyRWQJp"fhV^_aQ6caf �~Q"RyQ"_apJfi\^bgQ¦¿»na_�\^bhXYV^RWfgZWe VGfgZWQ"R]\£Z]fg�^V � {yfi\

Mi ,Pi

Qi

(Qi > 0) δi , maxθ{W |D −Mi|}

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W |DQ− P | − δiQ ≤ γ Qi

P ≥ 0minp,q,γ

γ=⇒Mi+1 ≥ 0

¸�À q Ÿ¨ À ¹

S�VYRWXYQ1na_v\`jlna_apJfhVY_aQ1e*V«canUbhV*mona\^caRWV _aVY_�_UQ"X�\�ZWfg��\ � Mi+1 = Pi+1/Qi+1 ≥ 0q Á f(VY{r¶

{WQJRy��\»�F�aQ¦\^c`VYX^_afvfgZWQ"R]\�p"fhV^_aQ&�YQJ_aXYVY_aVGnUZ]fgbhfgp"p�\£Z]fo{Wfi\»fhbUcaQ"_aVê fg_v\�Z]V^RWQQi��F�aQ�b��sQ"RWRyVYRWQ

δi

caQJbhb�� fwZ]Q"RW\^p"fgVY_aQGSaRWQJ�"Q"cUQ"_�Z]Q�� S~QJR»�"nafA�JfA{Wf|Sana§�\^{yS~Q[ZWZF\�RWQ*�F�aQ`fhb e*QJZWV«caV�JVY_��YQ"RyX�\R]\�Safhcv\�e QJ_�ZWQ^qAx�_Uj�\£ZWZ]f {Wf2cafge VY{}Z]RW\��F�aQ �

limi−→∞

δi = δ∗¸�À q Ÿ¨ �Y¹

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M0 ≥ 0�"VY_

Q0 > 0q

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