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120

100

80

60

40

20

0

20

Frequency

Mag

nitu

de R

espo

nse

(dB

)

Order N=4096


`D"B?#%((A#DT:X*FRX0>**

1>

0 0.01 0.02 0.03 0.04 0.05 0.06180

160

140

120

100

80

60

40

20

0

20

Frequency

Mag

nitu

de R

espo

nse

(dB

)

Order N=4096


`DN$1`?D#%#1(A#UD:QRX.>*$ B!X*"#$

%'&)(+*,.-0/21354-7698;:?(@=AG8H/I>.8HJK,.-0/ LM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1180

160

140

120

100

80

60

40

20

0

20

Frequency

Mag

nitu

de R

espo

nse

(dB

)

Order N=4096


NGOPQRTSU9VWXY.NR!S[Z\QS^]_^`baS[cTde]c!SAfOhgjilkXmnOhgo+Qp]qgjOrpsgjOe]tuvwxyUz{rt|}~axqk

0 0.01 0.02 0.03 0.04 0.05 0.06180

160

140

120

100

80

60

40

20

0

20

Frequency

Mag

nitu

de R

espo

nse

(dB

) Order N=4096


NGOPQRTS)U9VWU9YGNRTS[Z\QS[]~_^`baS[c!de]cTSAfOg!iksXmnOhgoQp]qgjOrpg!Oe]t.R!p]c!OhgjOe]"m#p]

')+.02579;[email protected] 9

## lKH+y~K+D5+0nHDGH~

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Mql!2[

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(*,/13679;=?3?BCD1FGHI?JKLM/LO/1

...

...

Counter

128192

255

CLK4fsa

165

640

x(n)

Barrel Shifter (32in-32out)

22

y(m)

8

Coefficient ROM (4096 x 22bits)

1 2 32

5

sa fmux Add/Acc

mux Add/Acc

mux Add/Acc

KYH;|;HYXKYYX?`YX` JHXYHHYX``lHlHH`XX XO;H

sh1sh2sh3sh0

sh1 sh2sh3sh0

sh0sh1sh2 sh3

sh0sh1sh2sh3

out0

out1

out2

out3

in0 in1 in2 in3

Shift Lines

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O!RTVJY\E^-Cbcdf\Vij

d4efhg(i^jkmlnpoqjArs*tvus`utwfyxpz^{1i|s;r}i~f-tvjs`ks`{}i^jk

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:=?A@CB5DFEGIHJBLK9MNPO!MQORNS=UTIVCWX@YMZK[@\=]NPB_^+MQD^+MQW[@CB5D `baced\fJgheiZc!jkbcmlnd'obp\pqfrobhsi>otc!ouhsf)vwotcmxzyf{v}|~v}khef{\f)j\l|Pd\x}lndobbf{i>ced\x}iPg\v}cef)h>khe\f{hj\f{l)f{ieimoths|d\fx}jcmf{hsytobv}vJf)\f{i~kbhced\fhef{%f)j\l{xwf{iPd\x}lndPx}vwv/qfkbvwSoblnobhsf 5Pdf)hefced\f2gheiZcuvwkbf Rt xwiRced\fukiscul{hexzcmx}lrobvkjfbjkbhe\f{hRcmkhsf)\\l{fcmdfdobhsobhefhsf)\x}hef{f{jcmi)Aoo9x}\ \f)l{xwotcexwkj_heotcmx}kkt ` dobif{f)jlnd\kbief)j~khPobv}vkv}vwkPx}j\obp\p\hskoblnd\f{i)d\fP\f{iex}j\f{l{kugvzcmf)hd\obif)j\kdotvwxwobihefQZf)lcmx}kjuxwjcmdfPobvwxwobisf)obj\\i{ oy%x}j\/opobisieotj\hef)\f{j\l{|Rkb >S ` > bced\fkheiZc5lrobisfPobvwxwobisxwj\Px}v}vk%l)l)hefotc r r tCob\v}fPvwx}iscei5ced\fuxwjxw3\obv}xotihefZZf{l{cexwkjkhgheiZcj\kbcelnd\f{cef)hsx}j\fced\f5ox}otlnd\xwfytob\v}fobv}xwobiXhefZZf{l{cexwkj+d\fLhef{ie\vzcmiYvwx}iscef)3xwj]+ot\vwfPobhef5kcmobx}j\f)|obpp\v}|%x}j\ced\f/f)otcexwkbj~kh#ced\f/vwf{j\bced9 l)k3'l)obiel)ob\f

% Pd\f{hef3x}icmd\f/j\qf)h>kbCl)k3'gvzcmf{hei{ j\kbcmdf)h>kpcmx}kj~xwi>cmk\ief3j\kj9 v}f)jbcmdl{k~gvzcmf)h>x}j'cmdf3iZcnobf{i!kt

AC5FIJL9P!QRSUICXYZ[\]P_+Q+Q[C5 !"#%$&')( $*#$+$&',(.-/%0123 4#56 4#$7(8%$*9%%:(0;+$=?5@ %01UV+%#0f%UkV0f,%RDND%#+$9&2A E9FHoJILF

>VMY#b)Nh./rbm`!6uX.X.!M #

LV|c VLc VLc cz|c z/|c cz/|c %)8JZ. +) Q. +,L Q`# %)8JZW %, ZW %) Z.#

Y #9 #9H )#H 9p7 9]7# ) 9H #) + 9Q# 9H7 9]7 #)p )# p 9 9p7) p7 H #9p 9H 9H7 #Q7 )H ,! ,! #H !+ + 79H Q9H #) 9p# #H # )p# # ] % ] , 9p #H +

pdZ919h++X3,# / V)#6; #1X% Z##

! #"%$&(')+*-,).,/*021%3456)7'89*-;:

!#%&(+-./02%56789-;

;A@CBEDGF2HJIKBMLONQP*RSN-R'PT=VUJWCXY@ZN)L[@&=P*B]\N-D^\N-X[@CBED _2`abdc2e&fhg:i4j _lk fmg"nhg"oqprnprs&g:tdg"o&c%prsOu _wv o4xytdg"o4czprsOur{}|~"28~ v nm~ v x&gzjM.s&gKprf v o&n)g"fe&o4~prbd2oybn v%[ { " {* " {*

i4j _ s&gfmg v2 bdn.prs&g:nm~ v tdbo&cl v ~Qpr2fj

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1140

120

100

80

60

40

20

0

20

Normalized Frequency [Omega]

Atte

nuat

ion

(dB

)

Merged comb6 Filter (1 z08)5 (1 z10)

aliased band

abcze&fmgi4j _4Mv nh~ v x&gzZ&2gtdg"o&czpmsOu _lv o4x2o4gtdg"o4czprsOur{}|~"28aCbtprgf

?ACEG2JKMOQ*S-'TVJCYZ)[&*]-^-[CE 2].}0h%Y0?&mm&*m460%hr"0 rd26zEm&"h. dm"h"O:4""d6%rd2wr%m%}KdE&2md2&h64 h&}*r4w2q2"qm2&z2 r"K*r&z4'%0qrml"4%m&%r&".&A"2zhz&Kdd2&hK4d d')rm4r4m?2&mdhm46%Ymd"&%rOr}'z&2&d"&%rO-28"2 dm"h".&md&2d2m&'"mdm&rqmrd}'m"0&O".4642m6zd"m0&"&Qy.mm?m .&rmz&)"A&4rd22Am&2q2"qm2&zY"2 rd ' } 4 *&ddC24m4d *24rzd&"yqy&&' ) - ) 4 *&md.r&:m"zd&lzQr2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5160

140

120

100

80

60

40

20

0

20


Atte

nuat

ion

(dB

)


aliased band

d2&h4 "!h 2m&$#2 ddm"'*r?"dwrd28% 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5160

140

120

100

80

60

40

20

0

20


Atte

nuat

ion

(dB

)


aliased band

d2&h.4d #zm"z&.zh'&zr}?%&2&:d"&%rO-2'"2ddm"

&')(+*-,/.10324657.98::?=@;A(CB6D/E,F:G8H,(I;.KJL:?0MJL:?EH,/.10 N OPRQTSVUKWXZY\[^]`_ba cedfQg_h]`_iLjlkmneo$p q OO$r otsjluwvyxIvyzo{ut|}~ow}vjl|}9ut|Rfvownz~jl}k}|jl}vowny{o r jvyob}|vuzowxtqzo1uw|uxyu r o1j}RiLjlkmnyop q O^ uw|}xejxGvx|xyow~ot}Row}kvz O p} r |}oow}kvyz uw|vownyxwqzo-xej}ko>|n r ownwotne|@jxxyzj\vo r $xyjbvy|l|^ownnyow`mot}uwjowxtq9zo-} |ny$ljwo rneot`mot} uw ORL neowownyxv| qLzovyn}xGotnm}u

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'@.Ux0Os~F?U4040 #xA%D$h$D*PA#&yx$F$x4$&xh&Dx$N9UpO&y@F$$$&t}9###&~~#?#$&%S* &t#&H#BOt$APiDh#$att9i;

'@

sU !

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1180

160

140

120

100

80

60

40

20

0

20


Atte

nuat

ion

(dB

)

Composite Frequency Response D1 = 8, D2 = 4, (length8/8)6(length4/4)5 Comb Filter

"#%$&('*),+.-%/ (02143!576839 #%:;)@&()?A(CD

FK|2(

Y =K,=F=4.

s;.?KOj?O%7bOWO.7B*%(!;.4!*!lV.B(*l(BV.;%B(t;({l{((%;7;H*I.;(?*%!?{p;O({wdl.RuBO!!=%;?B(O;!(*pBp((O%{!I;({w%(uBO!!=TOBFp!;(%H(b7.llFjT(=

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j-?)k#&% l l _ g` f b57) l _:m%nA%=-nB l l l F c _ l

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}u~D?$aC=e8v&,s8$G,X$&"8.!t&,G;&u;7$8!8s8,?$",;?,,&,*&,8At$8&G;,88,.e,dCC$.=*k=&,&7!="&$*b&&8"=>?,=&tIX,$&v8Iv|$&,X382Ak&&8C=

"!# $F"

C",2&%3I$,8&=s&|$"',",",=s,X&&?$&"I&=,?,?&tk?8I&$ps;8?$"s;&,"$G,"*&(?,=2=($t?*?$=,8|$W8s,,v)8"2,8 &,"8+*-,/.01,32 e;4v8=& ,#5 4p3&CC,7,$"3WW=$" A$.W8 II=2 M$g,87$"t$&d8;?,,&,&,&,;& ,*=,K76FC2$>v)8",8 &,&3,"e8,?$&C7$898;:?"6I*7;88&GA,8C&&&I|&s,FC$;?D&88=&=sF8s",8& &k=|8I,",7$A,2F&8$?=v,?)*8|A8F*! &H$7T$.$&," ;?,3&,2,?)*=8s8t$Ck6?,X"88$&$>=s8s8tD$C,,|$&

IKJLNMPORQSUT&V-WEQYX[ZD\];Z]^\_La`-bRcdO3ZXO/Le\QgfhZSifhZcORQS jlk&k

First decimationFilter Filter

Second decimation

x(n)

@ 5MHzFilter 2Filter 1

FIR FilterComb CombCompensator

y(m)

@ 156.25kHz8 : 1

D D

4 : 1@ 156.25kHz@ 625kHz

mhn(o&p/qsrtFuwv&v[xy9z({|~}ENno&q7{7FrK9{ s7o&rY9{&^7s|#7Frnmh9{&>r?/{&q

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1180

160

140

120

100

80

60

40

20

0

20


Atte

nuat

ion

(dB

)

Merged Comb Filter D1 = 8 (1 z08)5 (1 z09)

aliased bandpassband droop

mhn(o&p/qsrt[uGv7[x^mqr?pFr/|Drs>{7/rCe/r{&p[/pF{/r4qss9{&mhn(zrqF{&q7zn(r?{4CEa&;

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1180

160

140

120

100

80

60

40

20

0

20


Atte

nuat

ion

(dB

)

Composite Frequency Response D1 = 8, D2 = 4, (length8/9)6(length4/4)5 Comb Filter

mhn(o&p/qsrtFuwv/x_9{&{&nrmqsr"p/rF|?rD{&/srNYFr;{&p[/pF{7/rr|?{&/9{7mhn(z(rq j qsr?rqs{+CE j vuG}C

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140


Atte

nuat

ion

(dB

)

Frequency Response of the FIR Compensator Filter vs desired Response, NFIR = 645

mhn(o&p/qsr4tFuwv&[xmqr?"p/r/|Drs>{&Fr4{7/rmh 9{7rF{&q7rqsp//rsn$qsrmqr"p/r?/|?rs>{&Fr

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

150

100

50

0

50

100

Frequency Response of the CombFIR Filter vs FIR Compensator, NFIR = 645


Atte

nuat

ion

(dB

)

mhn(o&p/qsrtFuwv7tFxmqr?pFr/|Dr{&/sr{7Fr9{& ~7o7r/ 9{&r/n({& mhn(zr?q

KNPRU&-EY[D;^_a-Rd3/eghihR lF

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1180

160

140

120

100

80

60

40

20

0

20


Atte

nuat

ion

(dB

)

Overall Frequency Response of the CombFIR Filter Cascade, NFIR = 645

h(&/s[G&P&7(K?F/Ds>&F^7h/h$?E7s#7F&/(NhF7/(7s_7"

/Cd/ 77/?7//~$&7/&($/$DF7/($^$?/7 ?& !(?#"/7///?$7 7P&/(/7 $ ?+7 7$s% s(s 7 &('*)+,- /. -0 '1. - 0 + !- 0 ' /. -032 '3)+54/. - 0 + F 76"/([/8(&"/(?9+//(&73/7/~+7#9?(h(&/Fw 6 :"/(//(Ns&/sE/ % -s^F( ;17/;( % "/?/(?=(/?s?$D F/s/&(71/~?"//s/7(719/7~$Y$#C49A@B F&7(C-?s"/?/?7="/;/(&/FPl&?7(&s/ED/(sYs?&&/F-#BF$/$G/F7 /s/ H-?/I"/$(//7$^F&/ % (/? Fe7(/7~7/ 7~/(?7/(&/?;$$7KJ?$7L"/e7Fs/F?(&

MEN1OQPSR:TLUWVAXZY[T>\]^/_(]K_`^aOcbZd:efR]g\hR*Oi^/TkjI]KUljI]KehR:TLU monqprtsAuwvyx*z[{3uy|gv1}~sA?{38vz,u/C|

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1tE$ m/ m/ m/ m/ m m pBp x*z,uyz A F|vx*z|y} F * r }vsAu *uz p$ |x*s |=vyx*z(Cs} B u E sA u * vx*F| **us }x Fvyx9vxz1|gvu*},vy*uyz*uz~|z vz zrtsAuyz *uyz pm z} B |z~z?vx vvyx*z}~sA*Fz8vZ }~uz |z~|, x*CFzx * vx*z| z *z,u;sBr}~sAk{3Fvyz~u| Q z Fv z z~ Z 8v B* s zB FvyCs 3 B z~u uz z~}~z,|y| u

1 - z -1

x(n)

1 - z -1

5

1 - z -1

(1/8)6

(1/4)

1 - z5 D = 4 FIR Filter

Compensator

y(m)-4

1 - z -8 1 - z -8

z -8

+

D = 8

5

uyz pBL Fs}Q B u rtsAuwvx*zs F{*z sA w |y} B z

E1QS:LWAZ[>/(K`acZ:fgh*i/kIKlIKh:L oA(t (Q Qf*f*3B,*~a~Ak3y~*B;y~;*A,C,;~A*,~C#$CB;yg~FAaB*3$yy*B*yAA*~HACBCBCC3$*Fy~~*3~1 !*BB"y~ oA #a$,3FA%C*iBCBy,3BL:9BF&B'i,837$FA*()* By*,=3By ~~,A H3y~+tA=i*,FHFAyF-,'F>y.Fy,/:,tA0F;*,F9$FA 1 f 1 )* A*~[3B,*~ ~A *y~3B2y? y~B3,~546!*1 879 y*;:/*3BC*C@?BADCE3$F C46!* 1 879B* 1 B F/*,**~%C*Gt.*~&H*AHF?H

A CIJ>FK F JLMNPOoF J M Q C 1 R7 FBSF*T3U*AHF"H A CV3$FK N O W 3$X LW 3 XY Z C 1 R7 ( F*;A,BF-yB*t,[E**&CB%F!B~H

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~`2g+!GraD|th-u66-+ ;

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1160

140

120

100

80

60

40

20

0

20


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y(m)

2 : 1 2 : 1 @ 156.25kHzH (z) 3H (z) 2

CompensatorStage

FIR Filter

Front-EndDecimator

f = 625kHz1 f = 312.5kHz 2z){|}~=IS w2b3# {2|~2 2r 2~. Iz)+vz){ ~2 2

100 dB

0 dB

1/4 1/2 3/4 1

Frequency

Mag

nitu

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30 dB

H(e )jW

W

z){|}~=.7 r {~4z~=K} D= 2r 3 z`{ 1 ~ 2|0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

140

120

100

80

60

40

20

0

20

Mag

nitu

de (d

B)


FIR Filter using Remez Exchange Algorithm, N = 11

z){|}~=. z~K} " = 2F ww

"L`bdg@WID+,lng`qrs-tD9)+v)+-`b &

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1120

100

80

60

40

20

0

20FIR Filter using Chebyshev Window, N = 136


Mag

nitu

de (d

B)

)=.2 K D=.2"FF 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

160

140

120

100

80

60

40

20

0

20

Mag

nitu

de (d

B)


Overall Frequency Response of H2(z) N1 = 11, N2 = 136

)=.7Il41=K.D="F5s@F#"F5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1160

140

120

100

80

60

40

20

0

20


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nuat

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aliased band

)=ISI=92==I2=)WDt5DL:0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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Loss in Attenuation Sharpened Comb vs desired FIR Compensator

)=.7I43.D=2Q.I:) =+,:

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20

15

10

5

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FIR Compensation Filter, D2 = 2, N31 = 46

12435+687:9=?

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combFilter

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@ 5MHz

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half-band filter

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@ 156.25kHz

y(m)f1

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5160

140

120

100

80

60

40

20

0

20


Mag

nitu

de R

espo

nse

(dB

)

HalfBand FIR Filter designed with Remez Exchange Algorithm

N = 22 fs = 0.125 fp = 0.375

)X%eV/VP/QD/:eQd%6%6Qe/%D/"?V&eQ6%b/IK/V:I?6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.148

6

4

2

0

2

4

6

8x 105


Mag

nitu

de R

espo

nse

(dB

)


)X%e8ml/DeF/%Q?XNCD%8eQ%NXKM/%\)XXDQ

!"#$%&))$

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|cI`Dx^iZDIdG^UiUiGI` 0

0 5 10 15 20 25 30 35 40 450.2

0.1

0

0.1

0.2

0.3

0.4

0.5

Coefficients

Val

ue

Impulse Response h(n) of the HalfBand Filter of order N = 44

I%! %0%`+0! # c*I! # c 2-i *eU 2 #U(x

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5160

140

120

100

80

60

40

20

0

20


Mag

nitu

de R

espo

nse

(dB

)


N = 600 fs = 0.25 fp = 0.27

U+3 %0%O #! 3#WK^ 3; 0 ! +!Pc 2-i *eU 2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

1.5

1

0.5

0

0.5

1

1.5x 104


Mag

nitu

de R

espo

nse

(dB

)


U+3 %%U 3! 0% L# 06! ! 3c +-h 0PU + #

|cI`DxîZDIdGÛiUiGI`

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hh

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hh

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]?êk>_>_?à _8h p?_>_>_>_>_>_>_>_>_>_>_>_>_

uwv6xzyA{/|~}.O|t59?cx/{'{"x5|$?}?{/|~} *!

0 100 200 300 400 500 6000.2

0.1

0

0.1

0.2

0.3

0.4

0.5

Coefficients

Val

ue

Impulse Response h(n) of the HalfBand Filter of order N = 600

" 9!/8 A %? " % 9O

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080.1757

0.1757

0.1756

0.1756

0.1755

0.1754

0.1754

0.1754


Mag

nitu

de R

espo

nse

(dB

)


!"3 t~ !"6" !/ 4 9 %? !/ %

w6zA/~.Ot59?c/'"5$??/~ *!

> zzUbz~?9zz&' "

"9"!

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}!FO8iM,5fk!.:,[??!F A

on g ! MO 4O5} '?O

4Fl+4l4

++. i{15.8o}mK88

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0

0

0

x (n)1

x (n)

x (n)

2

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y (m)

y (m)

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1160

140

120

100

80

60

40

20

0

20


Mag

nitu

de (d

B)

Target Filter using Chebyshev Window, fs = 0.125, fp = 0.1375, N = 478

,A(O2XQO@OQI9US(Q(QOjK(9K%eE)KOU,AA9QUISK

0 50 100 150 200 250 300 350 400 450

0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1160

140

120

100

80

60

40

20

0

20


Mag

nitu

de (d

B)

Radix3 encoded Coefficients of the Target Filter, fs = 0.125, fp = 0.1375, N = 478, q = 8, c {1, 0}

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1160

140

120

100

80

60

40

20

0

20


Mag

nitu

de (d

B)

Radix3 encoded Coefficients of the Target Filter, fs = 0.0625, fp = 0.06875, N = 478, q = 16

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100

80

60

40

20

0

20


Mag

nitu

de (d

B)

Radix3 encoded Coefficients of the Target Filter, fs = 0.125, fp = 0.1375, N = 188, q = 8

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Documents

Decimation Low Pass Filter for Sigma-Delta Modulators