2. Transfer function. Kanazawa University Microelectronics Research Lab. Akio Kitagawa

Transcription

1 . ransfr funion Kanazawa Univrsiy Mirolronis Rsarh Lab. Akio Kiagawa

2 . Wavforms in mix-signal iruis

3 Configuraion of mix-signal sysm x Digial o Analog Analog o Digial Anialiasing Digial moohing Filr Prossor Filr AF ADC DP DAC F y x x x q ampling im LB las signifian bi Coninuous-im Disr-im Digial Disrim an quaniz 3

4 Exprssions of isr-im signal x x amplr = funion p sampling ampl & Hol or /H Impuls sampling Puls Ampliu Moulaion or PAM x u x n { u n u n } n p amplr x x n n Impuls amplr NOE: h isr analog signal is praially obain by /H, bu h signal an b hanl similar o h impuls squns sli 7. 4

5 Lapla an Z ransform Coninuous-im signal x Disr-im signal x Z x 0 x x x 3 x 4 s 5

6 Chararisi of funion p funion f f P f { u P P u } f P 0 lim 0 P f P 0 Dla funion Imporan

7 Z ransform of isr-im signal PAM signal Lapla ransform /H signal Lapla ransform ransfr funion of sp samplr PAM signal 7

8 Z ransform of PAM signal Coninuous-im signal Lapla ransform Impuls sampling Disr-im signal Z ransform A isr-im signal an b asily ransform by using Eq.. You o no n o alula wih Lapla ransform. 8

9 s-plan an z-plan s-plan im omain z-plan frquny Lf half plan s-plan Righ half plan s j 4n 4 n z 3 4n 4 - s z-plan Im[z] Uni irl = 0 os R[z] n j sin 9

10 prum of PAM signal prum of oninuous-im signal prum of isr-im signal X X A by impuls-sampling / Appnix

11 Anialiasing an smoohing of signals X X Aliasing nois X AF isrizaion Ban limiaion f u-off X Original X / isrizaion Ronsru From sampling horm, / NOE: AF an F an b implmn in oninuous-im iruis. F Ban limiaion

12 Appnix Fourir sris of impuls samplr n n n jn jn jn n n jn n 0 Inrval of ingraion

13 3 Appnix prum of PAM n n jn s X jn s X s X jn n n x x n x x ranslaion horm } { n n j X X L us say, s = j X X Coninuous-im signal PAM signal

14 Appnix 3 prum of /H Lapla ransform of /H signal li 7 X u H u PAM PAM signal ransfr funion of sp samplr H u s H s s sin s s j j s s j j j s j u s j in Phas Ampliu NOE: h sprum of /H signal is via from h sprum of PAM signal by h sp samplr H u s. hrfor, h smoohing filr afr DAC mus has h in - hararisi. 4

15 . ransfr funion of oninuous-im analog iruis 5

16 Ingraion an Diffrniaion Cirui quaion in im omain Arihmi opraion Ingral opraion Diffrnial opraion s Cirui quaion of s-variabl Arihmi opraion ransfr funion olvabl wih arihmi opraions 6

17 Dfiniion of ransfr funion ransfr funion s = + j : ransfr funion s = j : Frquny ransfr funion Pol an Zro /Hs p = 0, h omplx numbr s p is a loaion of pol in s-plan. Hs z = 0, h omplx numbr s z is a loaion of zro in s-plan. Cornr frquny of pol an zro A ornr frquny in Bo iagram is obsrv as a onsqun of pol an zro. Pol frquny: h ornr of ampliu rspons is onvx ownwar. Zro frquny: h ornr of ampliu rspons is onvx upwar. 7

18 -pol ransfr funion H s a s b s a, b, = ral numbr b as b s yp of frquny rspons a 0, b 0 a 0, b 0 LPF HPF Im Hs of LPF H s b s Im h Bo iagram is rprsns h ampliu rspons in h imaginary axis. R pol H H 8

19 Bo iagram of s orr LPF ransfr funion b H s s Im Ampliu Pol ornr frquny p -0B/D -3B zro なし p Aw w 00 pol R Phas g -90 Pw 90 H Hs w 00 9

20 Posiional rlaion bwn pol an b H s s b H j j b H j ornr frquny Im -3B lvl lin - = p Cornr frquny R Pol DC gain b H j b H j b H j 3B Hj b/ p = -3B 0

21 -pol, -zro ransfr funion a s b H s a, b, = ral numbr s Hs of HPF yp of frquny rspons a 0, b 0 a 0, b 0 LPF HPF Im H s a s s Im R pol zro H H

22 Bo iagram of s orr HPF ransfr funion a s H s s Ampliu Pol ornr frquny p Im 0-3B Aw 40 +0B/D pol zro R p Phas Pw w +45g H Hs 0.0 w 00

23 Posiional rlaion bwn pol an as H s s a j H j j a H j ornr frquny Im -3B lvl lin - = p Cornr frquny R Pol DC gain =0 a H j a H j H j a 3B Hj a p = -3B 3

24 -pol ransfr funion H s a s b s s s Hs of LPF H s s s oluion of h pol D s s s s 0 j 4 a, b, = ral numbr Complx numbr of pols yp of frquny rspons a b 0, 0 a 0, b 0 b 0, a 0 b 0, a 0, 0 LPF BPF HPF BEF No: If h nominaor is faorabl, h ransfr funion has ral pols. 4

25 Bo iagram of n orr LPF H s s の場合 s Im Im pol R H H 5

26 Bo iagram of n orr LPF H s s s Rsonan frquny r Im Ampliu r R Aw w -40B/D 00 pol Phas H Hs Pw w 00 6

27 7 Posiional rlaion bwn pol an ornr frquny 4 } { 4 } { } }{ { j H j j j j j H r r i j j s s s s s s H 4 0 のとき 4 j H r Loaion of h pol in s-plan h smallr auss h highr pak. R Im -/ r r is los o i i Hj If = r, h ampliu rah a maximum.

28 .3 ransfr funion of isrim analog/igial iruis 8

29 Dfiniion of ransfr funion on z-plan h non-linar funions of z-variabl ar u from h raional funions inluing an ingraion an a iffrniaion of s-variabl. h omplxiy of h irui implmnaion is avoi by using a ranslaion horm an som approximaions. 9

30 ranslaion horm in z-ransform Z Z Z 3 x in Dlay s x ou X in z z - X ou z h quaion 3 shows ha h mulipliaion of z -m in z-plan is quivaln o h lay of m s in im omain. h z - opraor is all "Dlay lmn". 30

31 Approximaion of Z ransform Forwar Eulr ransformaion FE Powr sris xpansion of z a s = 0 s z s s s!! 3! s s z - Im[z] 3 Uni irl = 0 Exlln approximaion R[z] z-plan 3

32 Approximaion of Z ransform Bakwar Eulr ransformaion BE Powr sris xpansion of z - a s = 0 s z s s s!! 3! s s z Im[z] 3 Uni irl = 0 Exlln approximaion - R[z] z-plan 3

33 Approximaion of Z ransform 3 Bilinar ransformaion Powr sris xpansion of log za s = 0 s ln z z z s [ z z z z 3 z z z z z z 5 5 ] Im[z] Uni irl = 0 = frquny bu non-linar sal Exlln approximaion - R[z] z-plan 33

34 Ingraion of oninuous-im signal v v in Ingraor ou v 0 in Lapla ransform v ou Frquny ransfr funion s = j H I j s j H I [ B] 0log H H I B I 0log 0log V ou s V s V s in in s s s 0 v in Prioi funion -0B g. log sal -90 log sal 34

35 Ingraion of isr-im signal Ingraion approxima wih BE H I s s z v v ou ou v 0 N n0 in Disriz v in n Z v in n = 0 V ou z N n0 N z z n V in Gomrial sris of z - V in z n = N z H I s 35

36 Ingraion rror u o approximaion Frquny ransfr funion of ingraor s = 0 z s j H s j s H w Hs H w H3 w 0. 0 Goo agrmn BE Bilinar w xa Normaliz Frquny / 5 BE of Ingraor j z j Bilinar ransformaion of ingraor z z j j h approximaion is xlln in << s /. j j 36

37 Diffrniaion of oninuous-im signal v in iffrniaor v ou v V V ou ou ou v in Lapla ransform s sv s sv in in s v s in 0 = 0 で信号がない場合 Frquny ransfr funion s = j H D H s s j D [ B] 0log H D 0log H D B 0B log sal g. +90 log sal 37

38 Diffrniaion of isr-im signal Diffrniaion approxima wih BE H D s s z v in v in v v ou Disriz ou v in v in v in Z V ou z V in z z z V in Dlay lmn V z in z H D s 38

39 Diffrniaion rror u o approximaion Frquny ransfr funion of ingraor H s s j s = 0 z s j Hs 0 H4 w H5 w H6 w 0 0. Bilinar Goo agrmn xa BE Normaliz Frquny / 0.0 w 0 BE of iffrniaor z j j Bilinar ransformaion of iffrniaor z z j j j j h approximaion is xlln in << s /. 39