Explicit Transfer Function ofRC Polyphase Filter for Wireless Transceiver Analog Front-End

H. Kobayashi, J. Kang, T. Kitahara,

S. Takigami, H. Sadamura,

Y. Niki, N. Yamaguchi

EE Dept. Gunma University, Japan

2

Contents

� Research Goal

� Roles of RC Polyphase Filter

� Transfer Function Derivation of

RC Polyphase Filter

- 1st-order filter

- 2nd-order filter

- 3rd-order filter

� Summary

3

Research Goal

� To establish systematic design and analysis

methods of RC polyphase filters.

� As its first step,

　　to derive explicit transfer functions of

　　the 1st-, 2nd- and 3rd-order　

　　RC polyphase filters.　

　　

4

Roles of RC Polyphase Filter

in Wireless Transceiver

5

Features of RC Polyphase Filter

� Its input and output are complex signal.

� Passive RC analog filter

� One of key components in wireless transceiver analog front-end

- I, Q signal generation

- Image rejection

� Its explicit transfer function has not been derived yet.

6

First-order RC Polyphase Filter

Differential Complex Input: 　 Vin = Iin + j Qin

Differential Complex Output: 　Vout = Iout + j Qout

R1C1

Iin+

Qin+

Iin-

Qin-

Iout+

Qout+

Iout-

Qout-

C1

C1

C1

R1

R1

R1

7

I, Q signal generation from single sinusoidal input

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

vo

lta

ge [

V]

time [us]

Iout Qout

11

1

CRLO =ω

Polyphase

FilterQin = 0

Iin = cos (ωLO t)Iout = A cos (ωLO t+θ)

Qout = A sin (ωLO t+θ)

8

Cosine, Sine Signals in Receiver

IF Analog

Bandpass

Filter

cos(ω0 t)

I

Q

　AD

Converter

- sin(ω0 t)

analog digital

　AD

Converter

They are used for down conversion

9

Pure I, Q signal generation

3rd-order harmonics rejection

Iin=

cos(ωLOt)+B cos (ωLOt)3

With 3rd-order harmonics.

Without3rd-order harmonics.

Iout =

A cos(ωLOt+θ)　　　Polyphase

FilterQout =

A sin(ωLOt+θ)

Qin=

sin (ωLOt)+B sin (ωLOt)3

10

Simulation of3rd-order harmonics rejection

-8

-6

-4

-2

0

2

4

6

8

20 22 24 26 28 30 32 34 36 38 40

vo

lta

ge

[V

]

time [ns]

Iout Qout

)(sin)sin()(

)(cos)cos()(

3

3

tattQ

tattI

LOLOin

LOLOin

ωω

ωω

+=

+=

)sin()(

)cos()(

θω

θω

+=

+=

tAtQ

tAtI

LOout

LOout

-8

-6

-4

-2

0

2

4

6

8

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

voltage

[V

]

time [us]

Iin Qin

11

13

CRLO

=ω

11

　Image Rejection Filter

tjtjBeAe

ωω −+

tjAe

ω

Polyphase

Filter

Iin =

(A+B) cos(ωt)

Qin =

(A-B) sin(ωt)

Iout =

Acos(ωt)

Qout =

Asin(ωt)

12

Problem when ωLO≠１/R1C1

Amplitudes of I,Q signals differ significantly.

-3

-2

-1

0

1

2

3

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

vo

lta

ge

[V

]

time [us]

QoutIout

11

2

CRLO

=ω

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

voltage

[V

]

time [us]

Iout Qout

11

1

CRLO =ω

13

2nd-order RC Polyphase Filter

Iout Qout

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

voltag

e [

V]

time [us]

　

11

2

CRLO

=ω

The problem of large

difference between

Iout, Qout amplitudes

can be alleviated

R1

R1

R1

R1

C1

C1

C1

C1

C2 C2

C2

C2

R2

R2

R2

R2

Iin+

Qin+

Iin-

Qin-

Iout+

Qout+

Iout-

Qout-

14

3rd-order RC Polyphase Filter

11

2

CRLO

=ω

Iout Qout

The amplitude

difference problem

is further alleviated.

R1

R1

R1

R1

C1

C1

C1

C1

C2

C2

C2

C2

R2

R2

R2

R2

R3

C3

R3

C3

R3C3R3

C3

Iin+

Qin+

Iin-

Qin-

Iout+

Qout+

Iout-

Qout-

15

Transfer Function Derivation

of RC Polyphase Filter

16

Transfer Function of RC Polyphase Filter

� Complex Signal Theory

� Complex input

� Complex output

� Complex

Transfer Function

)(

)()(

ω

ωω

jV

jVjG

in

out=

outoutout

ininin

QjIjV

QjIjV

⋅+=

⋅+=

)(

)(

ω

ω

17

Transfer Function of1st-order RC Polyphase Filter

)()()(

)()()(

)()()(

)()()(

)()()(

)()()(

tjQtItV

tjQtItV

tQtQtQ

tItItI

tQtQtQ

tItItI

outoutout

ininin

outoutout

outoutout

ininin

ininin

+=

+=

−=

−=

−=

−=

−+

−+

−+

−+

Differential signal

Complex signal

R1

C1

Iin+

Qin+

Iin-

Qin-

Iout+

Qout+

Iout-

Qout-

C1

C1

R1

R1

R1C1

18

Transfer Function of1st-order RC Polyphase Filter

21

1

)(1

1)(

.1

1)(

RC

RCjG

RCj

RCjG

ω

ωω

ω

ωω

+

+=

+

+=

•Transfer Function

•Gain

RC

1−

19

Nyquist Chart of G1(jw)

G1(j ω) = X(ω) + j Y(ω)

Symmetric with respect to a line of Y = -X.

20

Explanation of I, Q signal generation by G1(jω)

][2

1)cos()( tjtj

in eettVωω

ω−

+==

)4

sin(2

2)

4cos(

2

2

])()([2

1)(

))((

1

))((

111

πω

πω

ωωωωωω

−+−=

−+=−∠+−∠+

tjt

ejGejGtVjGtjjGtj

out

4)(,2)(,0)( 11111

πωωω

ωω−=∠==

=−=jGjGjG

RCRC

),cos()(,0)( ttItQinin

ω=≡

21

Output Load (CL) Effects

G1(jω) =

1+ωR1C1

1+jωR1(C1+CL)

Iin+ Iout+

CL

CL

CL

CL

R1

C1

Qin+

Iin-

Qin-

Qout+

Iout-

Qout-C1

R1

R1

R1

C1

22

Input Impedance

Input Impedance Zin =

Complex Input Voltage

Complex Input Current

Zin =1+jωR1C1

2 jωC1 [1+j(1+ωR1C1)]

R1C1

Iin+

Qin+

Iin-

Qin-

Iout+

Qout+

Iout-

Qout-

C1

C1

R1

R1

R1C1

23

Component Mismatch Effects

VinXEVinout

VinXEVinout

⋅⋅+⋅

⋅⋅+⋅

δ

δ

1

1

１

１

Ｇ＝Ｖ

＝ＧＶ

211

111

)1(2

)1(:

ω

ω

CjR

CRjE

+

+=where

Mismatches among R’s, C’s

Image signal Vout is caused.

Xδ

Image transferfunction

R1C1

Iin+

Qin+

Iin-

Qin-

Iout+

Qout+

Iout-

Qout-

C1

C1

R1

R1

R1C1

24

Transfer Function of2nd-order RC Polyphase Filter

)2(1

)1)(1()(

21221122112

22112

CRRCRCjCRCR

CRCRjG

+++−

++=

ωω

ωωω

Transfer Function

11

1

CR−

22

1

CR−

Derivation is very complicated, so we used “Mathematica.”

Gain |G2(jω)| characteristics

25

Nyquist Chart of G2(jω)

G2(j ω) = X(ω) + j Y(ω)

Symmetric with respect to a line of X = 0.

26

Features of 2nd-order RC Polyphase Filter

For arbitrary a, .)/()(211211

ajGjaG ωωωω =

,0)(

,1)(lim

,0)()(

,)()(

21

2

2

2212

22

=

∂

∂

=

=−=−

−≠

=

±∞→

ωωω

ω

ω

ω

ω

ωω

ωω

jG

jG

jGjG

jGjG

,1)0(2

=jG

,)()(2212

ωω jGjG =

2

21

2

21

212

212

)(

)(

|)(|

|)(|

ωω

ωω

ωω

ωω

−

+=

− jG

jG

27

Flat passband design of2nd-order RC Polyphase Filter

Passband: ω1 ～　ω2

Stopband: -ω1 ～　-ω2

where ω1 := 1/R1C1,

ω2 := 1/R2C2

●　To make gain in passband flat,

|G2 (jω1)| = |G2 (jω2)| = |G2 (j ω1ω2)|.

●　Image Rejection Ratio =ω2 + ω1

ω2 - ω1

2

11

1

CR−

22

1

CR−

passband

stopband

28

Explanation why a 2nd-order filter reduces I, Q amplitude difference.

)(2

1)cos()(,0)( tjtj

inin eettItQωω

ω−

+==≡

)sin())(

)(1()()cos()

)(

)(1()(

])()([2

1)()(

1

1

1

11

1

1

1

1

1

1

111

θωω

ωωθω

ω

ωω

ωωθωθω

+−

−++−

+=

−+=+−−+

tjG

jGjGjt

jG

jGjG

ejGejGtjQtItjtj

outout

Input signal

Output of a 1st-order filter

29

)sin())(

)(1()()cos()

)(

)(1()(

])()([2

1)()(

1

2

2

22

2

2

2

222

222

θωω

ωωθω

ω

ωω

ωωθωθω

+−

−++−

+=

−+=+−−+

tjG

jGjGjt

jG

jGjG

ejGejGtjQtItjtj

outout

)(

)(

)(

)(

2

2

1

1

ω

ω

ω

ω

jG

jG

jG

jG −>>

−

Explanation why a 2nd-order filter reduces I, Q amplitude difference.

Output of a 2nd-order filter

●　According to the transfer functions,

then, the amplitude difference is reduced.

LOωω ≈at

30

Transfer Function of3rd-order RC Polyphase Filter

332211

3

313221332211

3

23122231331133222211

2

3

3322113

)](2[

)(

)](2[1

)(

)1)(1)(1()(

CRCRCRCRCRCRCRCRCR

D

CRCRCRCRCRCRCRCRCRCR

D

CRCRCRjN

I

R

ωω

ω

ω

ω

ωωωω

−+++++

=

+++++−

=

+++=

)()(

)()(

33

33

ωω

ωω

IR jDD

jNjG

+=

where

31

Gain, Phase of3rd-order RC Polyphase Filter

Gain:

Phase:

11

1

CR−

22

1

CR−

33

1

CR−

Gain characteristics

)(

)())(tan(

)()(

)()(

3

33

23

23

33

ω

ωω

ωω

ωω

jD

jDjG

jDjD

jNjG

R

I

IR

−=∠

+

=

32

Nyquist Chart of G3(jω)

G3(j ω) = X(ω) + j Y(ω)

[Y(ω

)]

[X(ω)]

In case

R1C1=R2C2

=R3C3

Symmetric with respect to a line of Ｙ= X.

33

Summary

●　Explicit transfer functions of

　　 1st-, 2nd- and 3rd-order RC polyphase filters.

Systematic design and analysis are possible.

●　On-going projects

� Derivation of higher-order filter ones.

� Nonideality effects in higher-order filters.

� Systematic design method using Nyquist chart.