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.