Mixer
©James Buckwalter
Mixer References• “Noise in RF-CMOS Mixers: A Simple Physical Model” by Darabi and Abidi, IEEE
Transactions on Solid-State Circuits, Vol. 35, No. 1, 2000.
• “Noise in Current-Commuting CMOS Mixers” by Terrovitis and Meyer, IEEE Journal of Solid-State Circuits, Vol. 34, No. 6, 1999.
• “Behavioral Models for Noise in Bipolar and MOSFET Mixers” by Hu and Mayaram, IEEE Transactions on Circuits and Systems II, Vol. 46, No. 10, 1999.
• “A Class AB Monolithic Mixer for 900MHz Applications” by Fong and et al, IEEE Journal of Solid-State Circuits, Vol. 32, No. 8, 1997.
• “Monolithic RF Active Mixer Design” by Fong and Meyer, IEEE Transactions on Circuits and Systems II, Vol. 46, No. 3, 1999.
• “A 12mW Wide Dynamic Range CMOS FrontEnd for a Portable GPS Receiver” by Shahani et al, IEEE Journal of Solid-State Circuits, Vol. , No. 12, 1997.
• “A Parallel Structure for CMOS Four-Quadrant Analog Multipliers and Its Application to a 2GHz Down-conversion Mixer” by Hsiao and Wu.
• “A Low Voltage Bulk Driven Down-conversion Mixer Core” by Kathiresan and Toumazou, 1999.
• “Low Voltage Mixer Biasing Using Monolithic Integrated Transformer De-coupling” by MacEachern and et.al, 1999.
©James Buckwalter
References• “A Charge-Injection Method for Gilbert Cell Biasing” by MacEachern and Manku,
1998.
• “Doubly Balanced Dual-Gate CMOS Mixer” by Sullivan and et al, IEEE Journal of Solid-State Circuits, Vol. 34, No. 6, 1999.
• “A 1.5GHz Highly Linear CMOS Down conversion Mixer” by Crols and Steyaert, IEEE Journal of Solid-State Circuits, Vol. 30, No. 7, 1995.
• “Micro-power CMOS RF Components for distributed wireless sensors” by Lin and et al, IEEE Radio Frequency Integrated Circuits Symposium, 1998.
• “A Zero DC-Power Low Distortion Mixer for Wireless Applications” by Kucera and Lott, IEEE Microwave and Guided Wave Letters, Vol. 9, No. 4, 1999.
• “A 900MHz/1.8GHz CMOS Receiver for Dual-Band Applications” by Wu and Razavi, IEEE Journal of Solid-State Circuits, Vol. 33, No. 12, 1998.
• “The MICROMIXER: A Highly Linear Variant of the Gilbert Mixer Using a BiSymmetric Class-AB Input Stage” by Gilbert, IEEE Journal of Solid-State Circuits, Vol. 32, No. 9, 1997.
• “A 2V, 1.9GHz Si Down-Conversion Mixer with an LC Phase Shifter” by Komurasaki and et al, IEEE Journal of Solid-State Circuits, Vol. 33, No. 5, 1998.
©James Buckwalter
References• “A Low Distortion Bipolar Mixer for Low Voltage Direct Up-Conversion and High IF
Systems” by Behbahani and et al, IEEE Journal of Solid-State Circuits, Vol. 32, No. 9, 1997.
• “A 2GHz Balanced Harmonic Mixer for Direct-Conversion Receivers” by Yamaji and Tanimoto, Custom Integrated Circuits Conference, 1997.
• “A Blocker-Tolerant, Noise-Cancelling Receiver Suitable for Wideband Wireless Applications,” David Murphy et al. JSSC 2012”
©James Buckwalter
Tradeoffs in RX
• Noise Figure(Sensitivity)
• Distortion (Linearity)
• Phase Noise (Aliasing)
©James Buckwalter
Review: Homodyne versus Heterodyne
• Heterodyne
• Homodyne
©James Buckwalter
rf lo if
0rf lo
Heterodyne: High-side injection
• LO is above the RF tone
• Image frequency (IM) is above the LO tone
©James Buckwalter
, , 1rf N lo N
, , 1im N lo N
Heterodyne: Low-side injection
• LO is below the RF tone
• Image frequency (IM) is below the LO tone
©James Buckwalter
, , 1rf N lo N
, , 1im N lo N
Differential Receiver
• N-path filtering translates the baseband impedance to higher frequency (more on this later)
• Impedance loads LNA and provides gain to desired band while blocking on out-of-band signals.
• Mixer is a “voltage-mode” operation. ©James Buckwalter
Gm Receiver
©James Buckwalter
• Gm cell produces an RF current to go through the mixer.
• Mixer is a “current-mode” operation.
• Avoids large voltage gain at RF and filters out-of-band at baseband.
Mixer-First Architecture
©James Buckwalter©James Buckwalter
• N-path filtering presents a high-impedance to the antenna
• Avoids linearity problems of LNA by putting mixer at front-end
• Costs is noise figure.
Mixer Noise
• How do we cascade the noise contribution of a mixer?
• For homodyne,
• For heterodyne,
©James Buckwalter
FDSB
= 1+N
added
kTDfGLNA
FSSB
= 1+G
IM - IF
GRF-IF
æ
èç
ö
ø÷ 1+
Nadded
GIM -IF
+ GRF- IF( )kTDf
æ
èç
ö
ø÷
=G
RF- IF+ G
IM -IF
GRF- IF
+N
added
GLNA
kTDf
Notes: This is the single sideband noise figure (SSB NF) and is at least 3dB.Please remember that SSB noise figure is generally 3 dB higher than double sideband (DSB) noise figure.
Pulse Waveform
• Spectral Components of LO waveform
VLO
= vLO
p t( )p t( ) = a
0+ a
kcos kw
LOt( )
k³1
å
a0
= Ad ak
=2A
kpsin kp d( )
VLO
tA
d
©James Buckwalter
LO Waveform in Mixer
• Noise factor degraded by images around harmonics.
©James Buckwalter
VLO
t( ) =Vpk
2sin npd( )np
n=1odd
¥
å cos nwLO
t( )
Pulse Waveform
• Some special cases…
• d = 1/2
• d = ¼
• d -> 0 we approach an infinite train of delta functions.
VLO
tA
d
p t( ) =
1
2+
2
pcos w
LOt( ) -
4
3pcos 3w
LOt( )+
4
5pcos 5w
LOt( ) - ...
p t( ) =
1
4+
2
pcos w
LOt( )+
2
3pcos 3w
LOt( ) -
2
5pcos 5w
LOt( ) -
2
7pcos 7w
LOt( )...
©James Buckwalter
Mixers: Switching Operation
SRF wRF( )
1
1
Sout
LO Input
Sout = SRF cos wRFt( )´ .........................................}{
1
1
Sout
= SRF
cos wRF
t( )Ä4
pcos w
LOt( ) -
4
3pcos 3w
LOt( ) +
4
5pcos 5w
LOt( ) - ...
ìíî
üýþ
SW t( )
Note that “S” is a generic signal. Could be voltage or current?
©James Buckwalter
Mixer Gain
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
sig M RFI G V
0®T
LO
2:V
out= V
cc- I
DC+ I
sig( ) RL
éë
ùû-V
cc= - I
DC+ I
sig( ) RL
TLO
2®T
LO:V
out=V
cc- V
cc- I
DC+ I
sig( ) RL
éë
ùû= I
DC+ I
sig( ) RL
Vout
= IDC
+ Isig( ) R
L´ SW t( )
1
2
TM
S
GR
©James Buckwalter
Mixer Noise Analysis• Noise analysis of a single balanced mixer:
VLO
RL RL
VLO
VRF
Vout
,DC mix RF NoiseI I I
M1
M2 M3
wLO -wRF
LO RF wLO +wRF
VOUT
t
Instantaneous Switching
©James Buckwalter
Mixer Noise Analysis• Noise analysis of a single balanced mixer cont...:
• If the switching is not instantaneous, additional noise from the switching pair will be added to the mixer output.
• Let us examine this in more detail.
VLO
RL RL
VLO
VRF
Vout
,DC mix RF NoiseI I I
M1
M2 M3 VOUT
t
Finite Switching Time
©James Buckwalter
Mixer Noise Analysis• Noise analysis of a single balanced mixer cont...:
• When M2 is on and M3 is off:
– M2 does not contribute any additional noise (M2 acts as cascode)
– M3 does not contribute any additional noise (M3 is off)
VLO
RL RL
VLO
VRF
Vout
M1
M on2 M off3 VOUT
t
Finite Switching Time
,DC mix RF NoiseI I I
©James Buckwalter
Mixer Noise Analysis• Noise analysis of a single balanced mixer cont...:
• When M2 is off and M3 is on:
– M2 does not contribute any additional noise (M2 is off)
– M3 does not contribute any additional noise (M3 acts as cascode)
VLO
RL RL
VLO
VRF
Vout
M1
M off2 M on3VOUT
t
Finite Switching Time
,DC mix RF NoiseI I I
©James Buckwalter
Mixer Noise Analysis• Noise analysis of a single balanced mixer cont...:
• When VLO+ = VLO- (i.e. the LO is passing through zero), the noise contribution from the transducer (M1) is zero. Why?
• However, the noise contributed from M2 and M3 is not zero because both transistors are conducting and the noise in M2 and M3 are uncorrelated.
VLO
RL RL
VLO
VRF
Vout
M1
M on2 M on3 VOUT
t
Finite Switching Time
,DC mix RF NoiseI I I
©James Buckwalter
Mixer Noise Analysis• Optimizing the mixer (for noise figure):
• Design the transducer for minimum noise figure.
• Noise from M2 and M3 can be minimized through fast switching of M2/M3 by:
– making LO amplitude large
– making M2 and M3 small (i.e. increasing fT of M2 and M3)
• Noise from M2 and M3 can be increased by using large M2/M3 switches.
VLO
RL RL
VLO
VRF
Vout
M1
M on2 M on3
VOUT
t
Trise
...m DCg W fixed I 1
...T DCfixed IW
,DC mix RF NoiseI I I
©James Buckwalter
Mixer Noise Analysis• Noise Figure Calculation:
• Let us calculate the noise figure including the contribution of M2/M3 during the switching process.
VLO
RL RL
VLO
VRF
Vout
M1
M on2 M on3VOUT
t
Trise
,DC mix RF NoiseI I I
©James Buckwalter
Heterodyne Mixer Noise Analysis: RL Noise• Noise Analysis of Heterodyne Mixer (RL noise):
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
IF RF LO
,DC mix RF NoiseI I I
2 4 2noise RL Lv kT R
©James Buckwalter
Heterodyne Mixer Noise Analysis: Transducer Noise• Noise Analysis of Heterodyne Mixer (Transducer noise):
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
inoise- M1-switch
= inoise- M1
t( )SW t( )
= inoise- M1
t( )4
pcos w
LOt{ } -
4
3pcos 3w
LOt{ } +
4
5pcos 5w
LOt{ } - ...
æ
èçö
ø÷
VLO
t,DC mix RF NoiseI I I
©James Buckwalter
Heterodyne Mixer Noise Analysis: Transducer Noise• Noise Analysis of Heterodyne Mixer (Trans-conductor noise):
IF LO
inoise-M1
2 f( ) = 4kTg gdo,1
inoise- M1-switch
= inoise- M1
t( )SW t( )
= inoise- M1
t( )4
pcos w
LOt{ } -
4
3pcos 3w
LOt{ } +
4
5pcos 5w
LOt{ } - ...
æ
èçö
ø÷
3 LO
SW f( ) =
4
pd f - f
LO( )+4
3pd f -3 f
LO( )+ ...
inoise- M1
2 wIF( ) = 2
4
p
æ
èçö
ø÷
2
1+1
32+
1
52+ ..
é
ëê
ù
ûú4kTg g
do1
5 LO
inoise-M1
2 wIF( ) = 4 ×4kTg g
do,1
1
n2
n=1,odd
¥
å =p 2
8
©James Buckwalter
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise):
VLO VLOM on2 M on3id ,3
2id ,2
2
id
2 » 4kTg gm
id
2g vm gs g vm gs
vgn
2 =4kTg
gm
©James Buckwalter
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise):
• Show that:
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
out outi i
VLO
Gm
VLO
G
m= g
m2= g
m3= g
m2,3»
2IDC ,mix
DV
,DC mix RF NoiseI I I
0mG
©James Buckwalter
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise) cont...:
Gm
VLO
2,3n mv
iout
2,3.out m n mi t G t v t
2
LOT
T
©James Buckwalter
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise) cont...:
Gm t( )
Gm
t( ) =DTG
m0
TLO
/ 21+ 2
sin kDTw
p
2
æ
èçç
ö
ø÷÷
kDTw
p
2
æ
èçç
ö
ø÷÷
k=1
¥
å cos kwpt( )
é
ë
êêêêê
ù
û
úúúúú
Gm f( )
p 2 p 3 p
2,3n mv f
p 2 p 3 p
vn-m2,3
2 = 24kTg
gm2,3
2
/ 2p
LOT
2
LOT
T
2 2 2
2,3 2 3n m n m n mv v v
©James Buckwalter
VLO
= vLO
p t( )p t( ) = a
0+ a
kcos kw
LOt( )
k³1
å
a0
= Ad ak
=2A
kpsin kp d( )
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise) cont...:
Gm f( )
p 2 p 3 p
2,3n mv f
p 2 p 3 p
vn-m2,3
2 = 24kTg
gm2,3
iout
t( ) = Gm
t( )vn-m2,3
t( )
inoise- M 2,3
2 wIF( ) = v
n-m2,3
2 Gm0
DT
TLO
2
æ
èçö
ø÷
æ
è
çççç
ö
ø
÷÷÷÷
2
+ 2vn-m2,3
2 Gm0
DT
TLO
2
æ
èçö
ø÷
æ
è
çççç
ö
ø
÷÷÷÷
2
sinc kDTw
p
2
æ
èç
ö
ø÷
æ
èç
ö
ø÷
2
k=1
å
How do we solve this?©James Buckwalter
inoise- M 2,3
2 wIF( ) = v
n-m2,3
2 Gm0
DT
TLO
2
æ
èçö
ø÷
æ
è
çççç
ö
ø
÷÷÷÷
2
+ 2vn-m2,3
2 Gm0
DT
TLO
2
æ
èçö
ø÷
æ
è
çççç
ö
ø
÷÷÷÷
2
sinc kDTw
p
2
æ
èç
ö
ø÷
æ
èç
ö
ø÷
2
k=1
å
Heterodyne Mixer Noise Analysis: Switch Noise
inoise- M 2,3
2 wIF( ) = v
n-m2,3
2 Gm0
DT
TLO
2
æ
èçö
ø÷
æ
è
çççç
ö
ø
÷÷÷÷
2
+ 2vn-m2,3
2 Gm0
DT
TLO
2
æ
èçö
ø÷
æ
è
çççç
ö
ø
÷÷÷÷
2
-1+2p
DTwp
2
inoise- M 2,3
2 wIF( ) = v
n-m2,3
2 Gm0
DT
TLO
2
æ
èçö
ø÷
æ
è
çççç
ö
ø
÷÷÷÷
2
TLO
2DT= v
n-m2,3
2 Gm0
2 DT
TLO
2
æ
èçö
ø÷
sinc kDTw
p
2
æ
èç
ö
ø÷
æ
èç
ö
ø÷
2
k=1
å =
-1+2p
DTwp
2
©James Buckwalter
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise) cont...:
Gm f( )
p 2 p 3 p
2,3n mv f
Gm f( )
p 2 p 3 p
2,3n mv f
inoise- M 2,3
2 wIF( ) =
1
TLO
2
æ
èçö
ø÷
Gm0
2 DT vn-m2,3
2
©James Buckwalter
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise) cont...:
inoise- M 2,3
2 wIF( ) =
1
TLO
2
æ
èçö
ø÷
Gm0
2 DT vn-m2,3
2
,
0
2 DC mix
m
IG
V
DV = SlopeDT
V
LOt( ) = A
LOcos w
LOt( )
90
90LO
LO
LO
LO LOt
t
dV tSlope A
dt
G
m= g
m2= g
m3= g
m2,3»
2IDC ,mix
DV
vn-m2,3
2 = 24kTg
gm2,3
©James Buckwalter
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise) cont...:
inoise- M 2,3
2 wIF( ) =
Gm0
2 DT
TLO
/ 2v
n-m2,3
2 =G
m0
2 DT
TLO
/ 22
4kTg
gm2,3
æ
èç
ö
ø÷
= 4G
m0DT
TLO
4kTg( ) =4DT
TLO
2IDC ,mix
DV4kTg( )
= 42I
DC ,mix
TLO
DT
DV4kTg( ) = 4
2IDC ,mix
TLO
1
ALO
wLO
4kTg( )
= 4 ×4kTgI
DC ,mix
p ALO
æ
èç
ö
ø÷
Total Noise Contribution due to switches M2 and M3
G
m»
2IDC ,mix
DV
vn-m2,3
2 = 24kTg
gm2,3
inoise- M 2,3
2 wIF( ) =
1
TLO
2
æ
èçö
ø÷
Gm0
2 DT vn-m2,3
2
DV
DT= A
LOw
LO
©James Buckwalter
Heterodyne Mixer Noise Analysis: Total Noise• Noise Analysis of Heterodyne Mixer (total noise):
inoise-M1
2 wIF( ) = 4 ×g4kTg
m1= 4 ×g4kT
IDC ,mix
VGSQ
-VT 0( )
inoise- M 2,3
2 wIF( ) = 4 ×g 4kT
IDC ,mix
p ALO
æ
èç
ö
ø÷
2 4 2noise RL Lv kT R
vnoise- MIX
2 wIF( ) = 4kTR
L2 + 4g I
DC ,mixR
L
1
VGSQ
-VT 0
+1
p ALO
æ
èç
ö
ø÷
ì
íï
îï
ü
ýï
þï
0
1
2
DS short DS shortm short ox sat
GS GSQ T
dI Ig WC v
dV V V
2 2 2 2 2 2
1 2,3noise MIX IF noise RL L noise M L noise Mv v R i R i
©James Buckwalter
Heterodyne Mixer Noise Analysis: Total Noise• Noise Analysis of Heterodyne Mixer (total noise):
0 1 &GSQ TV V M linearity Noise
0... min 2 / 3GSQ T LOAs V V A to imize noise contribution from M M
2
noise MIX IFv
1.6GSQV V
0.8GSQV V
VLO
vnoise- MIX
2 wIF( ) = 4kTR
L2 + 4g I
DC ,mixR
L
1
VGSQ
-VT 0
+1
p ALO
æ
èç
ö
ø÷
ì
íï
îï
ü
ýï
þï
©James Buckwalter
Heterodyne Mixer Noise Analysis: Noise Figure
• This assumes that all of the “white noise” from Rs is folded down. In reality, there is some matching and filtering between the generator and mixer.
22
,20
1 11 1 4
noise MIX IF SDC mix L
L T LOGSQ Tnoise RS IF
v RNF I R
R AV Vv
2 2
2 1 416
2
T Tnoise Rs IF S
S S
kTi kT R
R R
©James Buckwalter
Load
No
ise
LO P
ort
No
ise
Heterodyne Mixer Noise Analysis: Total Noise• Noise Analysis of Heterodyne Mixer (total noise)--{Terrovitis and Meyer}:
F =a
c2+
g3+ r
g3g
m3( )gm3
a + 2g1G + R
LO+ 2r
g 2( )G2 +1
RL
c2gm3
2 Rs
sin2 2
2
LO
LO
T
cT
41
3LOTf
G =2I
p ALO
G2 = 4.64K
sq
1/2IDC ,mix
3/2
2p ALO
©James Buckwalter
Homodyne Mixer Noise Analysis: Transducer Noise• Noise Analysis of Heterodyne Mixer (Transducer noise):
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
inoise- M1-switch
= inoise- M1
t( )SW t( )
= inoise- M1
t( )4
pcos w
LOt{ } -
4
3pcos 3w
LOt{ } +
4
5pcos 5w
LOt{ } - ...
æ
èçö
ø÷
VLO
t,DC mix RF NoiseI I I
©James Buckwalter
Homodyne Mixer Noise Analysis: Transducer Noise• Noise Analysis of Heterodyne Mixer (Trans-conductor noise):
IF LO
inoise-M1
2 f( ) = 4kTggdo,1
inoise- M1-switch
= inoise- M1
t( )SW t( )
= inoise- M1
t( )4
pcos w
LOt{ } -
4
3pcos 3w
LOt{ } +
4
5pcos 5w
LOt{ } - ...
æ
èçö
ø÷
3 LO
SW f( ) =
4
pd f - f
LO( )+4
3pd f -3 f
LO( )+ ...
inoise-M1
2 0( ) =4
p
æ
èç
ö
ø÷
2
1+1
32+
1
52+ ..
é
ëê
ù
ûú4kTgg
do1
5 LO
inoise-M1
2 0( ) = 2 ×4kTggdo,1
1
n2
n=1,odd
¥
å =p 2
8
©James Buckwalter
Harmonic Rejection Mixers
• Harmonic aliases can be uniformly removed. Recall that we saw a duty cycle dependence on harmonic terms.
©James Buckwalter
Harmonic Rejection Mixers
• The LO can be synthesized to remove harmonic content.
©James Buckwalter
3 LOs separated by 45 degrees are weighted to generate desired spectral properties
sLO t( ) = ak p t - kT( )k=-N
N
å
T =1
fLO
p
2 N +1( )
Harmonic Rejection Mixers
• When the mixer is driven with the stepped waveform
and we eliminate any aliasing of the signal on to N-1 harmonics
©James Buckwalter
Harmonic Rejection Mixer Example
©James Buckwalter
LO Phase
Shift
No. of
Square
Waves
Square
Waves
Harmonics Can be
Cancelled
45o 3 √2*LO(t), LO(t-π/4), LO(t+π/4) 3rd, 5th
30o 5
2*LO(t), √3*LO(t-π/6),
√3*LO(t+π/6), LO(t-π/3),
LO(t+π/3)
3rd ,5th ,7th, 9th
22.5o 7
√2*LO(t)
√2*cos(π/8)*LO(t-π/8)
√2*cos(π/8)*LO(t+π/8)
LO(t-π/4), LO(t+π/4)
√2*sin(π/8)*LO(t-3π/8)
√2*sin(π/8)*LO(t+3π/8)
3rd, 5th, 7th, 9th,11th,13th
Harmonic Rejection Mixer Implementation
• Harmonic rejection can be realized as a set of parallel RF paths or through direct frequency synthesis techniques
©James Buckwalter
Harmonic Rejection Receiver
©James Buckwalter