Post on 25-Aug-2020
Mixer
Analog and Mixed-Signal Center, TAMU
ECEN 665Edgar Sánchez-Sinencio
A1
A2
Mixer Output
What Devices Perform Frequency Translation?
Linear, time-invariant systems can not generate spectral component not present in the input.
Mixer must be non-linear or time-variant system.
Historically, a lot of devices are being tried as mixers: electrolytic cells, magnetic ribbons, brain tissues, rusty scissors, vacuum tubes and transistors.
Virtually any nonlinear elements can be used as mixer. Some nonlinearities work better and more practical.
Mathematical ModelMost mixer implementations use some kind of multiplication of two signals in time domain:
RF×LO IF (down conversion)
IF×LO RF (up conversion)
( ) ( ) ( ) ( )tAAtAAtAtA 2121
2121
2211 cos2
cos2
coscos ωωωωωω ++−=×
A1
A2
Mixer Output
Up conversion filters out ω1- ω2 component.
Down conversion filter out ω1+ ω2 component.
Mixer MetricsMixer Metrics
In order to evaluate the performance of mixers, several metrics are defined:
Conversion gain/loss
Noise figure
Port isolations
Linearity
Power consumption
Mixer Metrics (cont’d)
Conversion gain or loss is the ratio of the desired IF output (voltage or power) to the RF input signal value ( voltage or power). More specifically:
Voltage Conversion Gain =r.m.s. voltage of the IF signalr.m.s. voltage of the RF signal
Power Conversion Gain =Gc = IF power delivered to the loadAvailable power from the source
The power gain definition is actually transducer power gain.
Conversion Gain
Mixer Metrics (cont’d)Mixer’s noise figure
f LOf RF1 f RF2
Ni
Desired Signal
2N iG
N noRF IF
LO
Si Si
f IF
2S iG
f LOf RF1 f RF2
Ni
Desired Signal
2N iG
N noRF IF
LO
Si
f IF
S iG
GNNF
i
noDSB 2
1+=
GNNF
i
noSSB += 2
1−+=′G
GFF imSSBSSB
DSBSSB FF 2=
DSBSSB FF 2<′
Double side-band (DSB)
Single side-band (SSB)
Single side-band with image rejection filtering
f LOf RF1 f RF2
Ni
N noRF IF
LO
Si
f IF
S iG
N i G(1+Gim)
Desired Signal
Where Nno, Ni, and G are the output noise power, the input noise power, and the gainof the system, respectively.
Mixer Metrics (cont’d)
LO-to-RF leakage, which will mix with LO again, causes self-mixing problem in direct conversion. Due to the nonzero reverse gain of LNA, the LO leakage may even reach the antenna through the LNA
LO
RF IFLO-to-IF feed through may cause desensitization of the blocks following the mixer. (recall that the LO power is usually greater than that of the desired IF signal.)
LO
RF IFRF-to-LO feed through allows interferers and spurs present in the RF signal interact with the LO.
RF-to-IF feed through may cause problems in direct conversion architecture due to the low-frequency even-order inter-modulation product.
Port isolation
FILTER FILTER
FILTER
LO
RF IF
A diode mixer concept
Cd(V)
RS
Equivalent Circuit
Mixer Topologies
Discrete implementations:
Single-diode and diode-ring mixers
Schottky barrier diode is preferred to regular diode due to its low junction capacitance and low series resistor
IC implementations:
MOSFET passive mixer
Gilbert-cell based mixer
Harmonic mixer
VIN VOUT
RCL
LO IF RF
Single-diode:
Single-balanced:
Double-balanced:
LO
IF
RF
Passive mixer topologies
LO
LO
VRF
RS/2
RS/2
C1 L1
C2 L2
L3
L4
RG C3
CL
M1 M2
M3 M4
LO
LO
VIF
CMOS Passive Mixer
SSB noise figure of a mixer is 3 dB higher than theDSB noise figure if the signal and image bands experience equal gains at the RF port of a mixer.
multipliers.ppt
MultipliersWhat is a multiplier?
How do you obtain a multiplier?
.dimensionssuitableith constant w a isK andoutput theis Zsignals,input thearey and x
WhereKxyZ =
x
y
Z
⎪⎩
⎪⎨
⎧
)t(v
)t(v)t(v
2
1
i)t(v)t(kvi 210 =
⋅⋅⋅++
+=
3i
2ii0
cv
bvavi
i.e.,Device
Nonlinear
scheme oncancellati
tyNonlineari
Fig. 1 Basic idea of multiplier
Transconductance-Mode Multiplier
1mG 1mG
1mG
2mG
1v 1v
1v
2v
1biasI1biasI
1biasI
2biasI2i 2i
0i 0i
122110 vkvvki +=
(a)(b)
(c)
1mG
2mG1v 2v 1biasI
2biasI
2i
1mG
1biasI
2110 vvki =
1mG
2mG
1biasI
2biasI2i
1mG
2i
1biasI
2110 vvk2i =
1v
2v
1v
(d) (e)
Fig. 2 Multiplication operation using programmable transconductor
How Does It Work?
)t(v)t(vk2)t(i2(e). . Figin dillustrate as inputs
aldifferentifully are vand vand ,uctor transcondaldifferentifully a becomes )(Guctor transcond third when theachieved ison cancellatiBetter . 2(d) . Fig
in shown as eliminated becan component This . )t(vk ,component unwantedan and , )t( vand )t( vsignals twooftion multiplica therepresents )t(i , Thus
)t(vk)t(v)t(vk)t(ior
V2
)t(vIV2V2
)t(v)t(vI)t(vV2
IV2
)t(v)t(vG)t(i
)t(vV2
)t(vGIvG)t(i
V2
IG
becomes G or,ansconductbipolar tr aFor )I(GG
where vGi
2110
21
m2
12
210
122110
t1bias1
tt212bias
1t1bias
t212m
0
1t
22m1bias11m0
t1bias
m1
m1
1bias1m1m
11m0
=
+=
+=+=
+==
=
=
= (1)
(2a)
(2b)
(3a)
(3b)
(3c)
(4)
ACTIVE MIXER211 vvkio =
For a mixer, the variables for the multiplier become
( )( )
( )( )tvfvGk
tvvtii
LO
o
RF
IFo
====
2
1
1
Thus A Sigmoidal Function
( ) ( ) ( )[ ] ( ) tcosAtv;tvfGtvti LOLOLOLOoRFIF ω==
We have two important cases:
( )( )( ) ( ) ( )
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
<<−−
<
>
=
+−−+
−
−
+
LOLOLOLOLOLO
LO
LOLO
LO
LO
VtvVtvmVVVt
Vv
v
tVf
for or v
- - - - - - - - - - - - - - - - -for 0
V for 1
LO
LO
( )tvLO
( )⋅f
1
m−
LOV +LOV
( ) ( )( ) ( )
( )
( ) ( ) ( ){ }
( )
( )2
filteringAfter 2
yields For
then Assume
LOoCLORFRFCIF
LORF
LORFLORFRFLOo
IF
LORFRFLOoIF
IFRFRFRF
LOLOoRFIF
LOLOLO
AGG;tcosAGi
;
tcostcosAAGti
tcostcosAAGtiti,tcosAtv
tcosAGtvti
V,VA
=ω−ω=
ω+ω
ω+ω+ω−ω=
ω⋅ω=ω=ω=
< −+
This case corresponds to a multiplier where iIF (t) is a function of ALO
( )( ) .
PP
RFin
IFout
ω=ωω=ωby given isgain conversion The
For this case, the maximum output noise occurs in comparison of large driving ALO. That is the case to be discussed next.
Now let us focus on the case of large ALO, that is
( )( ) ( )( ) ( )⎪⎩
⎪⎨⎧
<−
>=
−>
−
+
−+
LOLO
LOLO
LOLOLO
Vtv
Vtvt
V,VA
for10
for1vf
recall and
LO
RFv
RFv
t
LOv +1
-1
Time-Domain
RFω ω
LOω LOω3 LOω5 ω
π4
π4
31
π4
51
ωωLO - ωRF ωLO + ωRF 3ωLO ± ωRF 5ωLO ± ωRFc
To handle mathematically the product vRF(t) vLO(t) we resort to fourier series to express vLO(t) and its product with vRF(t).
( )[ ] tnsinn
tvF o,,,n
LO ω∑π
=∞
= L531
14
Then( ) ( ) ( )
L+ωπ
+ωπ
⎥⎦⎤
⎢⎣⎡ +ω
π+ω
πω==
tsintsin
tsintsintcosAGtvtvGti
LOLO
LOLORFRFoLORFoIF
7745
54
3344
Let us focus on the term
( ) ( )[ ]tsintsinAGtsintcosAG LORFRFLORFo
LORPRFo ω−ω+ω+ωπ
=ωπ⋅ω
2144
Using a suitable filter one can, ideally, end with
( ) ( )
π=
ωπ
=ω−ωπ
≅
oC
IFRFo
LORFRFo
IF
GG
tsinAGtsinAGti
2
22
MIXER IMPLEMENTATIONS
Unbalanced Mixer(see Figs 2(b) & (c))
Single Balanced Mixer(see Fig 2(d))
( ) ( )
( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
ωπ
π
∑+⋅ω=
=
∞
=tncosn
nsinGtcosAtv
Rtitv
on
oRFRFIF
LIFIF
2
221
1
• Undesirable vRF(t) feedthrough.
( ) tncosn
nsinGtcosAtv o
noRFRFIF ω
π
π
∑⋅ω=∞
=
2
221
• No even harmonics
• Undesirable vLO(t)feedthourgh
VDD
VDDRL
IIF
IRFM3
+ -vLO
vLO
vRF ( ) 13
1−π
=Lm
C RgG
VDDVDD
+LOv −
LOv
+IFi −
IPi
IFiRFv
DDV DDV+IFi −
IFi RR
1IFiM1 M2M1 M2
+LOv
M3 M3
−LOv
+RFV
SSI
−RFv
−LOv
−LOv
+RFv
+LOv
−LOv
−RFvER ER
EEI
EEV
π= Lm
CRgG 34
C MOS Version BJT Version
23
3 24 ⎟⎟⎠
⎞⎜⎜⎝
⎛ +=
e
eETIP r
rRvv
Double Balanced Mixer
The conversion gain with a differential load CL is
Ideal Square Wave Mixing
Non-Ideal Mixing Functions
L
mV RCs
RgG21
2+π
=
( )( )Ttr
TtrsinRCsRgG
L
mV 21
2+π
=
Bulk-Driven Mixer
−LR
+RFv
+LOv −
LOv−RFv
biasI
+RFv
+LR
A Possible Floating Gate Mixer
LR LR
RFv
LOv
SSI
RFv
LOvRFv
LOv
LR LR
2sR 2sR
−lov
+RFv
+LOv
−LOv
−RFv
sL sL
L C Minimum Supply-Headroom DB Mixer
Pluses
• Low Power Consumption• Low Power Supply• Good Conversion Gain• NF Reasonable• Acceptable IIP3 (poor)
Minus• High ALO (power)• Low Gain Compression
Distortion, Low Frequency Analysis of MOS Gilbert MixerIt can be proved that one can express
( ) ( ) ( )
( )LWcoxI
kII
vavavaI
vavavaI
Itv
o
SSSSSS
RFRFRFSS
RFRFRFSS
SSRF
n
nn
n
μ==
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡+++⎟
⎟⎠
⎞⎜⎜⎝
⎛++++=
22
2812
23
32
21
3
33
221 LL
Solving for these coefficients, it can be obtained the following:
( )
3
232
1
33
31
2
3221
33321
41
Thus
2810
2
HDIM
AI
LWcoxAaaHD
aI
a;a;I
a
RFSS
oRF
SS
SS
n
n
=
μ==
⎟⎟⎠
⎞⎜⎜⎝
⎛−===
Assuming we are interested that the amplitude of the inteference can be denoted as AINT. Then
( )22
33233
TGS
INT
VVAIM−
=
Furthermore, it can be shown that
( ) ( )22
2222
3
23
2
33
332
332
3
then
3
tGSerferenceint
erferenceintTGSINTIP
IP
INT
VVA
AVVIMAA
AAIM
−=−
==
=
Example
dBm.orV.AA,V.VV pINTRFTGS 47232120403 −===−
Assuming the vLO(t) is not large enough to force switching.
Then
( )
dBm...IMPIIP
V..V.A
dB.VV
AIM
dBdBmi
rmspIP
TGS
INT
dBm 3312815472321
3536030641230641
63132
3
33
3
2
2
33
=+−=−=
××==
−⇒−
=
Mixer Noise
vLO not switching case
LO switches behave like a regularDifferential pairMaximum output noise contribution
VLO switching caseLO switches behave like a Cascode transistorMinimal output noise contribution.
Noise factor for a mixer is given by
( )( )( )
( ) ( )
dBNFNF
dBSSBNDSBN
NN
F
SSBDSB
sourceosourceo
IFsourceo
IFtoto
3
and
3
where
−Δ
+=
ωω
=
vLO (instantaneous)
10
TYPICAL BJT MIXER NOISE AT VARIOUS vLO LEVELS
s/n around vLO = 0 is very poor (gain is also low)
8
6
4
2
-0.1 0 0.1
Output Noise
Total
Top Transistors
Bottom Transistors
Source resistance
RF
LO
IF
L C
M1 M2
M3 M4 M5 M6
Single or Double-balanced mixer
Good LO-IF isolation (40dB~60dB)
M1, M2 work as V-I converter
M3~M6 work as current commuting switches
LC tank: zero-headroom
Low IF noise figure problem
mc gGπ2
=Transconductance conversion gain:
A Popular mixer topology: IC implementation
Gilbert-Cell based mixer
LORF RF
VCC
RL RLIF
Two emitter-coupled BJT pairs work as two limiters.
The small RF signal will modulate the zero crossing point of the relatively large LO signal.
Harmonic mixing: Bipolar implementation
Sub-harmonic mixer
Design Considerations for Gilbert-cell Mixer
Noise contributionsLoad noise
V-I converter noise
whit noise: up/down conversion
flicker noise: up conversion
Switch pair noise
High frequency noise of switches or coming together with LO signal
flicker noise modulates switching point
Charging and discharging the parasitic capacitance at the common-source of differential pair makes flicker noise appearing at output
H. Darabi, A.A. Abidi, “Noise in RF-CMOS mixers: a simple physical model,” IEEE JSSC Vol. 35 Issue: 1 , Jan. 2000 pp. 15 -25
Design Considerations for Gilbert-cell Mixer (cont’d)
Linearity
V-I converter
The arguments of LNA linearity apply here
Switch pair
Low frequency: higher LO swing better linearity
High frequency: Optimal LO swing
M. T. Terrovitis, R. G. Meyer, “Intermodulation distortion in current-commutating CMOS mixers,” IEEE JSSC, Vol. 35 Issue 10 , Oct. 2000 pp. 1461 -1473
Mixer topologies for Narrowband Applications
Discrete implementations:
Single-diode and diode-ring mixers
IC implementations:
MOSFET passive mixer
Gilbert-cell based mixer
Harmonic mixer
Diode Mixers
VIN VOUT
RCL LO IF RF
Single-diode:
Single-balanced:
Double-balanced:
The single-diode mixer is the simplest and oldest passive mixer. The output RLC tank is tuned to the desired IF, and input is the sum of RF, LO and DC bias.This mixer can not provide any isolation and conversion gain. However, at very high frequency (millimeter-wave band) this kind of mixer is extremely useful.
LO is large enough to make the diodes work as switches, regardless of the level of RF signal. When the diodes are on, RF and IF are connected together, so the RF-IF isolation is poor. But the RF signal is common-mode for the transformer, so the RF-LO isolation is excellent.
Due to the symmetry of the circuit, isolations between each pair of ports are excellent, mainly limited by the device matching. The diode mixer is pretty much linear and the upper limit of the dynamic range is constrained by diode break-down.
Typically, double-balanced mixers can achieve conversion loss of around 6dB, isolation of at least 30dB.
LO
IF
RF
CMOS Passive Mixer
LO
LO
C1
L1
VR
F
RS/2
RS/2
C2
L2
L3
L4
RG C3
CL
M1
M2
M3
M4
LO
LO
VI
F
Input LC network provide matching and filtering. R1 sets the input common-mode level.
Due to the matching network, voltage conversion gain can be greater than 1. Noise figure and IIP3 are strong functions of LO drive level.
MOSFET M1~M4 are working as switches and are driven by LO in anti-phase. Only one diagonal pair of transistors is conducting at any given time. When M1 and M4 are on, VIFequals VRF, and when M2 and M3 are on, VIF equals –VRF. So it is equivalent to observe that the mixer multiplies the RF signal by square wave whose amplitude is alternating between +1 and –1 and whose frequency is that of LO.
Gilbert-Cell Based Mixer
RF
LO
IF
L C
M1 M2
M3 M4 M5 M6
This is a double-balanced mixer. Good LO-IF isolation (40dB~60dB) can be achieved due to the symmetry. M1 and M2 work as V-I converter and M3~M6 are driven by large enough LO, working as current commuting switches. LC tank is to create zero-headroom AC current source. If the power supply voltage is not a limitation factor, the LC tank can be replace with a transistor working as current source.
The linearity of the mixer is limited by the linearity of the V-I converter. For low IF, the noise figure is limited by the flicker noise of the current switches and for higher IF, the noise figure is limited by the thermal noise of the circuit.
mc gGπ2
=
Transconductance conversion gain:
Additional linearization techniques can be applied to improve the linearity of the mixer.
Sub-sampling Mixer
IF
RF
VC
M
VC
M
φ1_
b
φ1_
b
φ1
φ3
φ3
φ2
φ2
M1
M2
M3
M4
M5
M7
M6
Properly designed track-and-hold circuit works as sub-sampling mixer
The sampler must have good time resolution. So the clock’s absolute time jitter must be a tiny fraction of the carrier period.
Noise folding make the mixer present large noise figure.
The linearity of the mixer is high
Harmonic Mixer (I)
Harmonic mixer has low self-mixing DC offset. It is very attractive for direct conversion application.
The RF single will mix with the second harmonic of the LO. So the LO can run at half rate, which make the VCO design easier.
Because of the harmonic mixing, conversion gain is usually small (several
Harmonic Mixer (II)
LO
RF
RF
VCC
RL
RL
IF
Two emitter-coupled BJT pairs work as two limiters. The odd symmetry of their transfer function suppress even order distortion including LO self-mixing.
The small RF signal will modulate the zero crossing point of the relatively large LO signal. The output of the mixer is rectangular wave in the pulse width modulation fashion, a low pass filter will demodulate the
Harmonic Mixer (III)
Simulated waveforms of the harmonic mixer:
LO @ 1GHzRF @ 2GHz+100kHzIF @
100kHz
Calculate the input impedance looking at the bulk, and the output voltage expression of the simple bulk-driven amplifier. Note that the circuit has two small signal input signals. is injected to the bulk and to the gate. Also determine an expression for .
oV1inV
2inV Di
A Bulk-Driven, Gate-Driven, Multiplier
2M
1MoV
1inV
2inV
1biasV
2biasV
DV
+
For low frequency
bulkZin @
∞=
For high frequency
bulk
gggZ mbin
@
102011 ++
=
Di
The drain current containing the DC and small signal is
[ ]( )[ ]21 221
fDTGSpd VvlwKi φγ+−=
[ ]21112 22
21 γφφγ FinFDTbiasinpD vVVv
LwKi +−−−+=
We will use the following approximation for the square root containing by means of a Taylor Series:1inV
L+−≅−a
xaxa2 1&2; inF vxa == φ
( )
( ) ( )[ ]
( ) ( )
( ) ( )2
242
13212211
22
21
221
2111
1121
21
2212
2
21
21
21
21
21
22
221
222
221
21
2221
2222
21
ininininMinind
ininFF
inininTobiasin
F
Tobiasd
F
inTobiasinTobiasin
FF
inininD
dDD
TobiasF
ininTobiasD
f
inFinFTobiasD
vavavvavavai
vvvv
vVVvVV
LwKpi
vVvvVvvvvvLwKpi
iIi
VvLwKpvvVv
LwKpi
vvVvLwKpi
++++=
⎥⎦
⎤⎢⎣
⎡+++−+
−=
⎥⎦
⎤⎢⎣
⎡−+−+++=
+=
+−=⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛++−=
⎥⎥⎦
⎤
⎢⎢⎣
⎡+−++−=
φγγ
φγ
φγ
φγ
φγγ
φγ
φγ
φγφγφγ
L
Simulation Results
Vin1: Constant Input, Vin2: 20KHz Sine Input
Simulation Results
Vin1, Vin2 are sine signal with same frequency
Simulation Bulk-Driven Multiplier Results
Vin1: 18KHz, Vin2: 20KHz sine wave, Vout: Modulated Output
RF Front-end for Bluetooth Low IF Receiver
LNA
Vin+
Vo
Vin-
Vdd
M1 M1
M2M2
Ls Ls
Ldd Ldd
VbVb
Lg Lg
Gate NQS resistance Rgs=1/5gm
Gat induced noise modeled by gate NQS resistance gsg RkTV δ42 =
Voltage Gain: 18dB
Noise Figure:2.6dB
IIP3: 0dBm
Current: 4.4mA
Supply: 3V
RF Front-end for Bluetooth Low IF Receiver
Mixer
Double balanced mixerCurrent injection to alleviate the trade off between the linearity and power supply voltage
LO+
LO-
RF+
LO+
VDD
IF+ IF-
PAD
PAD
RF-
VGB
PAD
PAD
VGA
Voltage conversion gain: 26dBNoise figure: 12.4dBIIP3: -3dBm
RF Front-end for Bluetooth Low IF Receiver
Mixer layout considerations
The length of the poly gate should be kept short enough to reduce the effect of gate resistanceFor the layout of poly-poly capacitor, if the bottom plate is floating, the parasitic capacitance from it to substrate should be considered. It is about ¼ of the nominal capacitance.
De-coupling capacitor may be needed to prevent the circuits from oscillationMetal should be wide enough to carry large current. The current density allowed through metal is about 1mA/1μmGuarding rings are required around the circuits to provide isolation from other blocks
Front-end for BT/WiFi ReceiverLNA
LNA_cas_bias
Vo+
Vo-
VDD
LNA_bypass
Ls Ls
LdLd
LNA bypass switches and attenuator
LNA_rf_biasi_tail
CmVin+
Vin-
Inductor degeneration
Cascoded BJT: better matching
On-chip input matching
Noise figure: 1.6dB
Power/Voltage gain: 13dB
Power consumption: 16mW
NMOS attenuator for low gain(3dB)
Differential structure
MOS transistor is more linear
Front-end for BT/WiFi ReceiverS-parameters
Noise figure
1dB compression point
IIP3
LNA simulation results
Front-end for BT/WiFi Receiver
1nH
3nH
570um
580um
LNA Layout
Symmetrical layout
Deep trench lattice under spiral inductor
Inductors are placed far apart to avoid coupling
Differential inputs are decoupled by GSGSG pattern
Front-end for BT/WiFi ReceiverI/Q Mixer
IFI
LO_bias
LOI
IFQ
LO_bias
LOQ
VDD
imix_bias
RF_in
IIP3: 2dBm NF: 10.6dB
Gain: 18.5dB Power: 8.8mW
Fully-differential structure to suppress common-mode noise
I/Q branches share the same RF drive stage to achieve better matching between I and Q
Bipolar current switches requires low LO drive power (-10dBm)
Resistive loads have higher linearity and lower low-frequency noise compared to PMOS activeload.
Front-end for BT/WiFi Receiver
Noise Figure
IIP3
I/Q Mixer Simulation Results Conversion Gain
Front-end for BT/WiFi Receiver
130um
180um
I/Q Mixer Layout
Front-end for BT/WiFi Receiver
2mA 1.6
V2.2V
410uA 1.4
V12.8uA
Biasing Circuits for the Front-end
Front-end for BT/WiFi Receiver
LO drive: -10dBmNoise Figure: 5.5dBOverall gain: 33dBP 38 3 W
Overall Gain Noise
Figure
Performance of the front-end
Front-end for BT/WiFi Receiver
S11: High gain mode
S11: Low gain mode
I/Q mismatch: amplitude 2.3% phase 1.7 degrees
1dB compression
Circuit Implementation --- Merged LNA and Mixer (An alternative way)
VB_CAS
VB_RF
IFI
VB_LO
LOI
IFQ
VB_LO
LOQ
VDD
RF_IN
• Inter stage matching
• Inductive degeneration
• Current re-use
• Low power consumption
Power: 28.5mWNoise Figure: 3.6dBGain: 33.4dB
Simulation Results --- Merged LNA and Mixer
S11
Noise Figure
Gain
References
1. B. Leung, VLSI for Wireless Communication, Prentice HallUpple Saddle, NJ 2002, Chapter 4
2. T. Lee , The Design of CMOS Radio-Frequency Integrated Circuits, 2nd edition, Cambride University Press 2004, Chapter 13
3. B. Razavi, “ RF Microelectronics”., Prentice HallUpple Saddle, NJ 1998, Chapter 6.2
4. G. Han, E. Sanchez-Sinencio, “CMOS transconductancemultipliers: a tutorial ,” IEEE Transactions on Circuits and SystemsII: Analog and Digital Signal Processing , Volume: 45, Issue: 12 , pp. 1550 –1563, Dec 1998