665 Mixer 2008

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


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


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.


Noise Figure

IIP3

I/Q Mixer Simulation Results Conversion Gain


130um

180um

I/Q Mixer Layout


2mA 1.6

V2.2V

410uA 1.4

V12.8uA

Biasing Circuits for the Front-end


LO drive: -10dBmNoise Figure: 5.5dBOverall gain: 33dBP 38 3 W

Overall Gain Noise

Figure

Performance of the front-end


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

665 Mixer 2008

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Transcript of 665 Mixer 2008