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Berkeley
Mixers: An Overview
Prof. Ali M. Niknejad
U.C. BerkeleyCopyright c© 2014 by Ali M. Niknejad
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Mixers
cf
Information PSD Mixer
The Mixer is a critical component in communication circuits.It translates information content to a new frequency.
On the transmitter, baseband data is up-converted to an RFcarrier (shown above).
In a receiver, the same information is ideally down-convertedto baseband.
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Mixers Specifications
Conversion Gain: Ratio of voltage (power) at outputfrequency to input voltage (power) at input frequency
Downconversion: RF power in, IF power outUp-conversion: IF power in, RF power out
Noise Figure
DSB versus SSB
Linearity
Image Rejection, Spurious Rejection
LO Feedthrough
InputOutput
RF Feedthrough
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Mixer Implementation
f(x)
Non-linear Two tones
21
21
+ 2nd order IM 1, 2
We know that any non-linear circuit acts like a mixer
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Squarer Example
x yx2
Product component:
y = A2cos2ω1t + B2cos2ω2t + 2AB cosω1t cosω2t
What we would prefer:
vRF = vRF cosω1t
vLO = vRF cosω1t
vIF = 2vLO · vRF cos (ω1 ± ω2) t
LO
RF
IF
A true quadrant multiplier with good dynamic range isdifficult to fabricate
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LTV Mixer
LTI 21 ff + 21 ff + No new frequencies
LTV 21 ff + New tones in output
No new frequencies for a Linear Time Invariant (LTI) system
But a Linear Time-Varying (LTV) “Mixer” can act like amultiplier
Example: Suppose the resistance of an element is modulatedperiodically
vLO
R(t) iin(t)(RF )
vo(IF ) vo = iin · R (t)
= Io cos (ωRF t) · Ro cos (ωLOt)
=IoRo
2{cos (ωRF + ωLO) t+cos (ωRF − ωLO) t}
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Periodically Time Varying Systems
In general, any periodically time varying system can achievefrequency translation
v (t) = p (t) vi (t)p (t + T ) = p (t) =
∞∑n=−∞
cnejωontvi (t)
cn =1
T
T∫0
p (t) e−jωontdt
vi (t) = A (t) cosω1t = A (t)
(e jω1t + e−jω1t
2
)
vo (t) = A (t)∞∑−∞
cne j(ωont+ω1t) + e+j(ωont−ω1t)
2
Consider n=1 plus n=-1
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Desired Mixing Product
c1 = c−1
vo (t) =c12e j(ωot−ω1t) +
c−12
e−j(ωot+ω1t)
= c1 cos (ωot − ω1t)
Output contains desired signal (plus a lot of other signals).Must pre-filter the undesired signals (such as the image band).
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Convolution in Frequency
Ideal multiplier mixer:
p(t)
x(t) y(t)
y (t) = p (t) x (t)
Y (f ) = X (f ) ∗ P (f )
P (f ) =∞∑−∞
cnδ (f − nfLO)
Y (f ) =
∞∫−∞
∞∑−∞
cnδ (σ − nfLO)X (f − σ) dσ
=∞∑−∞
cn
∞∫−∞
δ (σ − nfLO)X (f − σ) dσ
=∞∑−∞
cnX (f − nfLO)
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Up-Conversion: n > 0
X(f)
fLO−fLO
f
fLO
n = 1
n = 2n = 3
f
Y (f)
n = −1
n = −2
2fLO 3fLO−fLO−2fLO
Input spectrum is translated into multiple points at the outputat multiples of the LO frequency.
Filtering is required to reject the undesired harmonics.
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Down-Conversion: n < 0X(f)
fLO−fLO
f
fLO
n = 1n = 2
n = 3
f
Y (f)
n = −1
n = −2
2fLO 3fLO−fLO−2fLO
n = −3
Output spectrum is translated from multiple sidebands (bothsides – image issue) into the same output.
This is the origin of the lack of image and harmonic rejectionin a basic mixer. We have already seen image rejectarchitectures. Later we’ll study harmonic rejection mixers.
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Balanced Mixer Topologies
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Balanced Mixer
An unbalanced mixer has a transfer function:
y(t) = x(t)× s(t) = (1 + A(t) cos(ωRF t))×{
0 LO < 01 LO > 0
which contains both RF, LO, and IF
For a single balanced mixer, the LO signal is “balanced”(bipolar) so we have
y(t) = x(t)× s(t) = (1 + A(t) cos(ωRF t))×{−1 LO < 0+1 LO > 0
As a result, the output contains the LO but no RF component
For a double balanced mixer, the LO and RF are balanced sothere is no LO or RF leakage
y(t) = x(t)× s(t) = A(t) cos(ωRF t)×{−1 LO < 0+1 LO > 0
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Current Commutating Mixers
I1 I2
M1 M2
RF
LO
M3 sB iI +
Io1 = I1 − I2 = F (VLO (t) , IB + is)
Assume is is small relative to IB and performTaylor series expansion.
Io1 ≈ F (VLO (t) , IB) +∂F
∂IB(VLO (t) , IB) · is + ...
Io1 = Po (t) + P1 (t) · is
+1
All current through M1
M2 Both on
( )tVLO
( )tP1
vx
-vx
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Current Commutating
i1 i2
M1 M2
is
i1is
=
1gm2
1gm1
+ 1gm2
i2is
=
1gm1
1gm1
+ 1gm2
p1 (t) =gm1 (t)− gm2 (t)
gm1 (t) + gm2 (t)
(=
i1 − i2is
)Note that with good device matching: p1 (t) = −p1
(t + To
2
)Expand p1(t) into a Fourier series:
p1,2k =1
TLO
TLO∫0
p1 (t) e−j2π2kt/TLOdt =
TLO/2∫0
+
TLO∫TLO/2
= 0
Only odd coefficients of p1,n non-zero
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Single Balanced Mixer
+LO −LO
+RF
IF+ −
Switching Pair
Transconductance stage (gain)
RF Current
Assume LO signal strong so that current (RF) is alternativelysent to either M2 or M3. This is equivalent to multiplying iRFby ±1.
vIF ≈ sign (VLO) gmRLvRF = g (t) gmRLvRF
g(t) periodic waveform with period = TLO
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Current Commutating Mixer (2)
g(t) = square wave =4
π(cosωLOt − cos 3ωLOt + ...)
Let vRF = A cosωRF t
Gain:
Av =vIFA
=1
2
4
πgmRL =
2
πgmRL
LO-RF isolation good, but LO signal appears in output (just adiff pair amp).
Strong LO might desensitize (limit) IF stage (even afterfiltering).
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Double Balanced Mixer
IEE
+LO
−LO
Q1 Q2 Q3 Q4
Q5 Q6
+LO
+RF −RF
RE RE Transconductance
LO signal is rejected up to matching constraints
Differential output removes even order non-linearities
Linearity is improved: Half of signal is processed by each side
Noise higher than single balanced mixer since no cancellationoccurs
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Mixer Design Examples
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Improved Linearity
Bias
RF
LO+ LO-
IF+ IF-
Role of input stage is to provide V -to-I conversion.Degeneration helps to improve linearity. Inductivedegeneration is preferred as there is no loss in headroom andit provides input matching.
An LO trap also minimizes swing of LO at output which maycause premature compression.
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Common Gate Input Stage
Iout
Vin
Vbias
Iout
Vin
Vbias
Iout
Vin
Vbias
Iout
Vin
Vbias
Vmirr
Iout
Vin
Vbias
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Gilbert Micromixer
LODRIVER
IF LOAD
RF INPUT
The LNA output is often single-ended. A good balanced RFsignal is required to minimize the feedthrough to the output.LC bridge circuits can be used, but the bandwidth is limited.A transformer is a good choice for this, but bulky andbandwidth is still limited.A broadband single-ended to differential conversion stage canbe used to generate highly balanced signals over very widebandwidths. Gm stage is Class AB.
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Gilbert Micromixer Gm StageI1 I3
IZ
IRF
+VRF
−
Q1
Q2
Q3QZ1
QZ2
Set IZ to bias for match: RIN = 2Vt/IZ .
Using “trans-linear concept” to derive currents by definingλ = IRF/2IZ
I 2Z = I1I2 = I1(I1 + IRF )
I1,2 = IZ
(√λ2 + 1∓ λ
)I1 − I3 = 2λIZ = IRF
RIN(λ) =Vt
2IZ
1√λ2 + 1
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Active and Passive Balun
Io(+) Io(−)
Vin
Vbias
Vbias
Vin
Vout(+)
Vout(−)
LC
Z1
Z2/2
L C Z2/2 Bias
RF
LO
IF
For broadband applications, an active balun is a good solutionbut impacts linearity. A fully passive balun cap be designedwith good bandwidth and balance, resulting in very goodoverall linearity. Watch out for common mode coupling inbalun.
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Bleeding the Switching Core
Bias
RF
LO+ LO-
IF+ IF-
Large currents are good for the gm stage (noise, conversiongain), but require large devices in the switching core ⇒ hardto switch due to capacitance and also requires a large LO(large Vgs − Vt)
A current source can be used to feed the Gm stage with extracurrent.
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Current Re-Use Gm Stage
VRF
VLO(+)
VLO(−)
Vbias,n
Vbias,p
Io1 Io2
Io = Io1 − Io2
Note that thePMOS currentsare out of phasewith the NMOS,and hence theinverted polarityfor the LO is usedto compensate.
CMOS technology has fast PMOS devices which can be usedto increase the effective Gm of the transconductance stagewithout increasing the current by stacking.
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Single, Dual, and Back Gate
VRF
VLO
Io
VRF
VLO
Io
Vbias
VRF VLO
Io
Vbias
Io
VRF
VLO
Note that for weak signals, the second-order non-linearity willmix the LO and RF signals.
We prefer to apply a strong LO to periodically modulate thetransconductance Gm(t) of the transistor.
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Ring or Gilbert Quad?+LO
−LO
+LO
M1
M4
M3
M2
+LO
M1
−LO
M2
+LO
M4
−LO
M3
+RF
−RF
+RF −RF+IF
−IF
+IF
−IF
Unfold the quad, we get a ring. They are the same!
Notice that a mixer is just a circuit that flips the sign of thetransfer function from ± every cycle of the LO. We can eithersteer currents or voltages.
The switches can be actively biased and switched with an LOof approximately a few Vgs − VT (or several times kT/q forBJT), or they can be biased passively and fully switched onand off.
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Passive Mixers (V )
LO
LO
+RF
−RF
IF
RIF
Devices act as switches andjust rewire the circuit sothat the plus/minus voltageis connected to the outputwith ±1 polarity.
Very linear but requires ahigh LO swing and largerLO power.
+RF
− RF
LO
LO
LO
LO
IF
L1
L2
C3 L3
Rs/2
Rs/2
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Passive Mixers (I )
+LO
−LO
+IF
−IF+RF −RF
+LO
Commutate currents using passive switches. Very linear(assuming current does not have distortion) but requireshigher LO drive.
Good practice to DC bias switches at desired operating pointand AC coupling the Gm stage.
Since we’re processing a current, we use a TIA to present alow impedance and convert I -to-V
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Sub-Sampling Mixers
ΦRF
VRF
VIF
X(f)
fRF
f
X(f)
fIF
f
Note that sampling isequivalent to multiplicationby an impulse train (spacingof TLO), which in thefrequency domain is theconvolution of anotherimpulse train (spacing of1/TLO).
This means that we cansample or even sub-samplethe signal. The problem isthat noise from allharmonics of the LO folds tocommon IF. Always true,but especially problematicfor sub-sampling.
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Rudell CMOS Mixer
VLO(+)
VLO(−)
Vbias Vbias
Vin(+) Vin (−)
VCM,out
ICMIgain
M16
Gain programmed using current through M16 (set byresistance of triode region devices)
Common mode feedback to set output point
Cascode improves isolation (LO to RF)
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Power Spectral Density
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Review of Linear Systems and PSD
Average response of LTI system:
y1 (t) = H1[x (t)] =
∞∫−∞
h1 (t) x (t − τ) dτ
y1 (t) = limT→∞
1
2T
T∫−T
y1 (t) dt
= limT→∞
1
2T
T∫−T
∞∫−∞
h1 (τ) x (t − τ) dτ
dt
=
∞∫−∞
limT→∞1
2T
T∫−T
x (t − τ) dt
︸ ︷︷ ︸
x(t)
h1 (τ) dt
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Average Value Property
y1 (t) = x (t)
∞∫−∞
h1 (t) dt
H1 (jω) =
∞∫−∞
h1 (t) e−jωtdt
y1 (t) = x (t)H1 (0)
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Output RMS Statistics
y12 (t) = limT→∞
1
2T
T∫−T
∞∫−∞
h1 (τ1) x (t − τ1) dτ1
∞∫−∞
h1 (τ2) x (t − τ2) dτ2
dt
=
∞∫−∞
∞∫−∞
h1 (τ1) h1 (τ2)
limT→∞1
2π
T∫−T
x (t − τ1) x (t − τ2) dt
dτ1dτ2
Recall the definition for the autocorrelation function
ϕxx (t) = x (t) x (t + τ)
= limT→∞
1
2T
T∫−T
x (t) x (t + τ) dt
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Autocorrelation Function
y12 (t) =
∞∫−∞
∞∫−∞
h1 (τ1) h2 (τ2)ϕxx (τ1 − τ2) dτ1dτ2
ϕxx (jω) =
∞∫−∞
ϕxx (τ) e−jωτdτ
ϕxx (τ) =1
2π
∞∫−∞
ϕxx (jω) e jωτdω
ϕxx(jω) is a real and even function of ω since ϕxx(t) is a realand even function of t
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Autocorrelation Function (2)
y12 (t) =
∞∫−∞
∞∫−∞
h1 (τ1) h1 (τ2)1
2π
∞∫−∞
ϕxx (jω) ejω(τ1−τ2)dωdτ1dτ2
=1
2π
∞∫−∞
ϕxx (jω)
∞∫−∞
∞∫−∞
h1 (τ1) h1 (τ2) ejω(τ1−τ2)dτ1dτ2
=1
2π
∞∫−∞
ϕxx (jω)
∞∫−∞
h1 (τ1) e+jωτ1dτ1
∞∫−∞
h1 (τ2) e−jωτ2dτ2
dω
y12 (t) =1
2π
∞∫−∞
ϕxx (jω)H1 (jω)H1∗ (jω) dω
=1
2π
∞∫−∞
ϕxx (jω) |H1 (jω)|2dω
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Average Power in X(t)
Consider x(t) as a voltage waveform with total average powerx2(t). Lets measure the power in x(t) in the band 0 < f < f1
Ideal LPF
1
1+
-
( )tx+
-
( )ty
The average power in the frequency range 0 < f < f1 is now
y12 (t) =1
2π
∞∫−∞
ϕxx (jω)|H1 (jω)|2dω
=1
2π
ω1∫−ω1
ϕxx (jω) dω =
f1∫−f1
ϕxx (j2πf ) df
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Average Power in X(t) (2)
= 2
f1∫0
ϕxx (j2πf ) df
Generalize: To measure the power in any frequency range,apply an ideal bandpass filter with passband f1 < f < f2
y12 (t) = 2
f2∫f1
ϕxx (j2πf ) df
The interpretation of ϕxx as the the power spectral density(PSD) is clear.
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Spectrum Analyzer
A spectrum analyzer measures the PSD of a signal
“Poor Man’s” spectrum analyzer:
VCO Sweep
generation 1f 2f
vertical
horiz. Linear wide tuning range
Wide dynamic range mixer
Sharp filter
CRT
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