Berkeley
Mixers: An Overview
Prof. Ali M. Niknejad
U.C. Berkeley
c 2014 by Ali M. Niknejad
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Mixers
Information
PSD
Mixer
fc
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 RF
carrier (shown above).
In a receiver, the same information is ideally down-converted
to baseband.
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Mixers Specifications
Conversion Gain: Ratio of voltage (power) at output
frequency to input voltage (power) at input frequency
Downconversion: RF power in, IF power out
Up-conversion: IF power in, RF power out
Noise Figure
DSB versus SSB
Linearity
Image Rejection, Spurious Rejection
LO Feedthrough
Input
Output
RF Feedthrough
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Mixer Implementation
Two tones
1,
2
f(x)
1
Non-linear
1
2
+
2
2nd order IM
We know that any non-linear circuit acts like a mixer
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Squarer Example
x2
x
y
Product component:
y = A2 cos2 ω1 t + B 2 cos2 ω2 t + 2AB cos ω1 t cos ω2 t
LO
What we would prefer:
vRF = vRF cos ω1 t
IF
vLO = vRF cos ω1 t
vIF = 2vLO · vRF cos (ω1 ± ω2 ) t
RF
A true quadrant multiplier with good dynamic range is
difficult to fabricate
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LTV Mixer
LTI
f1 + f 2
f1 + f 2
f1 + f 2
LTV
No new frequencies
New tones in output
No new frequencies for a Linear Time Invariant (LTI) system
But a Linear Time-Varying (LTV) “Mixer” can act like a
multiplier
Example: Suppose the resistance of an element is modulated
periodically
R(t)
vo (IF )
vo = iin · R (t)
iin (t)(RF )
= Io cos (ωRF t) · Ro cos (ωLO t)
vLO
=
Io R o
{cos (ωRF + ωLO ) t+cos (ωRF − ωLO ) t}
2
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Periodically Time Varying Systems
In general, any periodically time varying system can achieve
frequency translation
∞
X
p (t + T ) = p (t) =
cn e jωo nt vi (t)
v (t) = p (t) vi (t)
n=−∞
1
cn =
T
ZT
p (t) e −jωo nt dt
0
vi (t) = A (t) cos ω1 t = A (t)
vo (t) = A (t)
∞
X
cn
−∞
e jω1 t + e −jω1 t
2
e j(ωo nt+ω1 t) + e +j(ωo nt−ω1 t)
2
Consider n=1 plus n=-1
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Desired Mixing Product
c1 = c−1
vo (t) =
c1 j(ωo t−ω1 t) c−1 −j(ωo t+ω1 t)
e
+
e
2
2
= c1 cos (ωo t − ω1 t)
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
y (t) = p (t) x (t)
Ideal multiplier mixer:
Y (f ) = X (f ) ∗ P (f )
p(t)
y(t)
x(t)
P (f ) =
∞
X
−∞
Y (f ) =
Z∞ X
∞
−∞ −∞
=
∞
X
−∞

Z∞
cn 
−∞
=
cn δ (f − nfLO )
cn δ (σ − nfLO )X (f − σ) dσ

δ (σ − nfLO ) X (f − σ) dσ 
∞
X
−∞
cn X (f − nfLO )
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Up-Conversion: n > 0
X(f )
f
fLO
−fLO
Y (f )
n = −2
−2fLO
n = −1
n=1
n=2
−fLO
fLO
2fLO
n=3
3fLO
f
Input spectrum is translated into multiple points at the output
at multiples of the LO frequency.
Filtering is required to reject the undesired harmonics.
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Down-Conversion: n < 0
X(f )
f
fLO
−fLO
Y (f )
n=3
n=2
n=1
−2fLO
−fLO
n = −1
fLO
n = −2
n = −3
2fLO
3fLO
f
Output spectrum is translated from multiple sidebands (both
sides – image issue) into the same output.
This is the origin of the lack of image and harmonic rejection
in a basic mixer. We have already seen image reject
architectures. 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 < 0
1 LO > 0
which contains both RF, LO, and IF
For a single balanced mixer, the LO signal is “balanced”
(bipolar) so we have
−1 LO < 0
y (t) = x(t) × s(t) = (1 + A(t) cos(ωRF t)) ×
+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 so
there is no LO or RF leakage
−1 LO < 0
y (t) = x(t) × s(t) = A(t) cos(ωRF t) ×
+1 LO > 0
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Current Commutating Mixers
Io1 = I1 − I2 = F (VLO (t) , IB + is )
I1
I2
M1
Assume is is small relative to IB and perform
Taylor series expansion.
M2
LO
RF
M3
I B + is
Io1 ≈ F (VLO (t) , IB ) +
Io1 = Po (t) + P1 (t) · is
vx
-vx
∂F
(VLO (t) , IB ) · is + ...
∂IB
VLO (t )
+1
P1 (t )
All current
through M1
M2
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Current Commutating
M1
M2
i1
i1
=
is
i2
is
1
gm2
1
gm1
+
i2
=
is
1
gm2
gm1 (t) − gm2 (t)
p1 (t) =
gm1 (t) + gm2 (t)
1
gm1
1
gm1
+
1
gm2
i1 − i2
=
is
Note that with good device matching: p1 (t) = −p1 t +
To
2
Expand p1 (t) into a Fourier series:
p1,2k
1
=
TLO
TLO
Z
p1 (t) e
−j2π2kt/TLO
TZ
LO /2
dt =
0
+
0
Only odd coefficients of p1,n non-zero
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TLO
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TLO /2
=0
Single Balanced Mixer
+ IF
−
−LO
+LO
Switching
Pair
RF Current
Transconductance
stage (gain)
+RF
Assume LO signal strong so that current (RF) is alternatively
sent to either M2 or M3. This is equivalent to multiplying iRF
by ±1.
vIF ≈ sign (VLO ) gm RL vRF = g (t) gm RL vRF
g (t) periodic waveform with period = TLO
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Current Commutating Mixer (2)
g (t) = square wave =
4
(cos ωLO t − cos 3ωLO t + ...)
π
Let vRF = A cos ωRF t
Gain:
Av =
ṽIF
14
2
=
gm RL = gm RL
A
2π
π
LO-RF isolation good, but LO signal appears in output (just a
diff pair amp).
Strong LO might desensitize (limit) IF stage (even after
filtering).
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Double Balanced Mixer
+LO
Q1
Q2
Q3
Q4
+LO
−LO
+RF
Q5
Q6
RE
RE
−RF
Transconductance
IEE
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 cancellation
occurs
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Mixer Design Examples
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Improved Linearity
IF+
LO+
IF-
LO-
RF
Bias
Role of input stage is to provide V -to-I conversion.
Degeneration helps to improve linearity. Inductive
degeneration is preferred as there is no loss in headroom and
it provides input matching.
An LO trap also minimizes swing of LO at output which may
cause premature compression.
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Common Gate Input Stage
Iout
Vbias
Vin
Iout
Iout
Vbias
Iout
Vbias
Vin
Iout
Vbias
Vin
Vbias
Vin
Vmirr
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Vin
Gilbert Micromixer
IF LOAD
LO
DRIVER
RF INPUT
The LNA output is often single-ended. A good balanced RF
signal 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 and
bandwidth is still limited.
A broadband single-ended to differential conversion stage can
be used to generate highly balanced signals over very wide
bandwidths. Gm stage is Class AB.
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Gilbert Micromixer Gm Stage
I3
I1
IZ
Q1
IRF
+
VRF
−
Q3
Q2
QZ1
QZ2
Set IZ to bias for match: RIN = 2Vt /IZ .
Using “trans-linear concept” to derive currents by defining
λ = IRF /2IZ
IZ2 = I1 I2 = I1 (I1 + IRF )
p
I1,2 = IZ
λ2 + 1 ∓ λ
I1 − I3 = 2λIZ = IRF
Vt
1
√
RIN (λ) =
2IZ λ2 + 1
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Active and Passive Balun
Io (+)
Io (−)
IF
Vbias
Vin
Vbias
LO
Vout (+)
C
L
Z2 /2
L
C
Z2 /2
RF
Vin
Z1
Bias
Vout (−)
For broadband applications, an active balun is a good solution
but impacts linearity. A fully passive balun cap be designed
with good bandwidth and balance, resulting in very good
overall linearity. Watch out for common mode coupling in
balun.
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Bleeding the Switching Core
IF+
IF-
LO+
LO-
RF
Bias
Large currents are good for the gm stage (noise, conversion
gain), but require large devices in the switching core ⇒ hard
to 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 extra
current.
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Current Re-Use Gm Stage
Vbias,p
VLO (+)
Io1
VRF
Io2
VLO (−)
Note that the
PMOS currents
are out of phase
with the NMOS,
and hence the
inverted polarity
for the LO is used
to compensate.
Io = Io1 − Io2
Vbias,n
CMOS technology has fast PMOS devices which can be used
to increase the effective Gm of the transconductance stage
without increasing the current by stacking.
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Single, Dual, and Back Gate
Io
Io
VRF
VRF
VLO
VLO
Vbias
Io
VLO
VRF
VLO
VRF
Io
Vbias
Note that for weak signals, the second-order non-linearity will
mix the LO and RF signals.
We prefer to apply a strong LO to periodically modulate the
transconductance Gm (t) of the transistor.
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Ring or Gilbert Quad?
+LO
+IF
−
LO
+
−RF
LO
2
M
−
M3
M
4
+RF
−IF
LO
M4
+
−RF
3
+IF
−LO
M
M
1
M2
+RF
LO
M1
−IF
+LO
Unfold the quad, we get a ring. They are the same!
Notice that a mixer is just a circuit that flips the sign of the
transfer function from ± every cycle of the LO. We can either
steer currents or voltages.
The switches can be actively biased and switched with an LO
of approximately a few Vgs − VT (or several times kT /q for
BJT), or they can be biased passively and fully switched on
and off.
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Passive Mixers (V )
Devices act as switches and
just rewire the circuit so
that the plus/minus voltage
is connected to the output
with ±1 polarity.
LO
+RF
IF
−RF
RIF
Very linear but requires a
high LO swing and larger
LO power.
LO
L1
+RF
LO
Rs /2
C3
Rs /2
−
IF
L3
LO
RF
L2
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LO
Passive Mixers (I )
+LO
+IF
−LO
+RF
−IF
−RF
+LO
Commutate currents using passive switches. Very linear
(assuming current does not have distortion) but requires
higher LO drive.
Good practice to DC bias switches at desired operating point
and AC coupling the Gm stage.
Since we’re processing a current, we use a TIA to present a
low impedance and convert I -to-V
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Sub-Sampling Mixers
Note that sampling is
equivalent to multiplication
by an impulse train (spacing
of TLO ), which in the
frequency domain is the
convolution of another
impulse train (spacing of
1/TLO ).
ΦRF
VIF
VRF
X(f )
fRF
f
X(f )
fIF
f
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This means that we can
sample or even sub-sample
the signal. The problem is
that noise from all
harmonics of the LO folds to
common IF. Always true,
but especially problematic
for sub-sampling.
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Rudell CMOS Mixer
VLO (+)
VCM,out
VLO (−)
Vbias
M16
Igain
ICM
Vin (+)
Vbias
Vin (−)
Gain programmed using current through M16 (set by
resistance 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:
Z∞
y1 (t) = H1 [x (t)] =
h1 (t) x (t − τ ) dτ
−∞
1
y1 (t) = lim
T →∞ 2T
ZT
y1 (t) dt
−T
= lim
1
ZT

Z∞

T →∞ 2T
−T
−∞

h1 (τ ) x (t − τ ) dτ  dt


Z∞
ZT
limT →∞ 1
x (t − τ ) dt h1 (τ ) dt
=
2T
−∞
−T
|
{z
}
x(t)
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Average Value Property
Z∞
y1 (t) = x (t)
h1 (t) dt
−∞
Z∞
H1 (jω) =
h1 (t) e −jωt dt
−∞
y1 (t) = x (t)H1 (0)
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Output RMS Statistics
y1
2
1
(t) = lim
T →∞ 2T
ZT
=
Z∞

Z∞
h1 (τ1 ) x (t − τ1 ) dτ1  

−T
Z∞ Z∞

−∞

h1 (τ2 ) x (t − τ2 ) dτ2 dt
−∞

1

h1 (τ1 ) h1 (τ2 ) limT →∞
2π
−∞ −∞

ZT

x (t − τ1 ) x (t − τ2 ) dt  dτ1 dτ2
−T
Recall the definition for the autocorrelation function
ϕxx (t) = x (t) x (t + τ )
1
= lim
T →∞ 2T
ZT
x (t) x (t + τ ) dt
−T
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Autocorrelation Function
y1 2 (t) =
Z∞ Z∞
−∞ −∞
h1 (τ1 ) h2 (τ2 )ϕxx (τ1 − τ2 ) dτ1 dτ2
Z∞
ϕxx (jω) =
ϕxx (τ ) e −jωτ dτ
−∞
1
ϕxx (τ ) =
2π
Z∞
ϕxx (jω) e jωτ dω
−∞
ϕxx (jω) is a real and even function of ω since ϕxx (t) is a real
and even function of t
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Autocorrelation Function (2)
y1 2 (t) =
Z∞ Z∞
h1 (τ1 ) h1 (τ2 )
−∞ −∞
=
1
2π
Z∞
Z∞
Z∞ Z∞
ϕxx (jω)
ϕxx (jω) e jω(τ1 −τ2 ) dωdτ1 dτ2
h1 (τ1 ) h1 (τ2 ) e jω(τ1 −τ2 ) dτ1 dτ2
−∞ −∞

Z∞
ϕxx (jω) 
−∞
Z∞
−∞
−∞
1
=
2π
1
2π

h1 (τ1 ) e +jωτ1 dτ1  
−∞
y1 2 (t) =
1
2π
Z∞

h1 (τ2 ) e −jωτ2 dτ2  dω
−∞
Z∞
ϕxx (jω) H1 (jω) H1 ∗ (jω) dω
−∞
=
1
2π
Z∞
ϕxx (jω) |H1 (jω)|2 dω
−∞
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Average Power in X(t)
Consider x(t) as a voltage waveform with total average power
x 2 (t). Lets measure the power in x(t) in the band 0 < f < f1
+
x(t )
Ideal LPF
1
1
-
+
y (t )
-
The average power in the frequency range 0 < f < f1 is now
Z∞
1
y1 2 (t) =
ϕxx (jω)|H1 (jω)|2 dω
2π
−∞
1
=
2π
Zω1
Zf1
ϕxx (jω) dω =
−ω1
ϕxx (j2πf ) df
−f1
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Average Power in X(t) (2)
Zf1
ϕxx (j2πf ) df
=2
0
Generalize: To measure the power in any frequency range,
apply an ideal bandpass filter with passband f1 < f < f2
y1 2 (t) = 2
Zf2
ϕxx (j2πf ) df
f1
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:
Wide
dynamic
range mixer
Sharp filter
vertical
VCO
Linear wide
tuning range
Sweep
generation
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CRT
f1
f2