Berkeley
Mixer Noise
Prof. Ali M. Niknejad
U.C. Berkeley
c 2014 by Ali M. Niknejad
Copyright Niknejad
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Noise in Mixers
Prof. Ali M Niknejad and Dr. Osama Shana’a
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Noise in an Ideal Mixers
Consider the simplest ideal multiplying mixer:
RF
Noise
IF
LO
IF
RF
LO
IM
What’s the noise figure for the conversion process?
Input noise power due to source is kTB where B is the
bandwidth of the input signal
Input signal has power Ps at either the lower or upper
sideband (or both for direct conversion)
SNRi =
Ps
kTB
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Noise in Ideal Mixers
At the IF frequency, we have the down-converted signal G · Ps
and down-converted noise from two sidebands, LO − IF and
LO + IF
SNRo =
(G 0
G · Ps
+ G 00 ) kTB
For ideal mixer, G = G 0 = G 00
F =
Ps 2kTB
SNRi
=
=2
SNRo
kTB Ps
NF = 3dB
For a real mixer, noise from multiple sidebands can fold into
IF frequency and degrade NF
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SSB vs. DSB NF definition:
image
noise
LO
Ni+Gmix
Ni
S/Ni
image noise+Gmix
IF
IF
As we have seen, because of the image problem, a receive
mixer down converts both desired and the image bands to IF
frequency. This means folding the noise at the image
frequency on top of the desired band at IF.
Therefore, the total noise at IF is as follows:
The noise at desired RF band down converted to IF
The noise at image RF band down converted to IF
The noise added by the mixer noisy circuit itself.
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Single-Side Band (SSB) NF
The single-side band NF definition assumes that there is no
signal at the image frequency except the source noise. This
definition is useful in non-zero IF architectures, where the
image signal is suppressed by an image filter before reaching
the mixer.
The NF is the degradation of S/N at mixer output. Therefore,
one can write:
Sout = Sd Gmix,d
Nout = Nd Gmix,d +NI 0 m Gmix,I 0 m +Nmix,d Gmix,d +Nmix,im Gmix,I 0 m
where Sd is the desired signal, Nd is the noise in the desired
band, NI 0 m is the noise in the image band, Gmix,d and Gmix,im
is the mixer gain in the desired and image frequencies,
respectively.
Nmix,d and Nmix,I 0 m are the desired and image band noise due
to the mixer circuit itself referred to its input.
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Single-Side Band (SSB) NF (cont)
To simplify the analysis, we will assume that Nd = NI 0 m ,
Nmix,d = Nmix,im , and Gmix,d = Gmix,I 0 m , so that we can write
!
Sd
Sd
1
Sout
=
=
Nout
2Nd + 2Nmix
Nd 2 + 2NNmix
d
FSSB = 2 +
2Nmix
Nd
As seen from the SSB noise figure equation, if the mixer is
noiseless (Nmix = 0), the mixer SSB NF is 3dB because of the
image noise folding. It is important to know that this definition is
the one used by microwave mixer designers for years. It is also the
definition used in the SpectreRF simulator.
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IEEE Noise Definition
The IEEE has a slightly different definition for SSB NF. It
argues that the mixer should not be “penalized” by the image
source noise folding.
The only image noise folding that is allowed to count towards
calculating the mixer SSB NF is that which is due to the mixer
circuitry itself. The input image noise should not be counted.
Therefore, the IEEE SSB NF assumes there is a sharp
bandpass filter that passes the desired band with the source
noise and knocks down the image noise to the negligible level.
As a result, one can write
Sout = Sd Gmix,d
Nout = Nd Gmix,d + Nmix,d Gmix,d + Nmix,im Gmix,im
To simplify analysis, we will assume Nd = Nim and
Gmix,d = Gmix,im , therefore,
Nout = Nd Gmix,d + 2Nmix Gmix,d
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IEEE Noise Def (cont)
Image
noise filter
image
noise
LO
Noiseless
mixer
Nd+Gmix
Sd/Nd
Nd
IF
IF
#
"
⇒
Sout
Nout
=
⇒ FSSB
Sd
Nd +2Nmix
IEEE
=1+
=
Sd
Nd
1
1+
2Nmix
Nd
2Nmix
Nd
As seen from the IEEE SSB equation, if the mixer is noiseless,
the mixer SSB NF is actually 0dB, the spirit behind the new
definition.
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Double-Side Band (DSB) NF:
LO
Noiseless
mixer
image
Nd+Gmix_d
image
noise
Nd
Sd/Nd
Nim+Gmix_im
IF
IF
The double-side band NF definition assumes that the image
band contains both noise and an image signal identical to the
desired band signal. This definition is useful in
direct-conversion receiver where the image is the signal itself.
Therefore, one can write:
Sout = Sd Gmix
Nout = Nd Gmix d +Nim Gmix
d
+ Sim Gmix
im
im +Nmix d Gmix d +Nmix im Gmix im
where Sim is the image signal. To simplify analysis, we will
assumeSd = Sim , Nd = Nim , Nmix,d = Nmix,im , and
Gmix,d = Gmix,im . Therefore,
Nout = 2Nd Gmix,d + 2Nmix Gmix,d
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DSB NF (cont)
"
⇒
Sout
Nout
=
2Sd
2Nd +2Nmix
⇒ FDSB = 1 +
=
Sd
Nd
#
1
1+
Nmix
Nd
Nmix
Nd
It can be seen that the difference between the SSB NF and
DSB NF is exactly 3dB. However, with the SSB IEEE, the
difference is not exactly 3dB. In fact the difference between
the SSB IEEE and the SSB NF approaches 3dB as the mixer
NF is very high. The SSB IEEE noise factor can be related to
that of the DSB as:
FSSB
IEEE
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= 2FDSB − 1
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Noise in Active Mixers
Prof. Ali M Niknejad Based on the work of Dr. Manolis Terrovitis, “Analysis
and Design of Current-Commutating CMOS Mixers,” Ph.D. Dissertation,
Berkely, CA 2001.
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Recap: CMOS Mixer Operation
I1
I2
M2
Io1 = I1 − I2 = F (VLO (t) , IB + is )
∂F
≈ F (VLO (t) , IB ) +
(VLO (t) , IB ) · is + ·
∂IB
= p0 (t) + p1 (t) · is + · · ·
M3
LO
RF
IB + i s
M1
Periodic function:
p1 (t) =
gm1 (t) − gm2 (t)
gm1 (t) + gm2 (t)
p1 (t) = −p1 (t + TLO /2) with good matching, which means it
has only odd-order frequency components. Fourier Series
expansion
P1,2k ≡ 0
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Mixer Noise
AC signal at output is given by
Io1 = p1 (t) · is
Let X (f ) be the spectrum of is , Y (f ) be the spectrum of Io1
∞
X
Y (f ) =
n=−∞
P1,n X (f − nfLO )
If we fix the output frequency at fIF , then we see energy from
multiple bands folding into output spectrum
n
e
o is
no
−3fo
−2fo
fold
is
in g
o ld
ef
in g
−fs −fo
i s e f o l d i n g f ro m
no
3LO
n o i s e fo l di n
gf
rom
2L
O
from -3LO
fro
m - 2 LO
−fif
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fif
fo fs
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2fo
3fo
Mixer Noise (cont)
Y (fIF ) =
∞
X
n=−∞
p1,n X (fIF − nfLO )
n = +1
n = −1
n = +2
fLO − fIF
fLO + fIF
2fLO − fIF
The coefficient p1,k then represents the conversion gain from
frequency (k × fLO ± fIF ) to fIF .
If we assume hard switching, then p1 (t) is a square wave and
the coefficients are
2
p1,2k−1 =
π (2k − 1)
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Noise from Transconductance Stage
A cyclostationary process is a random process whose statistics
are periodic functions of time.
The PSD of a cyclostationary process is given by S(f , t).
If we measure a cyclostationary process over a bandwidth
< 1/period, we observe S (f , t), a stationary process.
Here we assume that the noise of the transconductor is
stationary. The output noise, though, is cyclostationary since
it’s multiplied by a periodic function.
n3 (t) WSS, PSD sn3 (f )
Yn3 = n3 (t) · p1 (t)
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PSD Calculation
Consider a white noise source n(t) multiplied by a periodic
deterministic function p(t):
y (t) = n (t) · p (t)
Ry (t1 , t2 ) = E [y (t1 ) y (t2 )]
= p (t1 ) p (t2 ) n (t1 ) n (t2 )
= p (t1 ) p (t2 ) δ (t1 − t2 ) · N
Ry (τ ; t) = p(t)2 δ (τ ) · N
Sy (f ; t) = Np 2 (t)
Z
Sy (f ; t) = N p 2 (t) dt
T
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PSD (cont)
If Sn3 (t) = Nn3 (white), then
α=
1
TLO
Sn3 (f ) = Nn3 · α
Z
∞
X
2
p1 (t) dt =
|p1,n |2
TLO
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n=−∞
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Square LO Waveform Noise
For a square wave p1 (t), α = 1
First sidebands (fLO ± fIF ) accounts for
2
2
≈ 81%
2×
π
(3fLO ± fIF ) accounts for
2×
2
3π
2
≈ 7%
The remaining harmonics account for the rest ∼ 10%
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Nearly Square LO Waveform
vo
vx
-vx
-vo
p1 (t )
+1
Assume p1 (t) is a straight line during period ∆ when both
devices on. Then
4
α = 1 − ∆fLO
3
−1 vx
π∆fLO = sin
vo
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Single Balanced Mixer Noise
Given the results so far, we can write the thermal noise due to
the transconductance stage as:
γ
Sn3 (f )
= 4kT Rs + rg 3 +
·
gm3 2
·α
|{z}
| {z }
gm3
|
{z
}
2
Noise due to M3
G for M3 with de(transconductor)
Input referred noise voltage
m3
generation
For a Gilbert Cell, we double the noise contribution due to the
presence of two devices with uncorrelated noise:
2γ
Sn3 (f ) = 4kT Rs + 2rgs +
Gm3 2 · α
gm3
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Thermal Noise of Switching Pair
I1
High
RF
M2
M3
M1
Low
When we switch hard, the noise is due to
M3 only (transconductor) (neglecting
capacitance and output impedance of M3)
When we switch hard, the noise is due to M3 only
(transconductor) (neglecting capacitance and output
impedance of M3)
Hard switching is good for low noise output. When both M1
and M2 on, during the switching period ∆, then the noise
PSD at output is

!2
!2 
1
gm1
1
gm2

SI1 = 4kT γ 
+
gm1
m2
gm1 1 + gm2
gm2 1 + ggm1
= 4kT γ
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gm1 · gm2
gm1 + gm2
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Thermal Noise of Switching Pair (2)
I3 = I1 + I2
Io1,n = 2I1,n ⇒ 4X power
Total transconductance of diff pair for LO
port to differential output current:
Sn1,2 (f , t) = 16kT γ
G (t) =
2gm1 gm2
gm1 + gm2
gm1 gm2
= 8kT γG (t)
gm1 + gm2

1
Sn1,2 (f ) = Sn1,2 (f , t) = 8kT γ 
TLO
TLO
Z

G (t) dt 
0
= 8kT γG
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Thermal Noise of Switching Pair (3)
vLO = vo sin (2πfLO t)
1
G=
πvo
Zvx
G (vLO ) r
1−
−vx
1
≈
πvo
Zvx
−vx
1
vLO
vo
2 dvLO
dIo1
2IB
· dvLO =
dvLO
πvo
16kT γ
Sn1,2 (f ) ≈
π
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IB
vo
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Noise from LO Port
Consider the noise that originates from the LO port, “outside”
of the mixer:
YnLO = G (t) · nLO (t)
SnLO (t) = NLO
∞
X
−∞
|Gn |2 = G 2 NLO
For example, if we assume the mixer port is driven with an
ideal voltage source, then only the gate resistances contribute
to the noise:
Sn,LO (f ) = 4kT (4rg 1 ) G 2
LO
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Flicker Noise
LO
DC
DC → LO in up-conversion from fLO and all odd harmonics
Flicker noise of switching transferred to output by
multiplication by G (t)
G (t) has period TLO /2, and Fourier series contains only
even-order harmonics of the LO.
Flicker noise from the switching pair will appear at output at
DC but not at fLO .
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Mixer Noise Figure
Neglecting Flicker Noise:
(NF )SSB
α (γ3 + rg 3 gm3 )gm3 α + 2γ1 G + (RLO + 2rg 1 )G 2 +
= 2+
2 R
c
c 2 gm3
s
And for the Gilbert cell is
(NF )SSB =
α 2(γ3 + rg 3 gm3 )gm3 α + 4γ1 G + (4rg 1 )G 2 +
+
2 R
c2
c 2 gm3
s
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1
RL
,
1
RL