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Noise Figure Optimization Tool for MillimeterWave Receivers at near-fmax Frequencies
J. Elkind, Student Member, IEEE, and E. Socher, Senior Member, IEEE

Abstract— This paper presents an evaluation tool of the noise
figure (NF) reduction that can be achieved by adding a low noise
amplifier (LNA) in a near fmax frequencies receiver. After
choosing a suitable topology and assessing its frequency
dependence, an analytical derivation is carried out and
preliminary frequency constrains are found. The analytical
assessment is followed by a practical example using the CMOS
65nm technology as reference for mm-wave (mmW) technologies,
where fmax = 234 GHz.
Index Terms— common-source, cross-coupling, LNA, lownoise, maximum-frequency, mixer, mmW, neutralization,
receiver.
I. INTRODUCTION
T
HE sensitivity of a receiver is highly affected by its noise
figure, and therefore one of the most important objectives
of a front-end LNA is to reduce and determine the overall
noise figure of the receiver. In mmW systems, as the
frequency increases and approaches fmax, the gain of the LNA
is highly diminished, down to the point where its contribution
to the overall noise figure reduction can be put to question.
For example, in [1], two types of receivers were proposed at
245 GHz - one with an LNA before the mixer and the other
without it. In this case the version that did not include an LNA
demonstrated a lower NF than the one that did. In [2], an LNA
was used only for the relatively lower frequency receiver, at
220 GHz, while at 320 GHz it was omitted due to the low
available gain. In [3] the LNA at 240 GHz it was completely
omitted, and [4] includes an LNA at 245 GHz as a part of the
receiver front-end.
Fig. 1(a) presents a typical receiver front-end block diagram
where the effectiveness of the LNA is not clear. In such cases,
the common approach is a trial and error approach, however,
to enhance efficiency, here we establish a method to clarify
and predict the frequency behavior limits of the LNA a priori
to the design process.
A theoretical investigation of the well-known receiver
design methodology, based on the Friis formula was first
performed. Then, once an appropriate topology for the LNA is
chosen, a large data base is created based on the core design
The authors are with the School of Electrical Engineering. Tel Aviv
University, Tel Aviv 69978, Israel (e-mails: jeniaelkind@mail.tau.ac.il,
socher@eng.tau.ac.il.)
parameters and used throughout the analysis in different
Matlab simulations. These simulations offer the designer a
preliminary evaluation tool of the maximum possible
Section II presents the chosen topology and frequency
dependence of the Gmax and NFmin of a single stage.
Section III provides an analytical derivation of the upper
frequency limits of the LNA for maximum NF reduction.
Section IV presents the considerations for choosing the bias
and sizing of the transistors. In Section V the final results for
N stages LNA are presented assuming lossless matching
networks. At the final implementation of the LNA a decrease
in maximum operation frequency is expected due to the use of
lossy matching networks. Therefore, to complete the picture
Section V.C provides an analysis and demonstration of the
operation frequency reduction caused by real matching
networks. Finally, Section VI concludes the results.
II. GMAX AND NFMIN FREQUENCY FITTING
When working at near fmax frequencies, special care for
topology choice must be taken. The commonly used cascode
topology suffers from intrinsically low fmax due to extra
parasitic capacitances and relatively high NF compared with a
common source (CS) topology. But, as can be seen in the
simple common source MSG expression derived in [5], Cgd is
a main source of gain degradation. Thus, as further discussed
in [6], a more appropriate choice for an LNA stage topology,
in terms of power gain at near fmax frequencies, is the
neutralized, cross-coupled common source differential pair, as
shown in Fig.1 (b). The capacitor neutralization technique
both increases the power gain and adds control over stability
and bandwidth [5]-[6].
After choosing the topology, a parameter investigation was
needed to evaluate the upper frequency bounds of the
technology and then find the highest NF reduction that can be
achieved near fmax frequencies. A parameter sweep of
neutralization capacitors, transistor sizes, and biasing was
performed, using a schematic model of the transistors and
ideal neutralization capacitors. The results constituted a data
base which was used throughout the entire paper.
In order to analyze the receiver NF at near fmax frequencies,
a simplified frequency model, separating the frequency from
all the other design parameters was needed. The neutralized
CS stage simulation was first used in order to obtain a
generalized frequency dependence of its NFmin and Gmax.
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(a)
(b)
Fig. 1. (a) Block diagram of receiver front-end; (b) LNA core schematic.
After simulating and extracting the frequency dependent
data, a parameter fitting was performed using the Matlab
Curve Fitting Toolbox. NFmin was reconstructed using a
polynomial fitting, and Gmax was reconstructed using an
exponential fitting. The Gmax curve was reconstructed using a
sum of two exponential curves: before and after the knee
frequency fc, at which Kf = 1; where Kf is the known stability
factor. Since we are interested in the high frequency limit
close to fmax, only the fitting above the knee point, i.e. f > fc
was eventually used.
The expressions of the simplified frequency model of Gmax
and NFmin, for f > fc are:
Gmax, dB  f   A log  f   B


2

 NFmin,dB  f   Cf  Df  E
; A  0, B  0
;D  0
(1)
Where A, B, C, D and E are constants that depend on the
sizing of the transistor and its biasing. Given this model, we
then focused solely on the behavior of the frequency
dependence.
The simplified model is validated in Fig. 2, as it nicely
confirms with the simulated results. Fig. 2(a) and 2(b) show
the simulated and reconstructed Gmax and NFmin of a constant
transistor width (5.4 µm) and biasing (272 µA/µm), with
neutralization capacitor sizes varying from 1 to 7 fF. For each
stage NFmin and Gmax, a set of parameters A, B, C, D and E
was found and then used to reconstruct the original plots using
(1). These expressions are revisited in Section III for deriving
the limits of the LNA NF reduction potential at high
frequencies.
(b)
Fig. 2. (a) Reconstructed NFmin and Gmax; (b) Simulated NFmin and Gmax.
(a)
Therefore, in order to benefit from adding an LNA, assuming
the LNA noise figure is less than the mixer noise figure, and
assuming GLNA>1, the total noise figure of the two cascaded
circuits should be less than the noise figure of the mixer alone:
Fmix  f   1 !
Ftot  FLNA  f  
 Fmix  f 
(3)
GLNA  f 
Separating Fmix from the LNA properties gives:
FLNA  f   GLNA  f   1
Fmix  f  
GLNA  f   1
(4)
In order to achieve a constant decrease in the mixer noise
figure, q:
F
1  q  mix
(5)
Ftot
A more general expression can be found:
Fmix  f  
FLNA  f   GLNA  f   1
GLNA  f 
q
; GLNA  f   q  1 (6)
1
III. MAXIMUM OPERATION FREQUENCY DERIVATION
A. Single Stage LNA Analysis
For a single stage LNA, the total noise figure of the LNA
and the mixer according to the Friis formula is:
Ftot  FLNA  f  
Fmix  f   1
GLNA  f 
Defining the Friis Expression, Fr:
Fr 
FLNA  f   GLNA  f   1
Gr  f   1
 Fmix
(7)
Where:
(2)
Gr 
G LNA
q
, q 1
(8)
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The minimization of Fr is equivalent to minimizing the NF
of the receiver. Fr depends solely on the properties of the LNA
and therefore enables a very convenient frequency dependent
NF optimization and maximum operation-frequency
evaluation tool.
Assigning the reconstructed expressions of NFmin and
Gmax, from (1) to the Friis Expression, (7), yields:
Fr  f  
B  f A 10Cf
B f
A
2
 Df  E

/ q 1
1
; A  0; B, D  0
Gmax (dB)
NF reduction ≥ qdB
NF reduction = 0
Gmax=qdB
f=fq
(9)
For a constant NF reduction, q, as the frequency increases
and GLNA diminishes to q, Fr goes to infinity and the condition
in (7) no longer holds:
lim Fr  f    Where: f q  f |GLNA  q
(10)
f  fq
It is clear now that the main limitation in NF reduction and
maximum operation frequency stems from the decrease in
GLNA. This means that an LNA stage cannot improve the NF
with a factor better than its gain, and the highest possible NF
reduction frequency is naturally fmax, where q=GLNA=1, as
seen in Fig. 3.
To verify the conclusions above, a Matlab simulation was
applied on the data base from Section II, assuming a typical
down-converting mixer at around 200 GHz has a NF of 20 dB.
The choice of a 20 dB NF mixer is arbitrary, and a similar
analysis can be applied on any given mixer.
Fr was calculated for a constant transistor size (5.4 µm)
and biasing (272 µA/µm), and for neutralization capacitor
sizes varying from 1 to 9 fF. The frequency dependence of Fr
compared to a mixer with a 20 dB NF for q=2 can be seen in
Fig. 4. Fig. 4(a) shows Fr for the reconstructed values of
NFmin and Gmax, and Fig .4(b) shows Fr for the simulated
values of NFmin and Gmax. As can be seen from the graphs,
the highest possible frequency for a 3 dB noise figure
reduction is f =206.4 GHz and the optimal Cn is 3 fF. In
addition, it can be seen that the reconstructed and simulated
plots produce the same results in terms of neutralization
selection, confirming again the frequency dependent model
(1).
B. N Stages LNA Analysis
For a cascade of N equal LNA stages with gain G and noise
figure of F :

 1 / G  N  1 
 F  1 ; G  1
 FLNA, N  1   F  1  
  F


N

1
/
G

1
G 1

 
 

; G  1 (11)
 FLNA, N  1   F  1  N  

N 1
F 1
F
1  i  
;G  1
 LNA, N
i 0 G

f=fmax
f
Fig. 3. LNA NF reduction potential and frequency limits as a function of its
gain.
Since Fr is only valid for Gr > 1, or GLNA > q, assigning FLNA,N
and GLNA,N derived above to Fr gives:
Fr , N 
GLNA, N FLNA, N  1
Gr , LNA, N  1
lim Fr

F 1 
 q   FLNA  LNA

N  
GLNA  1 

(12)
 q  FLNA, N
N , f  f q
Where:
GLNA, N   GLNA 
N
(13)
This suggests that the NF of the LNA, for a large enough
number of stages, converges to a constant independent of N;
therefore the Friis expression converges to q·FLNA,N. The
conclusion is that as long as GLNA,N > q the only condition that
needs to be met in order to benefit from adding the LNA
before the mixer is:
(14)
Fmixer  q  FLNA, N  FLNA, N
(a)
(b)
Fig. 4. (a) Fr based on reconstructed data for various Cn sizes; (b) Fr based on
simulated data for various Cn sizes.
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IV. CHOOSING BIAS AND SIZING
Since we are interested in the highest possible power-gain
contributing operation-frequency, the sizing and biasing of the
transistor should be aimed at achieving the maximum possible
fmax. According to [7]:
fT
f max 
(15)
2 Rg  g m CGD / CGS   g ds Rg  rch  Rs


From this approximate expression it is clear that fmax is
limited mainly by the gate resistance, Rg. Therefore it is best
to use small finger widths in the design. Furthermore, it can be
proven that fmax is independent of total transistor size, i.e. the
number of fingers, [8]. Another important fmax determining
factor is the biasing. According to [8], the transistor should be
biased in strong inversion.
The bias and finger width also determine NFmin, [6],
therefore the final biasing and finger width choices should
account for fmax and NFmin simultaneously. Fig. 5 shows the
fmax and NFmin as a function of the current density, Jds, and
the transistor finger width, W. It can be seen that a proper
choice is 272 µA/µm and W=900 nm, which corresponds to a
bias voltage of 0.85 V. These choices will ensure a minimum
noise figure of 6.1 dB at fmax and an fmax of 234.6 GHz.
According to these results, in order to maximize fmax while
maintaining an intermediate NFmin, it is best to work on the
edge of velocity saturation rather than strong inversion, as
recommended in [6] and [8], probably due to the proximity to
fmax operating frequencies.
Fig. 6 shows the maximum operating frequency for each
number of LNA stages and each q up to a 20 dB NF reduction.
It can be observed that the maximum operation-frequency for
a ~10 dB reduction in NF is ~200GHz where fmax=234 GHz,
which confirms the results of Fig. 4. In addition, this plot
shows that for more than 4 stages the benefit in terms of NF
reduction is minor.
B. N maximally stable stages LNA
As concluded in Section III, the gain of the stage is critical
for maximizing the NF reduction frequency. Fig. 7 shows Kf
and Gmax of the 2nd stage of a two stage LNA at the
maximum operation frequency, as a function of the
neutralization Cn for a 9 dB reduction in NF. Gmax peaks
when Kf is close to 1 and therefore it is clear why in order to
achieve maximum operation-frequency a choice of potentially
an unstable amplifier may be needed.
In other words, in order to maximize the stability of the
LNA a compromise in maximum operation-frequency should
be made. The maximally stable LNA operation frequency was
found by choosing the neutralization and number of fingers
that maximized the sum of the NF reduction q and the stability
factor µ for each frequency.
Fig. 8 shows the maximum operating-frequency for each
number of stages and each q up to 20 dB NF reduction for a
maximally stable LNA. It can be observed that the maximum
operation frequency for a ~10 dB reduction in NF is~130GHz.
Therefore, to maximize the LNA stability, a significant
compromise in maximum operating frequency of ~40% should
be made.
V. N-STAGE LNA CONSIDERATIONS
A. N stages LNA
In Section III we described a tool that enables to maximize
the operation-frequency for a given q, but in order to broaden
the understanding of the LNA NF reduction potential a direct
Matlab optimization was performed upon the data base in
Section II and the conclusions in IV. For each frequency and
each stage, the neutralization capacitor and number of fingers
that maximize the total NF reduction, q, are found.
Fig. 6. Maximum operation frequency for up to 20 dB NF reduction in a 20
dB NF mixer- for LNAs of 1 to 6 different stages.
Fig. 7. Kf and Gmax of a two stages LNA versus neutralization for q=9 dB.
Fig. 5. fmax and NFmin for different transistor finger widths and biasing.
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Fig. 8. Maximum operation frequency for up to 20 dB NF reduction in a 20
dB NF mixer for 1 to 6 maximally stable LNA stages.
C. N stages LNA – Including Interstage Matching Loss
The main source of gain loss in an LNA is the parasitic
resistance of its matching networks. Fig. 9 (top) shows a
scheme of an LNA stage and its surrounding matching
networks. The ohmic resistance of the inter-stage matching
transformers is represented using R1, R2, R3 and R4. Instead of
referring to Ri, i=1…4 as a part of the matching network, they
may be added to the LNA input and output series resistances
as shown in Fig. 9 (bottom). Adding R2 and R3 to the LNA
changes its stability factor Kf ; it changes Z11 and Z22 of the
original LNA [Z] matrix, while Z12 and Z21 remain the same:
Kf 



2  Re Z 22   R3  Re Z11  R2  Re Z12  Z 21
Z12  Z 21
(16)
As a result of changing Kf, Gmax changes as well, and it
may be directly expressed using the new Kf :
Gmax 
Z 21 

  K f  K 2f  1 
Z12 

(17)
With the use of [9] approximate values of the transformers
and their ohmic resistance were found (assuming a typical
Q=17, k=0.75 values) for each choice of LNA stage from the
data base in II. Then the new Gmax was calculated using (15)
and (16), and assuming NFmin remained unchanged, the new
maximum NF reduction, q, for each frequency was found.
Fig. 10 shows the maximum operating-frequency for each
number of stages and each q up to a 20 dB NF reduction,
including lossy matching networks. It can be observed that for
6 LNA stages the maximum frequency of operation is ~170
GHz for a 10 dB reduction in mixer NF. This is a 15%
decrease in maximum operation frequency due to matching
network loss.
To verify this result, the final chosen stage for 170 GHz and
an estimated 11 dB NF reduction was laid out. The
connections to the top metals of a 8x900nm transistor, with a
3fF MOM neutralization capacitor were simulated with
Keysight Momentum electromagnetic simulator and it was
found that the influence of the connections parasitics was only
0.1-0.2dB on Gmax and NFmin, therefore of little impact on the
analysis shown.
Fig.9. Top – a scheme of an LNA stage surrounded by its lossy matching
transformers; Bottom – an equivalent scheme of an LNA stage surrounded by
lossless matching transformers and added input and output series resistances.
Fig. 10. Maximum operation frequency for up to 20 dB NF reduction in a 20
dB NF mixer- for LNAs of 1 to 6 different stages including lossy matching
networks
LNA, but can be used for any receiver. It was found that with
a mixer NF of 20 dB, the NF reduction is possible with regard
to three different cases; lossless, maximally stable, and lossy
LNA. The maximum operation frequency for 1-10 dB NF
reduction was ~200 GHz, 120 GHz and 170 GHz respectively
(where fmax=234 GHz).
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
VI. CONCLUSION
We have presented an analytical and practical LNA NF
reduction pre-design evaluation tool at near fmax frequencies.
The final results were demonstrated using a CMOS 65 nm
K.Schmalz at al., "Subharmonic 245 GHz SiGe receiver with
antenna," Microwave Integrated Circuits Conference (EuMIC), 2013
European , pp.121,124, 6-8 Oct. 2013.
E. Ojefors, B. Heinemann, U.R. Pfeiffer, "Subharmonic 220- and 320GHz SiGe HBT Receiver Front-Ends," Microwave Theory and
Techniques, IEEE Transactions on , vol.60, no.5, pp.1397,1404, May
2012.
C. Bredendiek, N. Pohl, T. Jaeschke, K. Aufinger, A. Bilgic, "A 240
GHz single-chip radar transceiver in a SiGe bipolar technology with onchip antennas and ultra-wide tuning range," Radio Frequency Integrated
Circuits Symposium (RFIC), 2013 IEEE , pp.309,312, 2-4 June 2013.
Y. Mao, K. Schmalz, J. Borngraber, J.C. Scheytt, "245-GHz LNA,
Mixer, and Subharmonic Receiver in SiGe Technology," Microwave
Theory and Techniques, IEEE Transactions on , vol.60, no.12,
pp.3823,3833, Dec. 2012.
Z. Deng, A.M. Niknejad, "A layout-based optimal neutralization
technique for mm-wave differential amplifiers," Radio Frequency
Integrated Circuits Symposium (RFIC), 2010 IEEE , pp.355,358, 23-25
May 2010.
Z. Wang; P.Y Chiang, P. Nazari, C.C Wang, Z. Chen, P. Heydari, "A
CMOS 210-GHz Fundamental Transceiver With OOK Modulation,"
Solid-State Circuits, IEEE Journal of , vol.49, no.3, pp.564,580, March
2014.
A.M Niknejad, "Electromagnetics for High-Speed Analog and Digital
Communication Circuits", Cambridge University Press, 2007, pp. 24.
C.H. Doan, S. Emami, A.M. Niknejad, R.W. Brodersen, "Millimeterwave CMOS design," Solid-State Circuits, IEEE Journal of , vol.40,
no.1, pp.144,155, Jan. 2005.
E. Bloch, E. Socher, "Beyond the Smith Chart: A Universal Graphical
Tool for Impedance Matching Using Transformers," Microwave
Magazine, IEEE , vol.15, no.7, pp.100,109, Nov.-Dec. 2014