Polynomial Model of Blocker Effects on Small Signal
Gain and Noise Figure for LNA/Mixer Devices Used in Wireless
Receivers
Authors: William Domino, Nooshin Vakilian, &
Darioush Agahi
(all currently work for Skyworks Solutions, Inc.)
This article was previously published
in the June 2001 edition of Applied Microwave & Wireless
(which is no longer in print), and is reproduced here with the permission
of the authors. In designing today’s wireless handset receivers, it is important
to maximize both receive sensitivity and resistance to undesired
signals, also called “interferers”, or “blockers”. The starting
place for receiver design is the calculation of budgets for noise
figure and linearity, usually facilitated by a spreadsheet. While
it is straightforward to find the cascaded noise figure (NF) and
1dB compression point (P _{1dB}) using a spreadsheet calculation,
it is often not clear how to use these to predict the actual performance
of the receiver in the presence of a large blocker. To obtain a
reasonably accurate prediction may instead require an inconvenient
cosimulation of the system with circuit models embedded. However,
a simpler approach is possible, which is still performed at the
level of cascade calculations rather than simulation. For a digital cellular system such as GSM (Global System for
Mobile Communications), there are wellspecified blocking tests,
that include both inband and outofband interferers. In these tests,
the receiver front end must be able to reject the blocker while
amplifying the desired signal, without violating the maximum allowed
bit error rate. For outofband blockers, much of the rejection
comes from a receiveband filter placed in front of the lownoise
amplifier (LNA). Some architectures also place a similar filter
after the LNA and preceding the mixer, while others utilize an imagereject
mixer. In the latter case, the LNA/mixer combination is often implemented
as a single IC, and it must exhibit particularly good blocker resistance,
as there is only one receiveband filter placed ahead of it. In this article, the effect of blockers on LNA/mixer performance
is studied. The objective is to produce a model, based on empirical
measurements, of how an LNA/mixer’s smallsignal gain and noise
figure degrade in the presence of large blocking signals. Once such
a model is made, it can be used to accurately predict the amount
of receiver desensitization that results from varying blocker levels.
The LNA/Mixer device used as an example is the Conexant RF212 dualband
imagereject downconverter for GSM. Gain Compression
Typically, gain compression is modeled as only a 3^{rd}order
behavior, that is,
V_{o} = k_{1} V_{in} + k_{3} V_{in}^{3}
(1)
But gain compression is generally a saturationtype function
that is not well modeled by the 3^{rd}order component alone,
which causes a downward turn of the gain curve. Figure 1 is an illustration
of how the addition of higher order terms can produce a curve that
saturates. Of course, to truly model a saturation condition, one
would require an infinite number of terms, however a few terms
^{ }
Figure 1 Polynomial
Compression Models
are enough to produce an adequate model.
In all cases the use of the model must be stopped above the point
where it strays from true saturation.
With 1^{st}, 3^{rd}, 5^{th}, and 7^{th}
terms:
V_{o} = k_{1} V_{in} + k_{3} V_{in}^{3}
+ k_{5} V_{in}^{5} + k_{7} V_{in}^{7
}
(2)
To obtain
the nonlinear gain, the above equation is divided by the input
voltage, which yields
Gain = k_{1} + k_{3} V_{in}^{2}
+ k_{5} V_{in}^{4} + k_{7} V_{in}^{6
}
(3)
Note that the gain equation has only evenorder terms. An equation
of this type is particularly useful when V_{in} is a large
blocker, and the gain is the smallsignal voltage gain in the presence
of this blocker. Of course, with V_{in }always representing
the blocker, the coefficients will be different for the blocker
gain and the smallsignal gain, since the small signal always is
compressed faster than the large signal. (Refer to appendix A.) If the transfer function model is limited to 3^{rd}order
and 5^{th}order nonlinearity, and the coefficients are
defined as positive quantities, then one can write the following
equation for the large signal gain (LSG):
LSG = k_{1}  3/4 k_{3} (V_{Large})^{2}
+ 5/8 k_{5} (V_{Large})^{4 }
(4)
Where V_{Large} is peak amplitude of the large signal
blocker. Similarly the small signal gain (SSG) is:
SSG = k_{1}  3/2 k_{3} (V_{Large})^{
2} + 15/8 k_{5} (V_{Large}^{
})^{4 }
(5)
Note that the 2^{nd} order term is doubled for the small
signal gain, while the 4^{th} order term is tripled. The
k_{3} term causes the small signal to be compressed faster
than the large signal. The k_{5} term is the one that pulls
the gain curve back up and keeps it from turning downward. Even
though this term is tripled for the small signal gain, it is generally
not enough to keep the k_{3} term from causing the faster
compression. Below in Figure 2 and Figure 3 are given some example results
of compressions that occur according to these equations, where k_{1}
is unity.
Figure 2 Small Signal
Gain Compression with Coefficient K3 = .0025
Figure 3 Small Signal
Gain Compression with Coefficient K3 = .0050
A couple of things become noticeable. First is that when
there is only a 3^{rd}order term, the relation between
SSG and LSG compression is independent of the 3^{rd}order
term. For instance, it can be seen in the curves that a compression
of 1.0dB in LSG causes a compression of about 2.1dB in SSG, for
either case of k3, as long as k5 = 0. Secondly, the 5^{th}order
term has only a small influence on the relation between SSG and
LSG compression until the amount of compression becomes large. This
is why compression is often modeled using 3^{rd}order nonlinearity
only. In the method to be described, the coefficients
of the SSG compression are directly determined by measurement and
curve fit, rather than assuming they keep the precise relationship
with the LSG coefficients that is seen in eq. (4) and (5). The noise
figure increase that occurs is also measured and related to the
small signal gain.
System Gain and Noise Model
Figure 4 is a diagram of our model showing how the blocker affects
the gain and noise of the system.
Figure 4 Model for
Blocker Effects
Based on equation (5), the blocker amplitude affects the smallsignal
voltage gain, as:
SSG = a_{0}  a_{2}(Blocker)^{2} + a_{4}(Blocker)^{4}
(6)
where a_{0}, a_{2}, and a_{4} take the
place of k_{1}, 3/2 (k_{3}), and 15/8 (k_{5}),
respectively. The three noise contributors are the basic noise figure of the
system, the noise figure degradation due to the blocker, and the
LO phase noise mixed onto the blocker. It can be seen that the blocker
amplitude affects the last two of these. In our modeling exercise,
it is most convenient to relate the noise figure degradation to
the smallsignal gain, which is compressed by the blocker. This
relationship is best represented using 0^{th}, 1^{st},
and 2^{nd} order terms:
NoiseFactor = n_{0}  n_{1}(SSG) + n_{2}(SSG)^{2}
(7)
Actually the blocker causes the small signal
gain to be reduced while the system noise increases, causing a composite
degradation in the noise figure of the system. This is taken into
account when the above equations are fit to actual measurements.
Finally, after the blocker itself sees some
compression, the LO noise is reciprocally mixed onto it (refer to
appendix B), and the noise at the blockerdesired frequency offset
falls into the desired band. This noise is then summed in with the
other contributors.
Measurements
Figure 5 shows the test setup. Smallsignal gain and NF are
both measured on
Figure 5 Test Setup
the HP8970B noise figure meter. The blocker
generator is filtered so as not to emit noise in the desired band,
and the LO generator is filtered so as not to emit noise that would
mix with the blocker. Also, the IF output is filtered so the blocker
does not hit the NF meter with excessive power. The combiner has
isolators on either side of it. On the noise source side, the isolator
protects the noise source from the blocker. On the blocker side,
the isolator insures that the port sees 50W
both inside and outside the passband of the blocker filter.
The gain and NF with no blocker is measured,
then a blocker is applied and the measurement is repeated for different
blocker amplitudes. The measurements are stopped once the blocker
reaches the highest value expected in the GSM system, or once the
noise figure degrades by more than about 68dB.
It should also be noted that the frequencies
of the measured noise band and of the blocker need to be chosen
to avoid mixer spurs. If the blocker is located on or near a loworder
mixer spur, then the NF meter will register an incorrectly high
measurement.
Generating the Model from the Data
The gain data is converted to voltage gain, and the noise figure
data to noise factor, since the polynomial equations need to act
upon linear quantities rather than dB. A spreadsheet is used to
plot the data and produce bestfit curves. Figure 6 is the curve
for the smallsignal gain vs. blocker voltage for the Conexant RF212
LNA/Mixer in its GSM900 path. The lefthand side of the curve is
artificially constructed by folding the data over, resulting in
a fit that uses all even coefficients, to fit with equation (6).
Figure 6 Small Signal
Gain vs. Blocker Voltage
In the equation for the bestfit curve, only the even terms
are significant because the curve was forced to be even. The orders
of the significant terms are 0^{th}, 2^{nd}, and
4^{th}. The 0^{th}order term is forced to be the
same as the noblocker gain. The equation for this curve defines
the blocker dependence of the “small signal gain” box in Figure
4. Figure 7 is a plot of the noise factor vs. small signal gain
for the Conexant RF212 LNA/Mixer. The equation has terms of order
0^{th}, 1^{st} and 2^{nd}. This equation
defines (indirectly) the blocker dependence of the “noise figure
degradation” box in Figure 4.
Figure 7 Noise Factor
vs. Small Signal Gain
Making Use of the Model
Once these polynomial equations are generated, they are used
in a process where all of the noise contributions are summed, to
find the carrier/noise ratio (C/N) at the LNA/mixer output. An example
case is shown in Figure 8, where a desired signal of –102dBm and
a blocker of –16dBm are incident on the LNA/mixer input.
To find the composite C/N, the basic quantities
of gain, noise figure, and P_{1dB} are required, as are
the coefficients of the smallsignal gain polynomial (a0, a2, a4),
and the noise factor polynomial (n0, n1, n2). We find the small
signal gain and the noise figure based on the level of the blocker,
and then add the reciprocally mixed LO phase noise at the end.
We must apply the large signal compression to the blocker itself,
since in our model this compression occurs before the LO noise reciprocal
mixing is applied. A polynomial with higher orders can be applied
for the LSG just as for the SSG, but in our experience it is adequate
to apply 3^{rd}order distortion only for the LSG, as long
as the large signal does not go far beyond the P1dB point of the
system. Therefore the LSG coefficients can be derived either
in the same manner as the SSG coefficients, or they can be taken
from the measured P_{1dB} and the relationship in eq. (4)
where k5=0. Appendix C details the P_{1dB} derivation.
Figure
8 Example of Model's Prediction of C/N
(click to enlarge)
For our example, the –16dBm blocker at the input of the Conexant
RF212 LNA/Mixer causes the noise figure to degrade from 2.5dB (the typical
value with no blockers present) to 6.5dB. The small signal gain drops
in the process from 21.6dB to 16.75dB. Then the cumulative effects of
the compressed SSG and NF result in a 12.5dB C/N before the LO noise
is added, reciprocally mixed onto the blocker which has been amplified
by the LSG. The sum total C/N is 7.8dB. What makes
the approach powerful is that once the model is derived, in can be plugged
into the chain calculations for a complete receiver. Then, various whatif
analyses can be done with accurate results for any blocker level hitting
the LNA/mixer. For example, the effect of different frontend SAW filters
with different levels of stopband attenuation for outofband blockers
can be checked, and an accurate tradeoff can be made between the SAW’s
passband insertion loss and its stopband selectivity.
Furthermore, the approach can be used to accurately estimate the
sensitivity level for a receiver in the presence of a blocker. In the
typical GSM receiver, the sensitivity level is where the C/N drops to
about 6.0dB. For our example with a –16dBm blocker, the sensitivity
level estimate is –103.8dBm. The calculation can also show which contributors are the most important.
In this example, it turns out that even with the seemingly high noise
figure of 6.5dB, the LO phase noise floor of –150dBc/Hz is still the
most significant contributor to the system noise, due to the blocker’s
presence. If the LO phase noise improves by 1dB, then the sensitivity
improves by 0.7dB.
Conclusion
Presented in this article were steps in generating a polynomial
model of blocker effects on small signal gain and noise figure for LNA/Mixer
devices used in wireless receivers. Based on empirical measurements
of gain and noise figure of the device, the polynomials relating gain
compression and noise figure degradation to blocker level at the input
of device were generated. The polynomial coefficients then were
applied to the calculation of carriertonoise ratio in the presence
of a blocker. The method makes it possible for a variety of whatif
analyses to provide accurate results in predicting blockerdegraded
C/N. References:
1)
W. E. Sabin and E. O. Schoenike, “Single Sideband Systems and
Circuits”, second edition, McGraw Hill, 1995
2) Behzad Razavi, “RF Microelectronics”, Prentice Hall, 1998 Authors:
William Domino is Principal Engineer for GSM RF Systems Engineering
at Conexant Systems in Newport Beach, CA. He may be reached via email
at william.domino@conexant.com. Nooshin Vakilian is Systems Engineer for GSM RF Systems Engineering
at Conexant Systems. She may be reached via email at
nooshin.vakilian@conexant.com. Darioush Agahi is Director of GSM RF Systems Engineering at Conexant
Systems. He may be reached via email at darioush.agahi@conexant.com. Appendix A: NonLinear Analysis
for Large and Small Signals
Using a power series expansion, the output of a gain stage can be
related to its input as: V_{o}(t) = k_{1} v_{i}(t)
+ k_{2} v_{i}^{2}(t) + k_{3} v_{i}^{3}(t)
+ k_{4} v_{i}^{4}(t) + k_{5} v_{i}^{5}(t)
+ …………………….
(a1) Note that if the gain were perfectly linear, then k_{1}
would be the only nonzero coefficient, and the gain would be identical
to k_{1}. For a twotone input, V_{i}(t) is: V_{i}(t) = A Cos(w_{1}t)
+ B Cos(w_{2}t)
(a2) Inserting (a2) into (a1) and using the wellknown trigonometric
equalities, one can expand the expression for V_{o}(t) to the
following: V_{o}(t) = 1/2 k_{2}
A^{2} + 1/2 k_{2} B^{2} + { k_{1}
A + 3/4 k_{3} A^{3} + 3/2 k_{3} AB^{2}
+ 5/8 k_{5} A^{5} + 15/4 k_{5} A^{3}B^{2}
+ 15/8 k_{5} AB^{4} } Cos(w_{1}t)
+
{k_{1} B + 3/4 k_{3} B^{3} + 3/2 k_{3}
A^{2}B + 5/8 k_{5} B^{5} + 15/4 k_{5}
A^{2}B^{3} + 15/8 k_{5} A^{4}B}
Cos(w_{2}t)
+
{1/2 k_{2} A^{2} + 1/2 k_{4} A^{4}
+ 3/2 k_{4} A^{2} B^{2}} Cos(2w_{1}t)
+ { 1/2 k_{2} B^{2} + 1/2 k_{4} B^{4}
+ 3/2 k_{4} A^{2} B^{2}} Cos(2w_{2}t)
+
{k_{2} AB + 3/2 k_{4} A^{3}B + 3/2 k_{4}
AB^{3}} Cos((w_{1
}+_{ }
w_{2})
t ) + {k_{2} AB + 3/2 k_{4} A^{3}B + 3/2 k_{4}
AB^{3}} Cos((w_{1
}_{ }
w_{2})
t ) +
{1/4 k_{3}A^{3} + 5/16 k_{5}A^{5}
+ 5/4 k_{5}A^{3}B^{2}} Cos(3w_{1}t)
+ {1/4 k_{3}B^{3} + 5/16 k_{5}B^{5}
+ 5/4 k_{5}A^{2}B^{3}} Cos(3w_{2}t)
+
^{ }
{3/4 k_{3} A^{2}B + 5/4 k_{5}A^{4}B
+ 15/8 k_{5} A^{2}B^{3}} Cos((2w_{1
}±_{
}w_{2})
t ) +
{3/4 k_{3} AB^{2} + 5/4 k_{5}AB^{4}
+ 15/8 k_{5} A^{3}B^{2}} Cos((w_{1
}±_{
}2w_{2})
t ) + 1/2 k_{4}
A^{3}B Cos((3w_{1
}±_{
}w_{2})
t ) + 1/2 k_{4} AB^{3} Cos((w_{1
}±_{
}3w_{2})
t ) + 3/4 k_{4} A^{2}B^{2} Cos((2w_{1
}+_{ }2w_{2})
t ) + 1/8
k_{4}A^{4} Cos(4w_{1}t)
+ 1/8 k_{4}B^{4} Cos(4w_{2}t)
…………………….
(a3) The two terms that are of interest are the firstorder gain terms.
At frequency w_{1}
the output voltage is: V_{o}(w_{1})
= k_{1} A + 3/4 k_{3} A^{3}
+ 3/2 k_{3} AB^{2} + 5/8 k_{5} A^{5}
+ 15/4 k_{5} A^{3}B^{2} + 15/8 k_{5}
AB^{4}
(a4) And similarly at w_{2}
the output voltage is: V_{o}(w_{2})
= k_{1} B + 3/4 k_{3} B^{3}
+ 3/2 k_{3} A^{2}B + 5/8 k_{5} B^{5}
+ 15/4 k_{5} A^{2}B^{3} + 15/8 k_{5}
A^{4}B
(a5) Dividing both sides of equations (a4) and (a5) by their respective
inputs would yield gain at the corresponding frequencies: G(w_{1})
= k_{1} + 3/4 k_{3} A^{2} + 3/2 k_{3}
B^{2} + 5/8 k_{5} A^{4} + 15/4 k_{5}
A^{2}B^{2} + 15/8 k_{5} B^{4}
(a6) G(w_{2})
= k_{1} + 3/4 k_{3} B^{2} + 3/2 k_{3}
A^{2} + 5/8 k_{5} B^{4} + 15/4 k_{5}
A^{2}B^{2} + 15/8 k_{5} A^{4}
(a7) Let’s assume A represents the large signal blocker and B the small
desired signal. This means A>>B, therefore we can approximate
the above gain terms by letting B go to zero: G(w_{1})
= k_{1} + 3/4 k_{3} A^{2} +
3/2 k_{3} B^{2} + 5/8 k_{5}
A^{4} + 15/4 k_{5} A^{2}B^{2}
+ 15/8 k_{5} B^{4}
(a8) G(w_{1})
= k_{1} + 3/4 k_{3} A^{2} +
5/8 k_{5} A^{4}
(a9) Similarly we have; G(w_{2})
= k_{1} + 3/4 k_{3} B^{2}
+ 3/2 k_{3} A^{2} + 5/8 k_{5}
B^{4} + 15/4 k_{5} A^{2}B^{2}
+ 15/8 k_{5} A^{4}
(a10) G(w_{2})
= k_{1} + 3/2 k_{3} A^{2} + 15/8
k_{5} A^{4}
(a11) Equations (a9) and (a11) relate the large signal gain and small
signal gain respectively of an amplifier. The interesting point is that
both gains depend on the large signal amplitude, and further that under
large signal interference, the small signal gain suffers faster. This
is apparent by comparing the coefficients of
A^{2} and
A^{4} in the above equations. Appendix B: Reciprocal Mixing
In a receiver, a small desired signal may be accompanied by a large
blocking signal at some frequency offset. The local oscillator that
is used to mix the desired channel to IF exhibits finite phase noise
at this frequency offset. When the two signals are mixed by the LO,
each one has the LO’s phase noise spectrum modulated onto it. This process
is referred to as “reciprocal mixing”. Therefore the downconverted band
consists of two overlapping spectra, with the wanted signal suffering
from significant noise overlap due to the tail of the
blocking signal’s spectrum.
Figure B1 RX Input Signals,
and LO with Phase Noise Spectrum
Figure B2 Signals Downconverted
to IF, with Overlap of ReciprocallyMixed Noise
Appendix C: P1dB Point from
3^{rd}Order Coefficient
To obtain the value of singletone P_{1dB} gain compression
induced by only 3^{rd}order nonlinearity, equation a3 of Appendix
A is used, setting higher order terms as well as signal “B” to zero.
V_{o}(w_{1})
= k_{1} A + 3/4 k_{3} A^{3}
+ 3/2 k_{3} AB^{2} + 5/8 k_{5}
A^{5} + 15/4 k_{5} A^{3}B^{2}
+ 15/8 k_{5} AB^{4}
(c1) Setting the higher order terms equal to zero^{ }yields V_{o} = { k_{1} A
+ 3/4 k_{3} A^{3} + 3/2 k_{3}
AB^{2} }
(c2) Setting B = 0 and dividing both sides by A yields V_{o} = { k_{1} A
+ 3/4 k_{3} A^{3} }
(c3) G = V_{o}/A= k_{1}
+ 3/4 k_{3} A^{2}
(c4) Equation (c4) is the gain with thirdorder nonlinearity. At the P_{1dB} point, the overall gain is reduced by 1 dB
from the linear gain, that is, the voltage gain becomes k_{1}*(10^{1/20}).
To find the amplitude at the P_{1dB} point occurs, one needs
to solve the following: k_{1} + 3/4 k_{3}
A^{2} = 10^{1/20} k_{1}
Þ
A^{2} = 4/3 { 10^{1/20
}  1 } (k_{1}/k_{3})
(c5) Note that at this point, “A” represents amplitude at which P_{1dB}
occurs. For ease of manipulation, we set
a = 4/3 {
10^{1/20 }  1 }
(c6)
Then,
A =
√(a
k_{1}/k_{3})
(c7) Equation (c7) represents the amplitude at which gain is compressed
by 1dB. To find 1dB compression point in terms of power, we can start with
equation (c3) again: V_{o} = { k_{1} A
+ 3/4 k_{3} A^{3} } Next raise both sides to power 2, which yields V_{o}^{2 } =
{ k_{1} A + 3/4 k_{3} A^{3} }^{2}
(c8) After some routine arithmetic and replacing A with its equivalent
a, V_{o}^{2 } =
{ a + 3/2 a^{2}
+ 9/16 a^{3}}
(k_{1}^{3}/k_{3})
(c9) However a can be numerically
evaluated as a
= 4/3 { 10^{1/20 }  1 } =  0.145 And V_{o}^{2} becomes V_{o}^{2 } =
 0.11518 (k_{1}^{3}/k_{3})
(c10) Now V_{o} is a peak voltage (call it V_{p}). Converting
to dBm yields P_{dBm} = 10 Log {
(V_{p}^{2}/2R)
1000 } = 10 Log {V_{p}^{2} 50/R 10}
(c11) where R is the system source resistance
. Inserting (c10) into (c11) gives P_{1dB} in terms of the
amplifier’s coefficients: P_{1dB} = 10 Log {
0.11518 (k_{1}^{3}/k_{3}) 50/R 10
}
(c12) Since an amplifier’s k_{3} is a negative term, its sign
can be absorbed and the term can be presented in the absolute form,
which then results in: P_{1dB} = 10 Log { (k_{1}^{3}/k_{3})
50/R } + 0.614
(c13) In a 50 W system the R
term vanishes and P_{1dB} reduces to: P_{1dB} = 10 Log {k_{1}^{3}/k_{3}}
+ 0.614
(c14)
