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
co-simulation 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 well-specified blocking tests,
that include both inband and out-of-band 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 out-of-band blockers, much of the rejection
comes from a receive-band filter placed in front of the low-noise
amplifier (LNA). Some architectures also place a similar filter
after the LNA and preceding the mixer, while others utilize an image-reject
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 receive-band 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 small-signal 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 dual-band
image-reject 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 saturation-type 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 non-linear 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 even-order terms. An equation
of this type is particularly useful when V_{in} is a large
blocker, and the gain is the small-signal 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 small-signal 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 non-linearity, 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 non-linearity
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 small-signal
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 small-signal 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 blocker-desired frequency offset
falls into the desired band. This noise is then summed in with the
other contributors.
Measurements
Figure 5 shows the test setup. Small-signal 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 6-8dB.
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 low-order
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 best-fit curves. Figure 6 is the curve
for the small-signal gain vs. blocker voltage for the Conexant RF212
LNA/Mixer in its GSM900 path. The left-hand 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 best-fit 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 no-blocker 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 small-signal 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.