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 = k₁ V_in + k₃ V_in³                                                         (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 = k₁ V_in + k₃ V_in³ + k₅ V_in⁵ + k₇ V_in⁷                                  (2)

 

To obtain the non-linear gain, the above equation is divided by the input voltage, which yields

 

                                 Gain = k₁ + k₃ V_in² + k₅ V_in⁴ + k₇ V_in⁶                                   (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 =  k₁ -   3/4 k₃ (V_Large)² +  5/8 k₅ (V_Large)⁴                       (4)

 

Where V_Large is peak amplitude of the large signal blocker. Similarly the small signal gain (SSG) is:

 

                        SSG =  k₁ -  3/2 k₃ (V_Large) ² +  15/8 k₅ (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 k₃ term causes the small signal to be compressed faster than the large signal. The k₅ 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 k₃ 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 k₁ 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 = a₀ - a₂(Blocker)² + a₄(Blocker)⁴                                  (6)

 

where a₀, a₂, and a₄ take the place of k₁, 3/2 (k₃), and 15/8 (k₅), 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 = n₀ - n₁(SSG) + n₂(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.