RF Cafe Software

RF Cascade Workbook

About RF Cafe

Copyright

1996 -
2016

Webmaster:

Kirt Blattenberger,

BSEE
- KB3UON

RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling 2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas and reference material while performing my work as an RF system and circuit design engineer. The Internet was still largely an unknown entity at the time and not much was available in the form of WYSIWYG ...

All trademarks, copyrights, patents, and other rights of ownership to images and text used on the RF Cafe website are hereby acknowledged.

My Hobby Website:

AirplanesAndRockets.com

Try Using SEARCH

to
Find What You Need.

There are 1,000s of Pages Indexed on RF Cafe !

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

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

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

**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

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

If the transfer function model is limited to 3

LSG = k_{1} - 3/4 k_{3} (V_{Large})^{2} + 5/8 k_{5}
(V_{Large})^{4 }
(4)

Where V

SSG = k_{1} - 3/2 k_{3} (V_{Large})^{ 2} + 15/8 k_{5}
(V_{Large}^{ })^{4 }
(5)

Note that the 2

Below in Figure 2 and Figure 3 are given some example results of compressions that occur according to these equations, where k

**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

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

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

*
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

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

**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.

** Figure
8 Example of Model's Prediction of C/N (click to enlarge)**

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 what-if analyses can be done with accurate results for any blocker level hitting the LNA/mixer. For example, the effect of different front-end SAW filters with different levels of stopband attenuation for out-of-band blockers can be checked, and an accurate trade-off 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.

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

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.

Using a power series expansion, the output of a gain stage can be related to its input as:

V

Note that if the gain were perfectly linear, then k

For a two-tone input, V

V

Inserting (a2) into (a1) and using the well-known trigonometric equalities, one can expand the expression for V

V

{k

{1/2 k

{k

{1/4 k

{3/4 k

{3/4 k

1/2 k

1/8 k

The two terms that are of interest are the first-order gain terms. At frequency

V

And similarly at

V

Dividing both sides of equations (a4) and (a5) by their respective inputs would yield gain at the corresponding frequencies:

G(w

G(w

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

G(w

Similarly we have;

G(w

G(w

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

**Figure B1 RX Input Signals, and LO with Phase Noise Spectrum**

**Figure B2 Signals Downconverted to IF, with Overlap of Reciprocally-Mixed
Noise**

To obtain the value of single-tone P

V

Setting the higher order terms equal to zero

V

Setting B = 0 and dividing both sides by A yields

V

G = V

Equation (c4) is the gain with third-order non-linearity.

At the P

k

Note that at this point, “A” represents amplitude at which P

For ease of manipulation, we set

a = 4/3 { 10^{-1/20 } - 1 }
(c6)

Then,

A = √(a k

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

Next raise both sides to power 2, which yields

V

After some routine arithmetic and replacing A with its equivalent a,

V

However a can be numerically evaluated as

a = 4/3 { 10

And V

V

Now V

P

where R is the system source resistance .

Inserting (c10) into (c11) gives P

P

Since an amplifier’s k

P

In a 50 W system the R term vanishes and P

P