Linearization of Upcoming High-Efficient RF Power Amplifiers, Part 5: Modeling and Simulating PA for Digital Predistortion Applications

Good. So I'm going to talk about using power amplifier models for the development of digital pre-distortion algorithms, and tie together some of the things that we have seen so far. And I will talk a little bit about the generalized memory polynomials and binary polynomial models, and then go and show how use the models such as the high-frequency bilateral model from AMCAD to go beyond some of the limitations of memory polynomial models.

One thing that I want to make clear before we start is that we need power amplifier models, not for the development of power amplifiers, but for the development of everything around the power amplifiers, or the system which the power amplifiers end up being part of, and in particular, digital predistortion in this case. But to-- and also it's useful, extremely useful, to have models of power amplifiers during the development of the algorithms while you do lab testing to create scenarios, exploration of what you want to test, and to verify and debug possible issues.

But of course, the power amplifier models needs to be accurate or accurate enough. They need to be flexible, and they need to be fast to simulate, or faster than transistor-level simulation. Because most algorithmic models for linearization, such as digital predistortion, are developed in MATLAB, it's convenient to have a power amplifier model available in MATLAB as well. So one of the most simple things that we provide is general-- is memory polynomial models based on the-- I would say very famous paper from Dennis Morgan, where essentially, you fit the characteristics of the power amplifiers using a polynomial expression, where you have a certain memory depth and a degree of non-linearity.

In this case, y represents the output of the power amplifier, x represents the input. So you see that the output depends on the input, current values, and past values through a polynomial expression. And in this case, to extract the model, you need complex measurements, I and Q data, on the power amplifier.

So you need the input waveform, complex waveform, the output complex waveform, and essentially use a backslash operator to extract the model. And the model, nothing else consists of than a matrix of complex numbers that has as many rows as the memory length, and as many columns as the order or the degree of non-linearity.

So in MATLAB, you find a very simple white box, open procedure, or function that allows you to extract this model. It's fairly straightforward. And actually, in MATLAB, you also find something that is-- goes beyond that, and again, something that is captured in the paper of Morgan, where you can also model leading and lagging terms, meaning that the current output of the power amplifier depends on-- not only on certain values on the past, but with a certain skew in time before and after.

And this actually, this is what we call a generalized memory polynomial. And this implementation provides to actually capture a little bit better the asymmetry in the spectrum, or the memory effects of the power amplifier. And yeah, again, you can do all of this with MATLAB. One thing to keep in mind is that you-- often, there is maybe sometimes a source or a little bit of confusion. You use memory polynomial, or generalized memory polynomial, to fit power amplifiers, to create models of power amplifiers.

But you can also use the same approach for developing digital pre-distortion algorithms, where essentially, the digital pre-distortion algorithm is nothing else than the inverse characteristic implements the inverse characteristics of the power amplifier. In other words, I can express the digital pre-distortion again as a generalized memory polynomial. The only difference that I want to point out is that when you extract a generalized memory polynomial for the power amplifier, in general, you fit-- you have a way from the input waveform and the output, you extract the coefficients, and the coefficients are static.

They do not change because the power amplifier is essentially set in stone. On the contrary, the coefficients of the digital pre-distortion algorithm are not static. They tend to be adaptive based on the waveform that is fed to the power amplifier. And through the dynamic comparison of the input and the output of the power amplifier. And indeed, we-- in MATLAB and Simulink, you find an indirect learning architecture for the digital pre-distortion that is based on least square or recursive least square. What does it mean, that the coefficient changes and adapt over time.

Good. This also allows me to talk a little bit about AI and make a shout out, because this is a trend that is really becoming more popular nowadays. So you can use neural networks for digital predistortion. They do work, actually. And by the way, you find several examples in MATLAB starting from the augmented real value time-delayed neural network that is the classic, how can I say, architecture for neural network for DPD.

And by the way, if you look at how this neural network look, you will see that it's actually extremely similar in terms of architecture and structure compared to the generalized memory polynomial. It's just that the coefficients are found using a different, let's say, fitting procedure. But you also find a different type of neural network, like long, short term, bidirectional, gated, recurrent unit as well.

All these examples are available in MATLAB if you want to try out. And they do work. But what's the but? They are expensive, extremely expensive to implement. So for this reason, a lot of the AI methods are still, how can I say in the research phase, and not yet how can I say deployed in the field or as product. And to, how can I say, overcome some of the limitations, actually you also find examples on how to compress this neural network, or to reduce the computational complexity by using either projection or principal component analysis.

Very interesting topic and definitely something for research and that needs a exploration and it will become more actual in the future. In any case, let's talk a little bit about the specific use case of how we got-- how do we identify the PA model starting from the PA data that was kindly provided by Florian. We used a white box fitting procedure in MATLAB to extract a GNP model. Essentially, this matrix of complex coefficients. And we used these models essentially within a complex simulation framework to develop digital predistortion.

Before we get started, a small word about the shout out about the waveform that we used. It's a test model 1.1. However, this waveform has a very high peak-to-average power ratio, almost 11 dB, which is very unfriendly to the power amplifier itself. So what we decided was to experiment with different crest factor reduction techniques. And we came up to a waveform that has-- this is the waveform that ultimately was used, a peak-to-average power ratio, let's say 7, around 7.7 dB, and inherent EVM of 3.4% compared to the standard 3 GBP.

We applied essentially a clipping factor and the filtering to reduce the ACLR. A second-- so we generated this waveform. I passed it to Florian, Florian measured the PA, and then, he sent me back the data. And the first thing that I always do when I get data measured from a power amplifier is that I look at the data because I want to understand what I'm trying to fit. And this is very important.

I look at the data in the time domain, in the complex domain, in the power transfer curve, in the frequency domain. Why? Because I want to verify that the timing alignment is there in time, the time and the frequency, or the phase alignment, is there, and that the bandwidth is sufficient to extract a model. Now, in this case, I was blessed because I had, how can I say, data coming from Rohde & Schwarz, which means the best instruments in the world with the best operators in the world.

So the data looked perfect. I didn't have to do anything. But sometimes, you don't have such clean data. Sometimes, for example, your PA is embedded in a entire transceiver, and then you are affected by the frequency offset, for example, of the local oscillator on your transceiver. And you need to verify that the alignment is there in the time domain. And you may need to make sure that you capture the entire bandwidth, not only of the signal, but also of the sidebands on which you want to apply digital predistortion.

So I got this data. And please notice a few numbers. EVM is approximately 10%. ACLR is really not very good. I also got another set of data from Florian that is the one where digital, or direct DPD was applied. And this data, of course, is much better in terms of ACLR, much lower in terms of EVM. This is the base EVM associated with input waveform, where we apply the crest factor reduction. And I like this data a lot, because essentially, it was exerting the power amplifier on a larger dynamic range.

So you probably can't see it from there. But essentially, the original data exerted at a maximum voltage of the input of approximately 1.2-- 1.8 volts. And the direct DPD data exerted the power amplifier with an input voltage of 2.2, 2.3 volts. Larger dynamic range is good because then, it gives more room through the digital predistortion to linearize the device essentially. The model is only valid within the characterization range that you provide. So the larger characterization range, the better for the model. Good.

OK, so I chose the pre-distorted waveform, and then what? And then I need to extract the memory polynomial model with or without cross-terms. And what is the memory depth? And what is the order of, or the degree of non-linearity? So a good place for us to start is with a brute-force approach. So you just sweep the different possible values for memory, depth, and degree of non-linearity. And you try to find out if there is the sweet spot.

So in this case, these are the results of the grid search. Dark blue means smaller error, so better fitting of the model. So we see here that essentially, there's not much difference between cross terms and memory polynomial, but cross terms provide a slightly better model. And with a model with degree order of 7, for the non-linearity and memory order of 5, for the memory depth, provide good results. It's a bit conservative, by the way. And this is where you as a modeler can take action and try-- can try to find out a good trade off between, what is the complexity that I can accept for the models versus what are the quality of my results.

Good. Then what's next? Well, you test if-- what is the quality of the model? So you see here with the memory length of 5, and the degree length of 7, without cross terms. Actually, the model reproduces the EVM quite accurately. It doesn't quite capture well the ACLR. We'll comment a little bit about it. But it's also keep in mind, this is pre-distorted waveform. So the ACLR is extremely low.

And of course, good fitting in the time domain. The time domain, everything always looks very good. Good fitting also in the terms of power transfer curve. The more experiment, adding cross-term memory polynomial. Well, the fitting becomes a little bit better. And this case, I used this specific cross-term memory polynomial because I wanted to get the quality of the model as high as possible. So I wasn't particularly concerned about the cost of the model.

And then what? And then I used this model, and I tested it, or exerted it, in a simulation framework. In this case, it's very simple. It's just, you see here, this is the PA model standalone. But I excited the PA model with the non pre-distorted waveform. So remember, the model was fitted on the pre-distorted waveform because we had a larger dynamic range. The predistorted in terms of direct DPD coming from Rohde & Schwarz, this non-algorithmic DPD algorithm that provides the best performance for the PA.

But then, we compare the results of the output of the PA model obtained when the non pre-distorted waveform is passed through the device. And as you can see here, the model is actually doing a very good job. We reproduce the EVM to a very high accuracy, so at 10% EVM. This is, again, doesn't have any predistortion. And very good prediction of the ACLR, both in the near band, and the farther band apart. So this is important because essentially, we're using a different waveform to test the PA compared to the waveform that we use to characterize the model itself.

Now that we have a model, we can close it in a feedback loop with predistortion. And here, the fun begins because you can experiment with different type of DPD algorithms. So in this case, for example, you see in red, this is the output of the PA without DPD. This is in yellow, the output of the PA with an indirect learning architecture, like you were saying before. So this is adaptive. You see the feedback loops where we operate to linearize the PA. And in purple is the ultimate good results that you can obtain with direct DPD. That's the non-algorithmic approach implemented by Rohde & Schwarz.

And as you can see here, we can actually linearize the device and we get a much-improved EVM as well as ACLR. Now, we can do the same experiment with, for example, using a DPD that also implements cross terms, or with a higher memory order. And I like to compare rapidly the three slides. And you see that as the complexity of the DPD algorithm increases, the quality of the results increases. This is the type of tradeoff that you as a DPD designer have to take and to make decisions. Is it worth the complexity? Yes or no?

And here is a brief summary of the results, essentially without digital predistortion and with predistortion. And by the way, you will notice how the results are very, very close to what is shown by Florian as well. So what's next? This is all good. But so we're seeing actually that GNP model actually provides a relatively good local fidelity, let's say. But what if? What if your amplifier is actually part of a system where the load impedance matters?

So for this reason, we actually implemented the AMCAD model, the bilateral high-frequency model. And you see here a snapshot directly from the website of the equations that were implemented for simulation in MATLAB and Simulink. And this type of model allows us to go beyond the generalized memory polynomial. Why? Well, first of all, because with this model, we can model load pool effects. So you see here, load pools, circles, derived from different input power. And you see the distortion of the load pool.

And if you-- of the contours. And by the way, if you would use a generalized memory polynomial, the generalized memory polynomial is extracted for a given set of load conditions, which means that no matter the input power, the load contours, the load pool contours would always look the same. So this allows us to take into account the effects of the reflection coming back from the load onto the power amplifier. And of course, we validated the AMCAD model with the modulated waveform.

And this is interesting because as we some described, this model has been extracted using CW tones. But now, we are using a modulated waveform through the model, and verifying that indeed can predict the EVM, and the ACLR. And it does a fairly good job, especially in the bands. But what is more interesting is that we can use the same model at different operating center frequencies. So what we seen before was extracted at 3.6 gigahertz. Now, we are changing the center frequency to 3.45 gigahertz.

And you see that the measurements actually are the measurement says that the EVM is higher, essentially is 13.8%. And if you would use here a GNP model, the GNP model doesn't know anything about center frequency because it just operates on the envelope. So it would always give you an EVM of 10%. So in this case, you see how the AMCAD model actually is much more robust in different operating conditions, broader bandwidth, different center frequencies, as well as different loading conditions.

Then of course, we did the same experiments, closing the AMCAD model with the digital pre-distortion algorithm. And again, we can verify that our DPD with cross term is effective in linearizing the power amplifier.

And last but not least, we did something that is one step further. We used the AMCAD model of the curver device into essentially a beamforming configuration, where we put together a transmitter with eight parallel channels, with phase shifters as well, loaded on an antenna array with eight elements. So here, we're really taking into account the impedance mismatches, the coupling in between the antenna elements, the effects of reflection of the antenna back on the power amplifier, and as well while doing beam steering.

And of course, we validated the results without DPD and with DPD. And again, we could confirm that this particular DPD implementation, 7 by 7 with cross terms, could linearize our transmitter very effectively. In summary, this is a summary of the results, we have seen how to model power amplifiers using real-life characteristics, real-life-- real-life measurements to predict nonlinearity memory load pool effects.

We've seen different models. We've seen the memory polynomial, generalized memory polynomial, the high-frequency bilateral model from AMCAD, and we have seen how each of these models have pro, and cons, and how we are in control of finding the suitable trade off between accuracy and complexity for the model.

Then we did this use this model for digital predistortion and look at the whole closed loop for simulation, taking into account timing imperfection, RF imperfection, quantization error, and especially the antenna arrays. With this, I would like to thank you very much for your time and--

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