Parent Category: 2018 HFE

*By Boris Aleiner*

**Introduction**

Memory effects are caused by time variations in the amplifier’s transfer characteristics. Since the job of the RF power amplifier is to deliver stable and predictable power to the antenna port, any mention of characteristic variations may be of concern.

However, the situation is not as dire as it sounds. Memory effects do not affect the normal operation of a linear amplifier. They are realized only as a time-dependent variation in the amplitude of distortion. So, memory effects could be safely ignored even in nonlinear amplifiers whose distortions are below specification. It is important to take them into consideration when distortions exceed specifications and the amplifier needs linearization.

This paper discusses memory effects, reasons for their appearance, and methods to deal with them.

**1. Nature of Memory**

Memory is defined as a time difference between excitation (input) and response (output). It can happen only when the system to which excitation is applied have the means to slow it down. Those means are components capable of storing electromagnetic energy, e.g., inductors and capacitors. Since every amplifier has inductors and capacitors, at least parasitic ones, we should conclude that there are no memoryless amplifiers. However, the only way memory influences an amplifier’s operation is through its distortion.

A system is called “distortion-free” when its output is an exact replica of its input, except for a change of the output’s amplitude and a constant time delay [1]. In other words, a distortion-free system is repeating its input at the output without any modifications, and the only thing the memory is doing is introducing a constant phase shift between its input and output [2]. Clearly, the first definition is for Time Domain, and the second is for Frequency Domain. Distortions (modifications to the input signal) are introduced only when transistors start to operate outside the linear area of their I-V curve. In a nonlinear mode of operation, the phase shift between input and output, instead of being constant, is frequency dependent, and the time delay is also not a constant. Time dependence in the Time Domain translates to a phase shift’s frequency dependence in the Frequency Domain.

System memory is translated to a variation of phase with frequency. The specifics of how it is realized are given in the next section.

**2. Realization of Memory Effects**

Memory is affecting only the nonlinear mode of an amplifier’s operation. An amplifier’s nonlinearity is realized as a gain variation (compression or expansion) and as generation of additional (so called “intermodulation”) products. It is intuitively clear why additional products are generated (total power of the signal in the time domain is represented by spectral components in the frequency domain; their combination has to be constant and equal to the value of total power; so, if one of those components is reduced due to compression – the others should emerge) and it is confirmed by Parseval’s Theorem.

The way memory influences generated intermodulation products (IM) is by having a frequency dependent phase. The phase is important though, only when two or more products are to be added (which happens when they are landing at the same frequency) and the resulting amplitude would depend on their phases. What is happening in a nonlinear transistor is that several IM products indeed land at the same frequency. The reason for this is in the nature of a transistor’s nonlinearity.

Customary evaluations of an amplifier’s linearity are done with assumptions that the input signal is perfectly linear, a transistor is represented by a bulk nonlinearity, and all unwanted products which could be filtered out (that is, which are not too close to the carrier frequency) indeed are filtered out. In this case no additional IM products are appearing at the frequencies of initial ones, the transistor is thought to be infinitely thin (in other words – memoryless) and, as a result, it shows no amplitude variation between the same orders of IM products.

However, this is not a realistic model. A transistor itself must have memory because it is not infinitely thin, so it should slow input (due to parasitic capacitance), and memory should be realized when transistor is operating in a nonlinear mode. Its operation can be thought as a combination of two back-to-back diodes sharing a common base region (Gimmel–Poon model, for example, is based on this principle); a diode is essentially a nonlinear device; so instead of a single bulk nonlinearity, it should have two nonlinearities, both input and output one.

Input nonlinearity should be considered as memoryless, since a transistor’s base is very thin compared to the rest of its body. However, since there is no way to filter out unwanted IM input products, all of them are presumed to be present. Second order products mix with themselves and with one of the fundamentals to create a triple bit, which lands at the frequency of 3rd order product from output nonlinearity (see Appendix).

Output nonlinearity, however, should have memory, since the signal needs time to travel through the transistor. That memory, realized as phase dependency on frequency, affects all products landing at output IM. Thus, the resulting amplitude would depend on the relationship between phases of all products at that frequency.

Memory is also the reason for an amplitude variation of 3rd order products (IMD3) generated by two-tone input. Frequencies of IMD3 depend on delta frequency between initial tones: as shown in Appendix, IMD3 frequency is 2α-β; delta frequency between fundamentals is α-β; the difference in frequencies between IMD3 and fundamental is (2α-β) – (α) = α-β, that is the same as delta frequency between fundamentals. Since memory is realized as phase-frequency dependency, the phase of IMD3 would change with the change of that delta. Phase change would translate to the change of the relationship between phases of all products at that frequency, which means that resulting IMD3 amplitude would depend on delta frequency between tones.

So, memory affects the operation of a transistor through its dual nonlinearity. It is realized when IM products whose phases are frequency dependent are being combined. The result is an amplitude variation of IM products with frequency.

**3. Methods of compensation**

Memory effects are complicating the task of power amplifier linearization. The goal of linearization is to compensate for IM products popping up due to gain compression (as discussed in a previous section). Since memory effects make amplitudes of IM products to be frequency dependent, and the pattern of this dependency is difficult to predict, it is challenging to come up with a universal method for their compensation.

One of the popular linearization methods is based on feedback. With this technique the output is sampled, fed back, and then the input signal is modified to correct for output nonlinearities. To accommodate for memory, the sampling of output is done at several increments of a time frame, and corrections are to be introduced for each of sampled instances [3].

However, this method has its limitations. It is as good as the number of instances at which the sampling is done, and it is computation intensive. A better way to deal with memory effects is to get rid of what causes them – that is, to eliminate the contributions of other IM products falling on IMD3 frequencies. As mentioned in the previous section, those products are envelope and second harmonic. The amplitude of envelope products though, is twice as large as of second harmonic (see Appendix). So shunting envelope products should greatly reduce memory effects, as confirmed in the literature [4]. This result is memory dependent (that is, it should depend on transistor size), however having such significant difference in amplitude makes it highly probable that second harmonic products have less influence than envelope products for all instances causing memory effects.

**4. Formulas**

By now we have formed an idea of what memory is all about, how it is realized, and how to be rid of its effects. However, for accurate predictions of memory behavior and for specific means of its compensation – one needs to have a good mathematical description of its effects. This description usually comes in a form of transistor mathematical models which are based on Volterra Series.

The model of a transistor is an expression used for prediction of an output for a given input. Generally, it is a complicated nonlinear function; however, it could be simplified using a function fitting approach, that is – a representation of a function by series.

Any nonlinear mathematical function can be represented by Taylor series with the accuracy which is increased with increased order of the polynomial, the method used by mathematicians for centuries. The representation of transistor’s nonlinearity is customary done by this form of polynomials:

y=ax+bx^{2}+cx^{3}+… (1)

Where x is an input and y – output.

If we find coefficients a, b, c … we would be able to predict an output based on input. They are usually found from curve fitting extraction models.

The problem with this representation is that it does not have means to include memory. To capture memory effects, we need to use (1) at several time instances and then to combine the result (as described in the previous section). This is how Volterra series works.

Memory in a linear system is accounted for by the following convolution integral (a result of a unique system characterization by impulse response) [1]

(2)

Where: x is an input, y an output, and h an impulse response from input to output. Consequently, t is excitation time, τ is a response time, and t-τ is a system-memory time.

Volterra extended a linear system presentation (2) to nonlinear one. His approach was similar to a Taylor series (1) however instead of having a power series of the same input (x, x2, x3…) he used a series of integral operators at different time instances:

(3)

That is, instead of having the same input x at any time, Volterra series presents its values for specific times and integrates them. If there is no memory in the system, there would be no time dependency of x(τ_{i}), all of them would be the same, and a Volterra series would become a Taylor series.

As with Taylor series, the goal of Volterra series is to find coefficients, called Kernels. It is much more difficult than finding coefficients in the memoryless case and the reason for that is that they are mutually correlated. Additional problems are coming from the fact that not all the functions are converting to Volterra series. Solutions and workarounds do exist, however they are not discussed here since the goal of this paper is to explain how memory works in principle. Specifics of Volterra applications to RF power amplifiers are found in the literature (for example, [2], [4] or [5]).

**Conclusion**

The concept of memory effects in Power Amplifiers was presented and explained. It was shown, that they are realized as a time-dependant variation in distortions’ amplitude of nonlinear amplifiers. In other words, the only time it is important to take them into consideration is when the distortions exceed specifications and the amplifier needs linearization. In the frequency domain, memory effects are realized in a frequency dependent phase of intermodulation products.

It was shown that a transistor is a device with dual (input and output) nonlinearity. It cause some of input IM products to materialize at frequencies of output ones where they all add with their frequency-specific phases. This makes resulting IM amplitudes to be frequency dependent, which is the essence of memory effects.

The realization of memory effects is in dependence of IM products’ amplitudes on frequency. The best remedy for memory effects is in their compensation. It is shown that the largest input IM is the one at envelope frequency, so compensating it should be a priority in reducing memory effects.

**Appendix**

Any time-independent function can be represented by Taylor series. Usually transistor nonlinearity is presented in this form:

y=ax+bx^{2}+cx^{3}+… (A1)

Where: x – input, y – output, and a, b, c …– fitting coefficients.

Let’s consider a case when two CW signals with the same amplitude and different frequencies are applied to the input. Let their frequencies to be at α and β.

x = A{Cos(α)+Cos(β)} (A2)

This combination is called the two-tone test and used to create a simplest example of an amplitude-modulated signal. In a compressed amplifier the variation of input amplitude should create output harmonics, and the two-tone test clearly demonstrates it.

Substitution (A2) into (A1) shows Intermodulation Products (IM) content at the output. It is a result of straightforward manipulations with trigonometric identities, which was initially done by K.A. Simons [6]

In this paper IM products are grouped by order (see table above):

Let an input signal to consist not of two but of three CW tones with different amplitudes and frequencies, say, f1, f_{2} and f_{3} (the case considered in Simons’ paper). Then, in addition to 3rd order products at Fundamental, IMD3, and Third Harmonic – there will be one more product called “triple bit”. Its significance is that at certain conditions it lands at IMD3 frequencies, which are ±2α±β or ±2β±α. It happens if the one of the inputs, say f_{1}, is at envelope frequency (±α±β), the second, f_{2}, is at second harmonic frequency (2α or 2β), and the third, f_{3}, is at one of the fundamental frequencies (α or β).

This is a very realistic scenario, since a transistor has 2 nonlinearities (resulting from a transistor being in essence a combination of two back-to-back diodes), an input and output one. In the case of two-tone excitation, input nonlinearity produces IM products applied as inputs to the output nonlinearity. And among those products there would be envelope as well as second harmonic ones, which, together with the fundamental, would create a triple bit component.

As it follows from the table, the amplitude of IM products at the second harmonic is only half of the ones at envelope frequencies. So, in the ideal case of a memoryless transistor, the influence of envelope products on final IMD3 should be larger, than that of second harmonic ones.

**About the Author**

Boris Aleiner is an RF Engineer with many years’ experience at leading telecom companies. He has a number of patents, has published numerous papers, and now serves as a consultant. He can be reached at baleiner@gmail.com.

**Literature**

1. S. Hykin *“Communication Systems”,* Second Edition, John Wiley & Sons, 1983

2. Joel Vuolevi, Timo Rahkonen *“Distortion in RF Power Amplifiers”,* Artech House, 2003

3. F. Roger *“An Analog Approach to Power Amplifier Predistortion”* Microwave Journal, April 2011

4. Jeonghyeon Cha, Ildu Kim el al *“Memory Effect Minimization and Wide Instantaneous Bandwidth Operation of a Base Station Power Amplifier” *Microwave Journal, January 2007

5. J.Brinkhoff *“Bandwidth-Dependent Intermodulation Distortion in FET Amplifiers” *Macquarie University, 2004

6. K.A. Simons, *“The Decibel Relationships Between Amplifier Distortion Products,”* Proceedings of the IEEE, Vol. 58, No. 7, July 1970