Parent Category: 2014 HFE

*By Haydn Nelson and Sean Ferguson*

**Introduction**

RF transmitter designers face competing design trade-offs. Demand for greater data throughput in an overcrowded spectral environment leads to higher complexity modulation formats with challenging power amplifier (PA) linearity requirements. In addition, demand for improved energy efficiency and higher power output pushes designers to operate in the nonlinear region of these devices. A popular technique to address this trade-off is the use of digital predistortion (DPD) algorithms. DPD allows designers to operate in the efficient yet nonlinear region of an amplifier while retaining the transmitted signal linearity required of most digital modulation formats. This paper examines the driving factors of these trends, measurement techniques to characterize nonlinear devices, common DPD models, DPD instrumentation requirements, and measured results with a mobile device PA.

**Complexity of Wireless Modulation and the Need for Linearization**

Early wireless standards used simple modulation formats such as Gaussian minimum shift keying (GMSK) and phase shift keying (8-PSK). These systems had low data rates, and designers benefitted from their relatively narrow signal bandwidth and small peak to average power ratios (PAPRs). By contrast, modern wireless standards are designed to achieve greater data rates by using more bandwidth and higher order digital modulation techniques; examples of these are IEEE 802.11ac and LTE Advanced. These modern wireless standards use advanced techniques such as 256-QAM, orthogonal frequency division multiplexing (OFDM), and carrier aggregation. Modern digital communication signals feature not only wider bandwidth but also significantly larger PAPRs. The increase in data rates due to wider transmission bandwidths and higher order modulation schemes comes at the expense of higher linearity requirements and lower PA power efficiency. **Table 1** shows some typical parameters for these wireless signals.

Table 1 • Common Parameters for Wireless Standard Signals.

The RF PA is one major source of nonlinearity in an RF transmitter. **Fig 1** illustrates the in-band effect of PA nonlinearity on a 64-QAM modulated signal. The constellation at the output of the PA has visible compression that results in increased bit error rates and poor error vector magnitude (EVM) performance. In addition to the in-band distortion, nonlinearity degrades out-of-band performance, as shown in Fig 1. This figure shows the spectral regrowth at the output of a mobile device PA with a 5 MHz LTE stimulus.

**Amplifier Power Efficiency**

An important figure of merit for a power amplifier is its power added efficiency (PAE). PAE is defined as the percentage of power added to a signal relative to the power delivered via the power supply. In general, efficiency near 50% is considered to be excellent performance.** Fig 2a** shows the output power versus PAE of a typical PA; **Fig 2b** shows the typical nonlinear output power versus input power behavior.

Figure 2 • Typical Effect of PAPR on the Power Added Efficiency of a PA: (a) PAE versus Output Power and (b) Output Power versus Input Power.

Although the point of peak efficiency occurs when the amplifier is operated near its peak power, the device exhibits poor linearity at this operating point. A high PAPR signal with a peak output at the point of acceptable linearity pushes the average power of the signal to a region of poor efficiency. This concept is shown in **Fig 2** with a power versus time (PVT) graph of a common LTE signal. Note that depending on the design, the point of acceptable linearity can be several decibels below the point of peak efficiency. In this scenario, the maximum output power of the PA is not fully utilized, which creates additional efficiency capacity.

**AM-AM and AM-PM Representation of Device Nonlinearity**

**Fig 3** illustrates the typical gain compression characteristics of an amplifier viewed as a function of input power. Gain distortion is the leading contributor to device nonlinearity. However, amplifiers also introduce phase distortion as a function of input power, which further degrades modulation accuracy. The industry-standard method of quantifying these effects is with AM-AM and AM-PM data from an AMPM measurement.

(a) (b)

Figure 3 • (a) AM-AM (Gain) and (b) AM-PM Data for a Cellular PA with a 20 MHz, 100 RB LTE Stimulus: Each plot shows the measured data and a 7th order polynomial curve fit.

An AMPM measurement is performed by sweeping the input power to a device and measuring its response. The distorted output signal is then aligned to the stimulus waveform so that a sample-by-sample complex gain can be calculated. There are two ways to provide a test stimulus: a power-swept continuous wave (CW) signal or a wideband modulated waveform with a high enough PAPR to cover the power range of interest. A modulated waveform is generally preferred to a power-swept CW since it captures the memory effects of the device and therefore more accurately models the PA under actual operating conditions [1].

**Fig 3** shows the AM-AM (gain) and AM-PM characteristics of a cellular PA with a 20 MHz, 100 RB LTE signal applied. An ideal amplifier would have a constant gain as a function of input power. Similarly, the ideal AM-PM response of a PA would show a constant phase shift for all power levels. In Fig 3 the polynomial curve fit trace from both sets of data illustrates the static nonlinearity explained in [1]. The next section covers using DPD to correct for nonlinearity and its effect on the AMPM response of the device.

**Digital Predistortion Techniques**

A simple system for digital predistortion is illustrated in **Fig 4**, whereby a predistorter is cascaded with the digital-to-analog converter (DAC), RF upconversion circuitry, and a nonlinear PA. The predistorter applies an inverse distortion to the baseband signal so the cascaded system behaves linearly. **Fig 4** illustrates DPD from a modulation perspective. In the figure, symbols in a 64 QAM constellation that are near the peak magnitude need additional gain to counteract the compression characteristics of the PA. Although not easily visible, the symbols are also rotated to correct for the phase distortion introduced by the PA.

Figure 4 • Conceptual Operation of DPD on 64 QAM Constellation and System State Model for DPD Where x Is the Time-Domain Stimulus, h(x) Is the Predistorter, f(x) Is the AM-AM and AM-PM-Based Model of the PA, and y(x) Is the PA’s Output.

A wide range of literature examines both PA modeling and predistortion. Most DPD techniques can generally be classified as memoryless or with memory [1]. Memoryless predistortion is a nonlinear function where the output sample is only a function of the current input sample. Memoryless algorithms have the advantage of being simple to model and compute, but their performance often degrades as signal bandwidth increases. DPD models that correct for device memory are generally nonlinear functions of not only the current input sample but also some finite number of previous samples. Memory models have been shown to improve performance over memoryless models for wideband signals. However, the increased performance comes at the cost of added computational complexity. The following sections cover these two approaches in detail.

**Memoryless AMPM Lookup Table (LUT)**

Performing a modulated AMPM measurement yields both the AM-AM (gain) and AM-PM (phase) response of the device under test (DUT). The measured gain and phase information can be combined into a single complex polynomial model f(x). The most straightforward linearization approach is to find the predistorter h(x) that, when combined with the PA, results in a combined linear response. Such a system is illustrated in **Fig 4**, where x(t) is the time-domain stimulus, h(x) is the predistorter, f(x) is the model of the PA, and y(x) is the response. The combined response of the amplifier and predistorter to the stimulus waveform x(t) is given by y(x) = f(h(x)). If we represent our stimulus waveform x in polar form as|x|ejϴand the PA has a nominal linear gain G, then the ideal output is yideal (x)=G ∙|x|ejϴ. The predistorter h(x) that results in a linear output is therefore:

(1)

If the AM-AM and AM-PM responses are modeled by Kth-order polynomials, then the combined complex response f(x) can be represented as shown in (2).

(2)

Determining an analytic expression for f–1(g∙x) is quite difficult for high orders of K. As a result, the inverse is performed numerically over the domain of input powers for which the AM-AM and AM-PM model f(x) was computed and then stored in a LUT. It is then possible to fit a Kth order complex polynomial to the predistorter LUT resulting in an expression for h(x):

(3)

A graphical representation of this operation is shown in** Fig 5** where an exaggerated AM-AM response of an amplifier is plotted in addition to the ideal device characteristics. Here, it is assumed that the AM-AM response is modeled by a Kth order polynomial. Evaluating the polynomial for an input signal of magnitude Pin, we see that the output is compressed by dP dB compared to the ideal value Pout,ideal given by y = G∙x. Fig 5 illustrates that simply increasing the input power by dP dB to Pin’ would not result in a linear output. This behavior is expected since the PA has entered compression. However, by numerically calculating the inverse of the AM-AM response, the predistorted input Pin,DPD that results in a linear output can be found.

Figure 5 • System Model Used for an AMPM-Based LUT for DPD Where x is the Time-Domain Stimulus and h(x) Is the Predistorter.

**Fig. 6** shows that memoryless predistorters are effective at linearizing the average AM-AM and AM-PM responses of a nonlinear PA. However, the clouds of samples around the best fit curves indicate a significant level of residual memory in the corrected response. These memory effects deteriorate system performance but can be corrected with a more sophisticated DPD model. The next section examines DPD models that correct for memory.

Figure 6 • LTE Device Response Corrected with 7th Order Memoryless Polynomial DPD.

**Memory Models**

The most comprehensive model for nonlinear systems with memory is a Volterra series, shown in (4) [1],[3]. However, the complexity of the model increases drastically as the nonlinearity order K and memory depth M of the model increase, which makes the Volterra series computationally prohibitive to use in most practical applications.

(4)

In practice, nonlinear systems with memory, such as RF PAs, can be adequately modeled with only a subset of the terms in a full Volterra series [3]. Some common Volterra derivatives include the Memory Polynomial, Wiener, and Hammerstein models [1], [3], [4]. These models are considerably simpler to implement and offer comparable performance to the full Volterra series. The Memory Polynomial model is one of the more popular predistortion techniques [1], [3], [4], and is formed by taking a subset of the Volterra series, as shown in Equation (5):

(5)

Rewriting (5) using vector notation for a block of N samples produces:

Conceptually, the most straightforward method to create a model for a predistorter is to first model the PA and then calculate its inverse. A method for inverting a nonlinear system with memory, such as a memory polynomial, is the pth-order inverse [11]. This model results in another nonlinear system with memory but of a higher polynomial order. The higher order characteristics can lead to instability, so the preferred way of modeling the predistorter is to use what is known as the indirect learning architecture, shown in **Fig 7** [9].

Figure 7 • Indirect Learning Architecture Proposed in [11] and [3] for Determining the Coefficients of a Volterra-Based Predistorter.

At a high level, this architecture allows engineers to model the inverse distortion through the PA that, when applied to the transmit signal x, results in a combined linear output y. If the output y is scaled by the linear gain of the amplifier G, then u= y can be thought of as the transmit signal x with the residual distortion of the amplifier and predistorter combination. Then a model for the inverse distortion can be created by finding the mapping w between the distorted PA output u and the predistorted transmit signal z. At convergence u = x and z = ž so that

(7)

Where U can be constructed as in (6c) by replacing xk,q[n] with uk,q[n], where xk,q[n]=u[n–q]·|u[n–q]|k-1. The familiar least squares solution to (7) is

(8)

In this experimental setup, the vector z is initialized to the transmit signal x. The next section applies the results obtained in (8) to an actual LTE PA.

**Measured Performance Using a Mobile Device LTE PA**

The performance improvements that result from using DPD linearization are heavily dependent on the behavior of a particular device design.** Fig 8a** shows in-band distortion represented by EVM across a range of power levels. In this figure, the 10 MHz LTE waveform used features a PAPR of approximately 8 dB. Below +21 dBm, memoryless DPD improves EVM by 5 dB. Using a five-tap memory model further improves EVM by another 5 dB for an overall improvement of 10 dB. At average output powers between +21 dBm and +23 dBm, the device has significant compression and the benefit of both DPD algorithms converges to a point where the average power out plus the PAPR exceeds the maximum output power of the device. Above the maximum output power of the device, DPD can correct for only the phase distortion.

Figure 8 • LTE 10 MHz 100 Resource Blocks (a) EVM versus Average Power and (b) ACPR versus Average Power.

**Fig. 8b** shows the out-of-band distortion represented by ACPR. Similar to the scenario where the EVM is below +21 dBm, ACPR improves by more than 10 dB using memoryless DPD. By contrast, using 5 taps of memory only has a marginal improvement of 0.5 dB, which indicates that the memory effects of the device have a greater impact on in-band distortion than out-of-band distortion. **Fig 9** shows the PAE of the device across the same range of average output powers. As expected, PAE improves proportionally with output power. The small increase in PAE at a particular power is due to the gain expansion realized when DPD is applied. For this particular PA, if a system requires an EVM of less than -43 dB, the maximum output power would be +21 dBm without DPD. Using a 5-tap memory DPD correction, the output power range can be extended by 1.5 dB, which leads to an increase in PAE from 18% to 23%.

Figure 9 • PAE versus Power Out, LTE 10 MHz 100 Resource Blocks.

**Instrumentation Requirements for Digital Predistortion**

**Fig 10** shows a typical wireless PA test setup. Here, the use of DPD in the test system produces increased instrumentation requirements in four areas: synchronization, bandwidth, dynamic range, and linearity.

Figure 10 • Typical Mobile PA Instrumentation Test Setup.

Specific bandwidth requirements are closely coupled to DPD implementation and performance objectives. In some instances, a system configured to test a PA under DPD conditions requires 7X to 10X the bandwidth of the signal itself. **Fig 11a** shows the power spectrum of a signal before predistortion, and **Fig 11b** shows the same signal with DPD. For this particular signal, using a 7th order polynomial memoryless model resulted in more than a seven fold increase in bandwidth, as measured by signal power greater than -100 dBm/Hz. In addition to wider bandwidth, DPD increases the PAPR of the signal driving the PA. As a result, test instrumentation is required to provide additional dynamic range to accommodate the predistorted signal. In Fig 11, the PAPR of the predistorted signal increases by 3.7 dB compared to the signal’s original PAPR.

Figure 11 • LTE Signal Spectrum (a) before Predistorter Stage and (b) after Predistorter Stage.

The first step in any predistortion algorithm is to correlate the stimulus and response waveform of the DUT. Using instrumentation that can repeatably acquire a device response for a given stimulus waveform simplifies the correlation algorithms and reduces characterization time. Based on these system requirements, the PXI platform is well suited to meet the challenges of DPD test. PXI instrumentation is connected through a chassis backplane with shared clocks and triggers to enable tighter synchronization than nonintegrated instrumentation. In addition, the National Instruments PXIe-5646R vector signal transceiver has the unique characteristic of being an integrated vector signal generator and vector signal analyzer. Because both instruments can share a common oscillator, instrument phase noise is effectively removed from the acquired PA output, which improves AMPM measurements. The NI PXIe-5646R has up to 200 MHz of instantaneous bandwidth, which allows for the application of DPD to a wide range of wireless signals.

In addition to meeting the demands of DPD characterization, the NI PXIe-5646R allows users to extend functionality with a feature called instrument driver FPGA extensions. This feature enables users to add DPD IP to the FPGA using the LabVIEW FPGA graphical programming language. This reduces DPD test time considerably since waveforms don’t have to be re-downloaded every time a new DPD function is applied.

**Conclusion**

The evolution of wireless technology has forced mobile device designers to consider active linearization as a solution to meet the competing demands for high data rates and higher efficiency. Digital predistortion is becoming a popular technique to correct PA nonlinearity in mobile applications. Although the concept of correcting for imperfections is simple, the complexity of PA behavior forces engineers to better understand the techniques and driving forces behind digital predistortion. Because every device design is different, no single model or algorithm is applicable to all PAs. Going forward, the application of DPD techniques is quickly becoming common in wireless devices, which requires engineers to adopt test technology that can meet these stringent requirements.

**About the Authors**

Haydn Nelson is a product marketing manager for RF and Microwave test systems at NI. With over 10 years of experience in the automated test and RF areas, Nelson has contributed to the development of signal generating and RF and communication test systems. His areas of expertise include RF measurements, digital communication systems, and wireless technologies. Nelson currently supports the technical marketing efforts and product positioning for RF and Communication systems. Haydn holds bachelor’s and master’s degrees from the University of Texas at Austin where he graduated with highest honors.

Sean Ferguson’s work at NI has most recently been focused on on envelope tracking and digital predistortion applications for wireless front-end test. He holds bachelor’s and master’s degrees in electrical engineering, both from McGill University, Montreal. His primary research interests were in the areas of telecommunications and signal processing.

**References**

[1] Fadhel M. Ghannouchi, Oualid Hammi. “Behavioral Modeling and Predistortion,” IEEE Microwave Magazine, December 2009.

[2] Lei Ding, “Digital Predistortion of Power Amplifiers for Wireless Applications,” PhD Thesis, Georgia Tech University, March 2004.

[3] Dennis R. Morgan, Zhengxiang Ma, Jaehyeong Kim, Michael G. Zierdt, John Pastalan. “A Generalized Memory Polynomial Model for Digital Predistortion of RF Power Amplifiers.”

[4] Lei Ding, G. Tong Zhou, Dennis R. Morgan, Zhengxiang Ma, J. Stevenson Kenney, Jaehyeong Kim, Charles R. Giardina. “A Robust Digital Baseband Predistorter Constructed Using Memory Polynomials.”

[5] Frederick H. Raab, Peter Asbeck, Steve Cripps, Peter B. Kenington, Zoya B. Popovic, Nick Pothecary, John F. Sevic, Nathan O. Sokal. “RF and Microwave Amplifier Technologies,” High Frequency Electronics, May 2003.

[6] Joel H.K. Vuolevi, Timo Rahkonen, Jani P.A. Manninen. “Measurement Technique for Characterizing Memory Effect in RF Power Amplifiers.”

[7] Sung Won Chung, Jack W. Holloway, Joel L. Dawson. “Energy Efficient Digital Predistortion with Lookup Table Training Using Analog Cartesian Feedback,” IEEE Transactions on Microwave Theory and Techniques, vol. 56, October 2008.

[8] W.J. Kim, S. Stapleton, J.H. Kim, C. Edelman. “Digital Predistortion Linearizes wireless power amplifiers,” IEEE Microwave Magazine, vol. 6, no. 3, pp. 54-61, 2005.

[9] Changsoo Eun, Edward J. Powers. “A New Volterra Predistorter Based on the Indirect Learning Architecture.”

[10] O. Hammi, S. Carichner, B. Vassilakis, F.M. Ghannouchi. “Novel approach for static nonlinear behavior identification in RF power amplifiers exhibiting memory effects,” IEEE MTT-S Int. Microwave Symp. Dig., June 2008, pp. 1521-1524.

[11] M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems. New York: Wiley, 1980.

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