An Efficient Measure for Nonlinear Distortion Severity due to HPA in Downlink DS-CDMA Signals

Downlink direct sequence-code division multiple access (DS-CDMA) signals typically exhibit large dynamic range. This large dynamic range results in signal distortion for components falling in the highly nonlinear regions of the high power amplifier (HPA) characteristics. This nonlinear distortion (NLD) degrades the bit error rate (BER) and creates out-of-band emissions in adjacent channels known as spectral regrowth.

Over the decades, much research has been conducted to reduce the vulnerability of the amplifier input signal to nonlinearity. Such research often seeks to define a measure for NLD, which when reduced often allows NLD to also be reduced. Several measures were adopted to quantify NLD in relation to the input signal to the HPA such as peak-to-average power ratio (PAR) [1, 2], the instantaneous power variance (IPV) [3, 4], and the cubic metric (CM) [5, 6].

Reduction of such measures does achieve remarkable performance improvements that, in turn, enhance the system performance or the HPA efficiency. While the existing measures have their use, it is not mathematically clear how they relate to the system performance in terms of BER or spectral regrowth. Moreover, no close relation between these measures and IBO level of interest exists, which is important in determining the IBO required to work at according to design demands or service regulations. Consequently, reduction of such measures does not always lead to optimum performance as will be demonstrated later.

In [7, 8], we explored which signal characteristics at a predistorter-HPA's (PD-HPA) input are responsible for performance degradation (BER degradation and spectral regrowth). Based on such characteristics, in this paper, we introduce an efficient measure for characterizing NLD, namely, nonlinearity severity (NLS) measure, as an alternative to the existing measures, which is characterized by having a direct link to the system performance.

This paper is organized as follows. In Section 2, the DS-CDMA system under investigation is described. In Section 3, the signal characteristics established in [7, 8] that contribute to the BER degradation and spectral regrowth are presented, in order to provide the reader with the theory upon which the proposed measure is based. In Section 4, we present a brief survey on the most currently known nonlinearity measures highlighting their advantages and disadvantages. In Section 5, the proposed NLS measure is developed. In Section 6, the performance of the NLS measure is assessed in comparison with PAR. Finally, Section 7 summarizes the conclusions drawn from the paper.

The system under investigation is a downlink synchronous DS-CDMA system, where users' signals have equal power. The complex envelope of the th DS-CDMA symbol for active users is defined as [7]

where is the th user's signal energy per symbol, is the symbol duration, are the th user's symbol data, , , and

is the spreading waveform obtained with the Walsh-Hadamard matrix, is the spreading factor, is the chip duration, and is the impulse response of the pulse shaping filter. Without loss of generality, the spreading waveforms are assumed to have unit energy, that is, .

A PD-HPA pair is considered as the nonlinear amplifier chain, which has a zero AM-PM characteristic and an AM-AM characteristic given by

where is the saturation (clipping) threshold. In practice, the input/output characteristics of the PD-HPA are slightly different due to misalignment between the predistorter and HPA. However, such a slight difference will not greatly affect the performance, and, hence, the assumed ideal PD-HPA sufficiently serves the analysis approach. Finally, the output from the PD-HPA can be expressed as [7]

where is the clipped portion of the input signal , , and its envelope has the form

In this section, we shed light on the signal characteristics that contribute to the BER degradation and spectral regrowth presented in [7] and [8], respectively. The purpose of this section is to make sure the reader is familiar with the theoretical background, which we will rely upon for the formulation of the proposed measure NLS.

Several measures were adopted to quantify NLD in relation to the input signal to the HPA. PAR is the most commonly used measure of the potential nonlinearity due to HPAs, which when reduced decreases the dynamic range of the input signal to the HPA. In turn, the signal traverses a smaller range of the inherent nonlinearity of the HPA transfer function. Several techniques have been proposed to reduce the PAR, such as clipping [14], companding [15], selected mapping (SLM) [16], partial transmitted sequences [17], and Walsh code reassignment techniques [1, 18]. For a signal of duration , , the PAR is defined as

Recently, other measures have been adopted such as the instantaneous power variance (IPV) [3, 4, 6] and the cubic metric (CM) [5, 6]. The motivation for the IPV measure is to reduce the envelope fluctuations [6]. The IPV is defined normalized to remove the dependence on the average power as [4]

The CM measure was proposed with the motivation of reducing the third-order modulation product as it is the cause of the major distortion [5]. The CM is defined as

where is the root mean square (rms) value of .

While the above-mentioned measures have their use and have led to remarkable improvements, none of them has a close relation with the system performance. In other words, no clear mathematical relation exists between these measures and the resulting different forms of NLD (BER degradation and spectral regrowth). Accordingly, their reduction does not always lead to optimum performance. An example to illustrate the above idea is presented, where the BER is considered as the performance merit. The reader is reminded that it was established in [7], as presented in Section 3.1, that the signal characteristics contributing to BER degradation caused by amplifier nonlinearity are the threshold exceeding rate and the variance of the clipped signal portion .

Consider an example 16-user CDMA signal of duration with Walsh codes of length = 64 and filtered with an SRRC filter. Many representations of this signal have been generated. Three of them were selected in order to emphasize that the above-mentioned measures do not always lead to optimum performance. The three representations (R1, R2, and R3) of such a signal are generated with the same average power using the SLM technique [16]. Figure 3 shows the instantaneous power of the three representations , , and . The PAR, IPV, and CM are calculated for each representation. Also and are measured for each representation at different values of IBO and tabulated in Table 1.

It is clear in Table 1 that, among the three representations, R1 has the minimal PAR, R2 has the minimal IPV, and R3 has the minimal CM. Looking to the calculated parameters in Table 1, it can be observed that a representation with a minimal of one of the considered measures might have higher values of and at certain IBO values. Hence, such a representation, according to the analysis in Section 3.1, will be more vulnerable to NLD leading to worst BER degradation. For instance, R1 has the lowest values of and only at IBO = 1 dB, while has higher values than those of R2 and R3 at other IBO values. Similarly, R2 has the lowest values of and at IBO = 3 dB and 4 dB, while has higher values than those of R1 and R3 at other IBO values. Also, R3 has the lowest values of and at IBO = 0 dB, while has higher values at other IBO values. Moreover, at SNR_AWGN of 5 dB and using the measured values of and , the BER for all representations at the considered IBO levels are computed based on the BER in (16). It is evident, from the BERs in Table 1, that a representation with a minimal value of one of the nonlinearity measures of interest may achieve the best performance at certain IBO threshold values and fail at others. Therefore, it is concluded that the considered measures are not closely related to the system performance and do not always lead to the optimum performance. Also, an important conclusion should be clear here, which is the necessity to involve the IBO threshold level in the formulation of any nonlinearity measure.

It is worth mentioning another measure that is close to PAR is the peak-to-mean envelope power ratio (PMEPR), which is defined in [19] for Rayleigh-distributed envelopes as

where is the envelope value that is exceeded with probability , resulting in .

Regarding the PAR (or PMEPR) issue particularly, it can be justified as follows. In the presence of a PD-HPA, the signal dynamic range is already determined by its threshold, after which the output signal is clipped (distorted). Accordingly, using the dynamic range in terms of peak power as a measure for NLD loses its importance, and in this case, it is better to sacrifice high peaks by letting them be clipped in favor of keeping larger portions of the signal in the linear region below the PD-HPA threshold.

Based on the above discussion, we propose an efficient measure to quantify the severity of NLD, as an alternative to the existing measures, namely, nonlinearity severity (NLS) measure.

We define the NLS measure in a manner with similarities to the threshold crossing intensity parameter of [20], but explicitly using the signal characteristics directly related to BER performance and spectral regrowth ( , , , and ).

For the sake of having a simplified measure, it is useful to use the interesting relation between and , where their product gives [13, 21]. Thus, it can be argued that the signal characteristics affecting the overall system performance, in relation to NLD, are limited to and . Thus, the NLS measure is defined as

To emphasize the relation of the NLS measure and the signal characteristics contributing to BER degradation, recall the example drawn in Section 4 with adding four other representations R4, R5, R6, and R7, which are selected such that they have a minimal NLS at IBO = 0, 1, 2, and 3 dB, respectively. The NLS measure is measured for all representations and shown along with the other parameters of interest in Table 2.

As can be seen in Table 2, for each of the IBO levels of interest, the values of and and measured for the representations with the minimum NLS are the minimal or very close to the minimal values of and and measured for the other representations. Also, the BER measured for all representations at all IBO levels is computed at SNR_AWGN of 5 dB and shown in Figure 4, where it is clear that the representations with minimal NLS adjusted to a particular IBO level lead to the best BER performance at that level.

It appears evident here is another advantage of the NLS measure, in addition of being directly linked to the BER performance and spectral regrowth, which is the correlation between the NLS measure and the IBO level. This means that we now have an effective measure for NLD commensurate with the IBO level required to work at according to design requirements or service regulations. Accordingly, the effects of NLD can be reduced by minimizing such a measure as will be shown in the next section.

Two performance merits are used in the assessment: BER and adjacent channel power ratio (ACPR) as a measure for spectral regrowth. The SLM technique is adopted as a platform for performance assessment. In SLM, several representations of the signal to be transmitted are generated, where there should be a selection criterion, upon which the best representation is selected for transmission.

Although any of the nonlinearity measures (PAR, IPV, and CM) described in Section 4 can be used as a selection criterion in SLM, PAR is the most commonly used and appears to be the state of affairs until now. This may be because of its simpler form and the numerous investigations and discussions conducted on it in the literature. Accordingly, in this paper, the performance of the NLS is assessed in comparison with PAR as selection criteria in SLM.

The concept behind the SLM technique is based on creating equivalent representations of the same signal by rotating the phases of the data symbols such that [16]. Among the representations, the representation that has the minimum PAR is selected for transmission. In our approach, the representation that has the minimum NLS is selected for transmission as follows:

In this scenario, we have considered a CDMA system with Walsh orthogonal codes of length . An SRRC pulse shaping filter is used with roll-off factor of and upsampling factor of 4 in order to obtain an adequate signal representation in a nonlinear environment. Three different representations for the CDMA signal are generated as follows: with no SLM optimization, with SLM using the minimum PAR as the selection criterion, and with SLM using the minimum NLS adjusted to the IBO level of interest as the selection criterion.

The presented BER performance results represent the average BER of 100 frames of each CDMA signal representation of interest. Each frame consists in 10,000 random QPSK data symbols. The BER curves for 16-user CDMA signals in the presence of a PD-HPA at IBO of 1 dB and 2 dB are plotted in Figure 5. It is clear from the figure that using the minimum NLS measure, compared to the minimum PAR as selection criteria to select the signal to be transmitted, improves the BER performance. This was expected because the NLS measure is based on factors linked directly to the BER performance, and , as we showed in (16).

The PSDs of the signal representations with minimum PAR and minimum NLS are plotted in Figure 6. The PSDs of the signal

representations with no SLM optimization are omitted from the plots to allow for clearer comparisons between the two

representations we are interested in. However, the results of the computed ACPRs for all representations are tabulated in Table 3.

It is clear from the PSDs in Figure 6 and the computed ACPRs that using the minimum NLS measure adjusted to the IBO level of interest, compared to the minimum PAR as selection criteria to select the signal to be transmitted, leads to less out-of-band

emissions. In turn, lower spectral regrowth is achieved.

It is worth mentioning that a representation selected based on the minimum NLS criterion may also have the minimum PAR. In this case, both criteria will lead to the same performance.

In this paper, it is shown that the existing measures for NLD are not correlated well with the overall system performance. We concluded that there are two reasons for such an uncorrelation: first, the absence of a clear direct relation between these measures and the system performance, second, the disinvolvement of the IBO level in the formulation of such measures. Hence, based on the established signal characteristics contributing to BER degradation and spectral regrowth, we introduced a new alternative measure, NLS, to characterize NLD effects on CDMA signals. When formulating the NLS measure, we were keen to avoid the above-mentioned drawbacks by relying on the signal characteristics contributing to BER degradation and spectral regrowth. Additionally, being a function in the IBO level gives the NLS measure a superior advantage as it can be adjusted to the IBO level required to work at according to design demands or service regulation.

Using such a measure, an efficient CDMA system is achieved through the provision of a potential estimate of NLD effects on the transmitted signal and the ability to minimize such effects as shown in Section 6, where the NLS measure showed an outperformance over PAR when used as a selection criterion in the SLM technique.

Finally, it seems that the NLS measure would be more complex than PAR. However, nothing is priceless. A little bit more complexity is the price paid for more improved efficiency.