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

- Tarek K. Helaly
^{1}Email author, - Richard M. Dansereau
^{1}and - Mohamed El-Tanany
^{1}

**2010**:945427

https://doi.org/10.1155/2010/945427

© Tarek K. Helaly et al. 2010

**Received: **22 June 2010

**Accepted: **4 October 2010

**Published: **10 October 2010

## Abstract

This paper deals with the nonlinear distortion (NLD) effects of high power amplifiers (HPAs) on direct sequence-code division multiple access systems. Such a distortion drastically degrades the system performance in terms of bit error rate (BER) degradation and spectral regrowth. Much effort has been conducted to minimize NLD. A key requirement to do so is to define a certain measure for the HPA nonlinearity, which when reduced often allows NLD to also be reduced. Several measures were proposed such as peak-to-average power ratio, instantaneous power variance, and cubic metric. In this paper, we show that such measures are not closely related to NLD and their reduction does not always lead to optimum performance. Hence, we introduce an efficient measure, namely, nonlinearity severity (NLS), to characterize NLD effects, as an alternative to the existing measures. The NLS is characterized by having direct link to the system performance as it is formulated based on the signal characteristics contributing to BER performance and spectral regrowth. Additionally, a major advantage of the NLS measure is that it is linked to the IBO level allowing the possibility of improving performance at all IBO levels of interest.

## Keywords

## 1. Introduction

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.

## 2. Existing Nonlinearity Measures

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, .

## 3. Signal Characteristics Contributing NLD Effects

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.

### 3.1. BER Degradation

Consider the CDMA signal in (1), where the symbols are assumed equiprobable i.i.d. with zero mean and variance of , and belong to alphabet of size with .

For large number of users and assuming the pulse shaping filter corresponds to a square root raised cosine (SRRC) filter with small roll-off factor, can be regarded as a band-limited zero-mean complex Gaussian process with variance and an envelope having a quasi-Rayleigh pdf , [9].

It is more convenient in the context of this paper to use the IBO level instead of the threshold , where the IBO, , is defined as the ratio of the input power at the saturation level to the signal average power . That is, .

where
is the
th user power,
is the variance of the AWGN component [12, equation (15.3*‒* 29)],
is the variance of the nonlinear component
, and
is the variance of the interference from the other users. For sufficiently large
and
,
and
can be assumed Gaussian distributed given by
and
(for synchronous CDMA and equal users' powers), respectively (see [10] and the references therein). Since we are considering the Walsh orthogonal codes,
vanishes. The issue of the other user interference was handled in [7] using a decorrelating detector, where it is eliminated at the expense of noise enhancement.

where is the SNR per bit at the th detector due to the AWGN only (without nonlinearity).

The importance of such a BER expression is that it is formulated based on the signal characteristics in relation to the PD-HPA characteristics. This new characterization opens new avenues to minimize NLD via controlling such characteristics as will be demonstrated later; as and increase, the overall SNR decreases. In order to visualize the impact of the CDMA signal characteristics and on the system performance, the SNR in (15) is computed at of 5 dB and plotted against and in Figure 1. Computations are done at different IBO levels ranging from 0 dB up to the level that passes almost all of the signal without clipping, that is, .

### 3.2. Spectral Regrowth

where is the PSD of the input signal to the PD-HPA, and are the cross PSDs of the input signal and the clipped signal portion, and is the PSD of the clipped signal portion.

Without loss of generality, spectral regrowth can be defined for the upper channel as the additional out-of-band power at the output of the HPA, that is, . As shown in Figure 2, the PSD of the clipped signal portion almost coincides with the output distorted PSD outside the signal bandwidth, that is, . Also, since the PSD of the baseband filtered CDMA signal is almost rectangular, it can be assumed constant over the entire bandwidth and zero elsewhere, that is, . Therefore, , that is, spectral regrowth depends primarily on the PSD of the clipped signal portion, and, in turn, as the variance of the clipped signal portion increases, spectral regrowth increases.

where
and
is the duration of the *i* th piecewise segment.

where is the convolution operator.

In the frequency domain, the rectangular windows , , form a set of sinc functions with main lobe widths that are inversely proportional to the durations 's of the clipped subintervals in the time domain. At each new clipping event , the sinc function , when convolved with the frequency response of the pulse shaping filter with arbitrary bandwidth as in (23), results in with bandwidth . Consequently, given the power spectrum of the clipped signal portion in (22), it is concluded that as the number of segments increases, additional windowing with slower fall-off is introduced causing an increase in the out-of-band power of the clipped signal, and so to the spectral regrowth. In fact, the number of segments determines the number of zero-departures and zero-arrivals in the clipped signal portion. These zero-departures/arrivals in the clipped signal portion represent the threshold crossings in the input signal to the PD-HPA.

Therefore, it is deduced that the signal characteristics at the input of the PD-HPA that mainly contribute to spectral regrowth in relation to the PD-HPA clipping threshold are the variance of signal portion exceeding the threshold , the threshold crossing rate , and the time durations of the crossing events 's. Unfortunately, no convenient method of exact calculation of is available as yet [8, 13]. Therefore, in such an approach, the mean threshold crossing duration is used as an indicator to the durations of crossing events.

## 4. Existing Nonlinearity Severity Measure

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 .

Signal Characteristics of R1, R2, and R3.

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.

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.

## 5. Nonlinearity Severity Measure

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 ).

Signal Characteristics of R1, R2, R3, R4, R5, R6, and R7.

R1 | R2 | R3 | R4 | R5 | R6 | R7 | |||
---|---|---|---|---|---|---|---|---|---|

PAR (dB) | 4.46 | 5.73 | 8.85 | 5.59 | 5.52 | 5.28 | 5.37 | ||

IPV | 0.61 | 0.51 | 1.51 | 0.56 | 0.59 | 0.55 | 0.51 | ||

CM | 0.36 | 0.36 | 0.12 | 0.36 | 0.34 | 0.29 | 0.32 | ||

0.432 | 0.489 | 0.337 |
| 0.451 | 0.447 | 0.496 | 0 | ||

0.280 | 0.341 | 0.322 | 0.281 |
| 0.360 | 0.386 | 1 | ||

0.246 | 0.164 | 0.212 | 0.193 | 0.189 |
| 0.193 | 2 | ||

0.140 | 0.083 | 0.155 | 0.076 | 0.095 | 0.061 |
| 3 | ||

0.058 | 0.125 | 0.048 |
| 0.058 | 0.052 | 0.047 | 0 | ||

0.027 | 0. 033 | 0.142 | 0.034 |
| 0.028 | 0.035 | 1 | ||

0.015 | 0.015 | 0.097 | 0.022 | 0.018 |
| 0.014 | 2 | ||

0.006 | 0.004 | 0.063 | 0.010 | 0.008 | 0.006 |
| 3 | ||

NLS (dB) | 0 | ||||||||

1 | |||||||||

2 | |||||||||

3 |

_{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.

## 6. Performance Assessment

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.

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.

To compute the ACPR, the PSD of each time domain signal representation of length 10*L* (Walsh code length) is computed. The PSDs are evaluated using the Welch estimation method with the following parameters: Hamming window, segment length of 32, and 50% overlap between segments.

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

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.

## 7. Conclusion

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.

## Authors’ Affiliations

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