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.
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
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 
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 
where is the clipped portion of the input signal , , and its envelope has the form
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  and , 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 , .
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 , , for all , , and is the linear gain given by
The variance of the distorted signal is given by , where is the variance of the nonlinear component . As far as is considered as a zero-mean Gaussian process, can be calculated as
At the th user's receiver, complex zero-mean additive white Gaussian noise (AWGN) with power spectral density of is introduced, and the received signal is expressed as . Assuming perfect synchronization, after coherent demodulation and phase recovery, the received signal is match-filtered with the th user's spreading waveform and passed to the detector. Thus, the signal-to-noise ratio (SNR) per bit at the th detector input is given by
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  and the references therein). Since we are considering the Walsh orthogonal codes, vanishes. The issue of the other user interference was handled in  using a decorrelating detector, where it is eliminated at the expense of noise enhancement.
Therefore, in (9) will be
where is the SNR per bit at the th detector due to the AWGN only (without nonlinearity).
Let us define two important characteristics of the input signal in relation to the PD-HPA. First, the threshold exceeding rate is defined as the total time intervals where the signal exceeds the PD-HPA threshold or equivalently the probability that the signal exceeds the threshold
Second, the variance of the clipped signal portion , , is given as
Rearranging (11) and (12) and substituting in (7) and (8), respectively, give
Substituting (13) and (14) in (10), we obtain
It is concluded from (15) that the signal characteristics and represent the main contributors to NLD effects. In order to visualize the impact of such characteristics 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, . Finally, the BER performance for the considered QPSK modulation is given by
Figure 1. versus and at dB.
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
Since the modulated CDMA signal in (1) is cyclostationary, the PD-HPA output signal in (4) is also cyclostationary. Cyclostationarity is a special case of stationary, which describes a probabilistic model for certain random data that involves certain periodicity model. For instance, a cyclostationary signal has a periodic autocorrelation function, that is, . Such a periodicity advantage is not of a major importance for the purpose of analysis of this paper. Therefore, we use the ordinary autocorrelation function of stationary random signals for , which is given by
where represents the cross-correlation function of the arbitrary signals and . The Fourier transform of the autocorrelation function gives the PSD of the output signal as
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.
Figure 2 shows the PSDs in (18) for a 16-user baseband CDMA signal of 100 symbols duration with an SRRC filter with roll-off of 0.22. The PSDs are evaluated using the Welch estimation method with the following parameters: Hamming window and 50% overlap between segments. In order to have an adequate trade-off between estimate reliability and frequency resolution, the segment length is set to 32.
Figure 2. PSDs in (19) for an SRRC-filtered, 16-user baseband CDMA signal.
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.
Then, a piecewise analysis approach for is adopted to determine what other signal characteristics at the PD-HPA input, in addition to , would contribute to spectral regrowth. In such an approach, can be represented as a piecewise signal, where each piecewise segment is realized in the interval with to accommodate segments. Clearly, the segments alternate between zero for the unclipped regions and for the clipped regions. Without loss of generality, it can be assumed that even values of correspond to time intervals with clipping and odd values of correspond to time intervals with no clipping (i.e., ). Following this convention and with arbitrarily assigned as sec, can be represented as
where are represented by rectangular windows as
where and is the duration of the ith piecewise segment.
Using the definition of in (1), can be written as
Taking the Fourier transform of (21), , the power spectrum of the clipped signal portion can be written as
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
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 , companding , selected mapping (SLM) , partial transmitted sequences , 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 . The IPV is defined normalized to remove the dependence on the average power as 
The CM measure was proposed with the motivation of reducing the third-order modulation product as it is the cause of the major distortion . 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 , 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 . 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.
Table 1. Signal Characteristics of R1, R2, and R3.
Figure 3. Instantaneous powers , , and of representations R1, R2, and R3, respectively, at different IBOs .
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 SNRAWGN 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  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.
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 , 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.
Table 2. Signal Characteristics of R1, R2, R3, R4, R5, R6, and R7.
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 SNRAWGN 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.
Figure 4. BER of (16) computed for R1–R7 versus IBO levels.
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.
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 . 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).
Figure 5. BER of 16-user CDMA signals in presence of PD-HPA.
To compute the ACPR, the PSD of each time domain signal representation of length 10L (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.
The PSDs of the signal representations with minimum PAR and minimum NLS are plotted in Figure 6. The PSDs of the signal
Figure 6. PSDs of 16-user CDMA signals in presence of PD-HPA at (a) IBO = 1 dB and (b) IBO = 2 dB.
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.
Table 3. ACPR for 16-OFDM in Presence of PD-HPA.
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.
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