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

  • Tarek K. Helaly1Email author,

    Affiliated with

    • Richard M. Dansereau1 and

      Affiliated with

      • Mohamed El-Tanany1

        Affiliated with

        EURASIP Journal on Wireless Communications and Networking20102010:945427

        DOI: 10.1155/2010/945427

        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.

        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

        The system under investigation is a downlink synchronous DS-CDMA system, where users' signals have equal power. The complex envelope of the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq1_HTML.gif th DS-CDMA symbol for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq2_HTML.gif active users is defined as [7]
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ1_HTML.gif
        (1)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq3_HTML.gif is the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq4_HTML.gif th user's signal energy per symbol, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq5_HTML.gif is the symbol duration, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq6_HTML.gif are the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq7_HTML.gif th user's symbol data, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq9_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ2_HTML.gif
        (2)

        is the spreading waveform obtained with the Walsh-Hadamard matrix, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq10_HTML.gif is the spreading factor, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq11_HTML.gif is the chip duration, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq12_HTML.gif is the impulse response of the pulse shaping filter. Without loss of generality, the spreading waveforms are assumed to have unit energy, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq13_HTML.gif .

        A PD-HPA pair is considered as the nonlinear amplifier chain, which has a zero AM-PM characteristic http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq14_HTML.gif and an AM-AM characteristic given by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ3_HTML.gif
        (3)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq15_HTML.gif 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]
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ4_HTML.gif
        (4)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq16_HTML.gif is the clipped portion of the input signal http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq17_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq18_HTML.gif , and its envelope http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq19_HTML.gif has the form
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ5_HTML.gif
        (5)

        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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq20_HTML.gif are assumed equiprobable i.i.d. with zero mean and variance of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq21_HTML.gif , and belong to alphabet http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq22_HTML.gif of size http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq23_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq24_HTML.gif .

        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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq25_HTML.gif can be regarded as a band-limited zero-mean complex Gaussian process with variance http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq26_HTML.gif and an envelope http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq27_HTML.gif having a quasi-Rayleigh pdf http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq28_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq29_HTML.gif [9].

        It is more convenient in the context of this paper to use the IBO level instead of the threshold http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq30_HTML.gif , where the IBO, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq31_HTML.gif , is defined as the ratio of the input power at the saturation level http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq32_HTML.gif to the signal average power http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq33_HTML.gif . That is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq34_HTML.gif .

        The output of the PD-HPA can be represented as the sum of two uncorrelated components: a scaled linear component and a nonlinear component, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq35_HTML.gif [911], that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ6_HTML.gif
        (6)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq37_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq38_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq39_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq40_HTML.gif is the linear gain given by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ7_HTML.gif
        (7)
        The variance of the distorted signal http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq41_HTML.gif is given by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq42_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq43_HTML.gif is the variance of the nonlinear component http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq44_HTML.gif . As far as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq45_HTML.gif is considered as a zero-mean Gaussian process, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq46_HTML.gif can be calculated as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ8_HTML.gif
        (8)
        At the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq47_HTML.gif th user's receiver, complex zero-mean additive white Gaussian noise (AWGN) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq48_HTML.gif with power spectral density of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq49_HTML.gif is introduced, and the received signal is expressed as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq50_HTML.gif . Assuming perfect synchronization, after coherent demodulation and phase recovery, the received signal is match-filtered with the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq51_HTML.gif th user's spreading waveform http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq52_HTML.gif and passed to the detector. Thus, the signal-to-noise ratio (SNR) per bit at the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq53_HTML.gif th detector input is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ9_HTML.gif
        (9)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq54_HTML.gif is the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq55_HTML.gif th user power, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq56_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq57_HTML.gif is the variance of the AWGN component [12, equation (15.3 29)], http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq58_HTML.gif is the variance of the nonlinear component http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq59_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq60_HTML.gif is the variance of the interference from the other users. For sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq61_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq62_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq63_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq64_HTML.gif can be assumed Gaussian distributed given by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq66_HTML.gif (for synchronous CDMA and equal users' powers), respectively (see [10] and the references therein). Since we are considering the Walsh orthogonal codes, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq67_HTML.gif 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.

        Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq68_HTML.gif in (9) will be
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ10_HTML.gif
        (10)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq69_HTML.gif is the SNR per bit at the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq70_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq71_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq72_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ11_HTML.gif
        (11)
        Second, the variance of the clipped signal portion http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq73_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq74_HTML.gif , is given as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ12_HTML.gif
        (12)
        Rearranging (11) and (12) and substituting in (7) and (8), respectively, give
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ13_HTML.gif
        (13)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ14_HTML.gif
        (14)
        Substituting (13) and (14) in (10), we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ15_HTML.gif
        (15)
        It is concluded from (15) that the signal characteristics http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq75_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq76_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq77_HTML.gif of 5 dB and plotted against http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq79_HTML.gif 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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq80_HTML.gif . Finally, the BER performance for the considered QPSK modulation is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ16_HTML.gif
        (16)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Fig1_HTML.jpg
        Figure 1

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq81_HTML.gif versus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq82_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq83_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq84_HTML.gif  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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq85_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq86_HTML.gif increase, the overall SNR decreases. In order to visualize the impact of the CDMA signal characteristics http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq88_HTML.gif on the system performance, the SNR in (15) is computed at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq89_HTML.gif of 5 dB and plotted against http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq90_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq91_HTML.gif 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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq92_HTML.gif .

        3.2. Spectral Regrowth

        Since the modulated CDMA signal in (1) is cyclostationary, the PD-HPA output signal http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq93_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq94_HTML.gif has a periodic autocorrelation function, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq95_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq96_HTML.gif , which is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ17_HTML.gif
        (17)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq97_HTML.gif represents the cross-correlation function of the arbitrary signals http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq98_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq99_HTML.gif . The Fourier transform of the autocorrelation function gives the PSD of the output signal as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ18_HTML.gif
        (18)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq100_HTML.gif is the PSD of the input signal to the PD-HPA, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq101_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq102_HTML.gif are the cross PSDs of the input signal and the clipped signal portion, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq103_HTML.gif 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.
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Fig2_HTML.jpg
        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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq104_HTML.gif . 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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq105_HTML.gif . Also, since the PSD of the baseband filtered CDMA signal is almost rectangular, it can be assumed constant over the entire bandwidth http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq106_HTML.gif and zero elsewhere, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq107_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq108_HTML.gif , 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq109_HTML.gif increases, spectral regrowth increases.

        Then, a piecewise analysis approach for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq110_HTML.gif is adopted to determine what other signal characteristics at the PD-HPA input, in addition to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq111_HTML.gif , would contribute to spectral regrowth. In such an approach, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq112_HTML.gif can be represented as a piecewise signal, where each piecewise segment http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq113_HTML.gif is realized in the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq114_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq115_HTML.gif to accommodate http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq116_HTML.gif segments. Clearly, the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq117_HTML.gif segments alternate between zero for the unclipped regions and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq118_HTML.gif for the clipped regions. Without loss of generality, it can be assumed that even values of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq119_HTML.gif correspond to time intervals with clipping and odd values of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq120_HTML.gif correspond to time intervals with no clipping (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq121_HTML.gif ). Following this convention and with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq122_HTML.gif arbitrarily assigned as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq123_HTML.gif  sec, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq124_HTML.gif can be represented as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ19_HTML.gif
        (19)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq125_HTML.gif are represented by rectangular windows as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ20_HTML.gif
        (20)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq126_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq127_HTML.gif is the duration of the i th piecewise segment.

        Using the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq128_HTML.gif in (1), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq129_HTML.gif can be written as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ21_HTML.gif
        (21)
        Taking the Fourier transform of (21), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq130_HTML.gif , the power spectrum of the clipped signal portion can be written as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ22_HTML.gif
        (22)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ23_HTML.gif
        (23)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq131_HTML.gif is the convolution operator.

        In the frequency domain, the rectangular windows http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq132_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq133_HTML.gif , form a set of sinc functions with main lobe widths that are inversely proportional to the durations http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq134_HTML.gif 's of the clipped subintervals in the time domain. At each new clipping event http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq135_HTML.gif , the sinc function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq136_HTML.gif , when convolved with the frequency response of the pulse shaping filter http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq137_HTML.gif with arbitrary bandwidth http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq138_HTML.gif as in (23), results in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq139_HTML.gif with bandwidth http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq140_HTML.gif . Consequently, given the power spectrum of the clipped signal portion in (22), it is concluded that as the number of segments http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq141_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq142_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq143_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq144_HTML.gif , the threshold crossing rate http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq145_HTML.gif , and the time durations of the crossing events http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq146_HTML.gif 's. Unfortunately, no convenient method of exact calculation of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq147_HTML.gif is available as yet [8, 13]. Therefore, in such an approach, the mean threshold crossing duration http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq148_HTML.gif 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 [14], companding [15], selected mapping (SLM) [16], partial transmitted sequences [17], and Walsh code reassignment techniques [1, 18]. For a signal http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq149_HTML.gif of duration http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq150_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq151_HTML.gif , the PAR http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq152_HTML.gif is defined as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ24_HTML.gif
        (24)
        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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq153_HTML.gif is defined normalized to remove the dependence on the average power as [4]
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ25_HTML.gif
        (25)
        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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ26_HTML.gif
        (26)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq154_HTML.gif is the root mean square (rms) value of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq155_HTML.gif .

        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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq156_HTML.gif and the variance of the clipped signal portion http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq157_HTML.gif .

        Consider an example 16-user CDMA signal of duration http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq158_HTML.gif with Walsh codes of length http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq159_HTML.gif = 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq160_HTML.gif using the SLM technique [16]. Figure 3 shows the instantaneous power of the three representations http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq161_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq162_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq163_HTML.gif . The PAR, IPV, and CM are calculated for each representation. Also http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq164_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq165_HTML.gif are measured for each representation at different values of IBO http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq166_HTML.gif and tabulated in Table 1.
        Table 1

        Signal Characteristics of R1, R2, and R3.

         

        R1

        R2

        R3

          

        PAR (dB)

        4.46

        5.73

        8.85

          

        IPV

        0.61

        0.51

        1.51

          

        CM

        0.36

        0.36

        0.12

          

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq167_HTML.gif

        0.432

        0.489

        0.337

        0

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq168_HTML.gif (dB)

         

        0.280

        0.341

        0.322

        1

         
         

        0.246

        0.164

        0.212

        2

         
         

        0.140

        0.083

        0.155

        3

         

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq169_HTML.gif

        0.058

        0.125

        0.048

        0

         
         

        0.027

        0.033

        0.142

        1

         
         

        0.015

        0.015

        0.097

        2

         
         

        0.006

        0.004

        0.063

        3

         

        BER

        0.137

        0.146

        0.129

        0

         
         

        0.124

        0.129

        0.131

        1

         
         

        0.121

        0.115

        0.121

        2

         
         

        0.113

        0.109

        0.116

        3

         
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Fig3_HTML.jpg
        Figure 3

        Instantaneous powers http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq170_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq171_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq172_HTML.gif of representations R1, R2, and R3, respectively, at different IBOs http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq173_HTML.gif .

        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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq174_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq175_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq176_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq177_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq178_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq179_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq180_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq181_HTML.gif at IBO = 0 dB, while has higher values at other IBO values. Moreover, at SNRAWGN of 5 dB and using the measured values of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq182_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq183_HTML.gif , 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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ27_HTML.gif
        (27)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq184_HTML.gif is the envelope value that is exceeded with probability http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq185_HTML.gif , resulting in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq186_HTML.gif .

        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 ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq187_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq188_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq189_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq190_HTML.gif ).

        For the sake of having a simplified measure, it is useful to use the interesting relation between http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq191_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq192_HTML.gif , where their product gives http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq193_HTML.gif [13, 21]. Thus, it can be argued that the signal characteristics affecting the overall system performance, in relation to NLD, are limited to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq194_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq195_HTML.gif . Thus, the NLS measure is defined as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ28_HTML.gif
        (28)
        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.

         

        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

          

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq196_HTML.gif

        0.432

        0.489

        0.337

        0.323

        0.451

        0.447

        0.496

        0

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq197_HTML.gif (dB)

         

        0.280

        0.341

        0.322

        0.281

        0.285

        0.360

        0.386

        1

         
         

        0.246

        0.164

        0.212

        0.193

        0.189

        0.170

        0.193

        2

         
         

        0.140

        0.083

        0.155

        0.076

        0.095

        0.061

        0.057

        3

         

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq198_HTML.gif

        0.058

        0.125

        0.048

        0.050

        0.058

        0.052

        0.047

        0

         
         

        0.027

        0. 033

        0.142

        0.034

        0.024

        0.028

        0.035

        1

         
         

        0.015

        0.015

        0.097

        0.022

        0.018

        0.011

        0.014

        2

         
         

        0.006

        0.004

        0.063

        0.010

        0.008

        0.006

        0.004

        3

         

        NLS (dB)

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq199_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq200_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq201_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq202_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq203_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq204_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq205_HTML.gif

        0

         
         

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq206_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq207_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq208_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq209_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq210_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq211_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq212_HTML.gif

        1

         
         

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq213_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq214_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq215_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq216_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq217_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq218_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq219_HTML.gif

        2

         
         

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq220_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq221_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq222_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq223_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq224_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq225_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq226_HTML.gif

        3

         
        As can be seen in Table 2, for each of the IBO levels of interest, the values of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq227_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq228_HTML.gif and measured for the representations with the minimum NLS are the minimal or very close to the minimal values of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq229_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq230_HTML.gif 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.
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Fig4_HTML.jpg
        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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq231_HTML.gif equivalent representations of the same signal http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq232_HTML.gif by rotating the phases of the data symbols such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq233_HTML.gif [16]. Among the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq234_HTML.gif representations, the representation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq235_HTML.gif that has the minimum PAR is selected for transmission. In our approach, the representation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq236_HTML.gif that has the minimum NLS is selected for transmission as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Equ29_HTML.gif
        (29)

        In this scenario, we have considered a CDMA system with Walsh orthogonal codes of length http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq237_HTML.gif . An SRRC pulse shaping filter is used with roll-off factor of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq238_HTML.gif 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: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq239_HTML.gif with no SLM optimization, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq240_HTML.gif with SLM using the minimum PAR as the selection criterion, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq241_HTML.gif 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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq242_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq243_HTML.gif , as we showed in (16).
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Fig5_HTML.jpg
        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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_Fig6_HTML.jpg
        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.

         

        ACPR (dB)

        IBO, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq244_HTML.gif (dB)

        1

        2

        No SLM

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq245_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq246_HTML.gif

        SLM (PAR)

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq247_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq248_HTML.gif

        SLM (NLS)

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq249_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq250_HTML.gif

        Linear

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq251_HTML.gif

        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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq252_HTML.gif a potential estimate of NLD effects on the transmitted signal and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F945427/MediaObjects/13638_2010_Article_2072_IEq253_HTML.gif 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

        (1)
        Department of Systems and Computer Engineering, Carleton University

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        © Tarek K. Helaly et al. 2010

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