Novel Low-Complexity Receivers for Constant Envelope ... - IEEE Xplore

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 63, NO. 17, SEPTEMBER 1, 2015

Novel Low-Complexity Receivers for Constant Envelope OFDM Ahsen U. Ahmed and James R. Zeidler, Fellow, IEEE

Abstract—Constant Envelope OFDM provides a solution to the issue of high peak-to-average power ratio in OFDM by using angle modulation to transform the OFDM signal to a constant envelope signal. The modulation index of Constant Envelope OFDM controls this transformation. In this paper, we develop novel receiver structures for Constant Envelope OFDM based on the Taylor series expansion, alleviating the need for angle demodulation at the receiver. This results in immunity from phase cycle slips due to phase wrapping and the threshold effect which would otherwise cause performance degradation. These receivers allow for a simpler implementation without the need to compute the arctangent at the receiver. We show that these novel receivers perform well when compared to the conventional arctangent based receiver for small and moderate modulation indices for the cases of additive white Gaussian noise and multipath fading. For frequency selective fading, we show that the application of a frequency domain equalizer results in good performance when these novel receivers are employed. Finally, we study the performance of these new receivers when error correction coding is employed and show that they not only provide excellent performance but also significantly outperform the conventional arctangent based receiver for coded Constant Envelope OFDM performance. Index Terms—OFDM, peak to average power ratio, receivers.

I. INTRODUCTION

O

FDM is widely used for wireless communication for both commercial and military applications. Despite its many advantages, OFDM also suffers from major limitations including a high peak-to-average power ratio (PAPR) [1], [2] which results in intermodulation distortion among subcarriers and out-of-band spectral radiation at the non-linear power amplifier. It is customary to use a significant backoff and operate in the linear region of the amplifier to reduce, but not completely eliminate, the unwanted distortion and the accompanying spectral broadening. Such a backoff not only reduces the transmit power but also results in low power amplifier efficiency [3], [4]. This is especially detrimental for mobile devices operating on battery power. For example, a class A amplifier operating

Manuscript received May 21, 2014; revised January 07, 2015 and April 01, 2015; accepted April 03, 2015. Date of publication July 10, 2015; date of current version July 22, 2015. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Jian-Kang Zhang. A. U. Ahmed is with the Space and Naval Warfare Systems Center Pacific, San Diego, CA 92152-5435 USA (e-mail: [email protected]). J. R. Zeidler is with the Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093-0407 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2015.2441036

with a 6 dB backoff has a maximum efficiency of only 12.5% [3]. In fact, being limited to a more linear amplifier such as a class A amplifier due to design constraints and operating with a backoff can consume five times or more power than an alternate amplifier feasible for a constant envelope signal [5]. Constant Envelope OFDM (CE-OFDM) provides one solution to the high PAPR issue in OFDM [6]–[9]. Various PAPR reduction techniques have been previously developed for OFDM but these don't completely eliminate the OFDM PAPR issue [2]. In comparison, CE-OFDM transforms the high PAPR OFDM signal to a constant envelope signal using phase modulation. This not only eliminates the need for a backoff at the power amplifier, maximizing both transmit power and efficiency, but it also enables the use of even more non-linear, power efficient and cost effective amplifiers that are not feasible for OFDM [3]. The various aspects of the performance and applications of CE-OFDM have been studied previously in [6]–[27]. Due to the major advantage of a constant envelope, CE-OFDM is being researched for applications as varied as wireless [6]–[18], optical [19]–[22], powerline [23] and satellite [24] communications as well as radar [25]–[27]. The advantages of CE-OFDM for energy efficiency in cellular wireless networks are discussed in detail in [28]. With the continuing rapid growth of wireless communication based applications and the expected increase in future demand, the need for further spectrum will remain high. The availability of 7 GHz of unlicensed spectrum around 60 GHz, which is much greater than the current combined allocation for radio, TV, cellular, satellite and WiMAX bands, makes it an ideal candidate for future high rate communication [29]. A further benefit of operating at such high frequencies is that the size of components including the antenna is very small enabling the transceiver implementation on a chip. Advances in CMOS technology have made CMOS an attractive choice for low cost, highly integrated transceiver implementation at high frequencies [30], [31]. However, component design, including the power amplifier, remains challenging at high frequencies with added requirements for linearity coming at the expense of performance and a further increase in complexity and cost [32]. CE-OFDM with its constant envelope reduces design complexity and cost while enabling maximum performance and power efficiency thereby providing a very attractive choice for communication at high frequencies including the 60 GHz band. These advantages more than compensate for the reduction in bandwidth efficiency for CE-OFDM compared to OFDM [10], especially with the vast amount of bandwidth available at high frequencies. Furthermore, spectral precoding techniques presented in [16] can be employed to improve CE-OFDM spectral efficiency. Due to these advantages,

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AHMED AND ZEIDLER: NOVEL LOW-COMPLEXITY RECEIVERS FOR CONSTANT ENVELOPE OFDM

Fig. 2. The conventional CE-OFDM receiver.

Fig. 1. The CE-OFDM transmitter.

CE-OFDM was proposed in [33] as an attractive option for millimeter band communications in support of 5G, the next generation wireless communication standard. CE-OFDM is based on angle modulation and therefore, under certain conditions during phase demodulation, it is susceptible to the well known threshold effect whereby the demodulated signal to noise ratio (SNR) falls off much more rapidly than the input carrier to noise ratio (CNR) [13], [18]. The threshold effect is encountered at low CNR due to the appearance of phase cycle slips at the output of the phase demodulator and results in a degradation of the demodulated SNR and, consequently, the performance. In this paper, we develop two novel receiver structures based on the Taylor series expansion of CE-OFDM for low and mod. These receiver strucerate modulation indices tures eliminate the need for a phase demodulator, thus also eliminating phase cycle slips due to the threshold effect or phase wrapping. Due to this reason, while employing these receivers, CE-OFDM is not affected by the threshold effect or any phase wrapping issues. We study the performance of these novel receiver structures for both additive white Gaussian noise (AWGN) and frequency selective multipath fading channels. We show that a frequency domain equalizer works well with the novel receiver structures to equalize the signal distortion due to the channel. Finally, we study the application of convolutional error correction coding to CE-OFDM and compare the coded performance of these new receivers with the conventional arctangent based receiver. Our results show that these new receiver structures significantly outperform the conventional arctangent based receiver for coded CE-OFDM performance. II. SIGNAL DESCRIPTION CE-OFDM is based on a simple transformation of the OFDM signal as shown in Fig. 1. The OFDM signal is embedded within the phase of a constant envelope signal using phase modulation. This results in a constant envelope signal with a 0 dB PAPR. The baseband CE-OFDM signal is given as (1) where is the signal amplitude. The phase signal, the embedded OFDM signal is given as

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, with

(2) , modulate the orthogonal The data symbols, for OFDM subcarriers to . is the CE-OFDM block period. The normalizing constant, , ensures that the phase variance

Fig. 3. Arctangent based phase demodulator.

is independent of the number of OFDM subcarriers . is the data variance with for binary data ([7], p. 47). The modulation index is the key parameter that controls the tradeoff between performance and bandwidth [7] for CE-OFDM. In this paper, the scaled modulation index is also simply referred to as the modulation index. CE-OFDM works well for a 2X or higher sampling rate at the receiver [6]. In this paper, an oversampling factor of is used, i.e., samples per CE-OFDM symbol are used for all simulation cases. III. CONVENTIONAL RECEIVER STRUCTURE The conventional receiver structure for CE-OFDM consists of a phase demodulator, to undo the phase modulation transformation, followed by a standard OFDM demodulator as shown in Fig. 2. This provides for a practical receiver implementation for CE-OFDM [6], [8]. The baseband received CE-OFDM signal is given as (3) where is the baseband Gaussian distributed noise with power spectral density (4) where is the system front-end bandwidth. The in phase and quadrature noise components are independent with autocorrela([34], p. 158). The tion is taken to equal the bandwidth sampling rate resulting in independent Gaussian noise samples ([7], p. 32) at the receiver. A. Arctangent Based Receiver An arctangent based phase demodulator as shown in Fig. 3 has been previously employed [6]–[10], [12]–[16], [18] for phase demodulation in CE-OFDM and shown to perform well. The phase of the baseband received signal is extracted by taking the inverse tangent of the quadrature baseband components. The arctangent provides the instantaneous phase which is restricted to the range. Phase excursions outside this range, such as for large OFDM signal peaks, result in phase wrapping. Therefore, it is necessary to use a phase unwrapper for larger modulation indices to reconstruct the original unrestricted phase. The phase unwrapper is prone to making errors resulting in phase cycle slips due to the presence of noise, especially at low CNR when significant noise is present [13], [18]. For low and moderate modulation indices , phase wrapping is less frequent and the arctangent based receiver can be employed without a phase unwrapper. In

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such cases, phase wrapping occurs less frequently and results in burst errors [12]. The need to implement a phase demodulator in the conventional CE-OFDM receiver results in increased receiver implementation complexity. The arctangent based receiver requires the computation of the arctangent at the receiver. This can be accomplished by using a lookup table at the receiver, however, this lookup table needs to be quite large for high accuracy. Additionally, the table address values still need to be computed at the receiver. Alternately, several algorithms are available for computing the arctangent but due to the fact that the arctangent is a highly nonlinear function, these algorithms require a high computational complexity. For example, the popular CORDIC algorithm requires 10–12 iterations to attain a 0.1 degree accuracy. Other lower complexity algorithms still require several multiplications and divisions [35], [36]. IV. BASIC LINEAR RECEIVER (BLR) In this paper, a novel linear receiver is developed that eliminates the phase demodulator in the conventional CE-OFDM receiver (Fig. 2). It is shown that this receiver provides excellent performance. This linear receiver is based on the Taylor series expansion of the CE-OFDM signal. With representing the normalized OFDM signal, the CE-OFDM signal can be given as . Without loss of generality, the amplitude is set to 1. The CE-OFDM signal can then be expressed as

(5) By employing the Taylor series expansion [34], the in phase and quadrature components of CE-OFDM are given as

Fig. 4. The Basic Linear Receiver (BLR) structure for CE-OFDM.

Fig. 5. The Enhanced Receiver (ER) structure for CE-OFDM.

V. ENHANCED RECEIVER (ER) The basic linear receiver works well for small modulation as the higher order terms are negligible. indices However the higher order terms become more significant for higher modulation indices. The first term of (7) is the desirable OFDM signal while the other higher order terms contribute to distortion with their contributions decreasing as their order increases for the case of modulation indices below 1 . It is clear that the second term, , contributes the largest amount of distortion. An improved receiver can be developed by using both the in phase and quadrature components of CE-OFDM to cancel out the most significant cubic distortion term. This improved receiver performs better than the basic linear receiver for moderate modulation indices due to the removal of the cubic distortion term. It is shown below that the cubic term degrades the performance as distortion only when it is present with a negative sign (as in (7)) and that it in fact contributes constructively at the receiver when it is additive to the OFDM component (first term). Therefore, further improvement can be obtained by using an enhanced receiver (ER) with an additive cubic term given as

(6)

(8) (7) Note that the first term of the Taylor series expansion of the quadrature component is the normalized OFDM signal. The contribution from the remaining higher order terms decreases with a decrease in the modulation index . Therefore, for small modulation indices when the higher order terms in the Taylor expansion are relatively negligible compared to the first term, a simple linear receiver can be devised by only employing the quadrature component of the received CE-OFDM signal as shown in Fig. 4. This basic linear receiver for small modulation indices is essentially a direct OFDM receiver applied to the quadrature component of the received CE-OFDM signal thereby reducing the overall CE-OFDM receiver complexity.

where is the CE-OFDM phase with the embedded OFDM signal. Fig. 5 shows the block diagram of this enhanced receiver structure. The performance of the basic linear receiver and the enhanced receiver for several modulation indices is shown in Fig. 6 where it is compared with the conventional arctangent based receiver for the case of an AWGN channel. While the basic linear receiver performs well for small modulation indices , the enhanced receiver outperforms it, as expected, especially for moderate modulation indices when the contribution from the cubic term is significant. The novel receivers also perform well compared to the arctangent based receiver. It should be noted that these ranges are mainly based on the performance comparison with the arctangent receiver with a more rigorous performance


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(10) Fig. 6. CE-OFDM simulation performance of the basic linear receiver and the . enhanced receiver compared to the arctangent based receiver for

analysis presented in Section VI. For higher modulation indices, while the arctangent receiver performs better at low BERs, the ER outperforms it at higher BERs. A. Contribution of the Cubic Term Let us further examine the cubic term and its contribution at the receiver. For the case of sine and cosine subcarriers, with and as the -th data symbols present on the sine and cosine subcarriers respectively, the cubic term is given as

This expression for the cubic term contains four sine terms and four cosine terms which result in sines and cosines at different locations for different combinations of , and over the triple summation (total ). Only sines and cosines that fall on the OFDM sine and cosine subcarrier locations have an impact on the CE-OFDM performance. Consider the -th cosine matched filter at the receiver corresponding to the subcarrier. As shown in Appendix A, the number of cosine/sine terms from the triple sum representing in (10) that impact the -th cosine matched filter is given as

(11)

(9)

It is also shown in Appendix A that the number of these terms that contribute constructively is given as (12)

By expanding out the terms and through repeated use of trigonometric identities, the cubic term can be represented as

The contribution due to the non-constructive terms at the subcarrier matched filter output can be shown to be well modeled as Gaussian distributed. Consider the coefficient of the first cosine term in (10). It is given as (13) The data coefficients are zero mean, independent, identically distributed with a uniform probability density. Thus, each of the terms in (13) is also distributed with a uniform probability density . Furthermore, the terms in (13) are uncorrelated Bernoulli random variables and thus pairwise independent [38] for any combination of that generates unique terms. The application of the central limit theorem is well known for the case of the sum of independent Bernoulli trials to model the resulting binomial distribution ([39], p. 325]. Although the central limit theorem has been shown to hold for certain cases of pairwise independent identically distributed random variables [40], [41], pairwise independence is not a sufficient condition in general ([42], p. 109] for the central limit theorem. Therefore, Monte Carlo simulations were performed to show that the

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A. Basic Linear Receiver (BLR) For the basic linear receiver, the output of the OFDM matched filter for the -th cosine subcarrier based on the quadrature component of the received CE-OFDM signal is

(15) is the signaling component due to the first term of where (7), which is simply the embedded OFDM signal. It is given as

(16)

Fig. 7. Histogram of the sum of the coefficients of all terms in (10) (excluding with . the scaling) that impact the -th matched filter for The Gaussian PDF with mean based on (12) and variance based on (14) is also plotted.

. is due to the component of the where cubic term arising from terms with the correct data bits. The total number of terms with the correct data symbol is given in (12). The signal component due to the negative cubic term based on (7) and (12) is given as (17)

sum of the coefficients in (10) is well modeled as Gaussian distributed. Further examination also shows that each coefficient term from (10) that affects the -th cosine or sine matched filter is generated 6 times. For example, for , the coefficient is generated 6 times from the combinations of term of (1,2,3), (1,3,2), (2,1,3), (3,1,2), (2,3,1) and (3,2,1). Therefore, based on this observation and from (11) and (12), there are unique distortion causing coefficient terms with unit variance, each appearing 6 times, at the -th cosine or sine matched filter. The sum of these coefficient terms is Gaussian distributed with variance, based on the sum of individual variances due to pairwise independence, given as

(14) Fig. 7 shows the histogram generated by simulating the coefficients of all terms from (10) that impact the -th cosine matched with . filter for The sum of the coefficients that appear at the -th cosine matched filter due to the cubic term (10) matches well with the plotted Gaussian distribution with a mean of as predicted by (12) and variance based on (14). The bias in the mean towards due to the terms that contribute constructively is the reason that the presence of the additive cubic term helps improve the performance of the enhanced receiver compared to the basic linear receiver. VI. PERFORMANCE IN AWGN Performance approximations for the basic linear receiver and the enhanced receiver in AWGN can be obtained by considering the dominant first and third order signaling terms from the Taylor series expansion. The higher order signaling terms make a smaller contribution for modulation indices below 1 .

is due to the baseband quadrature noise component, (18) It is Gaussian distributed with zero mean and variance ([34], p. 158). is the distortion component due to the cubic term which was previously shown to be well modeled as Gaussian distributed. Therefore, is Gaussian distributed with zero mean and variance where is given in (14). Without loss of generality, we can assume that . The performance of the basic linear receiver for the -th cosine subcarrier can then be given as (19) Similarly, it can be shown that the performance of the -th sine subcarrier is also given by (19). The overall performance over . This perforall subcarriers is then given as mance approximation is plotted against the simulation performance in Fig. 8. For low modulation indices , the performance approximation is very accurate. For the higher modulation index of , the higher order terms (fifth order and above) become more significant and therefore the approximation is not as accurate at low bit error rates (BERs). At low modulation indices when the cubic term is negligible and can be ignored, the performance approximation can be simplified as

(20)


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The output of the OFDM matched filter for the -th cosine subcarrier is then given as

(22) and are from the first two terms of and were where , and are computed in (16) and (17) respectively. zero mean noise terms due to , and respectively. is the distortion component due to the cubic term modeled as Gaussian distributed with zero mean and variance . Let us consider which is given as Fig. 8. CE-OFDM simulation performance in comparison to the analytical ap. proximation in AWGN for the basic linear receiver for

It should be noted that this performance approximation matches the performance approximation for the arctangent based receiver [6], [9]. Therefore, for low modulation indices, the AWGN performance of the arctangent and basic linear receivers are a close match, as was the case in Fig. 6. B. Enhanced Receiver (ER) The computation of the performance approximation for the enhanced receiver is more involved due to the large number of signal and noise terms. Therefore, for the enhanced receiver, we consider the case of low modulation indices when higher order terms are much smaller than the dominant signaling and noise terms and can thus be ignored. The received signal at the input of the OFDM demodulator in the enhanced receiver is

(21) where

(23) Since the noise samples are uncorrelated, , the variance of can be computed as (24), shown at the bottom of the page. The CE-OFDM phase, , with the embedded OFDM signal can be modeled as Gaussian distributed based on the central limit theorem [43] with variance ([7], p. 48). The higher order moments for the case of a zero mean Gaussian random variable are given as [44] (25) where the double factorial is defined as for odd . The variance of is then given as ([34], p. 158)

was previously defined in (8) and (26)

(24)

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The higher order terms are ignored. Similarly,

is given as

(27) with variance is the cross noise term given as Finally,

.

(28) Since the in phase and quadrature noise is uncorrelated, the variance of can be shown to be . The overall noise at the output of the -th cosine matched filter is then given as . Since , , and are uncorrelated, the variance of is simply the sum of the individual variances, . Furthermore, is the sum of non-identically distributed independent random variables, and as shown in Appendix B, it satisfies the Lindberg condition which is a sufficient condition for the central limit theorem to hold [42]. Therefore, the overall noise, , can be modeled as Gaussian distributed with variance . The performance approximation for the enhanced receiver for small modulation indices for the -th cosine subcarrier is then given as (29) Similarly, it can be shown that the performance of the -th sine subcarrier is also given by (29). The performance over all suband is plotted in Fig. 9. carriers is given as It shows good agreement with simulation performance. VII. PERFORMANCE IN MULTIPATH FADING The performance of the new CE-OFDM receivers in frequency selective multipath fading channels is studied in this section. It was previously shown in [6] that a frequency domain equalizer (FDE) provides good equalization performance for CE-OFDM in time-invariant frequency selective fading channels when the arctangent receiver is employed. Here we show that the FDE also works well for the cases of the basic linear receiver and the enhanced receiver. When a cyclic prefix is used, the linear convolution with the channel is transformed to a circular convolution. Taking advantage of this property, the FDE can be implemented by performing the discrete Fourier transform (DFT) on the received signal followed by single tap equalization per subcarrier to correct the effect of the channel, , at each subcarrier. Finally, the inverse DFT is employed to obtain the equalized signal. In this paper, we employ the minimum mean squared error (MMSE) criterion to obtain the equalizer taps as (30)

Fig. 9. CE-OFDM simulation performance in comparison to the analytical ap. proximation in AWGN for the enhanced receiver for

where . Simulations were performed for two channel models representing frequency selective fading channels [6]. The channel models were defined based on the statistics of the channel impulse response, of length , which was normalized as . Channel A is a two path model with equal power from both paths with the second path delayed by 5 s. It has a coherence bandwidth around 67 kHz. Channel B has an exponential power delay profile (PDP) with a s. delay spread of 9 s and It has a coherence bandwidth around 147 kHz. The performance of the BLR and ER as well as the arctangent receiver for channels A and B is presented in Figs. 10–11. It was evaluated with a CE-OFDM symbol period of s, a cyclic prefix of 10 s and FDE based equalization with perfect channel state information (CSI). An oversampling factor of results in a system front-end bandwidth of 1 MHz. Both the novel receivers, BLR and ER, perform well compared to the arctangent receiver for all cases. While the arctangent receiver provides better performance at low BERs for , the ER outperforms it at high BERs. For example, the ER provides over 1.2 dB better performance at a BER of 0.1 for channel B. This has a significant impact on the error correction coding performance as shown in Section VIII. VIII. APPLICATION OF CONVOLUTIONAL CODING As previously shown, the noise at the matched filter outputs of the BLR is Gaussian distributed. Also, for the ER and the arctangent receiver [7], [12], the noise is well modeled as Gaussian distributed. Therefore, the conditional probability density function (PDF) of the -th matched filter output given input data is given as

(31) where is the signaling component of the matched filter output and is the noise variance. When the matched filter outputs are exactly Gaussian distributed, the maximum likelihood (ML)


Fig. 10. CE-OFDM receiver performance comparison in frequency selective . fading (Channel A, two paths with equal power) using a FDE for

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Fig. 12. CE-OFDM performance of the BLR, ER and Arctangent receivers in . AWGN using a rate 1/2 convolutional code of constraint length 9 for

(34)

Fig. 11. CE-OFDM receiver performance comparison in frequency selective . fading (Channel B, exponential power delay profile) using a FDE for

sequence of length which maximizes the joint conditional PDF of the 's leads to the ML detector that minimizes the Euclidean distance metric. By modeling the matched filter outputs as Gaussian distributed, the decoding metrics for the arctangent, basic linear and enhanced receivers are obtained as below. It should be noted that the decoding metric can only be considered the maximum likelihood decoding metric for the case of the BLR for which the noise is exactly Gaussian distributed.

(32)

(33)

While the arctangent based receiver generally provides slightly better performance than the ER at low BERs for moderate modulation indices, the ER provides better performance at high BERs ( in most cases). Since the error control coding corrects all errors that fall within the code's minimum distance, the performance advantage of the ER at high BERs translates into a significant performance advantage in a coded system. Figs. 12–13 show the performance comparison of CE-OFDM in AWGN for the case of a rate convolutional code with a constraint length of 9 with Viterbi decoding. Both new receivers provide slightly better performance than the arctangent receiver for . For the case of , the performance advantage is much more significant, over 1.2 dB at a BER of . Finally, Fig. 14 provides a performance comparison for a frequency selective fading channel (Channel B) when an FDE is employed along with error correction coding. Both the BLR and ER receivers outperform the arctangent receiver in this case as well, with the ER providing better performance by over 1.6 dB at a BER of . IX. CONCLUSION In this paper, we developed novel receivers for CE-OFDM . Our refor low to moderate modulation indices sults indicate that the basic linear receiver performs well for small modulation indices while the enhanced receiver performs well for low and moderate modulation indices . These receivers do not employ a phase demodulator and are thus immune to the threshold effect and phase wrapping issues. We compared the uncoded performance of these new receivers to the conventional arctangent based receiver for AWGN and multipath fading channels and found the performance to be comparable for most modulation indices. For

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selective fading channel with an exponential power delay profile as considered in this paper, the enhanced receiver provided a gain of over 1.6 dB compared to the arctangent receiver at a BER of . The results for both AWGN and multipath fading channels with error correction coding show that these new receiver structures provide an attractive choice for any practical implementation of CE-OFDM. APPENDIX A ANALYSIS OF THE CUBIC TERM Consider only the first cosine term from (10), more specifically the cases of interest are

(A1) Fig. 13. CE-OFDM performance of the BLR, ER and Arctangent receivers in convolutional code of constraint length 9 for . AWGN using a rate

This represents all the cases for different combinations of that result in a cosine term that impacts the -th cosine matched filter. These cases are given by all possible combinations that result in . For example, for the case of , we have combinations of the following forms: For , there is 1 combination, . For , there are 2 combinations, . , there are 3 combinations, For . Continuing in this manner, for , there are combinations. Therefore, for the case of , the triple sum (over ) of the first cosine term in (10) results in cosine terms from all combinations that impact the cosine matched filter. Generalizing over any -th matched filter, the number of cosine terms generated due to the triple sum (over of the first cosine term in (10) is given as (A2)

Fig. 14. CE-OFDM performance of the BLR, ER and Arctangent receivers in frequency selective fading (Channel B, Exponential power delay profile) using . a rate 1/2 convolutional code of constraint length 9 for

higher modulation indices, the novel BLR and ER receivers outperformed the arctangent receivers at high BERs while the arctangent receiver provided slightly better performance at low BERs. There are several advantages for using these novel receivers over the arctangent receiver. The arctangent receiver requires the computation of the arctangent which can require up to 12 iterations for an efficient CORDIC algorithm implementation. Even a table lookup based approach at the expense of accuracy requires computation of table addresses and storage space for the stored vales. These receivers altogether alleviate the need for a phase demodulator at the receiver, resulting in a reduction in complexity. Finally, the performance of the new receivers was studied for the cases when error correction coding is applied. The novel BLR and ER receivers were found to not only provide excellent performance but they also significantly outperformed the arctangent based receiver. For example, for the case of a frequency

The second and third cosine terms in (10) also generate the same number of cosine terms that impact the -th matched filter. By going through a similar exercise, it can be shown that the fourth cosine term in (10) results in

cosine terms that im-

pact the -th cosine matched filter, with . Therefore, the total number of terms from (10) that impact the -th cosine matched filter is (A3) where is the unit step function that equals 1 for and is 0 otherwise. The identity can be used to further simplify this expression as

(A4)


This expression represents the number of terms from the triple sum representing the cubic term in (10) that impact the -th cosine matched filter at the OFDM receiver. This expression is also valid for the case of the sine terms from (10) that impact the -th sine matched filter. The number of these interfering terms that contribute constructively at the matched filter need to be determined i.e., all such terms that not only impact the -th matched filter but also and for the -th have the correct data symbols as well cosine and sine matched filters respectively). Considering the first cosine term from (10), the cases corresponding to and (with ) results in

where was employed. In these cases, are present on the terms and the correct data symbols these contribute constructively at the receiver. Similarly, it can be shown that the cases corresponding to and (with ) also have the correct data symbols. Finally, the case of also results in one term with the correct data symbol (since ). Therefore, the total number of terms that are generated from the first term of (10) for different combinations of with the correct data symbol is . It can similarly be shown that the second and third terms of (10) also contribute this number of terms with the correct data symbol at the -th cosine matched filter while the fourth term does not contribute any terms. Therefore, the total number of terms that contribute constructively at the -th cosine or the -th sine matched filter of the OFDM receiver is (A5)

APPENDIX B NOISE MODELING FOR THE ENHANCED RECEIVER The total noise at the output of the -th matched filter of the OFDM demodulator of the enhanced receiver is (B1) where and with

and

. 's, are For a given message signal, the random variables, independent since they are based on independent in phase and quadrature noise samples. The Lindberg condition [42] is given for as follows. For every ,

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where Since

and and and

since

is the indicator function. , we define with . Based on this and

's are identically distributed,

(B2) denote the random variable . In order to show that , it should be noted that is nonzero if and only . Since this if event has probability that tends to zero as , it can be concluded that by the definition of convergence in and , based on probability [42]. Since the dominated convergence theorem [42], and the Lindberg condition is satisfied.

Let

ACKNOWLEDGMENT The authors would like to express their gratitude to Dr. S. Thompson for his invaluable advice and guidance on all aspects of CE- OFDM. The authors would also like to especially thank Dr. R. Axford for his invaluable advice over the years on communications research and applications. REFERENCES [1] H. Ochiai and H. Imai, “On the distribution of the peak-to-average power ratio in OFDM signals,” IEEE Trans. Commun., vol. 49, no. 2, pp. 282–289, Feb. 2001. [2] S. H. Hang and J. H. Lee, “An overview of peak-to-average power ratio reduction techniques for multicarrier transmission,” IEEE Wireless Commun., vol. 12, no. 2, pp. 56–65, Apr. 2005. [3] F. H. Raab, P. Asbeck, S. Cripps, P. B. Kenington, Z. B. Popovic, N. Pothecary, J. F. Sevic, and N. O. Sokal, “Power amplifiers and transmitters for RF and microwave,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 814–826, Mar. 2002. [4] T. Svensson and T. Eriksson, “On power amplifier efficiency with modulated signals,” presented at the IEEE Veh. Tech. Conf., Taipei, 2010. [5] G. Breed, “The quest for power amplifier linearity and efficiency,” High Frequency Electron., vol. 10, no. 6, pp. 6–7, Jun. 2011. [6] S. C. Thompson, A. U. Ahmed, J. G. Proakis, J. R. Zeidler, and M. J. Geile, “Constant envelope OFDM,” IEEE Trans. Commun., vol. 56, no. 8, pp. 1300–1312, Aug. 2008. [7] S. C. Thompson, “Constant Envelope OFDM Phase Modulation” Ph.D. dissertation, Univ. California, San Diego, CA, USA, 2005 [Online]. Available: http://zeidler.ucsd.edu/students/thesis_sthompson.pdf, [Online]. Available: [8] S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “Constant envelope binary OFDM phase modulation,” presented at the IEEE Milcom Boston, MA, USA, 2003.

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Ahsen U. Ahmed received his B.S. from Texas Tech University, M.S. from Purdue University, and Ph.D. from the University of California, San Diego (UCSD), all in electrical engineering. He works at the Space and Naval Warfare Systems Center Pacific as a researcher in the area of wireless and satellite communications. His interests include innovative application of novel signal processing and analysis techniques.

James R. Zeidler (M'76–SM'84–F'94) is the president and CEO of Adaptive Dynamics and is concurrently an emeritus research scientist and senior lecturer in the Department of Electrical Engineering at UCSD. He is a faculty member of the UCSD Center for Wireless Communications and the California Institute for Telecommunications and Information Technology and has advised/co-advised 25 completed Ph.D. dissertations at UCSD. He has more than 250 technical publications and fourteen patents for communication, signal processing, data compression techniques, and electronic devices. Dr. Zeidler was elected Fellow of the IEEE in 1994 for his technical contributions to adaptive signal processing and its applications. He received the Navy Meritorious Civilian Service Award in 1991 and the Lauritsen-Bennett Award for Achievement in Science from the Space and Naval Warfare Systems Center in 2000. He was an Associate Editor of the IEEE Transactions on Signal Processing and was a member of the technical committee on Underwater Acoustic Signal Processing for the IEEE Signal Processing Society.