Abstract:
Orthogonal frequency division multiplexing (OFDM) is a spectrally efficient multicarrier modulation technique for high speed data transmission over multipath fading channels, but has low power efficiency. OFDM signals have large crest factors, or peak-to-average power ratios (PARs) which lead to power inefficiency in the RF portion of the transmitter. Selected mapping can be used to reduce the PAR of an OFDM signal and is distortionless. A technique is disclosed that links the index of a phase rotation sequence used in selected mapping to the location of pilot tones that are used to estimate the channel. Each pilot tone location-phase sequence selection produces a different PAR value for the time-domain OFDM signal, and the signal with the lowest PAR value is transmitted. The technique is “blind” in that the optimum pilot tone location-phase sequence index is not transmitted as side information. A technique to blindly detect the optimum index at the receiver is also disclosed.

Description:
This application claims the benefit of U.S. Provisional Application No. 60/663,148, filed Mar. 18, 2005. 

   STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
   This invention was made with Government support under Agreement No. CCR-0218778 awarded by the National Science Foundation, and the U.S. Army under contract No. DAAD19-01-2-0011. The U.S. Government may have a paid-up license in this invention and the right in limited circumstances to require the patent owner to license to others on reasonable terms as provided for by the terms of the Agreement. 

   BACKGROUND 
   The present invention relates to RF transmission systems and methods, and more particularly, to peak-to-average power ratio (or crest factor) reduction in orthogonal frequency division multiplexing (OFDM) systems using blind selected pilot tone modulation. 
   OFDM is a spectrally efficient multicarrier modulation technique for high speed data transmission over multipath fading channels. However, OFDM signals suffer from significant amplitude fluctuations; i.e., they exhibit large peak-to-average power ratios (PARs). Crest factor is the square root of PAR so they have the same value in dB. These two terms are used interchangeably in the literature and herein. High PARs require significant backoff of the average operating power of a RF power amplifier if the signal is to be linearly amplified. Power inefficiency leads to low battery life for a mobile user and high operating cost for the base station. According to G. Rabjohn and J. Wight, in “Improving efficiency, output power with 802.11a out-phasing PAs,”  CommsDesign.com  ( EE Times ), January 2004, the high power consumption and limited performance of traditional 802.11a OFDM systems had delayed the adoption of 802.11a and dual-band WLAN products. 
   Denote by {X l [k] k=0   N−1 } the lth block of the frequency domain OFDM signal drawn from a known constellation, where N is the number of sub-carriers. For the balance of this disclosure, the block index l will be dropped for notational simplicity, since OFDM can be free of inter-block interference with proper use of the cyclic prefix. The complex baseband OFDM signal can be written as 
                     x   ⁡     (   t   )       =       1     N       ⁢       ∑     k   =   0       N   -   1       ⁢       X   ⁡     [   k   ]       ⁢     ⅇ         j   ⁢   2   ⁢   π     ⁢           ⁢   kt       T   s                 ,     
     ⁢     0   ≤   t   ≤     T   s               (   1   )               
where T s  is the OFDM symbol period and j=√{square root over (−1)}. The PAR of x(t) is defined by H. Ochiai, for example, in “Performance analysis of peak power and band-limited OFDM system with linear scaling,”  IEEE Trans. Wireless Commun. , vol. 2, no. 5, pp. 1055-1065, September 2003, as
 
                     PAR   ⁡     (     x   ⁡     (   t   )       )       =       P   max       P   av         ,           (   2   )               
where P max =max 0≦l≦T|x(t)|   2  is the peak power, P av =Ē|x(t)| 2  is the average power of the OFDM symbol, and  E [.] denotes expectation, or time-averaged expectation if x(t) is nonstationary. Nyquist-rate sampled OFDM signal is given by x[n]=x(t)| t=nT,IN .
 
   According to an Altera Corporation white paper entitled “Accelerating WiMAX system design with FPGAs,”, October 2004, http://www.altera.com/literature/wp/wp_wimax.pdf, crest factor reduction (CFR) is an essential function for OFDM based systems such as WiMAX (IEEE 802.16). The topic of CFR has attracted a lot of attention in the recent years. Proposed techniques include (i) distortionless CFR, such as coding discussed by A. E. Jones, T. A. Wilkinson, and S. K. Barton, in “Block coding scheme for reduction of peak to mean envelope power ratio of multicarrier transmission scheme,”  Elec. Lett. , vol. 30, no. 25, pp. 2098-2099, December 1994, tone reservation discussed by J. Tellado, in  Multicarrier Modulation with Low PAR—Applications to DSL and Wireless , Kluwer Academic, 2000 and B. S. Krongold and D. L. Jones, in “An active-set approach for OFDM PAR reduction via tone reservation,”  IEEE Trans. Signal Processing , vol. 52, issue 2, pp 495-509, February 2004, tone injection discussed by J. Tellado, in  Multicarrier Modulation with Low PAR—Applications to DSL and Wireless , Kluwer Academic, 2000, selected mapping discussed by R. W. Bäuml, R. F. H. Fischer and J. B. Huber, in “Reducing the peak-to-average power ratio of multicarrier modulation by selected mapping,”  Elec. Lett. , vol. 32, no. 22, pp. 2056-2057, October 1996, M. Breiling, S. H. Muller-Weinfurtner, and J. B. Huber, in “SLM peak-power reduction without explicit side information,”  IEEE Commun. Lett.,  vol. 5, no. 6, pp. 239-241, June 2001, and A. D. S. Jayalath and C. Tellambura, in “A blind SLM receiver for PAR-reduced OFDM,” in  Proc. IEEE Vehicular Technology Conference—Fall , vol. 1, pp. 219-222, September 2002, and partial transmit sequence discussed by A. D. S. Jayalath and C. Tellambura, in “Adaptive PTS approach for reduction of peak-to-average power ratio of OFDM signal,”  Elec. Lett. , vol. 36, no. 14, pp 1226-1228, July 2000; (ii) CFR with distortion, such as deliberate clipping discussed by S. M. Ju and S. H. Leung, in “Clipping on COFDM with phase on demand,”  IEEE Commun. Lett. , vol. 7, no. 2, pp. 49-51, February 2003, transmit filtering discussed by S. B. Slimane, in “Peak-to-average power ratio reduction of OFDM signals using pulse shaping,” in  Proc. IEEE GLOBECOM  2000, vol. 3, pp. 1412-1416, November 2000, companding approaches discussed by T. Jiang and G. Zhu, in “Nonlinear companding transform for reducing peak-to-average power ratio of OFDM signals,”  IEEE Trans. Broadcast. , vol. 50, no. 3, pp. 342-346, September 2004; and (iii) various combinations of the above. These methods entail different tradeoffs involving CFR capability, complexity, and information rate. 
   The techniques described herein relate to a selected mapping (SLM) approach which was first proposed by R. W. Bäuml, R. F. H. Fischer and J. B. Huber in “Reducing the peak-to-average power ratio of multicarrier modulation by selected mapping,”  Elec. Lett . , vol. 32, no. 22, pp. 2056-2057, October 1996. SLM is distortionless and offers moderate to significant amount of CFR. Denote by φ k   (m)  0≦k≦N, 0≦m≦M−1, a set of M (random) phase sequences of length N each. In SLM, the phases of X[k] is rotated as described by
 
 Z   (m)   [k]=X[k]e   jφ     k       (m)   .  (3)
 
   It is clear that Z (m) [k] and X[k] contain the same information, but their time-domain counterparts z (m) (t) and x(t) can have very different PAR values. From the Mcandidate z (m) (t) signals, z (  m )(t) , which has the lowest PAR, is transmitted. The index  m  (log 2  M bits) may be transmitted as side information, which is of critical importance to the receiver for decoding and is generally protected by channel coding discussed by R. W. Bäuml, R. F. H. Fischer and J. B. Huber, in “Reducing the peak-to-average power ratio of multicarrier modulation by selected mapping,”  Elec. Lett. , vol. 32, no. 22, pp. 2056-2057, October 1996. 
   If {φ k   (m) } are independent identical distributed (i.i.d.) satisfying E[e jφ     k       (m)   ]=0, then the best SLM performance can be achieved; the corresponding complementary cumulative distribution function (CCDF) is given by
 
 Pr{PAR [ z   (  m ) ( t )]&gt;γ}=[ Pr{PAR ( x ( t ))&gt;γ}] M .  (4)
 
   The simplest and yet optimal phase rotation table is one that has 0 and π entries with equal probability. In that case, no multiplication is necessary in equation (3) since e j0 =1 and e jπ =−1. The phase rotation table is pre-determined and is stored at both the transmitter and the receiver, so real-time optimization of the phase sequence is not necessary. 
   To avoid the information rate loss caused by the transmission of the optimum phase sequence index  m , a few blind SLM schemes have been proposed. In a paper by M. Breiling, S. H. Muller-Weinfurtner, and J. B. Huber, entitled “SLM peak-power reduction without explicit side information,”  IEEE Commun. Lett. , vol. 5, no. 6, pp. 239-241, June 2001, a scrambling technique was described. A log 2  M-bit binary label is inserted as a prefix to the frequency-domain OFDM signal and passed through a scrambler. Since the selected label is used in the receiver implicitly during descrambling, an erroneous reception of the label bits does not affect the error performance. In a paper by A. D. S. Jayalath and C. Tellambura, entitled “A blind SLM receiver for PAR-reduced OFDM,” in  Proc. IEEE Vehicular Technology Conference—Fall , vol. 1, pp. 219-222, September 2002, a blind SLM receiver was proposed by employing a maximum likelihood (ML) decoder, which avoids the transmission of any side information. However, the complexity and the error rate of the ML decoder are rather high as we will show in the simulation section. 
   In OFDM, channel state information (CSI) can be acquired by modulating pilot tones onto predetermined sub-carriers; this is called pilot tone assisted modulation (PTAM), discussed by R. Negi and J. Cioffi, in “Pilot tone selection for channel estimation in a mobile OFDM system,”  IEEE Trans. Consumer Electron. , vol. 44, pp. 1122-1128, August 1998, and S. Ohno and G. B. Giannakis, in “Optimal training and redundant precoding for block transmissions with application to wireless OFDM,”  IEEE Trans. Commun. , vol. 50, no. 12, pp. 2113-2123, December 2002. 
   The concept of joint channel estimation and CFR was explored by M. J. Fernández-Getino García, O. Edfors and J. M. Páez-Borrallo, in “Joint channel estimation and peak-to-average power reduction in coherent OFDM: a novel approach,”  Proc. IEEE Vehicular Technology Conference—Spring , vol. 2, pp. 815-819, May 2001. The “diversity” offered by the pilot phase (as opposed to the pilot location—the focal point of the present method) was exploited and the transmission of side information was assumed by García et al. 
   It would be desirable to have an improved selected mapping CFR technique that avoids the transmission of any side information and entails a very accurate detection scheme in the receiver. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The various features and advantages of the present invention may be more readily understood with reference to the following detailed description taken in conjunction with the accompanying drawings, wherein like reference numerals designate like structural elements, and in which: 
       FIG. 1  illustrates an exemplary blind selected mapping system; 
       FIG. 2  illustrates a reduced-to-practice blind selected pilot tone modulation (BSPTM) transmitter; 
       FIG. 3  illustrates a reduced-to-practice BSPTM receiver; 
       FIG. 4  illustrates exemplary scenarios for shifting the pilot tones in a BSPTM transmitter; 
       FIG. 5  is a graph that demonstrates the CFR capability of BSPTM; 
       FIG. 6  is a graph that illustrates the magnitude response for the specific channel realization whose realization response is given by equation (24); 
       FIG. 7  demonstrates the robustness of the peak power detector in the presence of the frequency selective fading channel depicted in  FIG. 6 ; 
       FIG. 8  illustrates BSPTM-OFDM outperforms the conventional PTAM-OFDM in term of bit error rate (BER) for the fixed channel (given by equation (24)); and 
       FIG. 9  illustrates BSPTM-OFDM outperforms the conventional PTAM-OFDM in term of BER for the Rayleigh channel. 
   

   DETAILED DESCRIPTION 
   Disclosed herein are a novel crest factor reduction (CFR) technique and apparatus that provide for orthogonal frequency division multiplexing (OFDM) systems using blind selected pilot tone modulation. The technique combines the merits of PTAM and SLM, and is implemented using a novel joint channel estimation and crest factor reduction algorithm. Instead of fixing the pilot tone locations as in conventional PTAM, different pilot tone locations are employed, and movement of the pilot tones is synchronized with the choice of phase rotation sequence. The pilot tone/phase sequence combination that results in the lowest PAR of the time-domain signal is used for transmission. However, the optimum index is not transmitted as side information in order to maintain the information rate. At the receiver, by taking advantage of the disparity between the pilot tone and information signal powers, the optimum index is blindly detected by resorting to simple frequency-domain averages. A reduced-to-practice implementation of the technique using the blind selected pilot tone modulation is discussed in detail, along with a more generic framework not specifically linked to the use of pilot tones. 
   In order to better understand the present approach, the PTAM technique is reviewed for OFDM, and the blind selected pilot tone modulation (BSPTM) technique is described. Computer simulations are described that demonstrate the impressive CFR capacity of the algorithm and its robust BER performance over frequency selective fading channels. 
   Review of PTAM-OFDM 
   A conventional OFDM transmitter processes a frequency domain signal X[k] that is to be transmitted using an inverse discrete Fourier transform (IDFT) circuit, an upsampling circuit, and a digital-to-analog (D/A) converter. The analog signal output by the D/A converter is input to a power amplifier for transmission by way of a wireless (or wired) channel. A conventional OFDM receiver processes the analog signal received over the wireless (or wired) channel. An analog-to-digital (A/D) converter converts the analog signal to a digital one and couples it to a discrete Fourier transform (DFT) circuit. The output of the discrete Fourier transform circuit is processed by a channel estimator. The output of the channel estimator is processed by symbol detection circuitry that outputs an estimate of the frequency domain signal transmitted by the transmitter. 
   In the OFDM transmission system with PTAM, P pilot tones are inserted in the frequency domain in order to acquire the channel state information (CSI); P≧L is assumed where L is the length of the finite impulse response (FIR) channel. The transmitted frequency domain signal can be described by X[k]=B[k] for k ∈Ω 0  and X[k]=S[k] for k ∈Ω 0   195  , where Ω 0  is the set of the P pilot tone indices in ascending order, Ω 0   ⊥  denotes the complement of Ω 0  (i.e., the set of N−P information sub-symbol indices in ascending order), {B[k]} k∈Ù     0    are the pilot tones, and {S[k]} k∈Ù     0     ⊥  are the frequency-domain information sub-symbols. 
   According to S. Ohno and G. B. Giannakis, in “Optimal training and redundant precoding for block transmissions with application to wireless OFDM,”  IEEE Trans. Commun. , vol. 50, no. 12, pp. 2113-2123, December 2002, the optimal way to place the pilot tones is to modulate P=L pilot tones with equal power onto equi-spaced sub-carriers. For simplicity, it is assumed that the number of sub-carriers N is an integer multiple of P; i.e., R=N/P is an integer. Define a set of P equi-spaced pilot tone indices as
 
Ω 0 = Δ   {k   i   |k   i   =iR+θ   0 , 0≦ i≦P− 1, 0≦θ 0   ≦R− 1},  (5)
 
which may be characterized by θ 0  alone.
 
   At the receiver of the OFDM transmission system, after removing the cyclic prefix and performing a DFT, a set of N linear equations in the frequency domain is obtained 
                       Y   ⁡     [   k   ]       =         X   ⁡     [   k   ]       ⁢     H   ⁡     [   k   ]         +     V   ⁡     [   k   ]           ,     
     ⁢     k   =   0     ,   1   ,   …   ⁢           ,     N   -   1     ,     
     ⁢   where     ⁢     
     ⁢       Y   ⁡     [   k   ]       =       1     N       ⁢       ∑     n   =   0       N   -   1       ⁢       y   ⁡     [   n   ]       ⁢     ⅇ       -   j2π     ⁢     kn   N                 ⁢     
     ⁢   and   ⁢     
     ⁢     V   ⁡     [   k   ]       =       1     N       ⁢       ∑     n   =   0       N   -   1       ⁢       v   ⁡     [   n   ]       ⁢     ⅇ       -   j2π     ⁢     kn   N                       (   6   )               
are the normalized DFT of the received signal y[n] (after the removal of the cyclic prefix) and the zero-mean additive noise v[n], respectively, and
 
             H   ⁡     [   k   ]       =       ∑     n   =   0       L   -   1       ⁢       h   ⁡     [   n   ]       ⁢     ⅇ       -   j2π     ⁢     kn   N                   
is the frequency response of the composite channel (the convolution of the transmit filter, the frequency selective channel, and the receive filter).
 
   Since X[k]=B[k] for k ∈ Ù 0 , an estimate of H[k] at P points of Ù 0      obtained from equation (   6): 
                       H   ^     ⁡     [   k   ]       =       Y   ⁡     [   k   ]         B   ⁡     [   k   ]           ,     
     ⁢     k   ∈       Ω   0     .               (   7   )               
Since H[k] is constrained by P parameters {h[n]} n=0   P−1 , H[k] can be estimated at any k. Afterwards, the information sub-symbols can be estimated as
 
                       S   ^     ⁡     [   k   ]       =       Y   ⁡     [   k   ]           H   ^     ⁡     [   k   ]           ,     
     ⁢     k   ∈     Ω   0   ⊥       ,           (   8   )               
which are then decoded to yield the  S [k] estimates belonging to the symbol constellation.
 
   Blind Selected Pilot Tone Modulation 
   The blind selected pilot tone modulation (BSPTM) technique described below is a combination of channel sounding and effective crest factor reduction, at a low computational cost. The BSPTM technique may be advantageously employed in a mobile communication system comprising a blind selected pilot tone modulation (BSPTM) transmitter  20  and a blind selected pilot tone modulation receiver  30 . Referring to the drawing figures,  FIG. 1  illustrates an exemplary blind selected mapping (BSLM) system  10 .  FIG. 2  illustrates an exemplary educed-to-practice blind selected pilot tone modulation (BSPTM) transmitter  20 , and  FIG. 3  illustrates an exemplary reduced-to-practice BSPTM receiver  30 . 
   More particularly,  FIG. 1 , illustrates an exemplary generic BSLM communication system  10 . The generic BSPTM system  10  comprises a BSLM transmitter  11  and a BSLM receiver  12 . The BSLM transmitter  10  processes a frequency domain signal X[k] that is to be transmitted over a wireless channel using a tagging process  21  that uniquely identifies each block of data in a plurality of sub-channels. The signal output of the tagging process- 21  is then transformed by way of a differentiating process  22  that is used to differentiate the blocks of data in each of the plurality of sub-channels. The differentiating process  22  comprises an invertible transform for generating different sub-channel signals. Each of the blocks of data in each of the plurality of sub-channels is then conventionally processed by an inverse discrete Fourier transform (IDFT)  11 , and is upsampled  12 . The blocks of data in each of the plurality of sub-channels are then summed and processed  23  to select the channel (or block of data) having the minimum crest factor (CF), or peak-to-average power ratio (PAR). The block of data in the selected sub-channel is then transformed to an analog signal using a digital-to-analog (D/A) converter  13 . The analog signal output by the D/A converter  13  is input to a power amplifier  14  for transmission over the wireless channel to the receiver  30 . 
   At the receiver  30 , the received block of data is digitized using an analog-to-digital converter  15  and is transformed to the frequency domain using a discrete Fourier transform (DFF) circuit  16 . The tag generated by the tagging process  21  in the transmitter  20  is then detected using a tag detector  24 . This identifies the sub-channel that was used to transmit the block of data. Then, a channel estimator  17  estimates the effects of the wireless channel, and this signal is processed by a symbol detector  17  that outputs an estimate {circumflex over (X)}[k] of the frequency domain signal transmitted by the transmitter  20 . 
   With the above in mind, a reduced-to-practice implementation of the technique using blind selected pilot tone modulation will now be discussed. 
   Referring to  FIG. 2 , an exemplary reduced-to-practice BSPTM transmitter  20  processes the frequency domain signal X[k] using a tagging process  21  that shifts pilot tones associated with the blocks of data in each of the plurality of sub-channels. The blocks of data output by the tagging process  21  are processed using the differentiating process  22  to separately rotate the phases of each signal in the block of data. The differentiating process  22  may comprise a lookup table  26  containing a plurality of pseudo-random phase sequences that are used to rotate the phases of the frequency domain signal X[k] in the sub-channels. The combination of the tagging process  21  and differentiating process  22  uniquely identify each block of data and each sub-channel. 
   The individual phase rotated sub-channels (blocks of data) are each inverse discrete Fourier transformed  11  and upsampled  12 . Each of the inverse Fourier transformed and upsampled sub-channels are then summed and processed  23  to select the sub-channel (block of data) having the minimum crest factor (CF), or peak-to-average power ratio (PAR). In particular, the sub-channels are processed using a crest factor selection algorithm that selects the signal having the minimum crest factor. The selected signal (sub-channel) having the minimum crest factor is converted to an analog signal by a digital-to-analog (D/A) converter  13  and input to a power amplifier  14  for transmission over the channel. 
   As is shown in  FIG. 3 , an exemplary reduced-to-practice BSPTM receiver  30  comprises an analog-to-digital (A/D) converter  15  that converts a received analog signal into a digital signal corresponding to the sub-channel that was transmitted. The digital signal is discrete Fourier transformed  16 . The discrete Fourier transformed signal is processed by a peak power detector  24  that detects the peaks in it, thus identifying the shift of the pilot tones as well as the associated sub-channel that was transmitted. The peak power detector  24  outputs an index (  m ) that is indicative of the phase rotation used in the differentiating process  22  used in the transmitter  20 . The lookup table  26  outputs the  m  th phase sequence for the consequent inverse phase rotation. The discrete Fourier transformed signal is then inverse phase rotated  25  by using the output phase sequence from the lookup table  26 . The channel estimator  17  estimates the communications channel based on the pilot tones. The output of the channel estimator  17  is processed by a symbol detector  18  to produce an estimate of the frequency domain signal transmitted by the transmitter  20 . 
   Implementation details regarding the reduced-to-practice transmitter  20  and receiver  30  will be discussed below. 
   Disparity in the Pilot and Information Signal Powers 
   An interesting feature of PTAM is that the pilot tones generally have stronger average power than the information sub-symbols, and this forms the basis of the blind selected pilot tone modulation technique. 
   Denote β, as the power allocation factor, which is the ratio between the total power allocated to the pilot tones and the total transmitted power; i.e., 
                 β   =         P   ⁢           ⁢     σ   p   2           P   ⁢           ⁢     σ   p   2       +       (     N   -   P     )     ⁢     σ   s   2           .             (   9   )               
where
 
             σ   p   2     =       1   /   P     ⁢       ∑     k   ∈     U   0         ⁢            B   ⁡     [   k   ]            2               
is the average power of the pilot tones and σ s   2  is the variance of S[k]; i.e., σ s   2 =E|S[k]| 2 .
 
   When the pilot tones are equi-powered; i.e.,|B[k]| 2 =σ p   2 , ∀ k ∈Ω 0 , an optimal β was determined by S. Ohno and G. B. Giannakis, in “Optimal training and redundant precoding for block transmissions with application to wireless OFDM,”  IEEE Trans. Commun ., vol. 50, no. 12, pp. 2113-2123, December 2002 as 
                 β   =     1   -     1     1   +       1       N   /   P     -   1                       (   10   )               
by minimizing the mean squared error (MSE) of the source estimates Ŝ[k], k ∈ Ω 0   ⊥ . Combining equations (9) and (10),
 
                     σ   p   2       σ   s   2       =         N   P     -   1               (   11   )               
which depends on N/P only. Since N&gt;&gt;P, the pilot tones have much stronger power than the information sub-symbols. For example, for P≦16 and N≧160, (11) gives rise to σ p   2 /σ s   2 ≦3. On the other hand, if P≦8 and N≧296, then σ p   2 /σ s   2 ≧6. Both are realistic scenarios.
 
   The σ p   2 /σ s   2 &gt;&gt;1 relationship helps to detect the pilot tone location parameter θ 0 , which will be described later. 
   Crest Factor Reduction (CFR) Using BSPTM 
   According to S. Ohno et al., as long as the pilot tones are equi-powered and equi-spaced and the additive noise is white, channel estimation performance is not affected by the choice of θ 0 . Therefore, instead of using a pre-selected θ 0 , different frequency shifts θ 0   (m)  may be tried for the pilot tones. One aspect of the blind selected pilot tone modulation approach is to tie the location of the pilot tones to the different phase rotation sequences. This enables crest factor reduction without the transmission of side information. 
   Recall that m is used to index the rows of the phase rotation table. Use the same m to index the M candidate frequency shifts for the pilot tones; i.e.,
 
θ Δ ={θ 0   (0) , θ 0   (1) , . . . , θ 0   (m) , . . . , θ 0   (M−1) }.  (12)
 
   The maximum number of distinct pilot tone locations is R=N/P, in which case {θ 0   (0) =0, θ 0   (1) =1, . . . , θ 0   (R−1) =R−1}. However, since R can be quite large and for practical reasons, M does not need to be greater than 8, for example, there is some flexibility in designating è. For example, if R=8 and M=4, then {θ 0   (0) =0,θ 0   (1) =2, θ 0   (2) =4, θ 0   (3) =6} (see  FIG. 4 ) or {θ 0   (0) =0,θ 0   (1) =1,θ 0   (2) =3,θ 0   (3) =7}, and so on.  FIG. 4  illustrates exemplary scenarios for X (m) [k] with N=16, P=2, M=4. The M delays are preferably equally-spaced in order to minimize the detection error in  m ; i.e., the possible pilot shifts are {θ 0   (0) =0, θ 0   (1) =R/M, . . . , θ 0   (M−1) =R(M−1)/M}. In addition, both the transmitter  20  and the receiver  30  should have the knowledge of θ. 
   In the blind selected pilot tone modulation approach, the mth PTAM-OFDM signal is given by 
                     X     (   m   )       ⁡     [   k   ]       =     {             B   ⁡     [   k   ]       ,               k   ∈     Ω   0                   S   ⁡     [   k   ]       ,               k   ∈     Ω   0   ⊥                       (   13   )               
where 0≦k≦N−1, 0≦m≦M−1, and Ω m  is characterized by θ 0   (m)  similar to the way that Ω 0  is characterized by θ 0 .
 
   Next, phase rotations are performed
 
 Z   (m)   [k]=X   (m)   [k]e   jφ     k       (m)   .  (14)
 
   Similar to SLM, z (m) (t) and PAR(z (m) (t)) are evaluated and z (  m ) (t), which has the lowest PAR among {z (m) (t)}, is transmitted. In other words, the optimum pilot tone location—phase sequence index is 
   
     
       
         
           
             
               
                 
                   m 
                   _ 
                 
                 = 
                 
                   
                     
                       arg 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       min 
                     
                     
                       0 
                       ≤ 
                       m 
                       ≤ 
                       
                         M 
                         - 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     
                       { 
                       
                         PAR 
                         ( 
                         
                           
                             z 
                             
                               ( 
                               
                                 m 
                                 _ 
                               
                               ) 
                             
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         ) 
                       
                       } 
                     
                     . 
                   
                 
               
             
             
               
                 ( 
                 15 
                 ) 
               
             
           
         
       
     
   
   Based on the results of S. Wei, D. L. Goeckel, and P. A. Kelly, in “The complex envelope of a bandlimited OFDM signal converges weakly to a Gaussian random process: proof and application,” http://www.ece.Isu.edul/swei, for example, it can be shown that the CCDF of the PAR of the transmitted BSPTM-OFDM signal z (  m ) (t) is given by 
   
     
       
         
           
             
               
                 
                   
                     
                       Pr 
                       ⁢ 
                       
                         { 
                         
                           
                             PAR 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   z 
                                   
                                     ( 
                                     
                                       m 
                                       _ 
                                     
                                     ) 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               ) 
                             
                           
                           &gt; 
                           γ 
                         
                         } 
                       
                     
                     = 
                     
                       
                         [ 
                         
                           1 
                           - 
                           
                             ⅇ 
                             
                               
                                 - 
                                 
                                   ⅇ 
                                   
                                     - 
                                     γ 
                                   
                                 
                               
                               ⁢ 
                               N 
                               ⁢ 
                               
                                 
                                   
                                     
                                       λ 
                                       ~ 
                                     
                                     N 
                                   
                                   ⁢ 
                                   log 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   N 
                                 
                               
                             
                           
                         
                         ] 
                       
                       M 
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   where 
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     λ 
                     ~ 
                   
                   = 
                   
                     
                       4 
                       ⁢ 
                       
                         
                           π 
                           2 
                         
                         3 
                       
                     
                     - 
                     
                       
                         
                           
                             π 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 β 
                                 P 
                               
                             
                             ) 
                           
                         
                         2 
                       
                       . 
                     
                   
                 
               
             
             
               
                 ( 
                 16 
                 ) 
               
             
           
         
       
     
   
   Blind Detection of  m   
   At the receiver  30 , the optimum index  m  must be determined. Replace the X[k] in equation (6) by the Z (  m ) [k] of equation (14) and write: 
                         Y   ⁡     [   k   ]       =           Z     (     m   _     )       ⁡     [   k   ]       ⁢     H   ⁡     [   k   ]         +     V   ⁡     [   k   ]                     =     {                 B   ⁡     [   k   ]       ⁢     ⅇ     jϕ   k     (     m   _     )         ⁢     H   ⁡     [   k   ]         +     V   ⁡     [   k   ]         ,                 k   ∈     Ω     m   _         ,                     S   ⁡     [   k   ]       ⁢     ⅇ     jϕ   k     (     m   _     )         ⁢     H   ⁡     [   k   ]         +     V   ⁡     [   k   ]         ,                 k   ∈     Ω     m   _     ⊥       ,                           (   17   )               
The task here is to detect θ 0   (  m )  (or equivalently,  m ) from {Y[k]} k=0   N−1 , knowing the candidate set of locations in θ.
 
   The following assumptions are used in the description below: 
   1. s[n], v[n], and h[n] are mutually independent, 
   2. h[n] is i.i.d. zero-mean with variance σ h   2 , and 
   3. |B[k]| 2 σ p   2  is constant ∀ k ∈ Ω   m   (equi-powered pilots). 
   From assumption 2, it is inferred that H[k] has mean zero and variance Lσ h   2 , ∀ k. Furthermore: σ s   2 =E|S[k]| 2  and σ v   2 =E|V[k]| 2 . It follows from equation (17) that 
   
     
       
         
           
             
               
                 
                   E 
                   ⁢ 
                   
                     
                        
                       
                         Y 
                         ⁡ 
                         
                           [ 
                           k 
                           ] 
                         
                       
                        
                     
                     2 
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 σ 
                                 p 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     σ 
                                     h 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               σ 
                               v 
                               2 
                             
                           
                           , 
                         
                       
                     
                     
                       
                         
                           
                             k 
                             ∈ 
                             
                               Ω 
                               
                                 m 
                                 _ 
                               
                             
                           
                           , 
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 σ 
                                 s 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     σ 
                                     h 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               σ 
                               v 
                               2 
                             
                           
                           , 
                         
                       
                     
                     
                       
                         
                           
                             k 
                             ∈ 
                             
                               Ω 
                               
                                 m 
                                 _ 
                               
                               ⊥ 
                             
                           
                           , 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 18 
                 ) 
               
             
           
         
       
     
   
   Next, let k=iR+r, where 0≦i≦P−1 and 0≦r≦R−1, and denote by Y i [r]=Y[iR+r] the ith sub-record (of length-R) of Y[k]. It follows from equation (18) that 
                   ρ   ⁡     [   r   ]       =         1   P     ⁢     E   ⁡     [       ∑     i   =   0       P   -   1       ⁢              Y   i     ⁡     [   r   ]            2       ]         =     {                 σ   p   2     ⁡     (     L   ⁢           ⁢     σ   h   2       )       +     σ   v   2       ,                 r   =     θ   0     (     m   _     )         ,                     σ   s   2     ⁡     (     L   ⁢           ⁢     σ   h   2       )       +     σ   v   2       ,               r   ≠       θ   0     (     m   _     )       .                         (   19   )               
Since σ p   2 &gt;σ s   2 , it is inferred from equation (19) that
 
   
     
       
         
           
             
               
                 
                   θ 
                   0 
                   
                     ( 
                     
                       m 
                       _ 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                       arg 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       max 
                     
                     
                       r 
                       ∈ 
                       
                         e 
                         . 
                       
                     
                   
                   ⁢ 
                   
                     { 
                     
                       ρ 
                       ⁡ 
                       
                         [ 
                         r 
                         ] 
                       
                     
                     } 
                   
                 
               
             
             
               
                 ( 
                 20 
                 ) 
               
             
           
         
       
     
   
   In practice, estimate ρ[r] as 
                       ρ   ^     ⁡     [   r   ]       =       1   P     ⁢       ∑     i   =   0       P   -   1       ⁢              Y   i     ⁡     [   r   ]            2           ,           (   21   )               
and find the optimum pilot location via
 
                     θ   ^     0     (     m   _     )       =         arg   ⁢           ⁢   max       r   ∈     e   .         ⁢     {       ρ   ^     ⁡     [   r   ]       }               (   22   )               
Since the receiver has the knowledge of θ, from {circumflex over (θ)} 0   (  m ) , a simple lookup table search yields {circumflex over (  m .
 
   Even if |H[iR+r]| 2 ≧0 exhibits a deep null at r=θ 0   (  m )  for a particular sub-record i, since P sub-records are involved in the averaging in equation (21) and σ p   2 &gt;&gt;σ s   2 , {circumflex over (ρ)}[r] is still likely to peak at r=θ 0   (  m ) . The side information  m  is critical for decoding at the receiver  30 . If {circumflex over (  m  is inaccurate for a particular OFDM block, the BER will be high for that block. The finite alphabet nature of è makes it less likely for errors to occur in θ 0   (  m ) . When the SNR is so low that σ v   2  dominates the other terms on the RHS of equation (19), ρ[r] at r≠θ 0   (  m )  and ρ[r] at r ≠θ 0   (  m )  become less distinguishable, and hence accurate detection of  m  becomes difficult. As is shown in simulations discussed below, at medium to high SNR levels, the detector in equation (22) performs quite reliably, especially when σ p   2 /σ s   2  is high. 
   Simulations 
   In the examples below (except when specified otherwise), it is assumed that the number of sub-carriers N=128, the length of the FIR channel L=4, the number of pilot tones P=L=4, and the power allocation factor β=0.15 (c.f. (10)). Except for the discussion of the blind detection of  m , the phase table includes independent identical distributed (i.i.d.) {0, π} entries with equal probability; in other words, a {e jφ     k       (m)   } table containing i.i.d. {1, −1} entries with equal probability. Such a sign change table is predetermined and is stored at both the transmitter  20  and the receiver  30 . The N−P information sub-symbols are independently drawn from a QPSK constellation with Gray coding. Under the unit channel energy constraint 
                 ∑     n   =   0       L   -   1       ⁢     E   ⁢            h   ⁡     [   n   ]            2         =   1     ,         
the signal-to-noise ratio (SNR) is defined as
 
                   SNR   =       P   dc       σ   v   2         ,           (   23   )               
where P dc  is the total amount of DC power consumed by the power amplifier  14  and σ v   2  is the variance of the additive white Gaussian noise. The effective SNR, which directly affects the BER, may be expressed as SNRe =P r /σ v   2 , where P t  is the average output power of the power amplifier  14  with the input signal z (  m ) (t). If an ideal linear power amplifier  14  is used and the signal is to be amplified undistorted, P t  is proportional to P dc /PAR if linear scaling is employed as described by Ochiai (3002). Therefore, SNRe∞P dc /PAR, and the benefit of effective crest factor reduction is realized as an increase in SNRe.
 
   CFR Performance In this example, approximate the continuous-time PAR of equation (2) by evaluating the discrete-time PAR of the 8-times oversampled OFDM signal discussed by J. Tellado (2000). 10 6  independent Monte Carlo trials were conducted. 
     FIG. 5  shows empirical CCDF curves (solid lines) of the PAR of the transmitted signal z (  m ) (t) for different number of selections, M, along with the theoretical CCDFs (dash-dotted lines) obtained from (16). M=1 corresponds to the original PTAM-OFDM case.  FIG. 5  shows that the empirical and the theoretical CCDFs are quite close. It is observed that when M=8, the proposed algorithm could achieve 3.5 dB of PAR reduction (as compared with the M=1 case) at the CCDF level of 10 −4 . It is also seen from  FIG. 5  that the larger the M, the smaller the resulting PAR. On the other hand, the computational complexity increases as M increases. There is also a diminishing return in the effective crest factor reduction capability as M further increases. As a rule of thumb, it is desirable to use min{R, 4}≦M≦min{R, 8}. 
   Blind Detection of  m   
   Below is an example that illustrates the blind detection of  m  from |Y[k]| 2 . In this example, SNR =0 dB, R=32, è={0, 4, 8, 12, 16, 20, 24, 28}, and thus M=8. The channel taps are assumed to be i.i.d. complex Gaussian distributed with zero-mean and variance σ h   2 =1/L (i.e., Rayleigh fading), and L=4.  FIG. 6  shows |H[kj]| vs. k for one realization of the Rayleigh fading channel with time-domain coefficients
 
 h=[ 0.2774−j0.4545, −0.4988+j0.1837, 0.1189+j0.1105, −0.0751−j0.6340] T ,  (24)
 
which exhibits several deep nulls in the frequency domain. Table 1 shows the PAR{z (m) (t)} values for one particular OFDM block, with 0 ≦m≦7. It is observed that m=6 corresponds to the lowest PAR value, thus the optimum pilot tone location parameter was θ 0   (6) =24. At the receiver  30 , calculate |Y[k]| 2 . In  FIG. 7 , for each sub-record |Y,[r]| 2 , circles indicate the values corresponding to the M candidate locations r ∈ θ. From the {circumflex over (ρ)}[r] plot, {circumflex over (θ)} 0   (  m ) =24 (or equivalently, {circumflex over (  m =6 ), which was indeed the true {circumflex over (θ)} 0   (  m )  that was used during transmission.
 
   
     
       
             
           
             
             
           
             
             
             
             
             
             
             
             
             
           
             
             
             
             
             
             
             
             
             
           
         
             
               TABLE 1 
             
           
           
             
                 
             
             
               PAR{z (m) (t)} for one OFDM symbol, 0 ≦ m ≦ 7. 
             
           
        
         
             
                 
               m 
             
           
        
         
             
                 
               0 
               1 
               2 
               3 
               4 
               5 
               6 
               7 
             
             
                 
                 
             
           
        
         
             
               PAR (dB) 
               7.44 
               7.33 
               10.34 
               8.11 
               8.55 
               8.03 
               6.69 
               7.36 
             
             
                 
             
           
        
       
     
   
   To illustrate the performance on the blind detection of m, Table 2 lists detection error rates for varying values of β, M, and SNR, calculated by averaging over 10 5  Monte Carlo trials. It is evident from Table 2 that the larger the β, the smaller the error rate. This is because when βis larger, ρ[r] of equation (19) stands out better at r=θ 0     m       d   . Moreover, it is observed from Table 2 that the larger the M, the higher the error rate in detecting θ 0   (  m ) . This is because there are more competing candidate m&#39;s when M is larger. When β is not too small (e.g., β&gt;0.1), the error rate can be quite small for SNR &gt;0 dB. However, β cannot be too large either, since when too much power is devoted to the pilot tones instead of the information sub-symbols, the receiver  30  becomes vulnerable to channel distortions and additive noise. 
   
     
       
             
           
             
             
           
             
             
             
             
             
             
             
             
           
             
             
             
             
             
             
             
             
             
           
         
             
               TABLE 2 
             
           
           
             
                 
             
             
               Error rates in detecting  m  for varying β, M, and SNR. 
             
           
        
         
             
                 
               SNR 
             
           
        
         
             
                 
               0 
               5 
               10 
               15 
               20 
               25 
               30 
             
             
                 
               dB 
               dB 
               dB 
               dB 
               dB 
               dB 
               dB 
             
             
                 
                 
             
           
        
         
             
               M = 4 
               β = 0.1 
               16.38% 
               2.28% 
               0.07%   
               0% 
               0% 
               0% 
               0% 
             
             
               M = 4 
               β = 0.2 
               2.15% 
               0.06% 
               0% 
               0% 
               0% 
               0% 
               0% 
             
             
               M = 4 
               β = 0.3 
               0.42% 
               0.01% 
               0% 
               0% 
               0% 
               0% 
               0% 
             
             
               M = 8 
               β = 0.1 
               23.99% 
               3.38% 
               0.13%   
               0% 
               0% 
               0% 
               0% 
             
             
               M = 8 
               β = 0.2 
               2.99% 
               0.10% 
               0.004%    
               0% 
               0% 
               0% 
               0% 
             
             
               M = 8 
               β = 0.3 
               0.68% 
               0.01% 
               0% 
               0% 
               0% 
               0% 
               0% 
             
             
               M = 16 
               β = 0.1 
               31.35% 
               4.96% 
               0.22%   
               0% 
               0% 
               0% 
               0% 
             
             
               M = 16 
               β = 0.2 
               4.09% 
               0.15% 
               0.004%    
               0% 
               0% 
               0% 
               0% 
             
             
               M = 16 
               β = 0.3 
               0.90% 
               0.01% 
               0% 
               0% 
               0% 
               0% 
               0% 
             
             
                 
             
           
        
       
     
   
   Comparison with A. D. S. Jayalath et al. (2002) on the detection of  m   
   In this example, the performance of the blind selected pilot tone modulation technique is compared with that described by A. D. S. Jayalath and C. Tellambura, in “A blind SLM receiver for PAR-reduced OFDM,” in  Proc. IEEE Vehicular Technology Conference—Fall , vol. 1, pp. 219-222, September. 2002 in the presence of Rayleigh fading. The simulation parameters were the same as in the previous example, except that the phases {φ k   (m) } were i.i.d. uniformly distributed in [−π, π) (the method of Jayalath et al. (2002) does not work when the phases have a discrete distribution). The maximum likelihood (ML) decoder described by Jayalath et al. needs the channel state information (CSI) in order to detect the optimum phase sequence index  m , but the blind selected pilot tone modulation technique does not. Table 3 compares the error rates in detecting  m  between the method described by Jayalath et al. and the blind selected pilot tone modulation technique. Perfect CSI was assumed for the Jayalath et al. method, but no CSI power for the blind selected pilot tone modulation technique. Despite the favorable setup for the Jayalath, et al. method, the blind selected pilot tone modulation technique is clearly more robust. 
   
     
       
             
           
             
             
           
             
             
             
             
             
           
             
             
             
             
             
           
         
             
               TABLE 3 
             
           
           
             
                 
             
             
               Error rate in detecting m when N = 128, P = 4, M = 8, β = 0.15. 
             
           
        
         
             
                 
               SNR 
             
           
        
         
             
                 
               0 dB 
               5 dB 
               10 dB 
               20 dB 
             
             
                 
                 
             
           
        
         
             
               ML in Jayalath et al. (2002) 
               35.28% 
               5.02% 
               0.46% 
               0.02% 
             
             
               BSPTM 
               1.42% 
               0.09% 
                 0% 
                 0% 
             
             
                 
             
           
        
       
     
   
   Moreover, the ML decoder of Jayalath et al. (2002) has a higher computational complexity than the BSPTM technique. For example, if the frequency-domain OFDM sub-symbols are drawn from a 16-QAM constellation, the ML decoder requires 16MN magnitude-squared (|.| 2 ) operations, whereas the BSPTM technique only needs N of them. 
   BER Performance 
   The BER performance of BSPTM-OFDM is now compared to that of PTAM-OFDM for N=128, P=4, β=0.15, and M=8. The receiver uses a zero-forcing equalizer and a suboptimal but simple hard-decision decoder discussed by S. Ohno et al. Similar to S. Ohno et al., two types of channels were used: a fixed FIR channel with tap coefficients in equation (24) and a Rayleigh fading channel with i.i.d. complex Gaussian taps. The BER was evaluated by averaging over 10 5  Monte Carlo trials. 
     FIG. 8  shows the BER performance of the blind selected pilot tone modulation technique and that of PTAM-OFDM for the fixed channel case.  FIG. 9  shows a similar comparison for the Rayleigh fading case. It can be seen from both  FIGS. 8 and 9  that the PTAM-OFDM performance is only 1-2 dB away from the known channel case, which serves as a benchmark. However, the BSPTM-OFDM method offers even better BER performance, which approaches the performance of the known channel case for both the fixed and the Rayleigh fading channels. Such superior performance is possible, since the reduction in the PAR has been used to boost the average transmission power for the same amount of DC power. Specifically, the peak power of an OFDM block has been kept fixed, but the average power has been adjusted according to the actual PAR value of the block. This linear scaling approach described by H. Ochiai et al. ensures the most efficient utilization of the power amplifier; in other words, the average transmit power is made proportional to P dc /PAR. Eventually, the benefit of effective crest factor reduction is realized as a decrease in the BER. 
   Thus, combining the frameworks of pilot tone assisted modulation (PTAM) for OFDM and selected mapping (SLM), a novel joint channel estimation and crest factor reduction scheme, has been described, referred to as blind selected pilot tone modulation (BSPTM). The index for SLM is carried by the location of the pilot tones, which can be blindly detected at the receiver by capitalizing on the average power disparity between the pilot tones and the information sub-symbols. Since no side information needs to be transmitted, the blind selected pilot tone modulation method is power efficient and bandwidth efficient. Simulation results demonstrate the PAR reducing capability and the robustness of BSPTM-OFDM over frequency selective fading channels in the presence of additive noise. 
   Thus, apparatus and methods have been disclosed that provide for peak-to-average power ratio (or crest factor) reduction in orthogonal frequency division multiplexing (OFDM) systems using blind selected pilot tone modulation. It is to be understood that the above-described embodiments are merely illustrative of some of the many specific embodiments that represent applications of the principles discussed above. Clearly, numerous and other arrangements can be readily devised by those skilled in the art without departing from the scope of the invention.