WO2013005214A1

WO2013005214A1 - Combined power loading and papr reduction for ofdm with erasure decoder

Info

Publication number: WO2013005214A1
Application number: PCT/IL2012/050236
Authority: WO
Inventors: Rafi HALFON; Dov Wulich
Original assignee: Ben Gurion University Of The Negev, Research And Development Authority
Priority date: 2011-07-07
Filing date: 2012-07-04
Publication date: 2013-01-10

Abstract

System and method for combining equal Signal to Noise Ratio (SNR) Power-Loading (PL) with Hard Decision Erase (HDE) decoder in Orthogonal Frequency-Division Multiplexing (OFDM) using Bit-Interleaved Coded Modulation (BICM) communication system. An SNR threshold is calculated such that BER in the receiver is substantially minimized. Zero power level is assigned to sub-carriers having SNR before PL that is lower than SNR threshold, and a power level is assigned to other sub-carriers, such that SNR after PL seen at the receiver is substantially equal across the sub-carriers to which a non-zero power level was assigned. The HDE decoder at the receiver erases the bits with substantially zero energy level. Additionally, TR based PAPR reduction is combined with the equal SNR PL and HDE decoder. In the later case, sub-carriers having SNR before PL that is lower than SNR threshold are used as reserved tones.

Description

COMBINED POWER LOADING AND PAPR REDUCTION FOR OFDM WITH

ERASURE DECODER

BACKGROUND OF THE INVENTION

[001] Orthogonal Frequency-Division Multiplexing (OFDM) combined with Bit- Interleaved Coded Modulation (BICM) appears to be a robust technique for reliable communication over fading radio channels. When Channel State Information (CSI) - namely the complex attenuation of the sub-carriers - is available at the transmitter end, negative effects of fading can be further reduced by adaptation to the time- varying attenuations of sub-carriers.

[002] High Peak-to-Average-Power-Ratio (PAPR) is a main disadvantage of OFDM systems, which dictates working in the low-efficiency range of the Linear Power Amplifier (LPA), as described in D. Wulich, "Definition of efficient PAPR in OFDM", IEEE Communications Letters, September 2005, Vol. 9 pp. 832-834. The implications of low efficiency are less output power and more power that is spent on undesired heating. A decrease in PAPR increases the efficiency of LPA, yielding an increase in the output power emitted by LPA and consequentially a decrease in Bit Error Rate (BER).

[003] There are some adaptation strategies: Bit-Loading (BL) adapts the number of bits assigned to each sub-carrier chosen from a pre-defined constellation dictionary. Power Loading (PL), that needs partial knowledge of CSI, namely the absolute value of the complex attenuations, adapts the power distribution between sub-carriers to minimize BER or maximize capacity. Various PL algorithms for minimizing BER for BICM OFDM systems under a power constraint are known in the art. For example, a simple quasi-optimal PL algorithm for minimizing BER is presented in L. Goldfeld, V. Lyandres, D. Wulich, "Minimum BER power loading for OFDM in fading channel", IEEE Trans, on Commun, Nov 2002, Vol.50 pp. 1729- 1733. Alternatively, a maximum capacity Bit and Power Loading (BPL) strategy, known as water filling, may be derived under a power constraint.

[004] BL method requires the transmission of control information to inform the receiver of bit allocations, a rather complex and bandwidth-consuming task. For example, in case of relatively fast varying multipath channels, it would be required to update bit allocations very frequently, resulting in a net data rate reduction. PL, on the other hand, does not require forwarding control information since the receiver may interpret the variations in the sub-carriers power levels as being caused entirely by the channel. This is an important advantage of PL over BL, which enables its use in a standard protocol communication system without updating the standard. Both methods require the updating of CSI at the receiver to perform coherent demodulation. CSI at the receiver may be obtained by use of a preamble or pilot sub-carriers at a rate that depends on channel coherence time. One can argue that when both the transmitter and the receiver have exactly the same CSI it is then possible to perform BL without transmitting the control information. Nevertheless, due to estimation errors, CSIs at the transmitter and the receiver are different. Since bit allocation is very sensitive to CSI accuracy, a slight error in CSI may result in different bit allocations, which could cause total data corruption. Therefore the receiver's CSI knowledge cannot be used for bit allocations, regardless of what it is.

[005] In Tone Reservation (TR), some sub-carriers, called reserved tones, are used for PAPR reduction. In a conventional TR method, the existence of reserved tones reduces the payload bandwidth.

[006] When the received signal is affected by fading or interferences, erasure detection may be applied. The erasure detection works with coded modulation and uses a Hard Decision Erase (HDE) decoder. The decoder has three alternative outputs: "0", "1" and "erase". The "erase" decision is made when the reliability of the decoded bit is too low. This kind of decoder is well known in the field of anti-jamming systems. Another example of using an HDE decoder is a procedure called puncturing used in convolutional code, when part of the coded bits in known locations is not transmitted, and the receiver inserts "erase" in those locations. SUMMARY OF THE INVENTION

[007] According to embodiments of the present invention there is provided a method for reducing bit error rate (BER) in Orthogonal Frequency-Division Multiplexing (OFDM) using Bit-Interleaved Coded Modulation (BICM) communication system comprising a receiver with a Hard Decision Eraser (HDE) decoder. The method may include calculating an SNR threshold, SNR_thr , that will substantially minimize BER in a receiver; and assigning a power level to a sub-group of sub-carriers, such that SNR after PL may eb equal across the sub-group of sub-carriers, wherein the sub-group of sub-carriers includes sub- carriers having an SNR before Power Loading (PL) that is not lower than the SNR threshold.

[008] Furthermore, according to embodiments of the present invention, the method may include assigning zero power level to sub-carriers having SNR before PL that is lower than the SNR threshold.

[009] Furthermore, according to embodiments of the present invention, the method may include obtaining levels of partial Cannel State Information (CSI),

, wherein the power level is calculated by:

where ^s denotes total symbol energy, k and / are variables denoting a specific sub-carrier number and N_sc is the total number of sub-carriers.

[0010] Furthermore, according to embodiments of the present invention, finding the SNR threshold level may include defining a function G of variable D, the function G is an upper p

bound for the probability for error ^b , wherein the variable D is a function of the SNR threshold, and wherein minimizing the variable D i_s equivalent to minimizing the function p

G which is equivalent to minimizing the probability for error ^b which is equivalent to minimizing the BER; and finding the SNR threshold that minimizes the variable D .

[0011] Furthermore, according to embodiments of the present invention, finding the SNR threshold that minimizes the variable ^ may include setting a variable ^min _l0 infinity; setting a variable SNR_mp to zero; for SNR_mp = 0 up to ^SNR™* , with Δ5Μ? _steps, repeating: calculating the variable D as a function of SNR_mp ; and if the variable D is smaller than the variable ^min ₅ then setting the SNR threshold to equal SNR and the variable ™ⁿ to equal the variable D. [0012] Furthermore, according to embodiments of the present invention, the the variable D may be given by:

D = q + Zj(l - p - q)p

wherein P , the crossover probability is given by:

P ⁽« ϊ_ο^ Η-%)²

and

wherein M denotes size of a signal set, ^s denotes total symbol energy and " denotes variance of additive white Gaussian noise that is imposed on a transmitted

signal by a frequency selective channel, Q is the function

and ^ , the erasure probability is given by: q{SNR_thr ) =—∑U{SNR_thr - SNR_l )

^ SC 1=1 where ^^sc denotes number of sub-carriers and ^ is a step function.

[0013] Furthermore, according to embodiments of the present invention, the method may include using sub-carriers having SNR before PL that is lower than the SNR threshold as reserved tones.

[0014] Furthermore, according to embodiments of the present invention, the method may include obtaining levels of partial Cannel State Information (CSI), \\h\}^N ^^-sc1 , wherein the power level e_¾ may be calculated by:

where denotes total symbol energy k and / are variables denoting a specific sub-carrier number and N_sc is the total number of sub-carriers.

[0015] Furthermore, according to embodiments of the present invention, the calculating of the SNR threshold that minimizes the variable D _may include setting a variable to infinity; setting a variable SNR_mp to zero; for SNR_mp = 0 up to ^SNR™∞, with &SNR _step_S, repeating: calculating the variable D as a function of SNR_mp ; and if the variable D is smaller than the variable , then setting the SNR threshold to equal SNR and the variable D™, ⁿ to equal the variable D.

[0016] Furthermore, according to embodiments of the present invention, the method may include the variable D may be given by:

D = q + 2j(l - p - q)p wherein P , the crossover probability is given by:

1 L L „ \2

32

and

where M denotes size of a signal set, ^s denotes total symbol energy " denotes variance of additive white Gaussian noise that is imposed on a transmitted signal by a frequency

- /

selective channel, Q is the function ^x , and ^DPG denotes gain of the data sub-carriers due to PAPR reduction given by: _γ _ _v {^Es ^eTR )

^ DPG — ^PG ^' „ where ^^TR denotes symbol energy used by the reserved tones and ^^PG denotes power gain due to the TR based PAPR reduction,

and 1 the erasure probability is given by:

1 ^Nsc

q(SNR_thr ) =—∑u(SNR_thr - SNR_l )

M sc ι=ι where ^^sc denotes number of sub-carriers and ^ is a step function

[0017] Furthermore, according to embodiments of the present invention, there is provided a transmitter adapted for reducing bit error rate (BER) in Orthogonal Frequency-Division Multiplexing (OFDM) using Bit-Interleaved Coded Modulation (BICM) communication system comprising a receiver with a Hard Decision Eraser (HDE) decoder, the transmitter may include a processor; and a computer usable medium connected to the processor, wherein the computer usable medium may include a set of instructions for calculating an

SNR threshold, ^,hr , that will substantially minimize BER in a receiver, and for assigning a power level to a sub-group of sub-carriers, such that SNR after PL is equal across the sub-group of sub-carriers, wherein the sub-group of sub-carriers includes sub- carriers having an SNR before Power Loading (PL) that is not lower than the SNR threshold.

[0018] Furthermore, according to embodiments of the present invention, the set of instructions may include instruction for assigning zero power level to sub-carriers having SNR before PL that is lower than the SNR threshold.

[0019] Furthermore, according to embodiments of the present invention, the set of instructions may include instruction for using sub-carriers having SNR before PL that is lower than the SNR threshold as reserved tones. BRIEF DES CRIPTION OF THE DRAWINGS

[0020] The subject matter regarded as the invention is particularly pointed out and distinctly claimed in the concluding portion of the specification. The invention, however, both as to organization and method of operation, together with objects, features, and advantages thereof, may best be understood by reference to the following detailed description when read with the accompanying drawings in which:

[0021] Fig. 1 is a high-level diagram of an exemplary BICM -OFDM system according to embodiments of the present invention;

[0022] Figs. 2, 3A and 3B present exemplary simulation results of the above HDE decoder PL algorithm, according to embodiments of the present invention;

[0023] Fig. 4 which presents exemplary maximal PAPR values obtained with PAPR reduction, as a function of the tone reservation ratio, derived from a Monte-Carlo simulation, according to embodiments of the present invention;

[0024] Fig. 5 which presents exemplary values of the data power gain as a function of the tone reservation ratio, according to embodiments of the present invention;

[0025] Figs. 6, 7A and 7B present exemplary simulation results of the combined TR based PAPR reduction and PL algorithm with HDE decoder, according to embodiments of the present invention;

[0026] Fig. 8 presents exemplary simulation results of BER at the output of a Viterbi decoder as a function of the average signal to noise ratio (SNR) for adaptive PL with HDE decoder and adaptive PL with HDE decoder and TR based PAPR reduction, according to embodiments of the invention;

[0027] Fig. 9 presents a flowchart illustration of a method for combining equal SNR PL with HDE decoder in BICM OFDM communication systems according to embodiments of the present invention; and

[0028] Fig. 10 presents a flowchart illustration of a method for combining TR based PAPR reduction with equal SNR PL and HDE decoder in BICM OFDM communication systems according to embodiments of the present invention.

[0029] It will be appreciated that for simplicity and clarity of illustration, elements shown in the figures have not necessarily been drawn to scale. For example, the dimensions of some of the elements may be exaggerated relative to other elements for clarity. Further, where considered appropriate, reference numerals may be repeated among the figures to indicate corresponding or analogous elements. DETAILED DESCRIPTION OF THE PRESENT INVENTION

[0030] In the following detailed description, numerous specific details are set forth in order to provide a thorough understanding of the invention. However, it will be understood by those skilled in the art that the present invention may be practiced without these specific details. In other instances, well-known methods, procedures, and components have not been described in detail so as not to obscure the present invention.

[0031] Although embodiments of the invention are not limited in this regard, discussions utilizing terms such as, for example, "processing," "computing," "calculating," "determining," "establishing", "analyzing", "checking", or the like, may refer to operation(s) and/or process(es) of a computer, a computing platform, a computing system, or other electronic computing device, that manipulate and/or transform data represented as physical (e.g., electronic) quantities within the computer's registers and/or memories into other data similarly represented as physical quantities within the computer's registers and/or memories or other information storage medium that may store instructions to perform operations and/or processes.

[0032] Although embodiments of the invention are not limited in this regard, the terms "plurality" and "a plurality" as used herein may include, for example, "multiple" or "two or more". The terms "plurality" or "a plurality" may be used throughout the specification to describe two or more components, devices, elements, units, parameters, or the like. Unless explicitly stated, the method embodiments described herein are not constrained to a particular order or sequence. Additionally, some of the described method embodiments or elements thereof can occur or be performed at the same point in time.

[0033] According to embodiments of the present invention a HDE decoder may be used with BICM-OFDM while exploiting the partial knowledge of CSI at the transmitter. A PL scheme may be found using adaptive PL with HDE decoder. Alternatively, a TR based PAPR reduction method may be used in combination with adaptive PL and with HDE decoder, to further minimize BER. It will be shown that when combining TR-based PAPR reduction with adaptive PL and HDE decoder the payload bandwidth may not substantially decrease while BER may become much lower than BER obtained using optimal PL with

HDE decoder. The term BER describes the probability of error ^^b in terms of the number of erroneous bits per the number of transmitted bits. [0034] According to embodiments of the present invention it may be assumed that all sub- carriers have the same bit allocation and that the transmitter knows the absolute value of the complex attenuations of the channel, also referred to as partial CSI. Partial CSI at the transmitter may be updated by a reverse link, or any other applicable method, as known in the art. In a Time Division Duplex (TDD) transmission, the extra control information may not be necessary, due to channel reciprocity.

[0035] Reference is made to Fig. 1 depicting a high-level diagram of an exemplary BICM - OFDM system according to embodiments of the present invention. According to embodiments of the present invention, system 100 may comprise a transmitter 100, which may transmit data to receiver 170 over a channel such as frequency selective channel 120. CSI may be derived by CSI estimation block 122.

[0036] The data to be transmitted by transmitter 100 may be encoded by a convolutional encoder 102. According to the principle of BICM, a bit- wise interleaving may be performed by bit-wise interleaver 104 after encoding. The depth of bit- wise interleaver 104 may correspond to the number of encoded bits in a data burst. The permutation of bit-wise interleaver 104 may be randomly generated and is hereinafter referred to as ideal interleaving. The subsequent Serial to Parallel (S/P) converter 106 may transform the interleaved bits into a sequence of bit vectors which modulate the OFDM's sub-carriers, denoted as ^^sc . As a result, the coded bits may be spread randomly among the ^^sc sub- carriers. Equal BL is assumed, i.e., the number of coded bits associated with each one of the active sub-carriers may be constant and equal to m. It should be noted that equal BL is assumed for ease of mathematical formulation and presentation and that embodiments of the present invention are not limited to a specific BL scheme. Embodiments of the present invention may support other BL schemes, for example, unequal predetermined BL may be used. Bit to signal mapper block 108 may map the coded bits onto complex signals using, for example, a Gray encoding scheme. Let M = 2^m denote the size of the constellation, wherein m denotes number of bits. In the mathematical formulation presented hereinbelow Quadrature Amplitude Modulation (QAM) constellation with square lattice signal constellations like Quadrature Phase-Shift Keying (QPSK) or 16-QAM is assumed. However, it is noted that embodiments of the present invention are not limited to any specific constellation and may operate with any constellation operable with BICM-OFDM based communication system. Such constellation may include, for example, but not limited to, Binary Phase-Shift Keying (BPSK) and 8-PSK.

h h

[0037] CSI, also denoted as ¹ ' ^Nsc , may be the complex attenuations, gain and phase rotation of OFDM sub-carriers 1,...,N_SC , respectively, defined between the inputs to front end Inverse Discrete Fourier Transform (IDFT) block 112 at transmitter 100 (the signal at point 111), and the outputs of front end Discrete Fourier Transform (DFT) block 150 at receiver 170 (the signal at point 151), wherein N_sc denotes the number of sub-carriers. It is

\h }^Nsc

assumed that full CSI, denoted by *^{■ k} ' k=i _{) ma}y be known to receiver 170, while only partial CSI, denoted by \ ^u\h\} ^,Nk~scl , may be known to transmitter 100. Full CSI may be obtained at receiver 170 by analyzing of pilot sub-carriers transmitted from transmitter 100 to receiver 170, or by any other applicable method, as known in the art. Partial CSI may be obtained at transmitter 100 by reverse link burst reception in Time Division Duplex TDD systems or by feedback in Frequency Division Duplex systems, or by any other applicable method, as known in the art.

[0038] The power or energy adaptation may be performed by adaptive power loading (PL)

,...,e_N

block 110 by scaling the average energies, ^sc , of substantially all sub-carriers, such that are constant during a burst duration that includes preamble, pilots sub-carriers, if they exist and payload. Furthermore, it is assumed that:

N_sc

∑^ek = ^ES

k=^l · (1) Where E_s denotes the total symbol energy. The terms power and energy may be used interchangeably as the average energy equals the power times the symbol time duration.

[0039] Front end IDFT block 112 may yield a discrete-time domain signal. The IDFT may be followed by TR-based PAPR reduction block 114 that may increase linear power amplifier (LP A) 116 efficiency and thereby the transmitted power. A Cyclic Prefix (CP) may be inserted to eliminate Inter-Symbol-Interference (ISI) and LPA 116 may amplify the signal which may then be emitted from the transmit antenna 118. The signal may travel through a medium characterized as frequency selective channel 120 and may be received by antenna 148 of receiver 170. [0040] For the clearance of the mathematical formulation and presentation, it is assumed that the coherence time is larger than the OFDM burst, i.e., the channel varies slowly enough to be modeled as frequency-selective but time-invariant during the duration of the burst.

[0041] Front end DFT block 148 may yield a frequency domain signal. It is assumed that additive white Gaussian noise is imposed on the transmitted signal seen at the input of receiver 170. This additive white Gaussian noise may yield a noise term represented by a complex Gaussian random variable with zero mean and variance " at the output of front end DFT block 150. Consequently, the Signal-to-Noise Ratio (SNR) of the k-th sub-carrier may be given by:

SNR, = ^{fc |} 2 ^k

σ, n (2)

[0042] The factors V¾¾ £ = 1,..., N. sc , may be used at receiver 170 to restore the transmitted signals using zero-forcing frequency domain equalizer (FEQ) 152. After zero- forcing FEQ 152 the bits are detected by HDE decoder 154, which performs Maximum

Likelihood (ML) detection based on the factors ^k and yields the corresponding bit combination. The subsequent Parallel to Serial (P/S) converter 156 may transform the sequence of bit vectors into an interleaved bit stream. Finally, the bit stream may be de- interleaved at bit- wise de-interleaver block 158, and a Viterbi decoder 160 may provide estimations for the data bits.

[0043] It should be noted that embodiments of the invention are not limited to the specific implementation of transmitter 100 and receiver 170 as presented in Fig. 1, and that other implementations that may include an adaptive power loading block 110, a TR-based PAPR reduction block 114 and a HDE 154, as necessary, are also within the scope of the current invention.

[0044] According to embodiments of the present invention, PL algorithm of adaptive power loading block 110 of transmitter 100 may be adapted to take advantage of HDE 154 found in receiver 170, thus resulting in PL algorithm with HDE decoder. To substantially minimize BER, an equal SNR PL may be considered, which means that the power allocated to a specific subcarrier may be either zero or such for which SNR after PL (post-PL SNR) seen at receiver 170 may be substantially the same for the non-zero power subcarriers. In order to perform such PL algorithm, a threshold SNR (SNR_thr ) may be calculated, i.e., if a calculated SNR before PL (pre-PL SNR) of a specific subcarrier is not smaller than SNR_thr , then power may be allocated for that subcarrier, otherwise the power allocated for the subcarrier may be substantially zero. BER may be a convex function of SNR_thr which guarantees that there exists such optimal SNR_thr for which BER is minimal. A method for finding SNR_thr according to embodiments of the present invention is presented hereinbelow.

[0045] Due to the use of BICM, it may be assumed that the system may be modeled as a

Binary Symmetric Erasure Channel (BSEC). The binary inputs are T = 0 ' 1 while the outputs may have three possible values, ^{r _} ^⁾, 1 ^ "_erase" _ φ [_{ΐ8 w&} erased with the probability while P is the crossover probability that a given bit was not erased and resulted in an error. The probability of bit error in the k-Xh, not zero energy assigned sub- carrier, may be a function of SNR ^k and the size M _Qf the used constellation. As known in the art, the crossover probability that a given bit was not erased and resulted in an error for a given channel k may

where

and here and throughout this application Q is the function:

[0046] Due to the use of the HDE decoder, the PL algorithm with HDE decoder may have two parts. The first part may be to decide which sub-carriers should have zero energy, and the second part may be to assign non-zero-energy coefficients to all other sub-carriers. Given that transmitter 100 and the receiver 170 have the same partial CSI, zero energy allocations at the transmitter will result in erased bits at the receiver.

[0047] According to embodiments of the present invention, an approximation for power distribution that may substantially minimize error probability may be given by:

[0048] PL method based on (4) may result in quasi-optimal power distribution. Furthermore, according to embodiments of the invention, taking the HDE decoder into

| 2

account, sub-carriers with small may have zero power assignment. To determine which sub-carriers are considered as small, an SNR threshold, SNR, ^' , may be used. Using

SNR„

the constraint given in (1), the energy partition as a function of may become:

1

l ! SNR^SNR^ SNR,≥ SNR thr

0 SNR_k < SNR,

(5) where SNR_k is pre-PL SNR. For example, SNR_k may be calculated using equation (2).

[0049] For example, adaptive power loading block 110 of transmitter 100 may assign energies to sub-carriers according to equation (5).

[0050] From (5) it may follow that post-PL SNR at receiver 170 side may have two values:

SNR

substantially zero for all sub-carriers for which pre- PL SNR is below ^{Ji V}"*^ , and substantially the same, non-zero, value for all sub-carriers for which pre-PL SNR is equal or SNR

above ^thr . As a result, the probability of error of coded bits in substantially all non-zero power assignment sub-carriers may be substantially the same and equal to P .

[0051] Post-PL SNR of all non-zero power assignment sub-carriers be found by substituting ( into (2):

SNR

Substituting (6) in (3), P as a functio thr may be iven by

where

The erasing probability, ^ , as a function of ^thr may be given

1 "sc

l(SNR_thr) -- ∑u{SNR_thr - SNR_t )

N, (8) where U is a step function. The erasing probability may thus relate to the number of zero power assignment sub-carriers divided by the total number of sub-carriers. It should be noted that, due to the interleaving, the erased bits are randomly spread with probability ^ .

SNR_tl

[0052] As increases, the crossover probability P decreases, resulting in a lower probability of error, ^b . On the other hand, the erasure probability ^ increases, resulting in

P S R

a higher probability of error ^b . Both P and ^ are functions of ^thr , and therefore it

SNR P

would be desirable find an optimal ^thr for which substantially minimum ^b or BER may be obtained.

[0053] According to embodiments of the present invention, the probability of error as a

SMR

function of ^thr , for the quasi-optimal PL method given by (5), may be found according to the procedure described hereinbelow. According to the probability theory, the probability

P P

of error, ^b , of a coded system may be upper bounded by the Chernoff bound. ^b may be described by decoupling the code and the coding channel as follows:

P„≤ G(D)

(9) where D may depend on the nature of the coding channel, e.g. hard-decision or soft- decision, and ^(^') is a function that may be determined by the specific code. The value of D for HDE may be given by:

[0054] As mentioned above, function ^(^') may be fitted to the specific code. For example, the most commonly used code for coherent BPSK and QPSK modulation is the constraint length K = 7 rate 1/2 convolutional code. For such a code ^b may be given by: P_b≤ 0.5[36D¹⁰ + 21 ID¹² + 1404D¹⁴ + 11633D¹⁶

(11)

It should be noted that embodiments of the present invention are not limited to any specific code and may operate with any code operable with BICM-OFDM based communication system for which minimizing D i_s equivalent to minimizing ^^b .

[0055] Substituting (10) into (11) may give an upper bound of ^^b . For a low probability of error the upper bound of (9) is very tight, and therefore minimizing ^(^') may also minimize ^^b and BER. For ^(^') , described as a polynomial of D as in (11), minimizing D _may be equivalent to minimizing ^^b and BER.

[0056] According to (10) D _may depend on P and ^ , which in turn are functions of *thr . Thus, D may be expressed as a function of ^thr :

D = f{p{SNR_thr ), q{SNR_thr )) = f{SNR_thr ) ^

and

^D }

(12b)

SNR

[0057] p and q as functions of ^thr do not depend on the sub-carrier index and may be given by (7) and (8). Introducing (7) and (8) into (12b) determines the relationship between SNR SNR

D and ^thr . Finding ^thr that minimizes D analytically may be a very complex task,

D SNR

and therefore ™ⁿ may be calculated numerically by sweeping ^thr until a minimum is found. Thus PL algorithm with HDE decoder may be performed by the following procedure:

I. Set initial to infinity.

II. For ^SNRthr = 0 up to ^OTmas , with &SNR _stepS! calculate the following: a. P from (7) using ^SNRthr .

b. 1 from (8) using ^SNRthr .

c. O by substituting P and ^ in equation (12b).

d. If ^D < °πήη , then set ^SNR*w* = ^SNRthr _and D_mia = D

III. Calculate the energy partition ^e* by substituting ^^RtM%^et instead of ^^R'^hri (5).

[0058] Reference is now made to Figs. 2, 3A and 3B which present exemplary simulation results of the above HDE decoder PL algorithm, according to embodiments of the present

\h I² invention. To illustrate the above HDE decoder PL algorithm, a random set of ' ^ ' was

2

generated having % distribution with two degrees of freedom and an assumed average

SNR

SNR of 5 (not in dB). First ^thr was found for minimize D . Fig. 2 shows D as a function of ^NR_{thr ^} j_t -_{s c}j_ear D_Mn -_s ob ained for ^^^i¾r _™» Fig. 3 A depicts

SNR at the receiver without activating a PL procedure (gray line, y axis values are presented on the left-hand side) and the calculated energy coefficients, ¾ (dashed line, y axis values are presented on the right-hand side). Fig. 3B shows again SNR at the receiver without activating a PL procedure (gray line) together with the resultant post-PL SNR at the receiver as a result of the PL procedure (dashed line). As can be seen, post-PL SNR at the receiver as a result of the PL procedure has the values 0 or 5. may be calculated by substituting SNR = 5 _mlo (7^ _ancj 1 _may b_e calculated by substituting ^^^tars^et into (8). The values obtained are P ^{= 1} "^{26 ' 10} and « ⁼ °"⁴³.

[0059] According to embodiments of the present invention BER may be further reduced by reducing PAPR. As mentioned above, high PAPR in OFDM systems dictates working in the low-efficiency region of LPA 116. Low efficiency causes more power to be wasted on heating and less power to be emitted by antenna 118 of transmitter 100. Reducing PAPR may enable LPA 116 to work at a higher efficiency, resulting in higher transmitted power. Post-PL SNR of receiver 170 may improve accordingly and the BER may decrease. [0060] There are many methods known in the art for reducing PAPR. One of the most effective distortionless methods may be Tone-Reservation (TR). According to TR some of the sub-carriers, the reserved tones, are devoted to reducing PAPR and are not modulated for data transmission. An increase in the number of reserved tones leads to a decrease in PAPR. On the other hand having more reserved tones means having less data bandwidth. Usually the position of reserved tones is a-priori given. However, according to embodiment of the present invention, a power-loading algorithm may assign zero power to some sub- carriers and, the zero power sub-carriers may be used as reserved tones in the TR algorithm.

1/zJ,...,

The position of the reserved tones may depend on the specific distribution of ^{1 sc 1} and may be known to both transmitter and receiver. The use of TR together with PL may substantially not result in payload rate reduction, since payload data is distributed to all sub- carriers and hence may be restored by error correction algorithms, as known in the art.

[0061] As discussed above, data assigned to zero power sub-carriers may be erased by the receiver. When these sub-carriers are used as reserved tones, some power may be allocated to them and their phases may be adjusted to reduce PAPR. The pseudo-data carried by the reserved tones may substantially be erased at HDE decoder 154, as it was in the PL algorithm with HDE decoder of equations 3-12.

[0062] Thus, according to embodiments of the present invention PL algorithm with HDE decoder may be combined with TR based PAPR reduction. For example, data bursts may start with a preamble, which may be a known OFDM symbol such that all of its sub-carriers are known pilots. The preamble usually has low PAPR characteristics and therefore no PAPR reduction is needed. Then, following the preamble, data OFDM symbols may be transmitted. According to embodiments of the present invention the transmitter may already i\h \\^Nsc

know partial CSI, ^ ^k '> k=l _ h_e PL algorithm may compute ^e> according to (5). This power distribution may be applied to the entire OFDM burst, including the preamble. The zeroed sub-carriers may be used as reserved tones only for data OFDM symbols, not for the preamble. The preamble with zeroed sub-carriers may be used by the receiver to detect the position of the reserved tones. Then the data OFDM symbols may be received, equalized and the pseudo-data of the reserved tones erased.

[0063] Again, to substantially minimize BER, an equal SNR PL may be considered, which means that the power allocated to a specific subcarrier may be either zero or such for which post-PL SNR seen at receiver 170 may be substantially the same for the non-zero power subcarriers. In order to perform such PL algorithm, a threshold SNR ( SNR_thr ) may be found, i.e., if pre-PL SNR of a specific subcarrier is not smaller than SNR_thr , then power may be allocated for that subcarrier, otherwise the power allocated for the subcarrier may be substantially zero. BER may be a convex function of SNR_thr which guarantees that there exists such optimal SNR_thr for which BER is minimal. Since subcarriers with zero power allocation are used to reduce PAPR exploiting TR method, the LPA may increase its emitted power and BER may decrease. Thus, it is expected that SNR_thr of PL algorithm with HDE decoder and TR based PAPR reduction may be different than SNR_thr of PL algorithm with HDE decoder resulting in lower BER levels. A method for finding SNR_thr for PL algorithm with HDE decoder and TR based PAPR reduction according to embodiments of the present invention is presented hereinbelow.

[0064] Finding SNR_thr for PL algorithm with HDE decoder and TR based PAPR reduction according to embodiments of the present invention may be similar to finding SNR_thr for PL algorithm with HDE decoder as described above. Again, function ^(^') and variable as presented in equations (9-10) may be used, wherein minimizing D i_s equivalent to minimizing ^^b , however, the crossover probability P may be different due to TR, as demonstrated hereinbelow.

[0065] OFDM communication systems dictates the use of a power amplifier 116 at transmitter 100 that may be linear within the dynamic range of the transmitted signal. The drain efficiency, of power amplifier 116 may depend on PAPR, Y . The efficiency f? - t?(y) _may be _a decreasing function of Y . The specific relationship ^ ^~~ Ή^Υ) may depend, inter alia, on the class and the particular design of LPA 116. The output power of LPA 116 may be given by:

paut ^{= r}l^pdc _y (_{1 3}) p

where ^dc is the direct current (DC) input power which is the power taken from the power supply The total symbol energy, E_s , given in (1), may be related to the output power by:

wherein T i_s the time duration of an OFDM symbol. SNR may therefore depend on the efficiency, ^ , according to:

SNR = ^ = ^η - ^Ρ^ - ^Τ

N₀ N_{0 (15)}

[0066] PAPR, y , of OFDM systems, with or without reduced PAPR, may be a random variable. Maximal PAPR,

may be defined as that which fulfills the following equation:

Pr{y > γ^(ι)) = λ, I = 0,1

(16) where ^ is the exceed probability, for example - 10 ₎ Y i_s the maximal PAPR for

OFDM without PAPR reduction and ' is the maximal PAPR of OFDM transmitter with PAPR reduction. Thus it may be expected that Ύ may be smaller than , Ύ ^< Ύ . The respective SNRs are

and their ratio may be given by

K

^{PG ~} SNR^ ^~ _n (y(°

(18) η ( γ) ' Y _rW < _r(°)

may be a decreasing function of ' and ' ' , therefore from (18) it follows that

K > 1 K

PG PG _may he related to a power gain due to TR based PAPR reduction.

[0067] For example, the theoretical efficiency upper limits of a LPA may be described by:

= G - exp(- gy_dB ) _(19a) wherein G and g are constants related to the class and the particular design of the LPA. For example, for class A LPA ^ =58.7 , # =0.1247, and for class B LPA ^ =90.7%. =0.1202. Ύ ^dB stands for PAPR in dB. For this particular case:

[0068] The power gain introduced in (18) may depend on the value of maximum PAPR obtained via the TR algorithm. As known in the art, there are several TR algorithms, which differ in their complexity and performance. Embodiments of the present invention are not limited to a specific TR-based PAPR reduction algorithm. For example, embodiments of the present invention may utilize optimal method using Quadratically Constrained Quadratic Program (QCQP), simple gradient algorithm and kernel design.

[0069] As an example, an implementation of the simple gradient algorithm with fast convergence is described infra. The algorithm may be performed in the time domain and may have two stages:

(i) Kernel generation based on the selected reserved tones. Kernel generation may be performed periodically, for example, every time the channel coherence time elapses or once per burst. The kernel P may be computed by performing an ^ ^~ ^^sc points, over-sampled IFFT with the reserved tones set to T. Since only R sub-carriers are used for reserved tones, the complexity of generating P can be reduced to the order of magnitude of N * R .

(ii) Implementing the iterative algorithm in the time domain for each OFDM symbol that follows. If ^x is the time domain symbol in the i-th iteration and ^A is the destination maximum amplitude, the algorithm is given by: (20a) where:

—< 0 )

X = X

a_n ^{( i }} = x^{( i >} - A - sign{x^{( i >} } μ - convergence constan t (20b) [0070] Implementing the algorithm may require relatively low computing power, since the a⁽ⁱ⁾ Ό ²

term " may be a complex scalar, and the term ^{1 n} may be calculated by a circular shift of the basic kernel ¹ by ⁿ samples. The total complexity may depend on the maximum number of iterations and the destination amplitude ^ . A v

[0071] In order to evaluate the power gain, ^PG , using (19), the value of ' may have to be found. For example, the value of * may be found by a Monte-Carlo simulation as described herein hereinbelow:

I. For each simulation point, a set of reserved tones may be randomly generated with a Tone Reservation Ratio (TRR) of 2-50%. The TRR relates to the ratio between the number of reserved tones to the total number of active sub-carriers, namely

II. The procedure described in (20) may be performed, for example, with the following parameters: oversampled IFFT of order 256 with oversampling ratio 4 resulting in a 1024 points OFDM symbol; the number of active subcarriers may be set to N_sc = 200 . Such a number of active sub-carriers may be typical for WiMAX according to IEEE 802.16d. The destination amplitude A may be chosen empirically to result in the minimum PAPR.

III. For each simulation point, the TR algorithm may be performed for 2000 statistically independent OFDM symbols yielding the Complementary Cumulative Distribution Function CCDF of PAPR. Having the CCDF of PAPR, γ ^(l) for λ = 10^~3 may be computed according to (16).

[0072] Reference is now made to Fig. 4 which presents ^ as a function of TRR ₎ derived from the above described Monte-Carlo simulation, according to embodiments of the present invention. [0073] To calculate ^^PGirom (19), r can be derived from Fig. 4 while ' may be calculated using the CCDF of PAPR of OFDM without PAPR reduction that is widely available in the literature. The power gain relates to both data sub-carriers and reserved p

tones. Since * depends only on SNR of the data sub-carrier, the amount of energy used for the reserved tones should be deducted from the total power gain. Therefore, data power gain, ^DPG , as the gain of the data sub-carriers due to PAPR reduction may be given by:

v _ _v {^Es ^{~ E}TR )

^DPG - ^ΆΡΟ ^' _p

s (21) where ^TR denotes the symbol energy used by the reserved tones for PAPR reduction. [0074] Reference is now made to Fig. 5 which presents ^DPG as a function of TRR computed according to (21) while ^PG is computed using (19) and Fig. 4. The lower bound of ^DPG may be derived from the data depicted in Fig. 5 and further used to evaluate the data power gain. For example, the lower bound of ^DPG is depicted in Fig. 5 by solid line 502.

[0075] Notice that ^DPG may be a function of TRR ₎ and TRR _may depend substantially on ^SNR^r and therefore ^K DPG - K _DPG {SNR_thr )

[0076] The new, higher post-PL SNR, due to TR based PAPR reduction may be calculated by introducing into (6), resulting in:

E_sK_DPG(SNR_thr)

SNR

(22)

Substituting the increased post-PL SNR of (22) into (7b) may yield:

[0077] Thus, according to embodiments of the present invention, the combined TR based PAPR reduction algorithm with the PL algorithm may be performed by the following procedure:

I. Set initial D_min to infinity.

II. For SNR_thr = 0 up to SNR_max , with ASNR steps, perform the following:

a. calculate q from (8) using SNR_thr . b. find ^DPG from the lower bound of Fig. 5 using ^ as the TRR. ^ , the erasing probability may be an approximation of TRR.

c. calculate P from (7a) and (23).

d. calculate D by substituting P and ^ into equation (10). e. If ^D < ^D , then set = ^SNR^ and ^D™^« = ^D .

III. Calculate the energy partition e_k by substituting SNR_{t arg et} instead of SNR_thr in (5). [0078] Reference is now made to Figs. 6, 7A and 7B which present exemplary simulation results of the above combined TR based PAPR reduction and PL algorithm with HDE decoder, according to embodiments of the present invention. To illustrate this algorithm and to compare it to the PL algorithm with HDE decoder without the TR based PAPR reduction, the simulations performed for the PL algorithm with HDE decoder and presented

\h I²

in Figs. 2, 3A and 3B were repeated using the same set ' of channel coefficients, this time using the combined TR based PAPR reduction and PL algorithm with HDE decoder.

First, D _Was found as a function of ^thr . The results are shown in Fig. 6. Note that

^Dmin obtained for ^^NR'^h ^ ~ ^ and recall that ^^NR'^hr™ for the PL with HDE

SNR ~ 1 decoder without TR based PAPR reduction, as seen in Fig. 2. Having ^i¾r™ⁿ Fig. 7A depicts the SNR at the receiver without activating a PL procedure (gray line, y axis values are presented on the left-hand side) and the calculated energy coefficients, ^ek (dashed line, y axis values are presented on the right-hand side). Fig. 7B shows again the SNR at the receiver without activating a PL procedure (gray line) together with the resultant post-PL SNR at the receiver while activating the combined TR based PAPR reduction and PL algorithm with HDE decoder (dashed line). As can be seen, SNR increases as a result of TR based PAPR reduction from 5 to 6.32 for the non-zero sub-carriers. ^ is reduced to 0.3 and

P decreases to 6 · 10

[0079] Comparison of the performance, in term of BER, of the two exemplary algorithms discussed above, namely adaptive PL with HDE decoder (HDE-PL) and adaptive PL with HDE decoder and TR based PAPR reduction (HDE-PL-PR) was carried out by means of computer simulation. The simulations were performed under the following assumptions: (1) For each burst comprising 20 OFDM symbols which may be equivalent to the coherence h

time of the channel, the coefficients ^k were generated as mutually independent complex- valued Gaussian random variables with zero mean and unit variance, resulting in Rayleigh fading channel characteristics. (2) A BICM-OFDM scheme with ^^{sc ~} was used, employing a rate ½ convolutional encoder with generators 133 oct and 171 oct. QPSK constellation was used, thus 2 coded bits for 1 data bit were assigned to each sub-carrier. The bit interleaver permutations were randomly generated for each burst. [0080] Reference is now made to Fig. 8 which presents exemplary simulation results of BER at the output of Viterbi decoder 160 as a function of the average SNR for HDE- PL(gray line with diamond signs) and HDE-PL-PR (gray line with dot signs), according to embodiments of the invention. Also presented, by way of comparison, are the BER of (i) Uniform Power Loading (UPL) with hard decision (HD) decoding (black line), and (ii) an optimal power-loading for BICM-OFDM with HD decoder (HD-PL, gray line with plus signs) calculated as known in the art. Analyzing the simulation results presented in Fig. 8, it may be seen that for the same level of average SNR, HDE-PL-PR have the lowest BER comparing to the other methods. Additionally, for BER = 10^~6 it may be seen that the SNR gain of HDE-PL-PR as compared to the other methods is as follows: (i) 1.8dB for HDE-PL; (ii) 4.3dB for HD-PL and (iii) 8dB for UPL.

[0081] Reference is now made to Fig. 9 which is a flowchart illustration of a method for combining equal SNR PL with HDE decoder in BICM OFDM communication systems according to embodiments of the present invention. According to embodiments of the present invention, a receiver may include 910 an HDE decoder for erasing bit with substantially zero energy level. Partial CSI may be obtained 920. An SNR threshold, SNR_thr , may be calculated 930 such that BER in the receiver may be substantially minimized. Zero power level may be assigned to sub-carriers having pre-PL SNR that is lower than SNR_thr 940, and a power level may be assigned 950 to sub-carriers having pre- PL SNR that is not lower than SNR_thr , such that post-PL SNR seen at the receiver may be substantially equal across the sub-carriers having pre-PL SNR that is not lower than SNR_thr

[0082] Reference is now made to Fig. 10 which is a flowchart illustration of a method for combining TR based PAPR reduction with equal SNR PL and HDE decoder in BICM OFDM communication systems according to embodiments of the present invention. According to embodiments of the present invention, a receiver may include 1010 an HDE decoder for erasing bits allocated for TR. Partial CSI may be obtained 1020. An SNR threshold, SNR_thr , may be calculated 1030 such that BER in the receiver may be substantially minimized. Sub-carriers having pre-PL SNR that is lower than SNR_thr may be used 1040 as reserved tones, and a power level may be assigned 1050 to sub-carriers having pre-PL SNR that is not lower than SNR_thr , such that post-PL SNR seen at the receiver may be substantially equal across the sub-carriers having pre-PL SNR that is not lower than 5NR_t¾r .

[0083] Some embodiments of the present invention may be implemented in software for execution by a processor-based system, for example, adaptive power loading block 110 and TR-based PAPR reduction block 114. For example, embodiments of the invention may be implemented in code and may be stored on a storage medium having stored thereon instructions which can be used to program a system to perform the instructions. The storage medium may include, but is not limited to, any type of disk including floppy disks, optical disks, compact disk read-only memories (CD-ROMs), rewritable compact disk (CD-RW), and magneto-optical disks, semiconductor devices such as read-only memories (ROMs), random access memories (RAMs), such as a dynamic RAM (DRAM), erasable programmable read-only memories (EPROMs), flash memories, electrically erasable programmable read-only memories (EEPROMs), magnetic or optical cards, or any type of media suitable for storing electronic instructions, including programmable storage devices. Other implementations of embodiments of the invention may comprise dedicated, custom, custom made or off the shelf hardware, firmware or a combination thereof.

[0084] Embodiments of the present invention may be realized by a system that may include components such as, but not limited to, a plurality of central processing units (CPU) or any other suitable multi-purpose or specific processors or controllers, a plurality of input units, a plurality of output units, a plurality of memory units, and a plurality of storage units. Such system may additionally include other suitable hardware components and/or software components.

[0085] While certain features of the invention have been illustrated and described herein, many modifications, substitutions, changes, and equivalents will now occur to those of ordinary skill in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the invention.

Claims

is claimed is:

A method for reducing bit error rate (BER) in Orthogonal Frequency-Division Multiplexing (OFDM) using Bit-Interleaved Coded Modulation (BICM) communication system comprising a receiver with a Hard Decision Eraser (HDE) decoder, the method comprising:

calculating a Signal to Noise Ratio (SNR) threshold, denoted SNR_thr , that will substantially minimize BER in a receiver; and

assigning a power level, denoted ¾, to a sub-group of sub-carriers, such that an SNR after Power Loading (PL) is equal across the sub-group of sub- carriers, wherein the sub-group of sub-carriers includes sub-carriers having an SNR before PL that is not lower than the SNR threshold.

The method of claim 1 , further comprising:

assigning zero power level to sub-carriers having SNR before PL that is lower than the SNR threshold.

The method of claim 2, further comprising: obtaining levels of partial Cannel State Information (CSI) given by

wherein the power level ¾ is calculated by:

SNR_k≥SNR_thr

SNR_k < SNR_thr where ^s denotes total symbol energy, k and / are variables denoting a specific sub-carrier number and N_sc is the total number of sub-carriers.

4. The method of claim 3, wherein the calculating of the SNR threshold level comprises: defining a function G of variable D, the function G is an upper bound for the p

probability for error ^b , wherein the variable D is a function of the SNR threshold, and wherein minimizing the variable D is equivalent to minimizing the function G which is equivalent to minimizing the p

probability for error ^b which is equivalent to minimizing the BER; and finding the SNR threshold that minimizes the variable D .

The method of claim 4, wherein finding the SNR threshold that minimizes the variable ^ comprises: setting a first temporary variable denoted to infinity;

setting a second temporary variable denoted SNR_tmp to zero;

SNR

for SNR_tmp = 0 up to a predetermined value denoted ^max , with a step size of ASNR _^ repeating:

calculating the variable D as a function of the second temporary variable SNR_tmp ; and if the variable D is smaller than the first temporary variable , then setting the SNR threshold to equal the second temporary variable SNR, tmp and the first temporary variable to equal the variable D.

The method of claim 5, wherein the variable D is given by: D = q + 2j(l - p - q)p _^ wherein P , the crossover probability is given by:

and

wherein M denotes size of a signal set and ^σ" denotes variance of additive white Gaussian noise that is imposed on a transmitted si nal by a frequency

selective channel and Q is the function

, and ^ , the erasure probability is given by:

1 ^Nsc

q{SNR_thr ) =—∑u{SNR_thr - SNR_l )

^ SC 1=1 where ^ is a step function.

The method of claim 1, further comprising:

using sub-carriers having SNR before PL that is lower than the SNR threshold as reserved tones.

The method of claim 7, further comprising: obtaining levels of partial Cannel State Information (CSI), given by

wherein the power level ¾ is calculated by:

where ^s denotes total symbol energy, k and / are variables denoting specific sub-carrier number and N_sc is the total number of sub-carriers.

The method of claim 8, wherein the calculating of the SNR threshold level done by: defining a function G of variable D, the function G is an upper bound for the p

The method of claim 9, wherein finding the SNR threshold that minimizes the variable ^ comprises: setting a first temporary variable denoted to infinity;

setting a second temporary variable denoted SNR_tmp to zero;

SNR

The method of claim 10, wherein the variable D is given by: D = q + 2j(l - p - q)p _^ wherein P , the crossover probability is given by:

and

where M denotes size of a signal set, ^σ" denotes variance of additive white Gaussian noise that is imposed on a transmitted si nal by a frequency

selective channel, Q is the function

, and ^DPG denotes gain of the data sub-carriers due to PAPR reduction given by:

where ^TR denotes symbol energy used by the reserved tones and ^PG denotes power gain due to the TR based PAPR reduction, and ^ the erasure probability is given by:

1 ^Nsc

q{SNR_thr ) =—∑u{SNR_thr - SNR_l )

^ SC 1=1

i where ^ is a step function.

A transmitter adapted for reducing bit error rate (BER) in Orthogonal Frequency- Division Multiplexing (OFDM) using Bit-Interleaved Coded Modulation (BICM) communication system comprising a receiver with a Hard Decision Eraser (HDE) decoder, the transmitter comprising:

a processor; and

a computer usable medium connected to the processor, wherein the computer usable medium contains a set of instructions for calculating a Signal to

SNR

Noise Ratio (SNR) threshold, denoted >^hr , that will substantially minimize BER in a receiver, and for assigning a power level , denoted ¾, to a sub-group of sub-carriers, such that SNR after Power Loading (PL) is equal across the sub-group of sub-carriers, wherein the sub-group of sub- carriers includes sub-carriers having an SNR before PL that is not lower than the SNR threshold.

13. The transmitter of claim 12, wherein the computer usable medium contains a set of instructions further for assigning zero power level to sub-carriers having SNR before PL that is lower than the SNR threshold.

14. The transmitter of claim 13, wherein the computer usable medium contains a set of instructions further for: obtaining levels of partial Cannel State Information (CSI), given by wherein the power level ¾ is calculated by:

SNR_k≥ SNR_thr

15. The transmitter of claim 14, wherein the calculating of the SNR threshold level is done by:

defining a function G of variable D, the function G is an upper bound for the probability for error ^^b , wherein the variable D is a function of the SNR threshold, and wherein minimizing the variable D i_s equivalent to minimizing the function G which is equivalent to minimizing the p

16. The transmitter of claim 15, wherein finding the SNR threshold that minimizes the variable D _ls done by: setting a first temporary variable denoted to infinity; setting a second temporary variable denoted SNR_t to zero; for SNR_tmp = 0 up to a predetermined value denoted SNR ^max , with a step size of ASNR ₅ repeating:

calculating the variable D as a function of the second temporary variable SNR_tmp ; and if the variable D is smaller than the first temporary variable ^min ₅ then setting the SNR threshold to equal the second temporary variable SNR_tmp and the first temporary variable to equal the variable D.

17. The transmitter of claim 16, wherein the variable D is given by:

wherein P , the crossover probability is given by:

and

wherein M denotes size of a signal set, and ^σ" denotes variance of additive white Gaussian noise that is imposed on a transmitted si nal by a frequency

selective channel and Q is the function

and ^ , the erasure probability is given by:

1 ^Nsc

q{SNR_thr ) =—∑u{SNR_thr - SNR_l )

^ SC 1=1 where ^ is a step function.

18. The transmitter of claim 12, wherein the computer usable medium contains a set of instructions further for:

19. The transmitter of claim 18, wherein the computer usable medium contains a set of instructions further for: obtaining levels of partial Cannel State Information (CSI), given by \ ^u\\\} '^{N k~scl} , wherein the power level denoted ¾ is calculated by:

' SNR_k≥SNR_thr

20. The transmitter of claim 19, wherein the calculating of the SNR threshold level is done by:

21. The transmitter of claim 20, wherein finding the SNR threshold that minimizes the variable D i_s done by: setting a first temporary variable denoted to infinity; setting a second temporary variable denoted SNR_t to zero; for SNR_tmp = 0 up to a predetermined value denoted SNR ^max , with a step size of ASNR ₅ repeating:

22. The transmitter of claim 21, wherein the variable D is given by:

wherein P , the crossover probability is given by:

and

selective channel, Q is the function

, and ^DPG denotes gain of the data sub-carriers due to PAPR reduction given by: v _ _v {^Es ^{~ E}TR )

^^■DPG - ^-PG ^' „ where ^TR denotes symbol energy used by the reserved tones and denotes power gain due to the TR based PAPR reduction,

and 1 the erasure probability is given by:

1 ^Nsc

v{SNR_thr) =—∑U{SNR_thr-SNR_l)

^SC 1=1 where ^ is a step function.