Cocktail Intra-Symbol-Codes: Exceeding the Channel Limit of QPSK Input
aa r X i v : . [ c s . I T ] M a y Cocktail Intra-Symbol-Codes: Exceeding theChannel Limit of QPSK Input
Bingli Jiao, Mingxi Yin and Yuli Yang
Abstract —This paper presents a new method, referred to as thecocktail intra-symbol-code, to reveal the possibility of exceedingthe channel limit of QPSK input by transmitting independentsignals in parallel manner and separating them the receiver,repetitively. In the proposed modulation scheme, the rhombicconstellation is created to work with the modulated signal and thechannel code in Euclidean- and Hamming space cooperatively.The theoretical derivations are done under the assumption ofusing the capacity-achieving codes and the results are shown interms of the reliable bit rates.
Index Terms —reliable bit rate, channel capacity, Rhombicconstellation, mutual information.
I. I
NTRODUCTION
A significant underlying principle of the communicationusing the finite alphabet can be found at the additive whiteGaussian noise (AWGN) channel model y = x + n, (1)where x is a finite alphabet signal, y is the received signal and n is the random noise variable from a normally distributedensemble of power σ N , denoted by n ∼ N (0 , σ N ) .The highest bit rate at which information can be transmittedwith arbitrary small probability of error is up-bounded by [1][2] I( X ; Y ) = H( Y ) − H( N ) = ˜I x ( γ ) , (2)where I(X; Y) is the mutual information, H( Y ) is the entropyof the received signal and H( N ) = log ( p πeσ N ) is theentropy of the AWGN, and ˜I x ( γ ) is the mutual information ex-pressed with symbol energy to that of noise ratio, γ = E s /σ N ,as the argument. The result of (2) has been regarded as a lawof the channel limit.However, for exceeding this limit, there were still someconsiderations, e.g., on a mathematical incentive ˜I x ( γ ) < ˜I x ( γ ) + ˜I x ( γ ) , (3)where x = x + x is the signal of the parallel transmission,and x and x are two independent signals. The inequality istrue due to the down-concavity of the mutual information [3]and [4]. B. Jiao ( corresponding author ) and M. Yin are with the Department ofElectronics and Peking University-Princeton University Joint Laboratory ofAdvanced Communications Research, Peking University, Beijing 100871,China (email: [email protected], [email protected]).Y. Yang is with the Department of Electronic and Electrical En-gineering, University of Chester, Chester CH2 4NU, U.K. (e-mail:[email protected]).
Nevertheless, the great difficulties can be encountered whenwe try to squeeze out some effects in relation to (3) on thesignal designs.This work is motivated by the above inequality to separatethe signals of parallel transmission with respect to Euclidean-and Hamming space. Eventually, we achieved a gain in termsof the reliable bit rates in comparison that of QPSK input.As a preparation for introducing the proposed method, werecall the conventional coded modulation scheme of BPSKthat starts from the information sequence of K bits in vectorpresentation, c = [ c , c , ..., c K ] ∈ { , } K , which is encodedby the code matrix G = { g ij } into the channel code of N bits v = [ v , v , ...v N ] ∈ { , } N . The channel code is mappedonto BPSK symbols x = [ x , x , ..., x N ] for the channelrealization. Each symbol is drawn from a discrete constellation S = { + α, − α } , i.e., x i ∈ S and i = 1 , ..., N . In general, thenumber of possible independent source vectors and that of thechannel codewords are equal to K with R = K/N < ,where R is the code rate.Since the up-bound of the reliable communication bit rateis of interest in this research, we assume that there existthe capacity-achieving channel codes that allow the error-free transmission along with the above procedures of thedemodulation and decoding. This theoretical assumption isreasonable, because that the practical coded modulation ofBPSK plus LDPC can approach the channel limit at a smallgap of 0.0045dB [5].At the receiver, in the detection of each received symbol,the estimate of ˆ y > or ˆ y < provides an initial value tothe channel decoder which chooses the codeword having thehighest probability of being transmitted to recover the sourcecodes.It is noted that the procedures of QPSK input are same asthat of BPSK in principle. Thus, we don’t go through QPSKin the introduction.Throughout the paper, the lowercase bold letters denotevectors, uppercase bold letters denote matrices, e.g., p =[ p , p , ....p N ] . The operation of mo-2 plus of two vectors areexpressed by p ⊕ q = [ p ⊕ q , p ⊕ q , ...., p N ⊕ q N ] , andthe mutual information is expressed as ˜I( γ ) with SNR as theargument. In addition, we use ˆ z to represent the estimate ofthe transmit quantity z at the receiver.II. C OMMUNICATION S CHEME
In our theoretical approach, the derivations are done basedon the assumption of error free transmissions of BPSK- andQPSK input. When the Euclidean distance between any two j a- j a a- a ,(1) rh x ,(4) rh x a ,(3) rh x ,(2) rh x Fig. 1. Constellations of the rhombic symbol. constellation points is set equal to or larger than α , the up-bounds of the reliable bits rate can be calculated by R b = ˆI b ( α /σ N ) , (4)and R q = ˆI q (2 α /σ N ) , (5)where R b and R q are the reliable bit rates of BPSK andQPSK and ˆI b and ˆI q are the mutual informations, respectively.Actually, we assume that there exist the capacity-achievingcodes to work with the up-bounds.Let us consider a binary information bit sequence which ispartitioned into three independent subsequences expressed invector forms of c (1) , c (2) and c (3) , respectively.Using the capacity achieving code matrix to c (1) , c (2) yields v ( i ) m = K X n =1 g m,n c ( i ) n , for m = 1 , , .., N, and i = 1 , (6)where G = { g m,n } is the capacity-achieve encoder matrix, v ( i ) m is m -th competent of codeword v ( i ) , K and N are lengthof the information subsequences and that of the correspondingchannel-codeword, respectively.While, c (3) is encoded by a different capacity-achievingcode v (3) .To increase the energy efficiency of the proposed method,we create a rhombic constellation with four possible pointsin Euclidean space as shown in Fig.1. For simplicity, we referthis symbol to as the rhombic symbol x rh, ( k ) . To be specified,the rhombic symbol is expressed by x rh, (1) = 0 + j √ α , x rh, (2) = α + j , x rh, (3) = 0 − j √ α and x rh, (4) = − α + j ,where j = √− .The Euclidean distance between two adjacent points in theconstellation is α so that the error free transmission appliesas long as that of QPSK holds. It is noted that the averagesymbol-energy is also same as that of QPSK, i.e., E s = 2 α .Before the signal modulation, the transmitter calculates v (1) ⊕ v (2) = v ( p ) and recodes each index m ′ , of which v ( p ) m ′ = 1 . Then, the signal transmission is performed in thefollowing two steps.The first step uses the rhombic symbol by mapping v ( p ) m = 1 onto x rh, (1) m or x rh, (3) m , and v ( p ) m = 0 onto x rh, (2) m or x rh, (4) m . TABLE IS
IGNAL MODULATION RESULTS . v ( p ) m ′ v (1) m ′ v (2) m ′ v (3) m ′ Modulation Symbol x rh, (1) m ′ x rh, (3) m ′ v ( p ) m v (1) m v (2) m − Modulation Symbol − x rh, (2) m − x rh, (4) m The selection of x rh, (1) m or x rh, (3) m is determined by thecomponent of V (3) : when v (3) l = 0 uses x rh, (1) m and v (3) l = 1 uses x rh, (3) m for l = 1 , , ..L . While, the selection of x rh, (2) m or x rh, (4) m is determined by the component of V (1) : when v (1) m = 0 uses x rh, (2) m and v (1) m = 1 uses x rh, (4) m .The signal modulation results are listed in Table I, wherewe use the subscript m ′ to indicate the case of v ( p ) m ′ = 1 and m the case of v ( p ) m = 0 .In the second step, transmitter transmits v (1) m ′ by using BPSKmodulation of s m ′ ∈ { α, − α } , whereby v (1) m ′ = 0 is mappedonto s m ′ = α or v (1) m ′ = 1 onto s m ′ = − α . The transmission isperformed selecting components of subscripts m ′ in sequencemanner from to N , whenever v ( p ) m ′ = 1 is found. The numberof code components of { m ′ | v ( p ) m ′ = 1 } is N/ in average.To save the time resource, we layer two adjacent BPSKsymbols perpendicularly into one QPSK symbol. Thus, thesecond step transmission requires N/ symbol durations tocomplete the transmission.We refer the proposed method to as the cocktail intra-symbol codes (CISC) method because the transmitted signalsare composed of the different amplitudes of the independentsignals in connections with their channel codes essentially.At the receiver, the signals of the first step transmission,i.e., the rhombic symbols, are demodulated and decoded forrecovering c ( p ) by taking ˆ y m = x rh, (1) m or x rh, (3) m for v ( p ) m = 1 ,and take ˆ y m = x rh, (2) m or x rh, (4) m for v ( p ) m = 0 .The decoding results are c ( p ) = c (1) ⊕ c (2) , because of theproperty of the linear codes.Because of the error free transmission, we can use ˆ c ( p ) = c ( p ) to re-construct v ( p ) without any error by ˆ v ( p ) m = K X n =1 g m,n ˆ c ( p ) n , for m = 1 , , .., N, (7)where ˆ v ( p ) m is the re-constructed channel code component with ˆ v ( p ) m = v ( p ) m .By calculating ˆ v (1) ⊕ ˆ v (2) = ˆ v ( p ) , the transmitter can find m ′ , of which v ( p ) m ′ = 1 , which is used to the second steptransmission.Then the constellation of the x rhm can be decoupled inEuclidean space: when ˆ v ( p ) m ′ = 1 , the signal represents theBPSK modulation of v (3) l along the vertical axis, and when ˆ v ( p ) m = 0 , the signal represents the BPSK modulation of v (1) m with respect to the horizontal axis as shown in Fig. 2(a) and2(b), respectively. j a- j a ( )' pm v = ,(3)' rhm x ,(1)' rhm x (a) a- a ( ) pm v = ,(2) rhm x ,(4) rhm x (b)Fig. 2. Constellations of the x rhm with (a) v ( p ) m ′ = 1 (b) v ( p ) m = 0 . We use the BPSK x rhm ′ along vertical axis to the recovery of c (3) by ˆ y m ′ = 0+ j √ α for v (3) l = 0 and ˆ y m ′ = 0 − j √ α for v (3) l = 1 , respectively. Hence, the c (3) can be fully recovered.Now, we use the rhpmbic symbols and the signals ofthe second step transmission the complete signal set for therecovering c (1) as follows. The receiver uses the signals ofthe second step transmission and converts each QPSK symbolback to the original BPSK symbol and inserts each of themamong the signals of the first step transmission, of which theBPSK is along with the horizontal axis, at the correspondingposition of v ( p ) m ′ = 0 . The conversion from QPSK to BPSKabove does not suffer from any SNR loss, because both thesignal energy and that of the noise are slit by a half.By demodulating and decoding over the complete signal setof the combined BPSK in connecting v (1) , we obtain ˆ c (1) .Finally, ˆ c (2) can be obtained by ˆ c (2) = ˆ c ( p ) ⊕ ˆ c (1) . (8)where ˆ c (1) = c (1) and ˆ c (2) = c (2) can be guaranteed bythe assumption of the error-free transmission based on theEuclidean distance of α . III. T HEORETIC C ONTRIBUTION
In this section, we compare the reliable bit rate of the CISCmethod with that of QPSK input.Let us examine N T = N + N/ symbol durations, where N T is the total number of the transmitted symbols, N is thenumber of the transmitted symbols of the first step and N/ is that of the second. E s / N2 [linear] R e li ab l e B i t R a t e [ b i t s / s e c / H z ] QPSKChannel Capacity BPSKProposal
Fig. 3. ADRs of cocktail BPSK compared with the channel capacity andBPSK versus linear ratio of E s /σ N by (11). The summation bits’ number of c (1) and c (2) is K . Thus,the contribution of their bit rate over N T symbol durationscan be averaged by R q = NN T ˜I q (2 α /σ N ) = 45˜I q (2 α /σ N ) , (9)where R q is the reliable bit rate of QPSK input, ˜I q (2 α /σ N ) is the mutual information of QPSK.The bit rate of c (3) can be calculated by R b = N/ N T ˜I b (3 α /σ N ) = 25˜I b (3 α /σ N ) , (10)where R b is the reliable bit rate owning to the transmission of c (3) and ˜I b (3 α /σ N ) is the mutual information of BPSK.The total reliable bit rate of CISC method is found by R T (2 α /σ N ) = 45˜I q (2 α /σ N ) + 25˜I b (3 α /σ N ) , (11)where R T is the reliable bit-rate of the proposed method.The numerical results of (11) are plotted in Fig.3, whereatone can find clearly that the reliable bit rate of CISC approachis larger than that of QPSK input over full range of SNR.An analytic comparison can be made with the QPSK inputat very low SNR, i.e. γ → by lim α /σ N → R T (2 α /σ N ) = 75 (2 α /σ N ) log e, (12)which is larger than (2 α /σ N ) log e of the QPSK, as ex-plained in the Appendix. -2 0 2 4 6 8 10 12 14 E s / N2 [linear] D [ b i t s / s e c / H z ] Compared to the QPSK InputCompared to the Channel Capacity
Fig. 4. The difference between the proposed scheme and QPSK.
To show the spectral efficiency gain of CISC method, thenumerical results of D = R T (2 α /σ N ) − I q (2 α /σ N ) , (13)are plotted as shown in Fig. 4, whereat one can find clearlythe phenomenon of exceeding the spectral efficiency of QPSKinput. IV. C ONCLUSION
In this paper, the new method called the cocktail intra-symbol-code is introduced to work in Hamming- and Eu-clidean Space for separating the parallel transmission of threeinformation sources. The higher reliable bit rates are obtainedfrom the signal separations in Euclidean space with help ofthe channel codes V ( p ) on the created rhombic symbol. Theresults are found better than that of QPSK input.A CKNOWLEDGMENT
This work was supported in part by the National NaturalScience Foundation of China under Grant 61531004 and theBeijing Municipal Natural Science Foundation under GrantL172010. R
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The first-order approximation of the QPSK using Taylorexpansion is f ( x ) ≈ f (0) + ( df /dx ) x, (14)where f ( x ) is the first-order approximation around the regionof x = 0 .Because of ˆI q (0) = 0 , using (14) to calculate the first order-approximation of QPSK yields ˆI q ( γ ) ≈ ( d ˆI q /dγ | γ =0 ) γ, (15)and that of the channel capacity as C ( γ ) ≈ ( dC/dγ | γ =0 ) γ = (log e ) γ, (16)where ˆI q ( γ ) and C ( γ ) represent the first-order approximationof mutual information of QPSK and that of channel capacity.According to Theorem 1 in [2], we obtain ˆI q ( γ ) = C ( γ ) ≈ (log e ) γ, (17)when γ <<1