In variance-based logic (VBL), information is encoded by the change in the variance of a signal as opposed to the conventional mean-based logic (MBL) where the information is encoded by the change in the mean of the signal. In this pa- per, we compare the fundamental limits on the minimum energy per bit that can be achieved by VBL and MBL representations in high-dimensional signal space. We show that while for MBL representations, the trade-off between the energy- per-bit and the bit-error-rate (BER) is fundamentally constrained by the classical Shannon-limit, using VBL representations it is theoretically possible to achieve ar- bitrarily small BER while dissipating near zero energy-per-bit. This surprising re- sult has been experimentally veriﬁed for Additive-white-Gaussian-Noise (AWGN) channels using Monte-Carlo simulations. We believe that high-dimensional VBL based encoding could provide a new approach for designing ultra-energy-efﬁcient communication and sensing systems. 1 Introduction There exists a trade-off between the energy required for transmitting information across a noisy communication channel and the reliability of correctly decoding the transmitted information. This trade-off manifests itself as the celebrated Shannon-limit [1] which determines the minimum energy-per-bit ( Eb=No) required to achieve arbitrary small bit-error-rate (BER) for a speciﬁc type of communication channel. In this paper, we argue that an alternate signal representation can overcome this fundamental limit and theoretically achieve arbitrarily small BER while dissipating near zero energy-per-bit. The proposed approach is based on our previously reported variance-based logic (VBL) representation where the information bit is encoded as the change in signal variance [2]. This logic representation is contrary to the conventional mean-based logic (MBL) rep- resentation where the information is encoded as the change in the average or the mean of the signal parameter, for instance, amplitude, phase or frequency [3–5]. Both of 1 dEb(KT) BERMBL1 VBL1 MBLLB20 15 10 5 0 0.1 0.2 0.3 0.4 0.50 1 10 BER Eb/N0 (dB)Shannon Bound -1.59 dBMBL1 VBL1 MBL∞cb 1D MBL 1D VBLℒi′′ℒia 2D VBL 2D MBL OFigure 1: Comparison of MBL and VBL representations: (a) 1 dimensional (1D) and (b) 2 dimensional (2D) MBL and VBL code words LiandL0 iwith Gaussian pdfs respectively. (c) Comparison of thermodynamic energy-per-bit for 1-dimensional MBL and VBL as a function of average probability of error or BER; (d) BER- Eb=N0plots showing the Shannon-limit for MBL, and the proposal that using high-dimensional VBL it is possible to surmount this limit. these representations are illustrated in Fig. 1(a) and (b) using one-dimensional (1D) and two-dimensional (2D) probability density functions (pdfs) where the overlap be- tween the pdfs (shown as shaded regions) determines the probability-of-error ( BER ) for each of the respective representations. For a 1D MBL representation, we and others have shown that the thermodynamic energy dissipated per bit is bounded from below by 4KT Joules per bit and that this limit is achieved when the average probability of error approaches 0:5or for high BER [6, 7]. However, in the case of 1D VBL it was shown in [6] that when the average probability of error approaches 0.5, the energy dis- sipated per bit goes to zero Joules per information bit. Both the trends are illustrated in Fig. 1(c) using a cross-over point Owhere one type of representation is shown to be more energy efﬁcient than the other. Note that these lower-bounds only applies to the energy required to communicate across a noisy-channel and does not include the energy necessary for encoding and decoding at the source and the receiver respectively. In this paper, we investigate the lower bounds on Eb=N0when using high-dimensional VBL (HD-VBL) representations. Speciﬁcally, we will investigate if the cross-over point (as shown in Fig. 1(c)) can be shifted towards the origin which would then imply that an HD-VBL representation could potentially achieve zero BER while dissipating near-zero amount of energy-per-bit. In the case of high-dimensional MBL (HD-MBL) lower bound on Eb=N0can be derived from Shannon’s channel coding theorem, as summarized in Appendix 4.1, and is approximately equal to 0:6or1:6dB for an additive white gaussian noise (AWGN) channel. Thus, all existing communication 2 methods that use MBL representations are limited by this bound even when the di- mensions used for encoding approaches inﬁnity, as illustrated in Fig. 1(d). In the case of HD-VBL determining the fundamental limits on Eb=N0will involve understanding how the volume spanned by the overlap between the pdfs (as illustrated in Fig. 1(d)-(e) for 2D, 3D representation) scales as the number of dimensions become large. Rest of the paper is organized as follows, section 2 presents a geometrical real- ization of high-dimensional VBL (HD-VBL) and derive the analytical expressions to estimate the BER and the capacity of HD-VBL. These results were validated using monte-carlo simulations. Brief discussions and conclusion of our work are provided in section. 3. Additional results are provided in the appendix. 4. 2 Analysis and Results of HD-VBL We ﬁrst present our analysis corresponding to a particular VBL representation that can be easily visualized. We follow a standard communication model, as shown in Fig. 2(a), comprising of a source that has access of Mcodewords, denoted by the setL, which it needs to communicate over a noisy channel to a receiver. In the case of VBL, each codeword is encoded by modulating the variance of a zero-mean high- dimensional signal along a speciﬁc direction v2RDin aD-dimensional space. For an additive-white-Gaussian-noise (AWGN) channel, the codeword received at the re- ceiver can be illustrated using the contours of a multi-dimensional Gaussian pdf that is geometrically represented using D-dimensional hyper-ellipsoids. When the channel is not being used, the received pdf at the receiver symmetrical and is illustrated us- ing a D-dimensional hyper-sphere S0and is determined by the noise spectral density (N0). When the target codeword L0is transmitted, the received pdf at the receiver is a hyper-ellipsoidS1(as illustrated in Fig. 2(b)) whose major axis is oriented along the directionv. To detect whether the target code-word L0was transmitted, the receiver uses a decoder whose decision boundaries coincide with two hyper-cylinders T1and T2, as illustrated in Fig. 2(b). Both T1andT2are aligned along v(known a-priori) and terminated by spherical caps C1andC2, formed by the overlap between hyper-ellipsoid S1and the hyper-sphere S0. If the received noisy vector lies within the boundary of the hyper-cylinders, then the received vector is decoded as L0. Based on this decoder we can analytically estimate the probability-of-detection pdof the code-wordL0. Math- ematically, we show in section 2.1, estimating the probability mass that lies within the decoding boundaries reduces to solving a two-fold hypothesis testing formulation involving 2distributions. Then, for a given probability-of-detection pd, we could es- timate the number of non-overlapping spherical caps C1andC2that could be packed on the surface of the hyper-sphere S0, which gives us an estimate of the capacity of this speciﬁc VBL representation. 2.1 Estimation of BER for VBL with One Degree-of-Freedom For a VBL with one degree of freedom, as illustrated in Fig. 2(b), the signal variance, corresponding to a code-word L0, is encoded along a D-dimensional vector. Without loss of generality, multi-dimensional variance could be represented using a diagonal 3 db fec 𝜂= 1 𝜂= 4 𝜂= 10 𝜂= 35 𝜂= 100 𝐷𝐶𝛺𝐷𝑆0𝑇1 𝑆1𝜙𝜃 𝑇2𝐷1 𝑣 𝐸𝑏/𝑁0𝐵𝐸𝑅1 Shannon Bound VBL Lower Envelope Monte Carlo𝐷𝐵𝐸𝑅1 BERanalytical BERmonte -carlo 𝛼 𝜃𝑝𝜒21𝑝𝜒2D−1 𝜒2𝑝𝑑𝑆𝑜𝑢𝑟𝑐𝑒 𝐶ℎ𝑎𝑛𝑛𝑒𝑙 𝐷𝑒𝑐𝑜𝑑𝑒𝑟 (ℒ) (T) (𝐴𝑊𝐺𝑁) 𝑣 𝒀aFigure 2: Estimation of capacity and BER for a one degree of freedom VBL represen- tation and a speciﬁc type of decoder applied to (a) a general communication system. (b) Decision rule for estimating the probability of detection ( pd). (c) Estimation of BER reduced to a two-fold hypothesis testing based on a Chi-square distribution. (d) BER 1 with respect to number of dimensions (D) with parameters ( SNR ,). (e) Estimated channel capacity as a function of log(D) (f)BER 1-Eb=N0plot for the case k= 1. 4 covariance matrix ( D) as, D= [P+N0; N0; N0;:::; N 0] (1) wherePis the total signal power and N0is the channel noise power in each dimension. Thus, the multi-variate zero-mean normal distribution corresponding to the recieved vector Y= [y1; y2; :::; yD]is given by, fY(y1;y2;:::yD) =Aexp 1 2" y2 1 P+N0+DX i=2y2 i N0#! ; (2) whereAis the normalization factor. Note that this distribution can be parameterized us- ing[a; b] = y1;rPD i=2y2 i D1 where each of the parameters a2andb2assumes a 2distribution with 1andD1degrees of freedom respectively. The respective 2distributions with degrees of freedom 1andD1are illustrated in Fig. 2(c). The decision rule (or decoder) is then determined by the solid angle Dand in Fig. 2(b) and can be mapped into Fig. 2(c) as thresholds and. The probability of detecting the codeword pdis given by the total probability mass concentrated between the two thresholds, as shown in Fig. 2(c) and is given by pd= 1BER 1=P(Y2(T1[T2)) =P (a2)[(b2 ) =P 2 1 P+N0 [ 2 D1(D1) N0 =P 2 10 DSNR + 1 [ 2 D1(D1)0 = (D1 2;(D1)0 2) (D1 2) 1 (1 2;0 2(D SNR +1)) (1 2)! (3) Here0= N0is the normalization parameter, = = (tan )2andSNR =P DN 0. Also, (;)is the lower incomplete gamma function and ()is the ordinary gamma function [8]. Fig. 2(d) shows the corresponding bit-error-rate BER 1, where the sub- script corresponds to the VBL representation with one-degree of freedom. BER 1 asymptotically decreases with the increase in the number of dimensions D, for differ- ent levels of signal-to-noise ratio (SNR) and decoder parameters. Also, the analytical model forpdhas been validated using Monte-carlo simulations for multiple parameter realizations, as shown in Fig. 2(d). To calculate the capacity of HD-VBL, we estimate the number of non-overlapping spherical caps that can be packed on the surface of a D-dimensional hyper-sphere deter- mined the thresholds, as illustrated in Fig. 2(b). Estimation of optimal packing density for spherical-caps has been well studied in the context of spherical codes and in partic- ular, wrapped spherical codes and laminated spherical codes [9]. The number of code words that can be accommodated in the space is given by the ratio between the total 5 surface area, and the area spanned by one single codeword and is given by M1=1 Isin2(tan1p)D1 2;1 2 (4) where= . Details of the derivation can be found in the Appendix 4.2. Using the expression in equation 4, the capacity for this representation can be estimated as C= log2M1= log2 1 Isin2(tan1p)D1 2;1 2! : (5) Fig. 2(e) plots the capacity Cwith respect to the signal dimensions D, for different choices of the decoder hyper-parameters ( , ,= =). The estimated capacity in equation 5 can be used to estimate the energy-per-bit (Eb N0) for a speciﬁc value of signal powerPas Eb N0=SNR C(6) The results in Fig. 2(c) and (d) were used to obtain the BER 1-Eb=N0plot as shown in Fig 2(f). Note that every point in the VBL simulation in Fig 2(f) correspond to a speciﬁc choice of a decoder parameter set (; ). The result shows that for this VBL representation, the BER can be reduced, but at levels of energy-per-bit Eb=N0that are signiﬁcantly higher than the theoretical Shannon-limit. 2.2 Estimation of BER for VBL with k-degrees-of-freedom We now generalize the VBL representation to a topology where the variance is now en- coded along a speciﬁc sub-manifold of dimension k(<D), instead of a single direction ork= 1. The code-words are now represented by sub-manifolds that are spanned by a vectorsv0= [v1;v2;:::vD], such thatPD i=1vi=kwithvi2f0;1g8i= [1;D]. Sim- ilar to the analysis presented in section 2.1, the estimation of the modiﬁed parameters [ak; bk] =rPk i=1y2 i k ;rPD i=k+1y2 i Dk , which admits the pdfs a2 k 2 k andb2 k 2 Dk. The respective distributions corresponding to k= 3 andk= 10 are shown in Fig. 3(a)-(b), where the respective probability-of-detections p3 dandp1 d0are given by the total probability mass within the two thresholds and . Following similar arguments in section 2.1, the probability of decoding pk dcan be estimated as pk d= 1BERk=P(Y2(T0 1[T0 2)) = (Dk 2;(Dk)0 2) (Dk 2)0 @1 (k 2;k0 2(DSNR k+1)) (k 2)1 A (7) It can be seen in Fig. 3(b), as kincreases the region signifying Pdspans a large pro- portion of the probability mass as depicted in the BERkplots shown in Fig. 3(c)-(d). Note that sub-script kdenotes the number of degrees-of-freedom considered in deﬁning 6 a 𝐷𝐵𝐸𝑅3b𝐵𝐸𝑅3 𝐸𝑏/𝑁0 𝐸𝑏/𝑁0c Shannon Bound VBL BoundVBL 𝐵𝐸𝑅10 Shannon Bound VBL BoundVBLBERanalytical BERmonte -carlo 𝐷𝐵𝐸𝑅10 BERanalytical BERmonte -carlo 𝛼 𝜃𝑝𝜒23𝑝𝜒2D−3 𝜒2 𝛼 𝜃𝑝𝜒210𝑝𝜒2D−10 𝜒2 d ef𝑝𝑑10𝑝𝑑3Figure 3: (a) (b) Estimation of the Pdusing two-fold hypothesis testing for a 100 dimensional VBL representation with k=3, k=10 respectively. (c) (d) Veriﬁcation of analytical BER expressions derived in equation. 3,7 with monte-carlo simulations for different parameter values (SNR; ). (e) (f) Energy per bit ( EbN 0) estimated as de- ﬁned in equation. 6 vs BER plots for the case of VBL-HDL. 7 HD-VBL. The analytical results are given by equations 7 have also been veriﬁed using Monte-Carlo simulations, as shown in Fig. 3(c)-(d). Capacity for a VBL representation with k-degrees of freedom is estimated using spherical-cap packing techniques applied to aDkdimensional sphere as described in Appendix 4 and is given by Mk=1 Isin2 Dk 2;1 2 (8) and capacity is given by, C= log2Mk= log2 1 Isin2(tan1p)Dk 2;1 2! (9) The average bit-error rates ( BER ) and capacity ( C) were computed for different thresh- olds (; ) and the energy per bit ( Eb=N0) was estimated using equation. 6 as described in the methods section. Fig. 3(e)-(f) show the BER- Eb=N0plots corresponding to the casek= 3 andk= 10 . Again, each point in the BER simulation correspond to the results obtained for a speciﬁc choice of the decoder parameter ; . Also the limiting case ofBER 10is shown in Fig. 4(d) which clearly shows the lower bound for the case of HD-MBL where Ebcorresponding to HD-MBL approaching zero. The results clearly show that there exist HD-VBL representations for dimensions no more signif- icant than 100, that can easily surpass the classical Shannon-limit. However, whether there exist VBL representations that can asymptotically achieve arbitrarily small BER at arbitrarily smallEb N0still needs to be veriﬁed through extensive Monte-Carlo stud- ies. This would require an exhaustive search over different decoding parameters and decoder topologies which in turn requires signiﬁcant computational resources. 3 Discussion The analysis and results presented in this paper seem to indicate that by encoding in- formation in the variance of the signal we can surmount the limitation imposed by the classical Shannon-limit. The intuitive understanding of this effect is that VBL encodes information on the surface of a hyper-sphere, as shown in Fig. 2(b), whose volume asymptotically goes to zeros as the number of dimensions approaches inﬁnity. As a result, for VBL the tails of the distributions project orthogonally to the hyper-spherical surface, as shown in Fig. 4(a), whereas the packing of the centroids is achieved on the surface of the hypersphere, as shown in Fig. 4(b). In a high-dimensional space and for low SNR, these two effects (packing and error rates) are not perfectly correlated and hence once could be reduced without reducing the other. For a conventional MBL representation, the Shannon-limit arises because the overlap between the pdfs of the ad- jacent codewords (centroids) lie within the volume spanned by their pdfs, as illustrated in Fig. 4(c). In higher dimensions and for a given signal-to-noise ratio, the packing den- sity (or equivalently the channel capacity) scale linearly with the number of dimensions (as shown in the appendix section 4.1). As a result, the energy-per-bitEb N0, cannot be further reduced while achieving near zero BER, thus leading to the Shannon-limit [10]. In terms of thermodynamic limits HD-VBL indeed moves the cross-over point Oin 8 Fig. 1(c) towards the origin, which was the original motivation for this paper. This is clearly shown in Fig. 4(d) for different cases of VBL with 10-degrees of freedom and for dimensions D= 100 . One of the challenges in using VBL representation for practical communication system is the complexity in implementing the decoder at the receiver. While in this paper we showed that the asymptotic beneﬁts hold for a simple VBL representation where the signal energy is directed towards modulating the variance along one direc- tion (or axis), the convergence rates are slow and thus would require a large num- ber of dimensions for encoding. We, therefore, extended the VBL representation by spreading the signal energy across variances along k < D different directions. This clearly showed in Fig. 3(d) that such type of VBL encoding that surpasses the classi- cal Shannon-limit. However, the optimality is achieved only when the correct set of decoding hyper-parameters are chosen. In appendix 4.3, we show how to choose the decoder parameters for a one-dimensional VBL representation. However, for higher- dimensional VBL and when multiple directions are chosen for encoding, one has to resort to an exhaustive search over the decoder parameter space to achieve the optimal decoding performance. Practical consideration and the complexity of the optimization procedure would impose limits on the correct choice of these parameters, however, note that for stationary channels, the parameters need to be estimated only once. Adap- tive determination of the decoding hyper-parameters will be a topic of future research in this area. Also, note that even though we have shown the asymptotic property of near-zero BER at near-zero Eb=N0for a speciﬁc type of VBL representation, there might exist other VBL representations that might demonstrate superior convergence. In this spirit, there might exist some connection between VBL representations and stochastic resonance based decoding, where “noise” or variance is introduced into the system to improve the detection of a buried but harmonic signal [11]. Additionally, the practical realization of the VBL based communication system would be a topic for future research. In our original paper on VBL [6], we had highlighted that the energy- harvesting systems [2] and emerging valleytronic devices [12] could form a natural platform to implement some of the proposed communication techniques. Other ap- proaches towards a practical realization of a VBL communication system could also involve exploiting the thermal-noise of electronic devices (resistors or transistors) to modulate the variance of the channel, as reported in [13, 14]. 4 Appendix 4.1 Lower Bounds onEb N0for A WGN Channel. The classical Shannon-limit has been estimated using the expression of channel-capacity for an AWGN channel. The limit on the Kinformation bits that can be transmitted us- ing aDdimensional signal and an average signal power of Pover a channel with N0 9 a b c d 𝑃+𝑁0 Ԧ𝛴=(𝜎1,𝜎2,…,𝜎D) Shannon Bound VBL L L’ 𝐵𝐸𝑅10𝐸𝑏𝐾𝑇 𝐷𝑁0(𝑃+𝐷𝑁0) 𝑁0Figure 4: (a) Illustration of spherical cap and decoding cylinder oriented in the direction of vector v or variance vector deﬁned by (b) Surface Packing of high dimensional variance based codes and optimal packing density. (c) Sphere packing giving rise to the shannon bound on energy per bit ( Eb=N 0) for reliable transmission of code words with signal power ( P) over a noisy channel ( N0). (d) FOM plot indicating the region where VBL achieves encoding information with lower energy per bit than that of determined by the shannon bound. 10 noise power per dimension is given by, K < D log2(1 +P N0D) K < D log2(1 +EbK N0D) (10) Deﬁning rate as R=K Dgives, R< log2(1 +Eb N0R) (11) For a ﬁniteKand when dimensions D!1 ,R!0. Thus, Eb N0>lim R!02R1 R= ln 2 = 0:6931 =1:59dB: (12) This is the absolute minimum amount of signal energy to noise power spectral density required to transmit at least one bit of information reliably. Similar lower-bound on Eb=N0can be arrived at by using sphere packing arguments presented in [10] and illustrated in Fig. 4(c). 4.2 Geometrical approach for ﬁnding capacity of VBL. Firstly, we analyze for the case of HD-VBL with 1 degree-of freedom and later the results were extended for a much generic case (k DOF). As shown in section 2.1 the contours of a high dimensional pdf for a HD-VBL takes shape of an hyper-ellipsoid and the elapsed area on the surface decision boundary is given by a D1dimensional hyper sphere as shown in Fig. 4(a). Thus the maximum number of codewords (channel capacity) is be determined by counting the number of non overlapping D1hyper spheres on the surface of decision boundary as shown in 4(b). This hyper-spherical arc makes an angle withvin the parameter space, as shown in Fig. 2(b). Surface-area of the spherical-cap is obtained by integrating (D-1) sphere of radius rsinwith arc elementrdover the angle 0 to , ACap D(r) =Z 0AD1(rsin)D2rd (13) =2D1 2 (D1 2)rD1Z 0sinD2d (14) =1 2AD(r)Isin2 D1 2;1 2 (15) whereADis the total surface area of a D dimensional hyper-sphere and I(:;:)is the regularized incomplete beta function. Also, sin2 could be expressed as sin2(tan1p) where= and thus, ACap D(r) =1 2AD(r)Isin2(tan1p)D1 2;1 2 (16) 11 Since the solid angle spanned by the hyper ellipsoid is on either side of the origin and spans an equal solid angle of Din thevdirection the surface area occupied by each code word is given by 2Acap D(r)and the maximum number of code words that ﬁll the entire surface area is given by, M1=AD(r) 2Acap D(r) =1 Isin2(tan1p)D1 2;1 2 (17) and the capacity is given by, C= log2M1= log2 1 Isin2(tan1p)D1 2;1 2! (18) Note that this value is dependent only on dimensions, the choice of and is indepen- dent ofSNR . This result was extended for a generic case, where v= [v1;v2;:::;vD],PD i=1vi= kwithvi2f0;1g8i= [1;D], and similar to the BERkresult as shown in equation. 7 the capacity is given by estimating number of decoding cylinders on a D-k spherical surface and is given by C= log2Mk= log2 1 Isin2(tan1p)Dk 2;1 2! (19) 4.3 Optimal decoding threshold for VBL. For a given signal power and dimension, (SNR;D )pair, goal is to ﬁnd the optimal parameter values (; )or(0;)that would maximize both probability of correctness and the number of codewords in the signal space (pd;M). For the case of simple 1D representation, the error probabilities were asymmetric and are given by p0j1=erfc(p 20) (20) p1j0= 1erfc(p 21) (21) whereis the threshold, erfc(.) is the error function and 0,1are respective standard deviations representing 0or1. Thus average probability of error is given by, BER 1=p1p0j1+p0p1j0 (22) =1 2 erfc(p 20) + 1erfc(p 21) : (23) Optimal Threshold in this case is determined by, min BER 1= min 1 2 erfc(p 20) + 1erfc(p 21) (24) 12 Taking derivative w.r.t and equating it to zero we get, opt=s 22 02 1ln(1=0) 2 12 0: (25) The second derivative test shows that optgives the minimum probability of error for any given snr ( 0,1). Acknowledgements This work was supported in part by a research grant from the National Science Foun- dation CNS1646380.