Nonuniform probability modulation for reducing energy consumption of remote sensors Jarek Duda Jagiellonian University, Golebia 24, 31-007 Krakow, Poland. Email: [email protected] Abstract—One of the main goals of 5G wireless telecom- orthogonal family of functions (subcarriers), for example munication technology is improving energy efficiency, espe- in OFDM. cially of remote sensors which should be able for example These constellations are often QAM lattices of size to transmit on average 1bit/s for 10 years from a single AAA battery. There will be discussed using modulation up to 64 in LTE. There is assumed uniform probability with nonuniform probability distribution of symbols for modulation (UPM) - that every symbol is used with improving energy efficiency of transmission at cost of the same frequency. Generally, a stream of symbols reduced throughput. While the zero-signal (silence) has P having pss probability distribution ( s ps = 1) contains zero energy cost to emit, it can carry information if used asymptotically Shannon entropy h = Pm p lg(1=p ) alongside other symbols. If used more frequently than s=1 s s others, for example for majority of time slots or OFDM bits/symbol (lg ≡ log2), where m is the size of alphabet. subcarriers, the number of bits transmitted per energy unit Entropy is indeed maximized for uniform probability dis- can be significantly increased. For example for hexagonal tribution ps = 1=m, obtaining h = lg(m) bits/symbol, modulation and zero noise, this amount of bits per energy which is lg(64) = 6 bits/symbol for QAM64. unit can be doubled by reducing throughput 2.7 times, However, this natural choice of using uniform prob- thanks to using the zero-signal with probability ≈ 0.84. There will be discussed models and methods for such ability distribution is not always the optimal one. For nonuniform probability modulations (NPM). example when the channel has constraints, like forbid- ding two successive ones (’11’) in Fibonacci coding, I. INTRODUCTION then choosing Pr(xt+1 = 0jxt = 0) = Pr(xt+1 = The currently being developed 5th generation mobile 1jxt = 0) = 1=2 is not the optimal way. Instead, we network (5G) has many ambitious goals, like 10Gbps shouldp more often choose ’0’ symbol, optimally with peak data rates, 1ms latency and ultra-reliability. Another ' = ( 5−1)=2 probability, as this symbol allows to pro- high priority is reducing energy consumption, especially duce more entropy (information) in the successive step. to improve battery life of mobile and IoT devices. This For a general constraints the optimal probabilities can be goal is crucial for expected omnipresent fleet of remote found by using Maximal Entropy Random Walk [1]. sensors, monitoring all aspects of our world. Such sensor Another example of usefulness of nonuniform should be compact, inexpensive and has battery life probability distribution among symbols used for of order of 10 years, as battery replacement in many communication are various steganography-watermarking applications is economically infeasible. Hence, this is an problems, where we want encoding sequence to resemble asymmetric task: the main priority is to reduce energy some common data, for example picture resembling QR requirements of the sender, tolerating increased cost at codes. Surprisingly, generalization of the Kuznetsov- the receiver side. A crucial part of the cost of sending Tsybakov problem allows such encoding without arXiv:1608.04271v1 [cs.IT] 15 Aug 2016 information to the base station is establishing connection decoder knowing the used probability distributions (e.g. - their number and so energy cost can be reduced by picture to resemble) ([2], [3]). However, this lack of buffering information, or can be nearly eliminated if knowledge makes encoding more expensive. the sensor just transmits the information in time periods precisely scheduled with the base station. In this paper we will focus on a more basic reason We will discuss approach for reducing energy need of to use nonuniform probability distribution among the actual transmission of such buffered data, preferably symbols: that the cost of using various symbols compressed earlier to reduce its size. The information does not have to be the same. Assume Es is the is encoded in a sequence of symbols as points from a cost of using symbol s, then entropy for a fixed P chosen constellation: a discrete set of points in complex average energy (E = s psEs) is maximized for (I-Q) plane. This sequence of symbols can be used for Boltzmann probability distribution among symbols time sequence of impulses, or as coefficients for usually Pr(s) / e−βEs . For example in Morse code dash lasts much longer than dot, what comes with higher time II. CAPACITY AND ENERGY EFFICIENCY OF and energy cost. Designing a coding with more frequent NONUNIFORM PROBABILITY MODULATION (NPM) use of dot (Pr(dot) > Pr(dash)) would allow to lower In this section there will be first reminded why average cost per bit. Anther example of nonuniform Boltzmann distribution is the optimal choice from the cost is sending symbol ’1’ as electric current through a perspective of energy efficiency, then three modulations wire, symbol ’0’ as lack of this current - symbol ’1’ is will be analyzed, first without then with noise. more energy costly, hence should be used less frequently. For better intuition, Shannon entropy is measured in Pm bits: h = s=1 ps lg(1=ps) bits/symbol (lg ≡ log2). We will focus here on application for wireless commu- A. Probability distribution maximizing entropy nication modulation, where the cost related to emitting a symbol is usually assumed to be proportional to square Assume Es is the cost (energy) of using symbol s. 2 We want to choose the optimal probability distribution of its amplitude: Ex / jxj , hence we could improve energy efficiency by more frequent use of low amplitude fpsgs for some fixed average energy E: symbols. X X psEs = E ps = 1 (1) Basic theoretical considerations will be reminded, then s s used to analyze potential improvements especially for such that Shannon entropy is maximized: h ln(2) = P the situation of modulation for wireless technology: − s ps ln(ps). to reduce required energy per bit, especially for the Using the Lagrange multiplier method for λ and β purpose of improving battery life of remote sensors. The parameters: average amount of bits/symbol (entropy) is maximized ! ! for uniform probability distribution (UPM), hence using X X X L = − ps ln ps+λ ps − 1 +β psEs − E nonuniform distribution (NPM) means that more sym- s s s bols are required to write the same message, so the @L tradeoff of improving energy efficiency (bits per energy 0 = = − ln(p ) − 1 + λ + βE @p s s unit) is also reducing throughput (bits per symbol). s e−βEs e−βEs The use of nonuniform probability distribution of sym- p = = (2) s e1−λ Z bols requires a more complex coding scheme, especially 1−λ P −βEs from the perspective of error correction (channel coding). where Z = e = s e is the normalization Entropy coders allow to work with kind of reversed task: factor (called partition function). encode a sequence of symbols having some assumed The parameter β can be determined from average probability distribution into a bit sequence. Switching energy: P E e−βEs its encoder and decoder, we can encode a message E = s s P −βEs (a bit sequence) into a sequence of symbols having s e some chosen probability distribution. Due to low cost, As expected, Boltzmann distribution is the optimal a natural approach would be using a prefix code here, way to choose probability distribution of symbols: ps / for example 0 ! a; 10 ! b; 11 ! c. However, e−βEs . The standard way of evaluating cost of a signal it approximates probabilities with powers of 1=2 and in wireless telecommunication is square of its amplitude: cannot use probabilities 1=2 < p < 1, which turn 2 Es = jxj . Hence for x 2 R the optimal probability is out crucial in the discussed situations. Additionally, its Gaussian distribution with standard deviation σ2 = E: error correction would require some additional protection layer. Hence, a more appropriate recent entropy coding 1 −x2 ρG(x) = p e 2E will be discussed for this purpose: tANS coding ([4], 2Eπ [5]). While having cost similar to prefix codes (finite Z 1 1 state automaton, no multiplication), it operates on nearly HG := − ρG(x) lg(ρG(x))dx = lg(2πeE) (3) 2 accurate probabilities, including 1=2 < p < 1. Addi- −∞ tionally, its processing has an internal state, which can Let us compare it with uniform distribution, which is be exploited like the state of convolutional codes [6] usually used in practical modulation schemes. Take a for error correction purpose - thanks of it encoder does rectangular density function on some [−a; a] range with height 1 to integrate to 1. Its average energy is E = not need to apply another coding layer, saving energy 2a p R a 1 2 a2 required for this purpose. −a 2a x dx = 3 , getting a = 3E parameter for a 2 chosen average energy E. Now Z a 1 1 Hu := lg(2a)dx = lg(2a) = lg(12E) −a 2a 2 So the gain of using Gaussian distribution is 1 H − H = lg(πe=6) ≈ 0:2546 bits: (4) G u 2 There was used differential entropy (with integrals), which gets natural intuition when approximated with Riemann integration for some quantization step q: Z 1 X H = − ρ(x) lg(ρ(x))dx ≈ − qρ(kq) lg(ρ(kq)) = −∞ k2Z X X = − qρ(kq) lg(qρ(kq)) + qρ(kq) lg(q) k2Z k2Z The left hand side term is the standard entropy for probability distribution of quantization with step q, the right hand side term is approximately lg(1=q).