Simple Closed-Form Approximations for Achievable Information Rates of Coded Modulation Systems
aa r X i v : . [ c s . I T ] D ec Simple Closed-Form Approximations for AchievableInformation Rates of Coded Modulation Systems
Maria Urlea and Sergey Loyka
Abstract —The intuitive sphere-packing argument is used to ob-tain analytically-tractable closed-form approximations for achiev-able information rates of coded modulation transmission systems,for which only analytically-intractable expressions are availablein the literature. These approximations provide a number of in-sights, possess useful properties and facilitate design/optimizationof such systems. They apply to constellations of various cardinal-ities (including large ones), are simple yet reasonably accurateover the whole signal-to-noise ratio range, and compare favorablyto the achieved rates of recent state-of-the art experiments.
Index Terms —Achievable information rate (AIR), quadratureamplitude modulation (QAM), spectral efficiency, approximation.
I. I
NTRODUCTION
The growing demand for higher transmission rates andhigher spectral efficiency along with the development of pow-erful capacity-approaching codes has recently stimulated sig-nificant interest in coded-modulation optical transmission sys-tems, with multi-level modulation and capacity-approachingcodes [1][2]. For such systems, an adequate performancemetric is achievable information rate (AIR), which can beexpressed as mutual information (MI) or generalized mutualinformation (GMI), depending on the specifics of decod-ing/demodulation algorithms [3][4]. They have been widelyused for analysis, optimization and design of various commu-nication systems, including optical fiber systems.While achievable information rates have been studied inthe information-theoretic literature for multi-level modulationformats over the additive white Gaussian noise (AWGN)channel [22]-[24], the corresponding expressions cannot beevaluated analytically in a closed form, even for the simplestAWGN channel, since they include a number of analytically-intractable integrals over infinite intervals. Numerical inte-gration (e.g., via Monte-Carlo approach) remains the onlyalternative [4][22]. This makes it difficult to obtain insightsand also to perform system design and optimization (forexample, to find an optimal power allocation in a multi-streamtransmission system, as in e.g. [25]).In this paper, we present analytically-tractable closed-formapproximations to achievable information rates for M -PAMand M -QAM constellations (modulation) on the AWGN chan-nel (possibly including nonlinear interference in optical fibermodeled as additional Gaussian noise [5]) using the intuitivesphere-packing argument. These approximations are simpleyet reasonably accurate over the whole signal-to-noise ratio(SNR) range (not just at high or low SNR) and for various The authors are with the School of Electrical Engineering and Com-puter Science, University of Ottawa, Ontario, Canada, K1N 6N5, e-mail:[email protected] constellation cardinalities M (including large ones). Compar-ison to the back-to-back (B2B) or long-haul end-to-end (E2E)rates of recent state-of-the-art systems/experiments shows thatthe proposed approximations are in fact more accurate thanthe ideal AIR (via MI or GMI) since the latter ignores manyimperfections and limitations of real-world systems (e.g. non-zero overhead of realistic codes, guard bands, etc.).The proposed approximations also provide a number ofinsights unavailable from the exact (MI-based) expressions. Inparticular, they allow one to obtain the minimum M requiredto approach closely the (modulation-unconstrained) AWGNchannel capacity without using unnecessarily large constel-lations. The derivation of these approximations offers anadditional insight into a mechanism causing a rate loss in themodulation-constrained system: while the rate loss is additiveat low SNR, it is multiplicative, i.e., much more pronounced,at high SNR. The obtained approximations possess usefulproperties: they are differentiable and concave in the SNR sothat the respective optimization problems are convex and thusall powerful tools of convex optimization (see e.g. [28]) can beused to solve them. The main contributions of this paper arethe approximations of the modulation-constrained achievableinformation rates of M -PAM and M -QAM constellations in(11), (17), and (19), from which the minimum number ofconstellation points to approach closely the channel capacity(without significant rate loss due to limited M ) can be foundas in (12) and (18).II. C HANNEL M ODEL AND I NFORMATION R ATES
Following the widely-accepted approach, we use the Gaus-sian channel model, which has been extensively used in manyareas of digital communications [26]. This model was alsodemonstrated (via simulations and also experimentally) to besufficiently accurate for the optical fiber channel in many casesof practical interest (uncompensated transmission in the low-to-moderate nonlinearity regime) and offers a good balance ofaccuracy and simplicity [5][6]. The system block diagram isshown in Fig. 1, which includes an encoder, a modulator, achannel, a demodulator and a decoder [26]. The function of theencoder is to protect the transmitted message against channel-induced errors using forward error correction and the functionof the modulator is to transform the encoded signal to a formsuitable for transmission over the communication channel. Thebaseband discrete-time channel model (after matched filteringand sampling) is y i = x i + ξ i , (1)where x i and y i are transmitted and received (real-valued)symbols respectively at time i , ξ i is the additive white Gaus-sian noise. The capacity of this channel, i.e. the maximum achievable rate under the reliability criterion (arbitrary smallerror probability) and the power constraint, is [23] C = 12 log(1 + γ ) [bit/symbol] , (2)where γ = σ x /σ is the SNR, σ x and σ are the signaland noise power and all logarithms are base-2; σ can alsoinclude nonlinear interference in optical fiber when modeledas Gaussian noise, see e.g. [5].In practical systems, digital modulator makes use of acertain number of signals conveniently represented via aconstellation (in the signal space) [26]. Due to implementationcomplexity, the constellation cardinality (i.e. the number ofpoints/signals) is limited to M . On the other hand, recentdevelopment and adoption of powerful capacity-approachingcodes allows one to build nearly-optimal encoders and de-coders. Consequently, modulation-constrained achievable in-formation rates are widely used as a performance metric [1]-[4]. From the information-theoretic perspective, the part of thesystem between the encoder and the decoder can be interpretedas an induced channel , which includes the actual channeland the constrained (fixed) modulator/demodulator. This isillustrated in Fig. 1, where the induced channel is betweenA and B. Assuming symbol-wise decoding, the achievableinformation rate of this system is the mutual information ofthe induced channel. This quantity has been studied in theinformation-theoretic literature [22]-[24]. For the real-valued M -PAM constellation under equiprobable signalling, the per-symbol MI C M is [23] C M = log M (3) − X j M p πσ Z ∞−∞ e − z σ log X i e − d ij σ e − zdijσ dz, where d ij = a i − a j so that | d ij | is the distance betweenconstellation points a i and a j ; the summation is over allpoints. Unfortunately, the integrals over infinite intervals can-not be evaluated in closed-form, which makes it difficult toobtain insights and to use it in design/optimization process.Numerical evaluation of these integrals can be also difficult,especially when constellation cardinality is large and/or realtime evaluation is required (e.g. to compute optimal powerallocation for multi-stream transmission system, see e.g. [25]).From the information-theoretic perspective, achieving thechannel capacity requires selecting the best encoder/decoderand modulator/demodulator while a modulation-constrainedachievable information rate implies the optimization of theencoder/decoder only while the modulator is fixed, so that C M ≤ C . On the other hand, the AIR cannot exceed theentropy of the corresponding constellation [23], C M ≤ log M ,so that C M can be upper-bounded as follows C M ≤ min( C, log M ) . (4)While the former two bounds can be rather loose individually(at high and low SNR respectively), the latter is significantlytighter when considered over the whole SNR range (see Fig.4). To obtain further insights and to overcome the above-mentioned difficulties, we present below a simple closed-formapproximation for C M . (cid:3)(cid:3397) (cid:1876) (cid:3036) (cid:1877) (cid:3036) (cid:2022) (cid:3036) encoder modulator decoder demodulator Induced Channel channel A B Fig. 1. System Model and Induced Channel. codeword noise sphere received signal sphere codeword region (cid:1870) (cid:2868) (cid:1870)
Fig. 2. Sphere-packing argument: a codeword region is just slightly largerthan a corresponding noise sphere; r ≈ q nσ , r ≈ q n ( σ x + σ ) . III. AIR
VIA THE S PHERE P ACKING A RGUMENT
While a rigorous information-theoretic derivation of (2) isavailable and well-known [23], the sphere-packing argument,which originates back to Shannon, provides a more intuitiveunderstanding as to why this expression holds [27]. Since ourapproximation is also derived via this argument, we brieflyreview it below; see [27] for more details.
A. Sphere packing
During the transmission, n consecutive symbols are groupedinto codewords, so that x = [ x , x , ..., x n ] is transmittedwhile y = x + ξ is received, where ξ = [ ξ , ξ , ..., ξ n ] isthe noise vector. As n is getting large, the total noise power | ξ | = P i ξ i approaches nσ with high probability (dueto the law of large numbers), which is known as ”spherehardening” [27], so that y belongs to the noise sphere centeredon a transmitted codeword, as shown in Fig. 2, with highprobability. The decoder decides in favor of a particularcodeword when received signal y belongs to the codewordregion of that codeword (also known as ”decoding region”).For any transmitted codeword, the received signal y belongswith high probability to the received signal sphere of radius p n ( σ x + σ ) which encloses all individual noise spherescentered on respective codewords. For reliable decoding, (i)codeword regions must enclose corresponding noise spheres,and (ii) they must not overlap. These can be satisfied byselecting codeword regions as spheres of a radius slightlylarger than that of noise spheres and by packing all thosespheres into the received signal sphere as tightly as possibleallowing no overlaps.The number N of distinct codewords that can be reliablytransmitted over the channel is approximately equal to the number of non-overlapping noise spheres that can fit into thereceived signal sphere. It can be estimated from the ratio ofrespective volumes: N ≈ V /V = (1 + γ ) n/ , (5)where V = α ( nσ x + nσ ) n/ and V = α ( nσ ) n/ are thevolumes of the received signal and noise spheres respectively, α = π n/ / Γ( n/ and Γ( · ) is the gamma function, so thatthe channel capacity is C ≈ n log N ≈
12 log(1 + γ ) . (6)Remarkably, this heuristic argument gives the exact value ofthe channel capacity for the AWGN channel. B. Achievable information rates
The argument above applies if the number of codewordsof lenght n can be as large as necessary (i.e. is not con-strained). For a fixed constellation of M points, the numberof codewords of length n can be at most M n . To evaluate theimpact of this constraint, we present the per-symbol AIR inthe following form: C M = C − ∆ C, (7)where C is the (unconstrained) channel capacity (as above)and ∆ C ≥ is the rate loss due to a fixed constellation ofcardinality M . To estimate the latter, consider a hypotheticalsystem with noise power σ such that the number of distinctcodewords (noise spheres) is exactly M n : M n = VV = ( nσ x + nσ ) n/ ( nσ ) n/ , (8)where V = α ( nσ ) n/ is the volume of the hypothetical noisesphere, which is also the volume of a codeword region whenthere are exactly M n codewords. For this system, there is noloss in capacity due to a fixed constellation (within the spherepacking approximation) since the noise power is ”right” (i.e.noise spheres are the same as respective codeword regions sothat no more codewords can fit without increasing the errorprobability): σ = σ x M − ≈ σ x M , (9)where the last approximation holds when M is reasonablylarge, M ≫ . However, if the true noise power is less thanthe hypothetical one, σ < σ , more than one noise sphere canfit within the hypothetical noise sphere (codeword region), asshown in Fig. 3 (the central sphere). The resulting ∆ C can beinterpreted as the capacity of the fictitious channel with signalpower σ and the noise power σ , which can be estimated viathe ratio of volumes approach similarly to (6): ∆ C ≈ n log ( nσ + nσ ) n/ ( nσ ) n/ ≈
12 log (cid:18) σ x M σ (cid:19) . (10)Substituting this in (7), one finally obtains an approximation C a for the per-symbol AIR of M -PAM: C M ≈ C a = 12 log 1 + γ γ/M . (11) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:2)(cid:6)(cid:3)(cid:7)(cid:2)(cid:8)(cid:9)(cid:4)(cid:10)(cid:9)(cid:11)(cid:12)(cid:4)(cid:6)(cid:4)(cid:6)(cid:4)(cid:1)(cid:4)(cid:8)(cid:13)(cid:4)(cid:3)(cid:10)(cid:9)(cid:8)(cid:14)(cid:7)(cid:15)(cid:16)(cid:9)(cid:11)(cid:12)(cid:4)(cid:6)(cid:4)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:2)(cid:6)(cid:3)(cid:10)(cid:6)(cid:4)(cid:14)(cid:8)(cid:2)(cid:7)(cid:15)(cid:3)(cid:3)(cid:8)(cid:17)(cid:8)(cid:2)(cid:7)(cid:15)(cid:16)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:2)(cid:6)(cid:3) Fig. 3. The impact of the limited number of codewords: more noise spheresrepresenting additional codewords could be packed into existing codewordregions when noise is small. -5 0 5 10 15 20 25 30 35 40 45 50SNR [dB]01234567 A I R [ b i t/ sy m bo l / d i m en s i on ] CC M C a [7], QPSK [7], 16-QAMexperimental results [7], 64-QAM [9], 1024 & 4096-QAM4096-QAM, [10] 16-PAM(256-QAM)32-PAM(1024-QAM)64-PAM(4096-QAM)8-PAM (64-QAM)4-PAM (16-QAM)2-PAM (QPSK) [8], 4096-QAM [14], 16384-QAM [20], 64-QAM [19], 64-QAM [2,4], 256-QAM [18], 64-QAM [13], 64-APSK [12], 64-QAM [17], 256-QAM [11], 64-QAM [16], 1024-QAM [15], 256-QAM Fig. 4. M-PAM (QAM) AIR in [b/sym./dimension] evaluated via numericalintegration in (3), its approximation in (11) (or (17)) and the AWGN channelcapacity in (2) vs. SNR; recent experimental results are also shown forcomparison.
While this approximation is simple and analytically tractable,it is also reasonably accurate over the whole SNR range andfor various constellation cardinalities M (especially when M is large), as Fig. 4 shows, being somewhat less accurate in thetransition region between low and high SNR regimes and for2-PAM (for which more accurate approximation is obtained inthe next section). In all considered cases, the approximation in-accuracy does not exceed about 0.3 b/symb, which transformsinto 30% for 2-PAM, 10% and 5% for 8-PAM and 64-PAM,respectively. Hence, the approximation is more accurate forhigher-rate constellations and also significantly more accurateoutside of the transition region (between low and high SNR).Since the experimental results are well below the ideal AIR,this approximation is more accurate than the ideal AIR, withrespect to the rates of real-world systems/experiments (see alsothe next section).Not only this approximation is much simpler than its exactcounterpart in (3) and thus facilitates a numerical or even an-alytical optimization (e.g. to find an optimal power allocationin a multi-stream transmission system, for which no closed- form solution is known [25]), but it also captures importantproperties of the ideal AIR C M and provides additionalinsights unavailable from the known results:1) As M increases, both C a and C M approach C : C a ≈ C M ≈ C if M ≫ √ γ , from which one can estimate minimum M required to approach closely the channel capacity withoutusing unnecessarily large constellations : M min ≈ { , √ γ } . (12)In this regime, the upper bound in (4) is tight.2) At high SNR, both C a and the ideal AIR C M approachthe constellation entropy and are independent of the SNR, C a ≈ C M ≈ log M if γ ≫ M , (13)so that the upper bound in (4) is also tight at high SNR.3) At low SNR, C a can be approximated as follow: C a ≈ (cid:18) − M (cid:19) log e γ if γ ≪ , (14)so that the rate loss is small if M ≫ and never exceeds25%. Comparing this to (13), one concludes that the rate loss(due to constrained modulation) is additive at low SNR andmultiplicative at high SNR, i.e. much more pronounced in thelatter case.4) The approximation C a is a continuously-differentiable(to any order) and increasing function of the SNR: ∂C a ∂γ = log e M − γ )( M + γ ) > . (15)5) C a is a strictly-concave function of the SNR: ∂ C a ∂γ = log e − M )(1 + M + 2 γ )(1 + γ ) ( M + γ ) < . (16)The latter two properties are instrumental in numerical op-timization of multi-stream transmission systems (e.g. MIMO,as in [25]) by rendering the respective optimization problemsconvex so that all powerful tools of convex optimization, seee.g. [28], can be used. C. M -QAM While the approximation above was derived for the M -PAMconstellation (which is 1-D), it can also be extended to M -QAM (2-D), which is a popular choice for modern opticaltransmission systems [1][2] as well as wireless systems (e.g.WiFi, LTE, 5G). Indeed, M -QAM can be considered as 2 ×√ M -PAM operating on in-phase and quadrature channels [26](here we assume that √ M is integer), so that its per-symbolAIR can be approximated as follows: C M − QAM = 2 C √ M − P AM ≈ log 1 + γ γ/M , (17) This approximation demonstrates that the upper bound in [24], whichcan be put in the form M min ≤ √ γ , is actually tight. Note alsothat the upper bound was obtained in [24] via an elaborate information-theoretic analysis (which does not yield a capacity approximation) while ourapproximation to M min follows directly from the approximation in (11). and the properties above hold with the substitution M → √ M .In particular, the minimum constellation cardinality to ap-proach closely the channel capacity is M min ≈ { , γ } (18)Since 1st equality in (17) holds for both the approximationsand ideal AIR, the relative accuracy of the M -QAM approx-imation is the same as that of M -PAM, see Fig. 4, where therates are in [bit/symbol/dimension] (recall that M -PAM is 1-D while M -QAM is 2-D; experimental results are 4-D sincetwo polarizations are used). Comparing this approximationwith recent state-of-the art experimental results for back-to-back (B2B) or long-haul end-to-end (E2E) rates in Fig. 4,one concludes that the approximation is in fact closer to theexperimental rates than the ideal AIR. Note a particularlygood agreement for QPSK, 16-QAM and 64-QAM in [7],and 64-QAM in [20]. We attribute this to the fact that theapproximation C M − QAM in (17) underestimates the ideal AIR C M in (3) while the experimental B2B or E2E rates arealso below the ideal AIR by about 0.5-2 [b/sym.] (due tofinite blocklength/complexity codes with non-zero overheadand non-zero guard bands, in addition to other imperfectionsof practical systems [21]) Hence, the proposed approximationsare more accurate than the ideal AIR C M with respect to therates of real-world systems.IV. A N A SYMPTOTIC A PPROXIMATION
In order to improve the accuracy of the AIR approximationsabove for BPSK and QPSK, we obtain here an approximationvia the tools of asymptotic analysis (see e.g. [29] for moredetails on these tools). Applying the Laplace method to theintegrals in (3), the AIR of 2-PAM (BPSK) and 4-QAM(QPSK) can be approximated, after some manipulations, as C − P AM ≈ − log (cid:0) e − γ (cid:1) , C − QAM = 2 C − P AM (19)As Fig. 5 shows, this approximation is remarkably accurateover the whole SNR range: the inaccuracy does not exceed10%, being much smaller outside of the transition region. Notealso that this approximation is a continuously-differentiable (toany order) and concave function of the SNR, making it suitablefor convex optimization algorithms.An alternative approximation for the 2-PAM (BPSK) and4-QAM (QPSK) rates, which is somewhat more involved dueto its use of the error function, can be found in [30, (45), (46)].V. C
ONCLUSION
Based on the intuitive sphere-packing approach, the simpleand analytically-tractable approximations have been obtainedfor the achievable information rates of M -PAM and M -QAMconstellations over the AWGN channel, including opticalfiber channels when nonlinearity can be modeled by additiveGaussian noise. Not only these approximations are reasonablyaccurate over the whole SNR range and for various values of M (including large ones), they also provide additional insights It should be noted that the quoted experimental results were obtainedunder different conditions, e.g. single-channel vs. multi-channel systems etc.,and hence differ significantly. -20 -15 -10 -5 0 5 10 15 20
SNR [dB] A I R [ b i t/ sy m bo l ] (3)(19) QPSKBPSK
Fig. 5. 2-PAM (BPSK) and 4-QAM (QPSK) AIR and the approximations in(19) vs. SNR. unavailable from the known expressions. The sphere-packingbased derivation sheds additional light on the rate loss mech-anism due to a fixed (constrained) constellation cardinality:while the loss is additive at low SNR, it is multiplicative, i.e.much more pronounced, at high SNR.Comparison to the B2B or E2E rates of state-of-the art realsystems/experiments shows that the proposed simple approx-imations are in fact more accurate than the ideal AIR.The presented approach and approximations can also beused for polarization-multiplexed transmission systems. Sincethe difference between the MI and GMI is small for Gray-labeled constellations [4], the above approximations can alsobe used for the latter. R
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Maria Urlea received her B.A.Sc. (2012) and M.A.Sc (2015) in Telecommu- nications form the School of Electrical Engineering and Computer Scienceat the University of Ottawa. She is currently a Systems Engineer in Ottawa,Canada and focuses on network design and architectures for several local andnational entities.