Sum-Rate Channel Capacity for Line-of-Sight Models

Abstract

This work considers a base station equipped with an M-antenna uniform linear array and L users under line-of-sight conditions. As a result, one can derive an exact series expansion necessary to calculate the mean sum-rate channel capacity. This scenario leads to a mathematical problem where the joint probability density function (JPDF) of the eigenvalues of a Vandermonde matrixWWHare necessary, whereWis the channel matrix. However, differently from the channel Rayleigh distributed, this joint PDF is not known in the literature. To circumvent this problem, we employ Taylor’s series expansion and present a result where the moments ofmnare computed. To calculate this quantity, we resort to the integer partition theory and present an exact expression formn. Furthermore, we also find an upper bound for the mean sum-rate capacity through Jensen’s inequality. All the results were validated by Monte Carlo numerical simulation.

1. Introduction

Since the earliest stages of the development and prototyping of wireless communications, researchers introduced random matrix theory as a mathematical tool for modelling and analysing wireless multiantenna communications [1]. A fully detailed model of a multiantenna point-to-point link between a transmitter and a receiver requires the characterisation of a product of random matrices. For example, cellular networks present typical scattering phenomena whose entries are not necessarily independent random variables [2]. Usually, researchers analysed practical scenarios by adopting sampled matrices from the Gaussian Unitary Ensemble [3] or Polynomial Ensemble [4] channel matrices, which simplified the analysis of the wireless signal variations [5,6,7,8]. It is possible, for instance, to find an adequately modelled multiantenna setup through sums and products of random matrices [9]. Thus, the availability of explicit expressions for the channel matrices’ spectral statistics makes the performance analysis and compact design guidelines discussion possible.

Currently, the demand for more wireless services created urgency for larger bandwidth available beyond the 6 GHz spectrum. 5G [10], for instance, adopted mmWave frequencies to enable ultra-broadband communications, and the use of new technologies became viable. It granted large integrated antenna arrays and directional beamforming to exist due to the smaller wavelengths that allowed smaller element size. On the other hand, the main caveat of mmWave communications is the higher attenuation, meaning that few strong paths characterise a mmWave channel requiring better strategies to ensure the access node and the user terminal alignment [11]. Henceforth, the usual Gaussian Unitary or Polynomial Ensemble approach becomes an inappropriate fit in such a new propagation reality, motivating the search for novel tools to improve channel model and performance analysis.

Wireless performance studies usually employ Vandermonde matrices as mathematical channel models in applications such as system identification, harmonic analysis, direction-finding and precoding [12,13,14,15,16,17,18,19]. Specifically, given a particular wireless environment, the sum-rate channel capacity is better modelled by the Vandermonde matrix approach as presented in [20,21,22,23]. For the first time, the authors demonstrated that the free probability theory improves the accuracy of capacity calculations by including rows of the channel matrix corresponding to the weaker links [24]. Most of those efforts focused on the limit eigenvalue distribution of random Vandermonde matrices with unit magnitude and complex independently and identically distributed phase entries. In [22], the authors first introduced analytical methods for finding the moments of random Vandermonde matrices with elements on the unit circle and introduced the concept of Vandermonde mixed moment expansion coefficient. Later on, the authors of [21] investigated the behaviour of the matrices’ eigenvalues and their impact on system capacity. The work in [25] derives an estimation on the number of degrees of freedom for the multiple-input, multiple-output (MIMO) transmission assuming a line-of-sight environment. Again, they use the random matrices approach, dependent on a linear number of random variables. They characterise the number of most significant singular values and give an upper bound on the highest singular value’s size. Finally, the conclusions presented in [23] includes upper and lower bounds for the maximum eigenvalue of random unit magnitude Vandermonde matrices, which are essential tools to calculate an explicit expression for the asymptotic capacity of the Vandermonde channel.

In this work, there is a proposal for an upper bound and a method to derive an exact series expansion for calculating the mean sum-rate channel capacity. The setup scenario considers a base station equipped with an M-antenna uniform linear array and L users under line-of-sight condition. This scenario leads to a mathematical problem where the joint probability density function (JPDF) of the eigenvalues of a Vandermonde matrixW^WHare necessary, whereWis the channel matrix. Similar to what was exposed earlier, the usual Gaussian Unitary or Polynomial Ensemble approach becomes an inappropriate fit to the derived JPDF. The novelty of our work is on circumventing this problem by employing Taylor’s series expansion to derive the moments of_mn=E_trL^WHWnrequired to compose the calculation of sum capacity. The exploration of the integer partition theory properties is the solution to find the exact expression for all_mn. We also give examples of how to derive the formulation and find the resulting sum-rate capacity. Furthermore, we also find an upper bound for the mean sum-rate capacity by using Jensen’s inequality. The validation of the results comes from Monte Carlo numerical simulations.

The paper is organised as follows. Section 2 refers to the development of the methods for calculating the mean capacity when one considers a random Vandermonde matrix. Section 3 presents a numerical analysis that evaluates the methods and compares the results with simulations. Finally, the discussion is closed in the Section 4.

2. Channel Capacity

For a general matrixW with M lines and L columns consider the mean capacity defined as [20]

C=1LE_log2IL+γW^WH=1L∑k=1LE_log21+γ_λk=_{∫R log2}1+γtp(t)dt,

whereγis the signal-to-noise ratio (SNR),_λkis the instantaneous k-th eigenvalue of the matrixW^WHandp(t) is the marginal distribution of the eigenvalues. Usually, this model is generally represented by the diagram of Figure 1 with M antennas at the base station and L single-antenna users.

Sum-rate channel vectors arise in a uniform linear antenna array (ULA) at the transmitter under far-field, line-of-sight propagation conditions. Such conditions frequently manifest in realistic wireless backhaul scenarios [26]. Here, if uniformly distributed users transmit signals to the base station, then one can suitably represent the wireless channel by a random Vandermonde matrix with unit magnitude (i.e., users can control the power such that unitary magnitude is possible throughout all the matrix elements) and complex phase entries such that

V=1⋯1^e−j_ω1⋯^e−j_ωL⋮⋱⋮^{e−jM−1_ω1}⋯^{e−jM−1_ωL},

where_ωlis an independent and identically distributed random variable. Unlike the case of complex Gaussian entries, formulas for the capacity whenW is a Vandermonde matrix are still yet not well explored [20].

2.1. Approximation by Taylor Series

When applying Taylor series, one has that [27]

_log2(1+t)=1ln2∑k=1∞^(−1)k+1tkk,

so that it converges only whent<1 . If one substitutes (3) into (1), then

C=1ln2∑k=1∞^(−1)k+1k^γk∫^tkp(t)dt=1ln2∑k=1∞^(−1)k+1k^γk _mk,

where

_mk=∫^tkp(t)dt,

fork∈^Z+,_mkare the average moments ofW^WH . Following this method, the authors of [20] show that one can calculate capacity through a finite partial sum of (4). In that case, the calculations relied on up to seven terms in (4) using the method described in [21]. This approach is valid for low SNR values, which does not suffice in capacity estimation for Vandermonde matrices over a wide range of SNR values. As suggested in [20], an extension to the methods proposed in [21] or novel methods are needed.

In this manuscript, with the aim of coming up with a different Taylor expansion other than the one in (3), one considers a one-dimensional Taylor series as an expansion of a real functionf(t)about a point_γ0 so that [28]

f(t)=f(_γ0)+^f′(_γ0)(t−_γ0)+^f′′(_γ0)2!^(t−_γ0)2+...+^f(n)(_γ0)n!^(t−_γ0)n+...

one can express other form of (3) such that

log(1+γt)=1ln2log1+γ_γ0+∑k=1∞^(−1)k+1k^{(1+γ_γ0)kγk t−_γ0k}.

The expression (6) is only valid for the conditionγ(t−_γ0)1+γ_γ0<1 . If one substitutes (6) in (1), then the following can be written

C=1ln2(log1+γ_γ0+∑k=1∞^(−1)k+1k^{(1+γ_γ0)kγk}∫^t−_γ0kp(t)dt

Notice that in (7), the term∫^t−_γ0kp(t)dtmust be evaluated for each value of k. For instance, fork=2, the term∫^t−_γ0kp(t)dtcan be written as

∫^t−_γ02p(t)dt=∫^t2p(t)dt−2_γ0∫tp(t)dt+_γ02∫p(t)dt=_m2−2_{γ0 m1}+_γ02.

Fork=2 , three moments are required according to the previous algebraic expansion. Following the same rationale and using Newton’s polynomial expansion, (7) can be written as

C=1ln2log1+γ_γ0+∑k=1∞^(−1)k+1k^{(1+γ_γ0)kγk} _Ψk(γ),

where

_Ψk(γ)=∑i=0k^(−1)k−iki_{γ0k−i mi}

Regarding Equations (4) and (9), both require the availability of_mk moments that can impact on the desired quality of the capacity estimation. The full computation of the first seven lower-order moments described in [21] are difficult to derive, and higher-order moments add further burdens to this task as a tedious evaluation of many integrals is needed and, as alternative, the usage of numerical methods. Thus, here one deal with the computation of_mkby reframing the problem of the mean empirical eigenvalue distribution of^WHW depicted in [20].

2.1.1. Computation of_mk

As it is clear in (9), the computation of_mk is essential. As it is well known, the eigenvalue moments can be computed as [29]

_mk=E_trL^VHVk=E_trL^VHV·^VHV⋯^VHV,

where_trL·=^L−1Tr· is the normalised trace. Originally, there are other ways to calculate (11) where further details can be found in [20,21,23]. The most important conclusion taken from previous works is that (11) can be addressed as a counting problem according to what one can explore in Appendix A. The solution of (11), in terms of a combinatorics approach, depends on the understanding of partition of a set [30]. Thus, it is important to highlight the following definition:

Definition 1.

DefineP(n)as the set of all partitions of{_j1,_j2,⋯,_jn}. ρ is a partition inP(n)such thatρ={_ρi∣i∈^Z+}.P(n,k)={ρ∣∀ρ∈P(n),|ρ|=k}._ρiis a subset (also denominated as block) of ρ. The|·|operator when applied to sets gives their cardinality.

One can write the exact moments (see Appendix A for the proof), considering that0<ω<2πis uniform and identically distributed, as

_mn=E_trL^VHVn=∑k=1n^{Mn+1−k (MN)n}NkΓ(n,k),

where

Γ(n,k)=∑ρ∈P(n,k)|ρ|!n!∏i=1|ρ|(_ri!)^(i!)_ri ,

_rk=|_Rk|,∃_Rk0,otherwise,

and,_Rk={A∈ρ∣|A|=k}.

Here, one use partition sets to calculate the moments. Currently, there is no closed-form expression for calculating the moments using partition sets. However, it has both asymptotic expansions that accurately approximate it and recurrence relations by which calculates exactly. Further hints are given in Appendix A.

2.2. Example on How to Evaluate_mn

In this given example, makeM=N=4. For calculating_m1 , substitute directly the values in (12) such that

_m1=∑k=1n^{Mn+1−k (MN)n}NkΓ(n,k),=∑k=11^{41+1−1 (4.4)1}41Γ(1,1).

Here,Γ(n,k) presents a detailed evaluation because it involves an algorithmic calculation based on the partition sets’ analysis. Nevertheless, once the process is understood, calculating all moments is similar to calculating the first one. To do this, use Table 1, which shows the equivalence of integer partitions and all the possible partitions formed by_jk objects for some values of n. Furthermore, use Table 2, which shows all possible cardinalities forρand the subsets_ρk presented in the Table 1. These tables offer the necessary information used to evaluate Equations (12)–(14) in this specific example.

Given (13), the reader first need to evaluateρ∈P(n,k)to find the items of the summation term. Ifn=4, the only case where there are two summation terms is whenk=2 according to Table 1 (i.e., “3 + 1”, “2 + 2”). Notice that for other values of k, one obtain only one summation term in (13).

Next, evaluate_rifor each summation term defined byρ∈P(n,k) expressed by (13). Given a chosen partition in the setP(n,k),_ri refers to the number of subsets with cardinality equals to i. For instance, if the summation term corresponds to, for example, the decomposition “3 + 1”, then see that there is only one subset with three elements and one subset with one element in Table 1. Thus, it is straightforward to check Table 2 and conclude that_r1=1and_r3=1. On the other hand, if the summation term corresponds to the decomposition “2 + 2”, then two subsets have two elements and, henceforth,_r2=2. All values of_riup ton=4 can be obtained from Table 2.

Now, remind_m1and evaluateΓ(1,1) . A quick glimpse at Table 1 and Table 2 lets us know thatΓ(1,1)=1and, henceforth,_m1=1.

Next,

_m2=^{42+1−1 (4.4)2}41Γ(2,1)+^{42+1−2 (4.4)2}41Γ(2,2).

As it was previously done, if we use (13), and Table 1 and Table 2, thenΓ(n,k)is evaluated such that

Γ(2,1)=1!2!(_r1!)^(1!)_r1 ×(_r2!)^(2!)_r2 =1!2!(0!)^(1!)0×(1!)^(2!)1.

Similarly,

Γ(2,2)=2!2!(_r1!)^(1!)_r1 ×(_r2!)^(2!)_r2 =2!2!(2!)^(1!)2×(0!)^(2!)0.

Finally,

_m2=14+316=716.

Continue the same procedure such that

_m3=^{43+1−1 (4.4)3}41Γ(3,1)+^{43+1−2 (4.4)3}42Γ(3,2)+^{43+1−3 (4.4)3}43Γ(3,3),

and thenΓ(n,k)is evaluated such that

Γ(3,1)=1!3!(1!)^(3!)1,Γ(3,2)=2!3!(1!)^(1!)1×(1!)^(2!)1,Γ(3,3)=3!3!(3!)^(1!)3.

Finally,

_m3=116+964+3128=29128.

Now, evaluate the fourth moment such that

_m4=^{44+1−1 (4.4)4}41Γ(4,1)+^{44+1−2 (4.4)4}42Γ(4,2)+^{44+1−3 (4.4)4}43Γ(4,3)+^{44+1−4 (4.4)4}44Γ(4,4),

The procedure continues the same as all previous ways to evaluateΓ(n,k)despite one exception. As discussed earlier, ifn=4 , the only case where there are two summation terms in (13) is whenk=2 according to Table 1 (i.e., notice two partitions that|ρ|=2). Thus, evaluating these cases the same way we did earlier but considering summation terms such that

Γ(4,2)=2!4!(1!)^(1!)1×(1!)^(3!)1+2!4!(2!)^(2!)2.

Finally,

_m4=164+9256+364+9256+94096=5534096.

Using_m1,_m2,_m3and_m4 enables the reader to use (9), (10) and calculate the sum rate capacity as

C=1ln2log1+γ_γ0+γ716−_γ01+γ_γ0−^γ229128−7_γ08+_γ022^(1+γ_γ0)2+^γ35334096−87_γ0128+21_γ0216−^γ33^(1+γ_γ0)3+⋯,

2.3. Upper Bound

One can state that the matrix^WHWis bounded such that

det(^WHW)≤∏i=1K||_vi||.

IfM→∞,L→∞, andc=LM, then the products of〈_vi,_vj〉→0, fori≠j [31]. Thus, in the limit, equality in Hadamard’s inequality is achieved. Therefore, from (1), an upper bound for the sum rate capacity is expressed as

C≤_log2IL+γdiag(^WHW)C≤_log2IL+γ_ILL≤_log2^L+γLL.

3. Numerical Analysis

In Section 2, several methods on how to calculate the capacity whenWis considered as a Vandermonde matrix (with uniformly distributed phases). This section will evaluate the analytical development from the previous subsections using simulations.

In Section 2.1, this investigation proposed a different Taylor expansion other than (3), considering a Taylor expansion as a power series expansion forlog(1+γt)about the pointγ=_γ0 . Furthermore, Equations (4) and (7) require the calculation of moments_mk as described in the Section 2.1.1. In order to compute_mk , random phase arguments are generated and used in Equation (11). The calculation of the squared error (SE) is computed as follows.

• Generate K ensembles ofV according to Equation (2). ConsiderVa matrix with unit magnitude and_ωlan independent and identically distributed random variable.

• Calculate the momentsm˜n=1K∑i=1K trLViH Vinusing the previous generated ensembles.

• Employ Equations (12)–(14) to derive the analytical moments.

• Find SE by calculating^{(_m˜n−_mn)2}.

Figure 2 shows the square errors for moments of ordern={2,3,4,5}versus the number of sample points K, ranging from 10 to 70. It is important to observe the convergence of the simulated moments of the Vandermonde matrices (with uniformly distributed phases) towards the analytical moments_mkas the number of samples increases. Note that the error magnitude is as low as¹⁰⁻⁵for moment of ordern=2with only ten samples. Notice also that the error magnitude decreases as the order n of the moment increases.

Next, this investigation compares the methods for calculating capacity through Equations (4) and (7). The setup of simulation assumes the values of antennas and users asM=L={4,8,16,32}. The simulation uses the same steps required to generate random phase arguments for the matrixVdiscussed previously. The calculation of the moments_mk follows the description in the Section 2.1.1. Figure 3 illustrates the results comparing the simulation and the analytical methods. To compute the analytical mean sum-rate, one hundred moments has been used. Straight lines represent Equation (7) and dashed lines represent Equation (4). The coloured discrete symbol × represents the values yielded by the simulations. Notice the perfect matching between simulation and analytical results forM=L={4,8,16,32} . For Equation (4), as it is a series representation aroundγ=0, it is expected that when the curve moves away from the origin, the series representation will no longer be as good as in the region nearγ=0.

The analytical upper bound, given in Equation (16), is compared against the simulation using the steps for generating a random variableV described previously. Figure 4 illustrates the results comparing the simulation and the upper bound approximation. Here, straight lines represent the curve for Equation (16) and the coloured discrete symbol × represents the values yielded by the simulations. As the number of base station antennas and usersM=Lincreases, the gap between the upper bound and simulation decreases.

4. Conclusions

This work investigates a base station equipped with an M-antenna uniform linear array and L users under line of sight condition. An exact series expansion to calculate the mean sum-rate channel capacity is presented. This scenario led to a mathematical problem where the joint probability density function (JPDF) of the eigenvalues of a Vandermonde matrixW^WHwere a necessary model, whereWis the channel matrix. However, differently from the case where the channel is Rayleigh distributed, this joint PDF is not known. To circumvent this problem, one can employ Taylor’s series expansion and present a result where the moments of_mn=E_trL^WHWnare computed. To calculate this quantity, this investigation resorted to the integer partition theory and presented an exact expression for_mn. Furthermore, one can derive an upper bound for the mean sum-rate capacity by making use of Jensen’s inequality. All the results were validated by Monte Carlo numerical simulation.

Future Works

In this work, we have assumed that the phase distribution of the entries of matrixW are uniformly distributed. It would be interesting investigating other phase distributions, such as the Von Misses distribution, which have parameters that better translate the directivity of the user. Other possible point of future investigation would be further simplification of the expressions associated to the integer partition theory. Furthermore, it would be interesting to investigate the use of large intelligent surfaces (LIS) [32] to aid the communication between the MIMO base station and the users.

n	Integer Partition	EquivalentjkSets	Partitionρ
1	1	{{j1}}	{ρ1}
2	2	{{j1,j2}}	{ρ1}
	1+1	{{j1},{j2}}	{ρ1,ρ2}
3	3	{{j1,j2,j3}}	{ρ1}
	1+2	{{j1},{j2,j3}}	{ρ1,ρ2}
	1+1+1	{{j1},{j2},{j3}}	{ρ1,ρ2,ρ3}
4	4	{{j1,j2,j3,j4}}	{ρ1}
	3+1	{{j1,j2,j3},{j4}}	{ρ1,ρ2}
	2+2	{{j1,j2},{j3,j4}}	{ρ1,ρ2}
	2+1+1	{{j1,j2},{j3},{j4}}	{ρ1,ρ2,ρ3}
	1+1+1+1	{{j1},{j2},{j3},{j4}}	{ρ1,ρ2,ρ3,ρ4}

Integer Partition	|ρk|=1	|ρk|=2	|ρk|=3	|ρk|=4	|ρ|
1	1	0	0	0	1
2	0	1	0	0	1
	2	0	0	0	2
3	0	0	1	0	1
	1	1	0	0	2
	3	0	0	0	3
4	0	0	0	1	1
	1	0	1	0	2
	0	2	0	0	2
	2	1	0	0	3
	4	0	0	0	4

Author Contributions

C.F.D. participated in the tasks: conceptualisation, formal analysis, research, software, visualisation and writing-original draft. F.A.P.d.F. participated in the tasks: concept design and writing-review E.R.d.L. participated in the tasks: conceptualisation, funding acquisition, project administration and writing-review and editing. G.F. participated in the tasks: visualisation, project administration and writing-review and editing. All authors have read and agreed to the published version of the manuscript.

Funding

This work was partially supported by R&D ANEEL-PROJECT COPEL 2866-0366/2013. This work was partially funded by EMBRAPII-Empresa Brasileira de Pesquisa e Inovacao Industrial. E.R. Lima was supported in part by CNPq under Grant 313239/2017-7. G. Fraidenraich was supported in part by CNPq under Grant 301777/2019-5.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Acknowledgments

The authors would like to thank Edno Alan Pereira and Dafne Campos Lima Bessades Pereira for their constructive comments, which certainly helped to improve the manuscript.

Conflicts of Interest

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Appendix A. Vandermonde Moment Coefficient

Proceed with the proof of (12). Calculate

_mn=E_trL^VHVn=E_trL^VHV·^VHV⋯^VHV,

where_trL·=^L-1Tr·is the normalised trace. This expression can be expanded as

_mn=E^L-1∑_i1,...,_inj1,...,_jnV^(_j1,_i2)H×V(_i2,_j2)×V^(_j2,_i3)H×V(_i3,_j3)×⋮^×nV^(_jn,_i1)H×V(_i1,_j1),

so that_jn∈Zand_in∈Zindexes the lines and columns of matrixV.

Substitute the elements of (2) in (A2) to have a new representation of (A1), which can be expressed as

_mn=E^{N-n L-1}∑_i1,...,_inj1,...,_jn^ej_i2._ωj1 ×^e-j_i2._ωj2 ×^ej_i3._ωj2 ×^e-j_i3._ωj3 ×⋮^{×nej_i1._ωjn} ×^e-j_i1._ωj1 ,

or, for a more convenient representation, the following expression,

_mn=^{N-n L-1}∑_i1,...,_inj1,...,_jnE^{ej_i2-_i1ωj1} ×^{ej_i3-_i2ωj2} ×^{ej_i4-_i3ωj3} ×⋮^{×nej_in-_in-1ωjn} ,

where0<_ik<M,1<_jk<L. Notice that as n increases, the number of required_jkand_ikindexes also grows making the computation ofE_trL^VHVn more complicated. As earlier introduced in Section 2.1.1, one way to simplify the design problem is to treat the indexes_jkas elements of a set.

According to the properties given in Definition 1, suppose a given sequence(_j1,_j2,...,_jn)gives rise to a partitionρ={_ρk∣k∈^Z+}. The formation law of the subset_ρkis determined by repeated values assumed in some instance of the sequence(_j1,_j2,...,_jn) during the evaluation of the summations in (A4). Thus,_ρkis formally expressed as

_ρk={_jr∣(1≤r≤n)(∃s∈^Z+)[_jr=s]},

for some instance of the sequence(_j1,_j2,...,_jn) in (A4).

Using Definition 1, (A4) can be rewritten as

_mn=^{N-n L-1}∑ρ∈P(n)∑_i1,...,_in∑_j1,...,_jngivingrisetoρE∏k=1|ρ|^{ej_{∑r∈ρk ir-1}-_{∑r∈ρk irωρk}} .

Now, the indexes_jk presented in the summation of (A4) do not follow an ordered incremental sequence of values within the range1≤_jk≤L . It follows the order of the possible sets ruled by (A5). A further manipulation of (A6) allows us to find another convenient representation of (A6) as

_mn=∑ρ∈P(n)∑_j1,...,_jngivingrisetoρ∑_i1,...,_in^{N|ρ|-n-1 c|ρ|-1 L-|ρ|}E∏k=1|ρ|^{ej_{∑r∈ρk ir-1}-_{∑r∈ρk irωρk}} ,

wherec=LN.

As a support for understanding the role of the setρ, let us assume for this moment a partitionρ={{1,2,3,⋯,n}}for a certain instance of the sequence(_j1,_j2,_j3,⋯,_jn). In this case, the partitionρhas only one element as one can assume that_j1=_j2=_j3=⋯=_jnand, henceforth,_ωj1 =_ωj2 =_ωj3 =⋯=_ωjn . If one evaluates (A7) using the current instance ofρ, the argument of the expectation becomes_∏k=1|ρ| ^ej0_ωρk =1no matter what is the value assumed by the random variable_ωρk .

Now suppose a partitionρ={{1},{2},{3},⋯,{n}}for a certain instance of(_j1,_j2,_j3,⋯,_jn). In this case, the partitionρhas a total of n elements, which indicates that not a single pair of_jn has the same values in the sequence. Thus, the expectation in (A7) can be written as

E∏k=1|ρ|^{ej_{∑r∈ρk ir-1}-_{∑r∈ρk irωρk}} =∏k=1|ρ|_∫ωρk ^{ej_{∑r∈ρk ir-1}-_{∑r∈ρk irωρk}} f(_ωρk )d_ωρk ,

wheref(_ωρk )is the probability distribution function of_ωρk .

Assuming the probability density function asf(_ωρk )=1/2πfor0<_ωρk <2π, the following can be written

E^{ej_{∑r∈ρk ir-1}-_{∑r∈ρk irωρk}} =1,for∑r∈_ρkir-1=∑r∈_ρkir0,otherwise.

The reader can count how many times (A9) assumes the unity value by knowing the cardinality of_Sρ,N, which is given by

_Sρ,N=_i1,_i2,⋯,_in|∑k∈_ρjik-1=∑k∈_ρjik∀j∈1,2,⋯,|ρ|.

If one consider again that|ρ|=1, one have that_ωρ1 =_ωρ2 =⋯=_ωρn and, henceforth, the number of solutions of (A10) is equal to the number of permutations of the sequence(_j1,_j2,⋯,_jn)such that|_S1,N|=^Nn. On the other hand, if one consider that|ρ|=nfrom the previous discussion, then no permutation is possible and|_Sn,N|=N. For other values of|ρ|it is easy to see that|_Sρ,N|=^Nn+1-|ρ|.

Note that the mean calculation is equivalent to count the set of solutions for each instance ofρ. Furthermore, no closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations that allow us to calculate it precisely.