Noncoherent SIMO Pre-Log via Resolution of Singularities
We establish a lower bound on the noncoherent capacity pre-log of a temporally correlated Rayleigh block-fading single-input multiple-output (SIMO) channel. Our result holds for arbitrary rank of the channel correlation matrix, arbitrary block-length , and arbitrary number of receive antennas , and includes the result in Morgenshtern et al. (2010) as a special case. It is well known that the capacity pre-log for this channel in the single-input single-output (SISO) case is given by , where is the penalty incurred by channel uncertainty. Our result reveals that this penalty can be reduced to by adding only one receive antenna, provided that and the channel correlation matrix satisfies mild technical conditions. The main technical tool used to prove our result is Hironaka’s celebrated theorem on resolution of singularities in algebraic geometry.
It was shown in  that the noncoherent capacity111 Noncoherent capacity denotes capacity in the setting where transmitter and receiver know the channel statistics but neither of them is aware of the channel realizations. pre-log for single-input multiple-output (SIMO) correlated block-fading channels can be larger than the pre-log in the single-input single-output (SISO) case. This result was surprising as it disproved a conjecture in  on the pre-log in the SIMO case being the same as in the SISO case. The channel model analyzed in  assumes that the fading process is independent across blocks of length and temporally correlated within blocks, with the rank of the corresponding channel correlation matrix given by and the number of receive antennas . For this channel model, under additional technical conditions on the channel correlation matrix, a pre-log of was established in . In contrast, in the SISO case the pre-log is given by .
The assumption made in  is very restrictive, and the proof technique used in  heavily relies on this assumption. More precisely, the main result in  is based on a lower bound on the differential entropy of the channel output signal that is obtained by applying a change of variables argument [1, Lem. 3]. The proof is then completed by showing that the expected logarithm of the Jacobian determinant corresponding to this change of variables is finite. For , the Jacobian determinant takes a very involved form making it difficult to say anything about its expected logarithm. The main contribution of this paper is to resolve this problem by introducing a new proof technique based on a result from algebraic geometry, namely [3, Th. 2.3], which is a consequence of Hironaka’s celebrated theorem on resolution of singularities [4, 5]. Roughly speaking, this result allows us to rewrite any real analytic function [6, Def. 1.6.1] locally as a product of a monomial and a nonvanishing real analytic function. The proof of our main result, a lower bound on the pre-log for the correlated block-fading channel with arbitrary number of receive antennas , is then effected by using this factorization to show that the integral of the logarithm of the absolute value of a real analytic function over a compact set is finite, provided that the real analytic function is not identically zero. This method is very general and could be of independent interest when one tries to show that a certain differential entropy is finite.
We conclude by noting that the main result in this paper shows that the pre-log penalty incurred in the SISO case, which is due to channel uncertainty, can be reduced to by adding only one receive antenna (i.e., by taking ), provided that and the channel correlation matrix satisfies mild technical conditions. In the limit with constant, the penalty in the SIMO case becomes arbitrarily small, whereas the penalty in the SISO case remains unchanged.
Finite subsets of the set of natural numbers, , are denoted by calligraphic letters and we write for the cardinality of . We use to designate the set of natural numbers . Uppercase boldface letters denote matrices, lowercase boldface letters designate vectors. The superscripts and stand for transposition and Hermitian transposition, respectively. The all-zero matrix of appropriate size is written as . For a matrix , the entry in the th row and th column is denoted by and we write for its th row. If , we denote by the submatrix of obtained by retaining all rows of with row index . Similarly, for a vector we denote its th entry by . If we denote by the vector obtained by retaining the entries of with . We write for the th unit vector of appropriate size and for the identity matrix of size . For a vector , denotes the diagonal matrix that has the entries of in its main diagonal. For two matrices and of arbitrary size, is the block matrix that has the matrix as upper left block, as lower right block, and as upper right and lower left block. For matrices , we define . We designate the Kronecker product of the matrices and as ; to simplify notation, we use the convention that the ordinary matrix product precedes the Kronecker product, i.e., . For a function , we write if there exists a vector in the domain of such that . For two functions and , the notation means that . If , and with denoting the set of integers. The logarithm to the base 2 is written as . The expectation operator is denoted by . Finally, stands for the distribution of a jointly proper Gaussian random vector with mean and covariance matrix .
Ii System model
We consider a SIMO channel with receive antennas. The fading in each component SISO channel follows a correlated block-fading model , with input-output relation for a given block
where denotes the signal-to-noise ratio (SNR), is the transmitted signal vector, is the received signal vector corresponding to the th receive antenna, and is additive noise. Finally, is the vector of channel coefficients between the transmit antenna and the th receive antenna. Here,222 When , capacity is known to grow double-logarithmically in SNR , and, hence, the pre-log is equal to zero. If we can always achieve the same pre-log as for by simply using only receive antennas. and .
Without loss of generality, we assume that the row vectors of satisfy (). The vectors and are assumed to be mutually independent and independent across . It will turn out to be convenient to write the channel-coefficient vector in whitened form as , where with . Finally, we assume that and change in an independent fashion from block to block for all .
Setting , , , and , we can combine the individual input-output relations in (1) into the overall input-output relation
Iii Lower bound on the pre-log
The capacity of the channel (2) is defined as
where denotes mutual information [8, p. 251] and the supremum is taken over all input distributions on that satisfy the average power constraint . The pre-log is defined as .
The main result of this paper is the following theorem.
Suppose that satisfies the following
Property (A): There exists a subset of indices with cardinality
such that every row vectors of are linearly independent. Then, the capacity of the SIMO channel (2) can be lower-bounded as
The SISO pre-log is .
The pre-log is , provided that , even if only.
Property (A) in Theorem 1 is not restrictive and is satisfied for a broad class of correlation matrices.
Since we are interested in a capacity lower bound, we can evaluate the mutual information in (3) for an appropriate input distribution. Specifically, we take the input distribution to have entries that are independent and identically distributed (i.i.d.), zero mean, unit variance, and satisfy . This implies that [7, Lem. 6.7]
For example, we can take . The mutual information in (3), evaluated for any input distribution satisfying these constraints, is then lower-bounded as follows. We first upper-bound according to [1, Eq. (8)]
and then lower-bound as in [1, Eq. (12)]
where is a constant that is independent of , is independent of , with , and
for sets satisfying
provided that we can find sets such that . The remainder of the paper is devoted to identifying such a choice for and proving that the corresponding differential entropy is, indeed, finite. The main idea is to choose the sets such that can be related to with through a deterministic one-to-one mapping. The quantity is much easier to deal with than .
Condition (10) implies that the mapping
is between two vector spaces of the same dimension , which is a necessary condition for this mapping to be one-to-one. Note that the RHS of (11) also depends on , which is, however, taken to be fixed, reflecting the fact that the pilot symbols are known to both transmitter and receiver. Any dependence on will henceforth implicitly mean a dependence on only. We set and shall choose as follows:
If , we set .
If , we let
with and .
be the Jacobian of the mapping in (11). If this mapping is one-to-one on almost everywhere (a.e.), we can apply the change-of-variables theorem for integrals [9, Th. 7.26] in combination with [10, Th. 7.2] and find that
The proof is then concluded by establishing that the mapping in (11) is one-to-one a.e. and
This requires an in-depth analysis of the Jacobian in (13), which will be carried out in the next section. \qed
Iv Properties of the Jacobian
The following lemma provides important insights into the structure of the determinant of the Jacobian in (13).
The Jacobian in (13) can be decomposed as
The lemma follows by noting that
Based on (6), we can conclude that333We assume that for all .
To conclude the proof of (14) it therefore remains to show that . Direct computation reveals that each vector in (17) with contains only one nonzero element, which is given by . Applying the Laplace formula [11, p.7] to in (17) therefore yields the decomposition
The expectation of the logarithm of the second term on the RHS in (20) is finite because . It remains to show that . This is the most technical part in the proof of Theorem 1 and can be accomplished by applying methods from algebraic geometry, namely Theorem 2 in Appendix -A, which is a consequence of Hironaka’s celebrated theorem on resolution of singularities [4, 5]. A direct proof would require showing that the expected log of the determinant of the (high-dimensional) matrix is finite, which seems exceedingly difficult. Hironaka’s theorem drastically simplifies the proof as it tells us that implies that . We start by noting that is a homogeneous polynomial in of degree , i.e.,
which allows us to apply the following proposition (for and defined above):
Let be a homogeneous polynomial in of degree with . Then implies that
Writing as and using the fact that is a homogeneous polynomial of degree , we can upper-bound the absolute value of the expectation in (23) by
for the complex vector , we can further upper-bound according to
where is obtained from by changing to polar coordinates . Note that is a real analytic function [6, Def. 1.6.1], implies that , and we are integrating over a compact set . We can therefore apply Theorem 2 in Appendix -A to conclude that . \qed
It remains to show that for our specific choice of sets , which will be proved in the following lemma:
Property (A) in Theorem 1 implies that .
See Appendix -B \qed
In summary, we proved that (14) holds provided that Property (A) in Theorem 1 is satisfied. It turns out that also implies that the mapping in (11) is one-to-one a.e. on . The proof, which is along the lines of the proof of [1, Lem. 2], is omitted due to space limitations. This completes the proof of Theorem 1.
-a Resolution of singularities
In this appendix, we show how Hironaka’s theorem on resolution of singularities can be used to prove that provided that is a real analytic function and is a compact set.
We start by defining notation that will be used in this appendix. Let denote the open cube with side length centered at . For and let . If is a subset of the image of a map then denotes the inverse image of .
The following lemma is an immediate consequence of a modified version of Hironaka’s theorem [3, Th. 2.3]. This modified version originally appeared in . The main point of this lemma is that it allows us to rewrite any real analytic function [6, Def. 1.6.1] locally as a product of a monomial and a nonvanishing real analytic function.
Let be a real analytic function from a neighborhood of to . Suppose that . Then, there exists a triple , where
is an open set in with ,
is a -dimensional real analytic manifold [3, Def. 2.10],
is a real analytic map
that satisfies the following conditions:
The map is proper, i.e., the inverse image of any compact set is compact.
The map is a real analytic isomorphism between and .
For each point , there exists a coordinate chart such that , is a real analytic isomorphism for some with ,
where is a nonvanishing real analytic function on and , and the determinant of the Jacobian of the mapping satisfies
where is a nonvanishing real analytic function on and .
The main idea is to apply [3, Th. 2.3] to the function . We omit the details due to space limitations. \qed
We are now in a position to state the theorem that is needed to prove in the proof of Proposition 1.
Let be a real analytic function on an open set . Then
for all compact sets .
Let . If then Lemma 3 implies that there exists a triple where is an open set containing , is a real analytic manifold, and is a proper real analytic map. Moreover, for each there exists a coordinate chart such that , , and
for all , where and are nonvanishing real analytic functions on . We can choose sufficiently small so that and are bounded on . If the existence of a triple with the properties specified above is guaranteed by taking sufficiently small such that does not vanish on and by setting to be the identity map.
Now for each , we choose an open neighborhood and a compact neighborhood such that . Since is a compact set, there exists a finite set of vectors in such that
For each , set , , , and . Since the mapping is proper, each set is a compact set. Therefore, there exists a finite set of points in such that
where are positive real numbers, are bounded nonvanishing real analytic functions on , are vectors of nonnegative integers, and we changed variables according to (25). \qed
-B Proof of Lemma 2
We present a proof for and skip the (simpler) cases and .
Suppose that and . We can write in (17) as with and defined as
Property (A) in Theorem 1 implies that for arbitrary subsets
with , we can find vectors () such that
for all vectors with ;
for all vectors with .
This implies that for each choice of such sets (), there exists a set of vectors () such that the number of nonzero elements in each matrix satisfies
Moreover, we have
which implies that we can choose the subsets and the vectors () such that each column of contains precisely one nonzero element. Applying the Laplace formula [11, p. 7] iteratively, we therefore get
where is a positive constant and we used Property (A) in Theorem 1 in the last step. \endproof
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