1804
`
`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 48, NO 7, JULY 2002
`
`High-Rate Codes That Are Linear in Space and Time
`
`Babak Hassibi and Bertrand M. Hochwald
`
`Abstract—Multiple-antenna systems that operate at high rates
`require simple yet effective space–time transmission schemes to
`handle the large traffic volume in real time. At rates of tens of
`bits per second per hertz, Vertical Bell Labs Layered Space–Time
`(V-BLAST), where every antenna transmits its own independent
`substream of data, has been shown to have good performance
`and simple encoding and decoding. Yet V-BLAST suffers from
`its inability to work with fewer receive antennas than transmit
`antennas—this deficiency is especially important for modern
`cellular systems, where a base station typically has more antennas
`than the mobile handsets. Furthermore, because V-BLAST
`transmits independent data streams on its antennas there is no
`built-in spatial coding to guard against deep fades from any given
`transmit antenna. On the other hand, there are many previously
`proposed space–time codes that have good fading resistance and
`simple decoding, but these codes generally have poor performance
`at high data rates or with many antennas. We propose a high-rate
`coding scheme that can handle any configuration of transmit and
`receive antennas and that subsumes both V-BLAST and many
`proposed space–time block codes as special cases. The scheme
`transmits substreams of data in linear combinations over space
`and time. The codes are designed to optimize the mutual infor-
`mation between the transmitted and received signals. Because of
`their linear structure, the codes retain the decoding simplicity of
`V-BLAST, and because of their information-theoretic optimality,
`they possess many coding advantages. We give examples of the
`codes and show that their performance is generally superior
`to earlier proposed methods over a wide range of rates and
`signal-to-noise ratios (SNRs).
`
`Index Terms—Bell Labs Layered Space–Time (BLAST), fading
`channels, multiple antennas, receive diversity, space–time codes,
`transmit diversity, wireless communications.
`
`I. INTRODUCTION AND MODEL
`
`I T is widely acknowledged that reliable fixed and mobile
`
`wireless transmission of video, data, and speech at high rates
`will be an important part of future telecommunications systems.
`One way to get high rates on a scattering-rich wireless channel is
`to use multiple transmit and/or receive antennas. In [1], [2], the-
`oretical and experimental evidence demonstrates that channel
`capacity grows linearly as the number of transmit and receive
`antennas grow simultaneously.
`Early uses of multiple transmit antennas in a scattering en-
`vironment achieve reliability through “diversity,” where redun-
`dant information is sent or received on two or more antennas
`
`Manuscript received October 13, 2000; revised July 21, 2001. The material
`in this paper was presented in part at the 38th Annual Allerton Conference on
`Communications, Control, and Computing, Monticello, IL, Sept. 2000.
`B. Hassibi is with the Department of Electrical Engineering, California Insti-
`tute of Technology, Pasadena, CA 91125 USA (e-mail: hassibi@caltech.edu).
`B. M. Hochwald is with Bell Laboratories, Lucent Technologies, Murray Hill,
`NJ 07974 USA (e-mail: hochwald@bell-labs.com).
`Communicated by M. L. Honig, Associate Editor for Communications.
`Publisher Item Identifier S 0018-9448(02)05197-0.
`
`in the hope that at least one path from the transmitter reaches
`the receiver [3]–[6]. To keep the transmitter and receiver com-
`plexity low, linear processing is often preferred [3]. To achieve
`the high data rates promised in [2], however, new approaches
`for space–time transmission are needed. One such approach is
`presented in [7], [8] where a practical scheme, called V-BLAST
`(Vertical Bell Labs Layered Space–Time), encodes and decodes
`rates of tens of bits per second per hertz (b/s/Hz) with 8 transmit
`and 12 receive antennas. The V-BLAST architecture breaks the
`original data stream into substreams that are transmitted on the
`individual antennas. The receiver decodes the substreams using
`a sequence of nulling and canceling steps.
`Since then there has been considerable work on a variety
`of space–time transmission schemes and performance mea-
`sures [9] such as the space–time trellis codes of [10] and the
`space–time block codes of [11], [12] for the known channel
`and [13]–[17] for the unknown channel.
`At very high rates and with a large number of antennas, many
`of these space–time codes suffer from complexity or perfor-
`mance difficulties. The number of states in the trellis codes of
`[10] grows exponentially with either the rate or the number of
`transmit antennas. The block codes of [11], [12] suffer from rate
`and performance loss as the number of antennas grow, and the
`codes of [14]–[16] suffer from decoding complexity if the rate is
`too high. Although V-BLAST can handle high data rates with
`reasonable complexity, the decoding scheme presented in [7]
`does not work with fewer receive than transmit antennas.
`We present a space–time transmission scheme that has many
`of the coding and diversity advantages of previously designed
`codes, but also has the decoding simplicity of V-BLAST at high
`data rates. The codes work with arbitrary numbers of transmit
`and receive antennas.
`The codes break the data stream into substreams that are dis-
`persed in linear combinations over space and time. We refer
`to them simply as linear dispersion codes (LD codes). The LD
`codes
`
`1) subsume, as special cases, both V-BLAST [7] and the
`block codes of [12];
`2) generally outperform both;
`3) can be used for any number of transmit and receive an-
`tennas;
`4) are very simple to encode;
`5) can be decoded in a variety of ways including simple
`linear-algebraic techniques such as
`a) successive nulling and canceling (V-BLAST [7],
`square-root V-BLAST [18]),
`b) sphere decoding [19], [20];
`6) are designed with the numbers of both the transmit and
`receive antennas in mind;
`
`0018-9448/02$17.00 © 2002 IEEE
`
`Authorized licensed use limited to: Purdue University Fort Wayne. Downloaded on March 03,2023 at 00:51:01 UTC from IEEE Xplore. Restrictions apply.
`
`Smart Mobile Technologies LLC, Exhibit 2013
`Page 1 of 21
`
`

`

`HASSIBI AND HOCHWALD: HIGH-RATE CODES THAT ARE LINEAR IN SPACE AND TIME
`
`1805
`
`7) satisfy the following information-theoretic optimality cri-
`terion:
`— the codes are designed to maximize the mutual infor-
`mation between the transmit and receive signals.
`
`We briefly summarize the general structure of the LD
`transmit antennas,
`receive
`codes. Suppose that there are
`antennas, and an interval of
`symbols available to us during
`which the propagation channel is constant and known to the
`receiver. The transmitted signal can then be written as a
`matrix
`that governs the transmission over the
`antennas
`during the interval. We assume that the data sequence has been
`broken into
`substreams (for the moment we do not specify
`) and that
`are the complex symbols chosen from
`an arbitrary, say -PSK or
`-QAM, constellation. We call a
`rate
`linear dispersion code one for which
`obeys
`
`(1)
`
`where the real scalars
`
`are determined by
`
`The design of LD codes depends crucially on the choices
`of the parameters
`,
`and the dispersion matrices
`.
`To choose the
`we propose to optimize a nonlinear
`information-theoretic criterion: namely, the mutual information
`between the transmitted signals
`and the received
`signal. We argue that
`this criterion is very important for
`achieving high spectral efficiency with multiple antennas.
`We also show how the information-theoretic optimization has
`implications for performance measures such as pairwise error
`probability. Section IV has several examples of LD codes and
`performance comparisons with existing schemes.
`We now present the multiple-antenna model considered in
`this paper.
`
`A. The Multiple-Antenna Model
`In a narrow-band, flat-fading, multiple-antenna communica-
`tion system with
`transmit and
`receive antennas, the trans-
`mitted and received signals are related by
`
`also
`
`often (but not always) assume that the channel matrix
`has independent
`entries.
`The entries of the channel matrix are assumed to be known
`to the receiver but not to the transmitter. This assumption is rea-
`sonable if training or pilot signals are sent to learn the channel,
`which is then constant for some coherence interval. The coher-
`ence interval of the channel should be large compared to
`[21].
`When the channel is known at the receiver, the resulting channel
`capacity (often referred to as the perfect-knowledge capacity) is
`[2], [1]
`
`(3)
`the distribution of
`where the expectation is taken over
`the random matrix
`.1 The capacity-achieving
`is a
`zero-mean complex Gaussian vector with covariance matrix
`, where
`is the maximizing covariance
`matrix in (3). When the distribution on
`is rotationally
`invariant, i.e., when
`for any unitary
`matrices
`(as is the case when
`and
`has independent
`entries), the optimizing covariance is
`,
`and (3) becomes
`
`(4)
`
`This expectation can sometimes be computed in closed form
`(see, for example, [22]).
`When the channel is constant for at least
`may write
`
`channel uses we
`
`so that defining
`
`and
`
`(where the superscript denotes “transpose”), we obtain
`
`(2)
`
`It is generally more convenient to write this equation in its trans-
`posed form
`
`denotes the vector of complex received signals
`where
`during any given channel use,
`denotes the vector of
`complex transmitted signals,
`denotes the channel
`matrix, and the additive noise
`is
`(zero-mean,
`unit-variance, complex-Gaussian) distributed that is spatially
`and temporally white. The channel matrix
`and transmitted
`vector are assumed to have unit variance entries, implying that
`
`and
`
`are independent, the
`, and
`,
`Since the random quantities
`normalization
`is the signal-to-noise
`in (2) ensures that
`ratio (SNR) at each receive antenna, independently of
`. We
`
`(5)
`
`where we have omitted the transpose notation from
`simply redefined this matrix to have dimension
`matrix
`is the received signal,
`transmitted signal, and
`is the additive
`noise. In
`,
`, and
`, time runs vertically and space runs
`horizontally. We are concerned with designing the signal matrix
`obeying the power constraint
`.
`
`and
`. The
`is the
`
`1Equation (3) actually slightly generalizes [2], [1], which assume that H has
`independent CN (0; 1) entries.
`
`Authorized licensed use limited to: Purdue University Fort Wayne. Downloaded on March 03,2023 at 00:51:01 UTC from IEEE Xplore. Restrictions apply.
`
`Smart Mobile Technologies LLC, Exhibit 2013
`Page 2 of 21
`
`

`

`1806
`
`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 7, JULY 2002
`
`matrices
`We note that, in general, the number of
`needed in a codebook can be quite large. If the rate in bits per
`, then the number of matrices is
`.
`channel use is denoted
`For example, with
`transmit and
`receive antennas
`the channel capacity at
`20 dB (with
`distributed
`) is more than 12 bits per channel use. Even with a relatively
`, we need
`matrices at
`small block size of
`rate
`. The huge size of the constellations generally rules
`out the possibility of decoding at the receiver using exhaustive
`search.
`The LD codes that we present easily generate the very large
`constellations that are needed. Moreover, because of their struc-
`ture, they also allow efficient real-time decoding. In the next sec-
`tion, we briefly describe and analyze some existing space–time
`codes so that we may motivate the LD codes and explain how
`they are different.
`
`II. INFORMATION-THEORETIC ANALYSIS OF SOME
`SPACE–TIME CODES
`
`Since the capacity of the multiple-antenna channel can easily
`be calculated, we may ask how well a space–time code performs
`by comparing the maximum mutual information that it can sup-
`port to the actual channel capacity. The distribution for the
`matrix
`that achieves (4) is independent
`entries,
`but we cannot easily use this by itself as a guideline for con-
`structing and decoding a (random) constellation of
`ma-
`trices because of the sheer number of matrices involved. There-
`fore, a constellation of matrices that has sufficient structure for
`efficient encoding and decoding is generally needed. One such
`structure is that of an orthogonal design, originally proposed in
`[11] and later generalized in [12].
`
`A. Mutual Information Attainable With Orthogonal Designs
`An orthogonal design is introduced by Alamouti in [11] for
`and has the structure
`
`(6)
`
`are drawn from a particular
`and
`The complex scalars
`( -PSK or
`-QAM) constellation, but we may simply assume
`that they are random variables such that
`.
`We show that this particular structure can be used to achieve ca-
`pacity when there is one receive antenna but not when there is
`more than one. Portions of our argument may also be found in
`[23], [24].
`1) One Receive Antenna (
`comes
`
`, (5) be-
`
`): With
`
`We effectively have an equivalent matrix channel
`in (7) that
`is a scaled unitary matrix. Maximum-likelihood decoding of
`and
`is, therefore, decoupled [11].
`We may ask how much mutual information the orthogonal
`design structure (6) can attain? To answer this question we need
`to compute the mutual information between the transmitted and
`received vectors and in the equivalent channel model (7) and
`compare it with the capacity of an
`,
`multiple-
`antenna system.
`Since
`, the maximum
`has the power constraint
`mutual information in (7) is simply the channel capacity that is
`. If we denote this
`obtained with the structured channel matrix
`maximum mutual information by
`, using (3) we obtain
`
`in front of the expectation normalizes for the
`where the factor
`two channel uses spanned by the orthogonal design. Since, sub-
`ject to a trace constraint, the determinant of any positive-definite
`matrix is maximized when its eigenvalues are all equal, it is easy
`to see that the maximizing covariance matrix is
`, so
`that we obtain
`
`(9)
`
`The expression on the right symbolically denotes the capacity
`attained by a system with
`transmit antennas and
`receive antennas at SNR . This equation implies that the or-
`thogonal design (6) can achieve the full channel capacity of the
`,
`system, and there is no loss incurred by as-
`suming the structure (6) as opposed to a general transmit ma-
`trix .
`2) Two or More Receive Antennas (
`receive antennas, (5) becomes
`
`): With
`
`which can be reorganized as
`
`This can be rewritten as
`
`It readily follows that
`
`(10)
`
`(11)
`
`(7)
`
`We now readily see
`
`and
`, maximum-likelihood estimation of
`As with
`is decoupled. However, unlike with
`, the orthogonal
`design structure prohibits us from achieving channel capacity.
`
`(8)
`
`Authorized licensed use limited to: Purdue University Fort Wayne. Downloaded on March 03,2023 at 00:51:01 UTC from IEEE Xplore. Restrictions apply.
`
`Smart Mobile Technologies LLC, Exhibit 2013
`Page 3 of 21
`
`

`

`HASSIBI AND HOCHWALD: HIGH-RATE CODES THAT ARE LINEAR IN SPACE AND TIME
`
`1807
`
`Fig. 1. Maximum mutual information achieved by 2  2orthogonal design (6) compared to actual channel capacity for the M = 2, N = 2 system. Solid line:
`maximum mutual information for 2  2orthogonal design. Dashed line: capacity of the M = 2, N = 2 system.
`
`To see this, we compare the maximum mutual information be-
`tween
`and
`in (10) with
`, the actual
`channel capacity for the system.
`As before, the maximum mutual information in (10) is simply
`the channel capacity for the structured channel matrix
`. De-
`noting this maximum mutual information by
`, we ob-
`tain
`
`we take a loss with the structure (6). The amount of this loss is
`substantial at high SNR and is depicted in Fig. 1 which shows
`the actual channel capacity compared to the maximum mutual
`information obtained by the orthogonal design (6).
`For
`receive antennas, the analysis is similar and is
`omitted. We simply state that for
`transmit antennas
`and
`receive antennas the
`orthogonal design allows us
`to attain only
`, rather than the full
`
`.
`3) Other Orthogonal Designs: The preceding subsection
`focuses on the
`orthogonal design but there are also
`orthogonal designs for
`. The complex orthogonal
`designs for
`are no longer “full-rate” (see [12]) and
`therefore generally perform poorly in the maximum mutual
`information they can achieve, even when
`. We give an
`example of these nonsquare orthogonal designs [12], [25].
`For
`, we have, for example, the rate
`orthogonal
`design
`
`(12)
`
`(13)
`
`The last equation implies that the orthogonal design (6) is re-
`strictive and does not allow us to achieve the full channel ca-
`pacity of the
`,
`system, but rather the capacity of
`an
`,
`system at twice the SNR. Thus, when
`
`. It can
`ensures that
`The factor
`be shown that maximum-likelihood estimation of the variables
`is decoupled. Again using an argument similar to
`
`Authorized licensed use limited to: Purdue University Fort Wayne. Downloaded on March 03,2023 at 00:51:01 UTC from IEEE Xplore. Restrictions apply.
`
`Smart Mobile Technologies LLC, Exhibit 2013
`Page 4 of 21
`
`

`

`1808
`
`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 48, NO 7, JULY 2002
`
`Fig. 2. Maximum mutual information achieved by 4  3 orthogonal design (13) compared to actual channel capacity. Solid lines: maximum mutual information
`of 4  3 orthogonal design for N = 1; 2; 3 receive antennas. Dashed lines: capacity of M = 3, N = 1; 2; 3 systems.
`
`, it is straightforward to show
`the one presented for
`that the maximum mutual information attainable with (13) with
`receive antennas is
`which is
`(much) less than the true channel capacity
`. We
`omit the proof and refer instead to Fig. 2 which shows the actual
`channel capacity compared to the maximum mutual information
`obtained by the orthogonal design (13).
`
`substream). Thus, over a block of
`transmit matrix takes on the form
`
`channel uses, the
`
`...
`
`...
`
`. . .
`
`...
`
`(14)
`
`B. Mutual Information Attainable With V-BLAST
`
`We showed in Section II-A that, even though orthogonal de-
`signs allow efficient maximum-likelihood decoding and yield
`“full-diversity” (the appearance of the sum of the
`in the
`mutual information formulas attests to this), orthogonal designs
`generally cannot achieve high spectral efficiencies in a mul-
`tiple-antenna system, no matter how much coding is added to
`the transmitted signal constellation. This is especially true when
`the system has more than one receive antenna. An examination
`of the model (10) (and its counterparts for other orthogonal de-
`signs) reveals that the orthogonal design does not allow enough
`“degrees of freedom”—there are only two unknowns in (10) but
`four equations.
`We can conclude that orthogonal designs are not suitable for
`very-high-rate communications. On the other hand, a scheme
`that is proven to be suitable for high spectral efficiencies is
`V-BLAST [7]. In V-BLAST each transmit antenna during each
`channel use sends an independent signal (often referred to as a
`
`is an independent symbol drawn from a complex
`where each
`constellation. Since the transmitted symbols are not dispersed in
`time,
`is arbitrary. (We could, for example, take
`.)
`(the number of receive antennas is at least as
`When
`large as the number of transmit antennas), there exist efficient
`schemes for decoding the V-BLAST matrices. These are based
`on “successive nulling and canceling” [7], and its more efficient
`variants [18], as well as more recently on sphere decoding [19].
`All these decoding schemes essentially solve a well-conditioned
`system of linear equations. Successive nulling and canceling
`provides a fast approximate solution to the maximum-likeli-
`hood decoding problem with the benefit of cubic complexity
`in the number of transmit antennas
`. Sphere decoding, on
`the other hand, finds the exact maximum-likelihood estimate
`and benefits from avoiding an explicit exhaustive search. Recent
`work [20] has shown analytically that for a wide range of SNRs,
`the expected computational complexity of sphere decoding is
`also roughly cubic in the number of transmit antennas.
`
`Authorized licensed use limited to: Purdue University Fort Wayne. Downloaded on March 03,2023 at 00:51:01 UTC from IEEE Xplore. Restrictions apply.
`
`Smart Mobile Technologies LLC, Exhibit 2013
`Page 5 of 21
`
`

`

`Smart Mobile Technologies LLC, Exhibit 2013
`Page 6 of 21
`
`

`

`Smart Mobile Technologies LLC, Exhibit 2013
`Page 7 of 21
`
`

`

`HASSIBI AND HOCHWALD: HIGH-RATE CODES THAT ARE LINEAR IN SPACE AND TIME
`
`1811
`
`Fig. 3. Bit-error performance comparison for a random (fA ; B g drawn from a complex Gaussian distribution and normalized) and an optimized LD code for
`M = 3 transmit and N = 1 receive antenna for T = Q = 4, and rate R = 6 bits/channel use (obtained by transmitting 64-QAM on s ; . . . ; s ).
`
`for an SNR of interest, subject to one of the following
`constraints:
`i)
`ii)
`iii)
`where
`
`,
`
`,
`is given by (24) with the
`entries.
`
`having independent
`
`rates, the pairwise error probability for any two signals is ex-
`tremely small. In Section I-A we argue that even for the small
`test-case of
`transmit and
`receive antennas, we
`could theoretically have a constellation size of as many as
`signal-matrices at
`20 dB. It is therefore conceivable that
`the pairwise error probability between any two could be roughly
`. Trying to minimize a quantity such as (27) that
`is already so small can be numerically quite difficult.
`Fortunately, information theory suggests a natural alternative
`that is connected with minimizing (27) but is more fundamental.
`Recall from Section II-A that orthogonal designs are deficient
`in the maximum mutual information they support for
`or
`. We therefore choose
`to maximize the mu-
`tual information between
`and
`in (23). This guarantees that
`we are taking the smallest possible mutual information penalty
`within the LD structure (16). We propose to design codes that
`are “blessed” by the “logdet” formula (3).
`We formalize the design criterion as follows.
`
`The Design Method
`
`1) Choose
`
`2) Choose
`
`(typically,
`
`).
`
`that solve the optimization problem
`
`(28)
`
`; as mentioned
`Note that (28) is effectively (3) with
`in Section III, we may take the entries of
`’s and
`’s) to
`(the
`be uncorrelated with variance . Moreover, because the real and
`imaginary parts of the noise vector
`in (23) also have variance
`, the SNR remains
`. We also note that (28) differs from (3)
`by the outside factor
`because the effective channel is real-
`valued and the LD code spans
`channel uses.
`We now make some remarks.
`
`1) Clearly,
`
`.
`
`2) The problem (28) can be solved subject to any of the con-
`straints i)–iii). Constraint i) is simply the power constraint
`(18) that ensures
`. Constraint ii) is more
`restrictive and ensures that each of the transmitted signals
`and
`are transmitted with the same overall power
`from the
`antennas during the
`channel uses. Finally,
`constraint iii) is the most stringent, since it forces the sym-
`bols
`and
`to be dispersed with equal energy in all
`spatial and temporal directions.
`
`Authorized licensed use limited to: Purdue University Fort Wayne. Downloaded on March 03,2023 at 00:51:01 UTC from IEEE Xplore. Restrictions apply.
`
`Smart Mobile Technologies LLC, Exhibit 2013
`Page 8 of 21
`
`

`

`1812
`
`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 7, JULY 2002
`
`3) Since constraints i)–iii) are successively more restric-
`tive, the corresponding maximum mutual informations
`obtained in (28) are necessarily successively smaller.
`Nevertheless, we have found that constraint iii) generally
`imposes only a small information-theoretic penalty while
`having the advantage of better coding (or diversity)
`gains. Using symmetry arguments one may conjecture
`that the optimal solution to the problem with constraint
`i) should automatically satisfy constraint ii). But we
`have not experimented sufficiently with constraint i) to
`confirm this; we instead usually restrict our attention to
`constraints ii) and iii). We have empirically found that
`of two codes with equal mutual informations, the one
`satisfying the more stringent constraint gives lower error
`rates. Examples of this phenomenon appear in Section IV.
`
`4) The solution to (28) subject to any of the constraints i)–iii)
`is highly nonunique: simply reordering the
`gives another solution, as does pre- or post-multiplying all
`the matrices by the same unitary matrix. However, there
`is also another source of nonuniqueness which is more
`subtle. Equation (23) shows that we can always pre-mul-
`tiply the transmit vector
`by a
`orthogonal matrix
`vector
`tries that are still independent and
`Thus, we may rewrite (23) as
`
`to obtain a new
`with en-
`-distributed.
`
`as in (22) allows us to write
`, and
`,
`Defining
`the new equivalent channel
`as shown in (29)
`at the bottom of the page. Since the entries of
`and
`have the same joint distribution, the maximum mutual
`information obtained from the equivalent channels
`and
`are the same. This implies that the transformation from
`the dispersion matrices
`to
`
`(30)
`
`is an orthogonal matrix, pre-
`where
`serves the mutual information. Thus, the transformation
`(30) is another source of nonuniqueness to the solution of
`(28).
`This nonuniqueness can be used to our advantage
`because a judicious choice of the orthogonal matrix
`allows us to change the dispersion code through the
`transformation (30) to satisfy other criteria (such as
`space–time diversity) without sacrificing mutual infor-
`mation. Examples of this appear in Remark 7, where we
`construct unitary
`from the rank-one V-BLAST
`dispersion matrices (20), and in Section IV in some of
`the two and three-antenna LD code constructions.
`
`5) The constraints i)–iii) are convex in the dispersion ma-
`trices
`since they can be rewritten as
`i¢)
`ii¢)
`,
`iii¢)
`,
`all of which are convex. However, the cost function
`is neither concave nor
`convex in the variables
`. Therefore,
`it
`is
`possible that (28) has local maxima. Nevertheless, we
`have been able to solve (28) with relative ease using
`gradient-based methods and it does not appear that local
`maxima pose a great problem. Table I in Section IV-A
`gathers the maximum mutual informations obtained via
`gradient-ascent for a variety of different
`,
`, and
`.
`The results show that maximum mutual informations
`obtained are quite close to the Shannon capacity (which
`is clearly an upper bound on what can be achieved) and
`so they suggest that the values obtained, if not the global
`maxima, are quite close to them. (For convenience,
`we include the gradient of the cost function (28) in
`Appendix A.)
`
`...
`
`. . .
`
`...
`
`...
`
`. . .
`
`...
`
`...
`
`. . .
`
`...
`
`(29)
`
`Authorized licensed use limited to: Purdue University Fort Wayne. Downloaded on March 03,2023 at 00:51:01 UTC from IEEE Xplore. Restrictions apply.
`
`Smart Mobile Technologies LLC, Exhibit 2013
`Page 9 of 21
`
`

`

`HASSIBI AND HOCHWALD: HIGH-RATE CODES THAT ARE LINEAR IN SPACE AND TIME
`
`1813
`
`, one solution
`,
`,
`6) We know that for
`to (28), for any of the constraints i)–iii), is the orthogonal
`design (19). This holds simply because the mutual infor-
`mation of this particular orthogonal design achieves the
`actual channel capacity
`. We note
`that there are also many other solutions that work equally
`well.
`
`7) When
`one solution to (28), subject
`and
`to either constraints i) or ii), is given by the V-BLAST
`matrices (20) since these achieve the full capacity of the
`multiple antenna link. The V-BLAST matrices, however,
`are rank-one and therefore do not satisfy constraint iii).
`But it is also possible to obtain an explicit solution to (28)
`subject to iii). For
`, one such set of matrices is
`given by
`
`where
`
`(31)
`
`...
`
`...
`
`. . .
`
`. . .
`
`...
`
`The above code can be constructed by starting with the
`V-BLAST matrices (20) and applying the transformation
`(30) with a suitable
`. We do not give the full
`here, and
`only mention that, for
`, the transformation is
`
`. It can be readily
`with similar expressions for the
`checked that the matrix
`constructed from the coef-
`ficients relating
`to
`is orthogonal.
`Fig. 4 in Section IV presents a performance comparison
`of the LD code (31) with V-BLAST.
`
`8) The block length
`is essentially also a design variable.
`Although it must be chosen shorter than the coherence
`time of the channel, it can be varied to help the optimiza-
`tion (28). We have found that choosing
`often yields good performance.
`
`9) Although the SNR is a design variable, we have found
`that the optimization (28) is not sensitive to its value for
`large
`(
`20 dB). Once the optimization is performed,
`the resulting LD code generally works well over a wide
`range of SNRs.
`
`10) It does not appear that (28) has a simple closed-form
`solution for general
`,
`,
`, although we see in Sec-
`tion IV that, in some nontrivial cases, it can lead to so-
`lutions with simple structure. We have found that the so-
`lution to (28) often yields an equivalent channel matrix
`that is “as orthogonal as possible.” Although complete
`orthogonality appears not always to be possible, our ex-
`perience with optimizing (28) shows that the difference
`can be made quite
`small with a proper choice of
`(see Table I in Sec-
`and
`tion IV-A). Thus, there appears to be very little capacity
`penalty in assuming the LD structure (16).
`
`is orthogonal,
`11) When the equivalent channel matrix
`maximum-likelihood decoding and the V-BLAST-like
`nulling/canceling [7] perform equally well because the
`estimation errors of
`are decoupled.
`
`12) The design criterion (28) depends explicitly on the
`, both through the choice
`number of receive antennas
`of
`and through the matrix
`in (24). Hence, the
`optimal codes, for a given ,
`, and
`, are different for
`different
`.
`Nevertheless, a code designed for
`receive antennas
`can also easily be decoded using nulling/canceling or
`sphere decoding with
`antennas. Hence, if we
`wish to broadcast data to more than one user, we may
`use a code designed for the user with the fewest receive
`antennas, with a rate supported by all the users.
`
`13) The ultimate rate of the code depends on the number of
`signals sent
`, the block length of the code
`, and the size
`of the constellation from which
`are chosen.
`We assume that the constellation is
`-PSK or
`-QAM.
`Then the rate in bits per channel use is easily seen to be
`
`(32)
`
`14) A standard gray-code assignment of bits to the symbols
`-QAM constellation may be used.
`of the -PSK or
`
`15) We see that the average pairwise error probability (27)
`and the design criterion (28) have a similar expression. By
`interchanging the expectation and log in (28), we see that
`maximizing (28) has some connections to minimizing
`(27).
`On the other hand, our design criterion is not directly
`connected with the diversity design criterion given in [9]
`and [10], which is concerned with maximizing
`
`(33)
`
`A constellation attains full diversity if (33) is nonzero.
`This criterion depends only on matrix pairs, and there-
`fore does not exclude matrix designs with low spectral
`efficiencies.
`At high spectral efficiencies, the number of signals in
`the constellation of possible matrices is roughly ex-
`ponential in the channel capacity at a given SNR. This
`number can be very large—in Section IV we present a
`
`Authorized licensed use limited to: Purdue University Fort Wayne. Downloaded on March 03,2023 at 00:51:01 UTC from IEEE Xplore. Restrictions apply.
`
`Smart Mobile Technologies LLC, Exhibit 2013
`Page 10 of 21
`
`

`

`Smart Mobile Technologies LLC, Exhibit 2013
`Page 11 of 21
`
`

`

`HASSIBI AND HOCHWALD: HIGH-RATE CODES THAT ARE LINEAR IN SPACE AND TIME
`
`1815
`
`Fig. 4. The upper two curves are the bit error performance of V-BLAST (20) with nulling/canceling (upper), and with maximum-likelihood decoding (lower).
`(lower). For both these codes,
`The lower two curves are the LD code given by (31) for M = N = T = 2 and Q = 4 (upper) and the code (34) with  = e
`sphere decoding is used to find the maximum-likelihood estimates. The rate is R = 4, and is obtained by transmitting QPSK on s ; . . . ; s .
`
`but achieves only 7.47-bits/channel use mutual information at
`20 dB, while the LD code achieves the full channel capacity
`of 11.28 bits/channel use. The orthogonal design and LD code
`are maximum-likelihood decoded (using the sphere decoder in
`the case of the LD code). The orthogonal design is easier to de-
`code than the LD code because
`and may be decoded sep-
`arately, and its performance is better for SNR
`35 dB (where
`spectral efficiency is low compared with capacity).
`But we may obtain a code that is uniformly better at all SNRs
`by using (34) to improve the diversity of (31) without changing
`its mutual information. As shown in [26], setting
`is a good choice when transmitting 16-QAM. The performance
`of this constellation is also shown in Fig. 5. Its performance is
`better than the unmodified LD code at high SNR. Clearly, the
`best code satisfies both the mutual information and diversity
`criteria, if possible.
`
`,
`
`LD Versus OD:
`,
`transmit antennas and
`We present a code for
`receive antennas and compare it with the orthogonal design pre-
`sented in Section II-A3 with block length
`. The orthog-
`onal design (13) is written in terms of
`as
`
`and
`
`(35)
`
`It turns out that this orthogonal design is a l