Md
`
`IEEF. TRANSACTIONS ON INFORMATION THEORY. VOL. 44. N0. 1, MARCH E993
`
`Space—Time Codes for High Data
`Rate Wireless Communication:
`
`Performance Criterion and Code Construction
`
`Vahid Tarokh, Member, IEEE, Nambi Seshadri, Senior Member, IEEE, and A. R. Calderbank, Fettow. {BEE
`
`Absrrocri We consider the design of channel codes for im-
`proving the data rate andi'or the reliability of communications
`over fading channels using multiple transmit antennas. Data is
`encoded by a channel code and the encoded data is split into
`n streams that are simultaneously transmitted using 'H' transmit
`antennas. The received signal at each receive antenna is a linear
`superposition of the 'n transmitted signals perturbed by noise. We
`derive performance criteria for designing such codes under the
`assumption that the fading is slow and frequency nonselective.
`Performance is shown to be determined by matrices constructed
`from pairs of distinct code sequences. The minimum rank among
`these matrices quantilies the divemigf gain, while the minimum
`determinant of these matrices quantifies the coding gain. The
`results are then extended to fast fading channels. The design
`criteria are used to design trellis codes for high data rate wireless
`communication. The encodingidecoding complexity of these codes
`is comparable to trellis codes employed in practice over Gaussian
`channels. The codes constructed here provide the best tradeoff
`between data rate, diversity advantage, and trellis complexity.
`Simulation results are provided for 4 and 8 PSK signal sets
`with data rates of 2 and 3 bitsr'symbol, demonstrating excellent
`performance that is within 2—3 dB of the outage capacity for these
`channels using only 64 state encoders.
`
`index Terms— Array processing, diversity, multiple transmit
`antennas, space—time codes, wireless communications.
`
`I.
`
`INTRODUCTION
`
`A. Motivation
`
`URRENT cellular standards support circuit data and fax
`services at 9.6 kbt’s and a packet data mode is being
`standardized. Recently,
`there has been growing interest
`in
`providing a broad range of services including wire—line voice
`quality and wireless data rates of about 64—128 kbfs (lSDN)
`using the cellular (850—MHZ) and PCS (I .9—GHZ) spectra [2].
`Rapid growth in mobile computing is inspiring many proposals
`for even higher speed data services in the range of I44 1(be
`(for microcellular wide—area high—mobility applications) and
`up to 2 Mbts (for indoor applications) [1].
`The majority of the providers of PCS services have further
`decided to deploy standards that have been developed at
`cellular frequencies such as CDMA (IS-95), TDMA (lS-S4i’lS-
`136), and GSM (DCS—l900). This has led to considerable
`
`I8. 1997. The
`Manuscript received December 15. 1996: revised Augusl
`malerial
`in this paper was presenlcd in purl at
`the lEEE [ntemational
`Symposium on Information Theory. Ulm. Genuany. June 29—July 4. I997,
`The authors are with the AT&T Labs—Research, Florham Park. NJ UT933
`USA.
`Publisher Ilem Idenlitier S 00l8—9448(98)00933—X.
`
`effort in developing techniques to provide the aforementioned
`new services while maintaining some measure of backward
`compatibility. Needless to say, the design of these techniques
`is a challenging task.
`Band—limited wireless channels are narrow pipes that do not
`accommodate rapid flow of data. Deploying multiple transmit
`and receive antennas broadens this data pipe. Information the-
`ory [14], [35] provides measures of capacity, and the standard
`approach to increasing data flow is linear processing at
`the
`receiver [15], [44]. We will show that there is a substantial
`benefit
`in merging signal processing at
`the receiver with
`coding technique appropriate to multiple transmit antennas.
`In particular, the focus of this work is to propose a solution
`to the problem of designing a physical layer (channel coding,
`modulation, diversity) that operate at bandwidth efficiencies
`that are twice to four times as high as those of today’s systems
`using multiple transmit antennas.
`
`B. Diver‘sigr
`
`Unlike the Gaussian channel, the wireless channel suffers
`from attenuation due to destructive addition of multipaths in
`the propagation media and due to interference from other users.
`Severe attenuation makes it
`impossible for the receiver to
`determine the transmitted signal unless some less—attenuated
`replica of the transmitted signal is provided to the receiver.
`This resource is called diversity and it
`is the single most
`important contributor to reliable wireless communications.
`Examples of diversity techniques are (but are not restricted to)
`
`- Temporal Diversity: Channel coding in conjunction with
`time interleaving is used. Thus replicas of the transmit-
`ted signal are provided to the receiver in the form of
`redundancy in temporal domain.
`- Frequency Divemity: The fact that waves transmitted on
`different frequencies induce different multipath structure
`in the propagation media is exploited. Thus replicas of
`the transmitted signal are provided to the receiver in the
`fortn of redundancy in the frequency domain.
`- Antenna Diversity: Spatially separated or differently po—
`larized antennas are used. The replicas of transmitted
`signal are provided to the receiver in the form of redun—
`dancy in spatial domain. This can be provided with no
`penalty in bandwidth efficiency.
`
`When possible, cellular systems should be designed to encom-
`pass all forrns of diversity to ensure adequate performance
`OBIS—94489351000 © 1993 IEEE
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`TAROKI'I
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`er of: SPACE—TIME CODES FOR HIGH DATA RATE WIRELESS COMMUNICATION
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`7’45
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`mission to weight the amplitudes of the transmitted signals.
`Explicit feedback includes switched diversity with feedback
`[41] as well as techniques that use spatiotemporal—frequency
`water pouring [27] based on the feedback of the channel
`response. However, in practice, vehicle movements or inter-
`ference causes a mismatch between the state of the channel
`
`perceived by the transmitter and that perceived by receiver.
`Transmit diversity schemes mentioned in the second cat—
`egory use linear processing at
`the transmitter to spread the
`information across the antennas. At the receiver, information
`is obtained by either linear processing or maximum—likelihood
`decoding techniques. Feedforward information is required to
`estimate the channel from the transmitter to the receiver. These
`
`estimates are used to compensate for the channel response
`at the receiver. The first scheme of this type was proposed
`by Wittneben [43] and it includes the delay diversity scheme
`of Seshadri and Winters [32] as a special case. The linear
`processing approach was also studied in [15] and [44]. It has
`been shown in [42] that delay diversity schemes are indeed
`optimal in providing diversity in the sense that the diversity
`advantage experienced by an optimal receiver is equal to the
`number of transmit antennas. We can view the linear filter as
`
`instance, cellular systems typically use channel
`[26]. For
`coding in combination with time interleaving to obtain some
`form of temporal diversity [28]. In TDMA systems, frequency
`diversity is obtained using a nonlinear equalizer [4] when
`multipath delays are a significant fraction of symbol interval.
`In DSrCDMA, RAKE receivers are used to obtain frequency
`diversity. Antenna diversity is typically used in the up-link
`(mobile—to—base) direction to provide the link margin and
`coehannel interference suppression [40]. This is necessary to
`compensate for the low power transmission from mobiles.
`Not all forms of diversity can be available at all times. For
`example, in slow fading channels, temporal diversity is not an
`option for delay-sensitive applications. When the delay spread
`is small, frequency (multipath) diversity is not an option. In
`macrocellular and mierocellular environments, respectively,
`this implies that
`the data rates should be at
`least several
`hundred thousand symbols per second and several million
`symbols per second, respectively. While antenna diversity at
`a base—station is used for reception today, antenna diversity
`at a mobile handset is more difficult to implement because
`of electromagnetic interaction of antenna elements on small
`platforms and the expense of multiple down-conversion RF
`paths. Furthermore,
`the channels corresponding to different
`antennas are correlated, with the correlation factor determined
`by the distance as well as the coupling between the antennas.
`Typically, the second antenna is inside the mobile handset,
`resulting in signal attenuation at
`the second antenna. This
`can cause some loss in diversity benefit. All
`these factors
`motivate the use of multiple antennas at the base-station for
`transmission.
`
`In this paper, we consider the joint design of coding,
`modulation,
`transmit and receive diversity to provide high
`performance. We can view our work as combined coding and
`modulation for multi—input (multiple transmit antennas) multi—
`output (multiple receive antennas)
`fading channels. There
`is now a large body of work on coding and modulation
`for single—inputimulti—output channels [5], [10], [ll],
`[29],
`[30], [38], and [39], and a comparable literature on receive
`diversity, array processing, and beamforming. In light of these
`research activities, receive diversity is very well understood.
`By contrast, transmit diversity is less well understood. We
`begin by reviewing prior work on transmit diversity.
`
`C. Historical Perspective on Transmit Diversity
`
`Systems employing transmit fall
`gories. These are
`
`into three general cate-
`
`' schemes using feedback,
`-
`those with feedforward or training information but no
`feedback, and
`' blind schemes.
`
`The first category uses implicit or explicit feedback of
`information from the receiver to the transmitter to configure
`the transmitter. For instance, in time—division duplex systems
`[16], the same antenna weights are used for reception and
`transmission, so feedback is implicit in the appeal to channel
`symmetry. These weights are chosen during reception to
`maximize the signal-to-noise ratio (SNR), and during trans-
`
`a channel code that takes binary data and creates real-valued
`output. It is shown that there is significant gain to be realized
`by viewing this problem from a coding perspective rather than
`purely from the signal processing point of view.
`The third category does not require feedback or feedforward
`information. Instead, it uses multiple transmit antennas com-
`bined with channel coding to provide diversity. An example of
`this approach is to combine phase sweeping transmitter diver-
`sity of [18] with channel coding [19]. Here a small frequency
`offset is introduced on one of the antennas to create fast fading.
`An appropriately designed channel codeiinterleaver pair is
`used to provide diversity benefit. Another scheme is to encode
`information by a channel code and transmit the code symbols
`using different antennas in an orthogonal manner. This can be
`done either by frequency multiplexing [9], time multiplexing
`[32], or by using orthogonal spreading sequences for different
`antennas [3?]. A disadvantage of these schemes over the
`previous two categories is the loss in bandwidth efficiency due
`to the use of the channel code. Using appropriate coding, it is
`possible to relax the orthogonality requirement needed in these
`schemes and obtain the diversity as well as coding advantage
`ofl'er without sacrificing bandwidth. This is possible when the
`whole system is viewed as a multiple—inputfmultiple—output
`system and suitable codes are used.
`Information-theoretic aspects of transmit diversity were
`addressed in [14], [25], and [35]. We believe that Telatar
`[35] was the first
`to obtain expressions for capacity and
`error exponents for multiple transmit antenna system in the
`presence of Gaussian noise. Here, capacity is derived under the
`assumption that fading is independent from one channel use
`to the other. At about the same time, Foschini and Gans [14]
`derived the outage capacity under the assumption that fading
`is quasistatic;
`i.e., constant over a long period of time, and
`then changes in an independent manner. A particular layered
`space—time architecture was shown to have the potential to
`achieve a substantial fraction of capacity. A major conclusion
`
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`7’46
`
`IEEE TRJ‘XNSACTIONS UN [NFORMATICIN THEORY. VOL. 44. NO. 2, MARCH I993
`
`
`
`
`
`
`
`Modulator
`
`Fig. l. The block diagram of a delayr diversity transmitter.
`
`of these works is that the capacity of a multi-antenna systems
`far exceeds that of a single-antenna system.
`In particular,
`the capacity grows at
`least
`linearly with the number of
`transmit antennas as long as the number of receive antennas
`is greater than or equal to the number of transmit antennas.
`A comprehensive information—theoretic treatment for many of
`the transmit diversity schemes that have been studied before
`is presented by Narula, Trott, and Womell [25].
`
`D. Space— lime Codes
`
`We consider the delay diversity scheme as proposed by
`Wittneben [44]. This scheme transmits the same information
`from both antennas simultaneously but with a delay of one
`symbol
`interval. We can view this as a special case of the
`arrangement in Fig. l, where the information is encoded by
`a channel code (here the channel code is a repetition code of
`length 2). The output of the repetition code is then split into
`two parallel data streams which are transmitted with a symbol
`delay between them. Note that there is no bandwidth penalty
`due to the use of the repetition code, since two output—channel
`symbols are transmitted at each interval.
`that the effect of
`It was shown in [32], via simulations,
`this technique is to change a narrowband purely frequency—
`nonselective fading channel
`into a frequency-selective fad-
`ing channel. Simulation results further demonstrated that a
`maximum—likelihood sequence estimator at
`the receiver
`is
`capable of providing dual branch diversity.
`to ask if
`When viewed in this framework,
`it
`is natural
`is possible to choose a channel code that
`is better than
`
`it
`
`the R = 1/2 repetition code in order to provide improved
`performance while maintaining the same transmission rate?
`We answer the above question affirmatively and propose a
`new class of codes for this application referred to as the Space—
`Ylme Codes. The restriction imposed by the delay element in
`the transmitter is first removed. Then performance criteria are
`established for code design assuming that the fading from each
`transmit antenna to each receive antenna is Rayleigh or Rician.
`It is shown that the delay diversity scheme of Seshadri and
`Winters l32] is a specific case of space—time coding.
`In Section II, we derive performance criteria for design—
`ing codcs. For quasistatic flat Rayleigh or Rician channels,
`performance is shown to be determined by the diversity
`advantage quantified by the rank of certain matrices and by
`the coding advantage that is quantified by the determinants of
`these matrices. These matrices are constructed from pairs of
`distinct channel codewords. For rapidly changing ilat Rayleigh
`channels, performance is shown to be determined by the
`diversity advantage quantified by the generalized Hamming
`distance of certain sequences and by the coding advantage that
`is quantified by the generalized product distance of these se-
`quences. These sequences are constructed from pairs of distinct
`codewords. In Section III, this performance criterion is used to
`design trellis codes for high data rate wireless communication.
`We design coded modulation schemes based on 4-PSK, 8-PSK,
`and lfi-QAM that perform extremely well and can operate
`within 2—3 dB of the outage capacity derived by Foschini and
`Gans [14]. For a given data rate, we compute the minimal
`constraint length, the trellis complexity required to achieve a
`certain diversity advantage, and we establish an upper bound
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`TAROKI'I at (at: SPACE—TIME CODES FOR HIGH DATA RATE WIRELESS COMMUNICATION
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`
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`Pulse
`
`Shaper
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`Modulator
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`
`
`Modulator
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`Fig. 2. The block diagram of the transmitter.
`
`on the data rate as a functiori of the constellation size and
`
`diversity advantage. For a given diversity, we provide explicit
`constructions of trellis codes that achieve the minimum trellis
`
`complexity as well as the maximum data rate. Then, we revisit
`delay diversity and show that some of the codes constructed
`before have equivalent delay diversity representations. This
`section also includes multilevel constructions which provide an
`efficient way to construct and decode codes when the number
`of antennas is large (4—8). It
`is further shown that it is not
`possible for block-coded modulation schemes to outperform
`trellis codes constructed here at a given diversity advantage
`and data rate. Simulation results for many of the codes that
`we have constructed and comparisons to outage capacity for
`these channels are also presented. We then consider design
`of space—time codes that guarantee a diversity advantage of
`“-"1 when there is no mobility and a diversity advantage of
`1'2 2 'f'l when the channel is fast-fading. ln constructing these
`codes, we combine the design criteria for rapidly changing
`flat Rayleigh channels with that of quasistatic flat Rayleigh
`channels to arrive at a hybrid criteria. We refer to these
`codes as smart greedy codes which also stands for low-rate
`multidimensional spaccin'me codes for both slow and rapid
`fading channels. We provide simulation results indicating that
`these codes are ideal for increasing the frequency reuse factor
`under a variety of mobility conditions. Some conclusions are
`made in Section IV.
`
`[1. PERFORMANCE CRITERIA
`
`A. The System Model
`
`We consider a mobile communication system where the
`base—station is equipped with n. antennas and the mobile is
`equipped with m antennas. Data is encoded by the channel
`
`the encoded data goes through a serial—to~parallel
`encoder,
`converter, and is divided into n streams of data. Each stream
`of data is used as the input to a pulse shaper. The output
`of each shaper is then modulated. At each time slot t,
`the
`output of modulator i is a signal c; that is transmitted using
`transmit antenna (Ta: antenna) i for l S 'i S n. We emphasize
`that the to signals are transmitted simultaneously each from a
`different transmit antenna and that all these signals have the
`same transmission period T. The signal at each receive antenna
`is a noisy superposition of the n, transmitted signals corrupted
`by Rayleigh or Rician fading (see Fig. 2). We assume that the
`elements of the signal constellation are contracted by a factor
`of \/E_s chosen so that the average energy of the constellation
`is 1.
`
`At the receiver, the demodulator computes a decision statis-
`
`tic based on the received signals arriving at each receive
`antenna 1 S j S m. The signal
`it}? received by antenna j
`at time t is given by
`
`in.
`
`d“: =2 tY-:,jCi\/E_s+7fi
`1": l.
`
`(1)
`
`time t is modeled as independent
`where the noise 7],} at
`samples of a zero-mean complex Gaussian random variable
`with variance N0 / 2 per dimension. The coefficient (n, j is the
`path gain from transmit antenna i to receive antenna j. It is
`assumed that these path gains are constant during a frame and
`vary from one frame to another (quasistatic flat fading).
`
`B. The Case rgfltrdependem Fade (.‘oeflicrems
`
`In this subsection, we assmne that the coefficients (a, J- are
`first modeled as independent samples of complex Gaussian
`
`random variables with possibly nonzero complex mean Eng: _.,-
`and variance 0.5 per dimension. This is equivalent to the
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`7’48
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`IEEE TRANSACTIONS ON lNl—‘ORMATICIN THEORY. VOL. 44. N0.
`
`2. MARCH I993
`
`assumption that signals transmitted from ditferent antennas
`undergo independent fades.
`We shall derive a design criterion for constructing codes
`under this transmission scenario. We begin by establishing
`the notation and by reviewing the results from linear algebra
`that we will employ. This notation will also be used in the
`sequel
`to this paper [34]. Let a: = ($1, .133, ---,.1?;,.) and
`y : (:91,
`:92,
`-
`-
`-
`, yr.) be complex vectors in the its—dimensional
`complex space 03". The inner product of a: and y is given by
`.I‘.
`
`.1:
`
`- y : Z my,
`1:].
`
`where 1—,- denotes the complex conjugate of 92;. For any matrix
`A,
`let A‘ denote the Hermitian (transpose conjugate) of A.
`Recall from linear algebra that an n. X 11. matrix A is Hermiti‘an
`if A : A”. The matrix A is nonnegative dyinire if ach’“ 2 0
`for any l x n complex vector 3:. An 11. x n, matrix V is unitary
`if VV‘ 2 I where I is the identity matrix. An 'n. x 1 matrix B
`is a square root of an n X 11. matrix A if 88* = A. We shall
`make use of the following results from linear algebra [20].
`
`- An eigenvector 1.: of an “H. x 31 matrix A corresponding to
`eigenvalue A is a 1 x n. vector of unit length such that
`11A : A1: for some complex number A. The vector space
`spanned by the eigenvectors of A corresponding to the
`eigenvalue zero has dimension n — a", where 1' is the rank
`of A.
`
`' Any matrix A with a square root
`definite.
`
`.8 is nonnegative
`
`- For any nonnegativc-definite Hennitian matrix A, there
`exists a lower
`triangular square matrix B such that
`BB‘ 2 A.
`
`' Given a Hermitian matrix A, the eigenvectors of A span
`it”, the complex space of n dimensions and it is easy
`to construct an orthonormal basis of III” consisting of
`eigenvectors A. Furthermore, there exists a unitary matrix
`V and a real diagonal matrix D such that VAV“ = D.
`The rows of V are an orthonormal basis of {[3" given
`by eigenvectors of A. The diagonal elements of D are
`the eigenvalues Ar,
`1‘ : l, 2,
`n, of A counting
`multiplicities.
`' The eigenvalues of a Hermitian matrix are real.
`- The eigenvalues of a nennegalive-dcfinile Hermitian ma-
`trix are nonnegative.
`
`Assuming ideal channel state information (CSI), the proba-
`bility of transmitting C and deciding in favor ofe at the decoder
`is well approximated by
`
`P0: —J cluigj, 1' : 1,2, ---, n, j : 1,2,
`-, m)
`5 exp(—d2[c, e)E,/4No)
`
`where RIO/2 is the noise variance per dimension and
`Ti?-
`2
`
`«2m. «J= :2 i«m. —:«J
`j=1 i=1.
`i=1
`
`
`
`(2)
`
`.3.
`
`This is just the standard approximation to the Gaussian tail
`function.
`
`Setting fl,- : (cal _,-,
`“I
`N.
`
`«
`
`« -, an 4.), we rewrite (3) as
`
`ZZZfltJn‘t.)
`-=1 i=1 1.":—1
`
`flitX—Cr—(3:)
`
`After simple manipulations, we observe that
`
`(£203, 3) = Z 9,-1in
`i=1
`
`(4)
`
`where Am 21:? .13.;f and 9:?-— (CP— (31, cg—cég,
`for 1 S p, q S n. Thus
`
`ref—c?)
`
`P(c—>e|a,z,j,il= 1, 2, ---, n,_-j= l, 2,
`
`m)
`
`S H exp[—QjA(c, (3)9: 55/4530)
`i=1
`
`(5)
`
`where
`
`t'
`
`W:
`i=1
`
`Mali—ct)-
`
`Since A(c, e) is Hermitian, there exists a unitary matrix V
`and a real diagonal matrix D such that VA(c_._ e)V* = D. The
`rows {111, 112,
`-
`-
`-
`, 0”} of V are a complete orthonormal basis
`of CD” given by eigenvectors of A. Furthermore, the diagonal
`elements of D are the eigenvalues hi. i : l_._ 2, ---, n of .4
`counting multiplicities. By construction, the matrix
`
`ci—ci
`cf—cf
`
`cfi—cé
`cfi—cg
`
`ell—Ci
`cf—cf
`
`B(c. e) 2
`
`cf — c3?
`
`c3 — cg
`
`'.
`
`:
`
`c? — c?
`
`(6)
`
`Let us assume that each element of the signal constellation
`is contracted by a scale factor JET chosen so that the average
`energy of the constellation elements is 1. Thus our design
`criterion is not constellation—dependent and applies equally
`well to 4—PSK. 8—PSK, and lé—QAM.
`We consider the probability that a maximum-likelihood
`receiver decides erroneously in favor of a signal
`
`(5’1" — cf
`
`(33 — (:‘2‘
`
`.
`
`. .
`
`.
`
`. .
`
`(if — cf
`
`is clearly a square root of A(c, 6). Thus the eigenvalues of
`A(c. e) are nonnegative real numbers.
`Next, we express 032(c, e) in terms of the eigenvalues of
`the matrix A(6, 6).
`Let ([31,),
`(in) = Q-V“, then
`
`B2818?” Effie;- “(33---(:31(332-~c}’
`
`Q-A(e, eJQ‘l:
`
`ZAI/J'ul
`
`(7)
`
`assuming that
`
`(3:616?---c‘fc%cE---c§--«lief-“c?
`
`that (v.33- are samples of a complex Gaussian
`Next, recall
`random variable with mean Ear: j. Let
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`TAROKI'I et at: SPACE—TIME CODES FOR HIGH DATA RATE WIRELESS COMMUNICATION
`
`7’49
`
`Design Criteriafbr Rayieigh Space—Iime Codes:
`- The Rank Criterion:
`In order to achieve the maximum
`
`diversity nm. the matrix B(c, c) has to be full rank for
`any codewords c and e. [f B[c, c) has minimum rank
`r over the set of two tuples of distinct codewords, then
`a diversity of rm is achieved. This criterion was also
`derived in [15].
`- The Determinant Criterion: Suppose that a diversity ben—
`efit of rm is our target. The minimum of rth roots of the
`sum of determinants of all r x r principal cofactors of
`A(c, e) = 8(6, e)B”‘ (c, 6) taken over all pairs ofdistinct
`codewords e and 6 corresponds to the coding advantage,
`where r is the rank of A(e, e). Special attention in the
`design must be paid to this quantity for any codewords e
`and c. The design target is making this sum as large as
`possible. If a diversity of nm is the design target, then
`the minimum of the determinant of A(c_._ e) taken over all
`pairs of distinct codewords e and 6 must be maximized.
`
`We next study the behavior of the right—hand side of inequality
`(8) for large signal-to-noise ratios. At sufficiently high signal-
`to-noise ratios, one can approximate the right-hand side of
`inequality (8) by
`
`—?7?-
`
`
`
`P(c—>C)S (4i:0)— (H A?)
`
`i=1
`
`H H aim—Km)
`
`1:1 i=1.
`
`'
`
`(11)
`
`Thus a diversity of rm. and a coding advantage of
`
`‘
`
`l/r'm
`
`is an orthonormal
`11:1}
`Since V is unitary, {111, v2,
`basis of (13” and flit-J are independent complex Gaussian
`random variables with variance 0.5 per dimension and mean
`If3 Iii-5.. Let K71“; = lEfi-fijlg = lIf‘I'a I'Uilg. Thus Ifitdi are
`independent Rician distributions with pdf
`
`Idlthnl) = 21th, .a'l expi—lfi-é-JIZ —
`
`Ki. .‘iflfllzlfiid | va'H-i)
`
`for |;'5‘,-__‘J-| 2 0, where 100) is the zero-order modified Bessci
`fiinction of the first kind.
`
`Thus to compute an upper bound on the average probability
`of error, we simply average
`YT!
`
`H oxp(—(E3/4No) Z AilfihiP)
`i=1
`i=1
`
`with respect to independent Rician distributions of |fi._.-_?j| to
`arrive at
`
`(3)
`
`ragga-
`P(C—>e)<H Hl—‘ESAj-exp —T50
`
`J—l
`:1 + 41“)
`4N0
`
` 1
`
`We next examine some special cases.
`The Case afRayleigh Fading:
`In this case. Em; _.,- = 0 and
`as afortiori Km 2 0 for all i and j. Then the inequality (8)
`can be written as
`
`”I
`
`(Al/\g - - -/\,.)1/" H H (mm—Km)
`j=l {:1
`
`resag
`
`1
`n— (9}
`Hu+nanm)
`i=1
`
`is achieved. Thus the following design criteria is valid for the
`Rician space—time codes for large signal-to-noise ratios.
`Design Criteria for The Rieian Space—Time Codes:
`- The Rank Criterion: This criterion is the same as that
`
`Let 7' denote the rank of matrix A, then the kernel of A has
`dimension n — -r and exactly in — r eigenvalues of A are zero.
`Say the nonzero eigenvalues of A are A1, A2,
`, A,., then it
`follows from inequality (9) that
`
`given for the Rayleigh channel.
`- The Coding Advantage Criterion: Let A02, e) denote the
`sum of all the determinants of r x r principal cofactors
`of A(c._ e), where r is the rank of A(c, e). The minimum
`of the products
`
`l/r'm.
`
`Ptc—AeJS (H A)
`
`i=1
`
`(EH/worm.
`
`(10)
`
`M“:(3)1“ Hi:
`
`493')
`
`Thus a diversity advantage of mr and a coding advantage
`of (Al/\g -
`-
`- ATP/r is achieved. Recall that Jim? -
`-
`- A, is the
`absolute value of the sum of determinants of all the principal
`r x r cofactors of A. Moreover, it is easy to see that the ranks
`of A(c, e). and B(c_._ e) are equal.
`Remark: We note that the diversity advantage is the power
`of SNR in the denominator of the expression for the pairwise
`error probability derived above. The coding advantage is an
`approximate measure of the gain over an uncoded system
`operating with the same diversity advantage.
`Thus from the above analysis, we arrive at the following
`design criterion.
`
`taken over distinct codewords c and e has to be maxi-
`mized.
`
`Note that one could still use the coding advantage
`criterion, since the performance will be at least as good
`as the right—hand side of inequality (9).
`
`C. I he Case of Dependent Fade Coejjfieients
`
`In this subsection, we assume that the coefficients (12,; J- are
`samples of possibly dependent zero—mean complex Gaussian
`random variables having variance 0.5 per dimension. This is
`the Rayleigh fading, but the extension to the Rician case is
`straightforward.
`
`
`
` aiX-I
`%
` T 1018
`HUAW 4'.
`
`
`
`VS .
`SPH
`HUAW41
`
`
`
`000006
`
`HUAWEI EXHIBIT 1018
`HUAWEI VS. SPH
`
`000006
`
`

`

`'.-‘Sl'|
`
`IEEE TRJXNSACTIONS UN [NFORMATICIN THEORY. VOL. 44. NO.
`
`2, MARCH I993
`
`To this end, we consider the mm X ma. matrix
`
`D. The Case quupfd Fading
`
`./1(c, e)
`0
`
`o
`Abs, e)
`
`Y(c, e) =
`
`0
`
`0
`
`A(e, e)
`
`U
`0
`
`E
`
`u
`0
`
`U
`
`0'
`a
`a
`where 0 denotes the all-zero n. X 'n. matrix. If
`
`o Apia
`
`S2 : (£21,
`
`Sim)
`
`,then (S) can be written as
`
`P(c—+c|cr5,j.i=1.2. H-._n._j=l,2,---,rrt)
`
`gmm4wmqmanmynm
`
`Let 8 : EQ*Q denote the correlation matrix of Q. We assume
`
`that E) is full rank. The matrix (-J, being a nonnegalive-delinitc
`square Hermitian matrix, has a square root C which is an
`rim x rim lower triangular matrix. The diagonal elements of
`(-3! are unity, so that the rows of C are of length one. Let
`V = {BK/“)4, then it is easy to see that the components of
`.v are uncorrelated complex Gaussian random variables with
`variance 0.5 per dimension. The mean of the components of
`V can be easily computed from the mean of a“ and the
`matrix C. In particular, if the m, J- have mean zero, so do the
`components of :2.
`By (12), we arrive at the conclusion that
`
`When the fading is rapid, we model
`mathematical equation
`
`H
`
`the channel by the
`
`d:- = Zt.r.,,J-(t)r:: m + of.
`'f-=l
`
`(14)
`
`i, i = l, 2, m, n,
`The coefficients (rm-(t) for t : 1,2,
`j : 1, 2,
`m are modeled as independent samples of
`a complex Gaussian random variable with mean zero and
`variance 0.5 per dimension. This assumption corresponds to
`very fast Rayleigh fading but the generalization to Rician
`fading is straightforward. Also, iii are samples of independent
`zero-mean complex Gaussian random variables with variance
`Rio/2 per dimension.
`As in previous subsections, we assume that the coefficients
`a,,j(t)fort= 1, 2,
`f,i=1, 2,
`n,j= l, 2, ---, m
`are known to the decoder. The probability of transmitting
`
`a: ii 1‘125
`
`and deciding in favor of
`
`J
`(321$er «firing
`
`')
`(:3— tie;-
`
`(if
`
`at the maximum-likelihood decoder is well approximated by
`
`PkfidmfiflLLflSmMfiflqdflMM)
`
`mcaamds=1g,u.mj=tz,u,m)
`
`where
`
`S exI)(—r/C*Y(c, QC}? Es/4Nn).
`
`(13)
`
`We can now follow the same argument as in the case of
`independent fades with .r'1(e. 6) replaced by C*Y(c, (5)0.
`It
`follows that the rank of C’“Y(c. e)C has to be maximized.
`Since C is full rank, this amounts to maximizing
`
`r12(c, e) =
`
`
`
`
`
`This is just the standard approximation to the Gaussian tail
`function. Let
`
`rank [Y(e, (2)] = in rank [A(c, 6)].
`
`9:0?) = (Hedi): (mitt): ". Charm)
`
`and C(t) denote the n. x n. matrix with the element at pth row
`and qth column equal to (cf —e{’)(c—;‘ (sq). Then it is easy
`to see that
`
`1'”-
`
`d202, e):
`
`EZEZQW W
`y==1tl
`
`thus there exist a unitary
`The matrix C(t) is Hermitian,
`matrix V0?) and a diagonal matrix DH) such that C(t) =
`V(t)D(t)V*(t) [20]. The diagonal elements of DOE), denoted
`here by 13,56), 1 S i S n, are the eigenvalues of C(t) count—
`ing multiplicities. Since C(t) is Hermitian, these eigenvalues
`are real numbers. Let
`
`Wadi):
`
`remit» = Qi'ifJVit)
`
`t :
`n,j : l, 2, m,
`then f3,-_,j(t) for i = l, 2,
`l, 2, ---, i are independent complex Gaussian variables with
`mean zero and variance 0.5 per dimension and
`
`Thus the rank criterion given for the independent fade coeffi-
`cients holds in this case as well.
`
`Since (xi, J- have zero mean, so do the components of v. Thus
`by a similar argument to that of the case of independent fade
`coefl'icients, we arrive at the conclusion that the determinant
`
`of C‘*1"(c, e)C must be maximized. This equals to
`
`det. ((7)) det (1"(c, 6)) = det (6)[det (A(c, c))]’”.
`
`In this light the determinant criterion given in the case of
`independent fade coefficients holds as well. Furthennore. by
`comparing this case to the case of independent fade coeffi-
`cients, it is observed that a penalty of [IO/rim) lo“510(det(9))
`decibels in the coding advantage occurs This approximately
`quantifies the loss due to dependence.
`It follows from a similar argument that the rank criterion is
`also valid for the Rician case and that any code designed for
`the Rayleigh channel performs well for the Rician channel
`even if the fade coefficients are dependent. To obtain the
`coding adva