Space-time Diversity Codes for Fading Channels by Professor R. A. Carrasco School of Electrical, Electronic and Computing Engineering University of Newcastle-upon-Tyne

Summary Introduction Diversity: frequency, time and space Space diversity and MIMO channels Maximum likelihood sequence detection for ST codes Space-time block coding Capacity of MIMO systems on fading channels Space-time trellis codes Code design System block diagram Example BER performance Conclusions

1. Introduction Migration to 3G standards high data rate communications required high quality transmission and bandwidth efficient communications low decoding complexity Major obstacles to be solved: Multipath fading: signal is scattered among several paths, each path has a different time delay. Interference: + ISI in case of channels with memory + multi-user interference 2 Mbps indoors 144 kbps outdoors

2. Diversity Solution of the multipath fading problem, by transmission of several redundant replicas that undergo different multipath profiles Types: Frequency diversity: same information is transmitted on different frequency carriers, which will face different multipath fading. Time diversity: replicas of the signal are provided in the form of redundancy in the time domain by the use of an error control code together with a proper interleaver s1 s2 s3 … s1 time redundant of

2. Diversity Types: (cont.) Space diversity: redundancy is provided by employing an array of antennas, with a minimum separation of λ/2 between neighbouring antennas. Differently polarized antennas can also be used. 2 / l 2 / l

3. Space Diversity and MIMO channels where: ci(l) is the modulation symbol transmitted by antenna i at the time instant l. It is generated by a space-time encoder. gij is the path gain from Tx antenna i to Rx antenna j. ηj(t) is an independent Gaussian random variable (AWGN channel) ... g 11 22 12 21 1m nm n1 Tx 1 Tx 2 Tx n Rx 1 Rx 2 Rx m …

3. Space diversity and MIMO channels Taking the equation: the signals at the receiving antennas can be expressed in matrix form: Therefore we can create the MIMO channel matrix:

4. Maximum likelihood sequence detection for ST codes The probability of receiving a sequence if the code matrix Taking the likelihood function as the logarithm: has been transmitted is:

4. Maximum likelihood sequence detection for ST codes When maximising the log-likelihood we eliminate the constant term. After this, the problem is equivalent to minimising the following expression: This can be easily done with the Viterbi algorithm, using the above expression as a metric and computing the maximum likelihood path through the trellis.

5. Space-time block coding R E C I V x 1 k g ij ˆ 2 At time t, the signal , received at antenna j is given by

5. Space-time block coding where the noise samples jt are independent samples of a zero-mean complex Gaussian random variable with variance n/(2*SNR) per sample dimension. The average energy of symbols transmitted from each antenna is normalised to be one. Assuming perfect channel state information is available, the receiver computes the decision metric over all codewords and decides in favour of the codeword that minimizes the sum.

5. Space-time block coding Encoding algorithm A space-time block code is defined by a p x n transmission matrix H. The entries of the matrix H are linear combinations of the variable x1, x2, …, xk and their conjugates. The number of Transmission antennas is n.   We assume that transmission at the baseband employs a signal constellation A, with 2b elements. At time slot 1, Kb bits arrive at the encoder and select constellation signals s1,……,sK, setting xi = si for i = 1,2,….,K in H, we arrive at a matrix C with entries linear combinations of s1,s2,…..,sK and their conjugates. So, while H contains indeterminates x1,x2,….,xK C contains specific constellation symbols.

5. Space-time block coding Encoding algorithm  Examples: H2 represents a code that utilizes two antennas, H3 represents a code that utilizes three antennas and H4 represents a code that utilizes four antennas.

5. Space-time block coding Decoding algorithm Maximum likelihood decoding of any space-time block code can be achieved using linear processing at the receiver. Then maximum likelihood detection amounts to minimizing the decision metric over all possible values of s1 and s2. We expand the above metric and delete the terms that are independent of the code words and observe that the above minimization is equivalent to minimizing The above metric decomposes in two parts, one of which (1)

5. Space-time block coding is only a function of s1, and the other one is only a function of s2. Thus the minimization of (1) is equivalent to minimizing these two parts separately. This in turn is equivalent to minimizing the decision metric for detecting s1, and the decision metric for detecting s2. Similarly, the decoders for H3 and H4 can be derived.

5. Space-time block coding The decoder for H3 minimizes the decision metric for decoding s1. The decision metric for decoding s2. The decision metric for decoding s3, and the decision metric for decoding s4.

5. Space-time block coding For decoding H4, the decoder minimizes the decision metric for decoding s1. The decision metric for decoding s2. The decision metric for decoding s3, and the decision metric for decoding s4.

5. Space-time block coding There are two attractions in providing transmit diversity via orthogonal designs. There is no loss in bandwidth, in the sense that orthogonal designs provide the maximum possible transmission rate at full diversity. There is an extremely simple maximum-likelihood decoding algorithm which only uses linear combining at the receiver. The simplicity of the algorithm comes from the orthogonality of the columns of the orthogonal design.

6. Capacity of MIMO systems on fading channels For the single Tx/Rx channel the capacity is given by Shannon’s classical formula: where B is the bandwidth g is the fading gain (the realization of a complex Gaussian random variable) For a MIMO channel of n inputs and m outputs, the capacity is now given by: where Im is the identity matrix of order m snr is the signal-to-noise ratio per receive antenna G is the MIMO channel matrix * denotes the transpose conjugate bits/sec bits/sec

6. Capacity of MIMO systems on fading channels A particular case is when m = n and G = In (completely uncorrelated parallel sub-channels), then: Conclusion: Capacity can scale linearly with increasing snr Capacity can increase in almost n more bits/cycle for every 3 dB increase in the snr. bits/sec bits/sec/Hz

6. Capacity of MIMO systems on fading channels Average capacity of a MIMO Rayleigh fading channel

6. Capacity of MIMO systems on fading channels Channel correlation influence in the MIMO channel capacity Assume that all the received powers are equal. In this case we define: where R is the normalized channel correlation matrix ( ) whose components are Therefore , Where r = correlation coefficient

6. Capacity of MIMO systems on fading channels In the case of n >> 1 and r < 1, we finally obtain When n → ∞ When r = 0 (H = I) and

Channel Capacity for 3dB & 7dB

Channel Capacity for 5dB & 9dB

Channel Capacity for 11dB & 30dB

7. Space-time trellis codes The matrix C is called the code matrix, whose element ci(l) is the symbol transmitted by antenna i at the instant l. and l = 1, …, L The system model is: Ant 1 ..… ⇒ Ant 2 ..… …. …. …. Ant n ..… , where Es is the average symbol energy ηj(t) is an independent sample of a complex Gaussian random variable with variance No/2 per dimension. gij(t) is a complex Gaussian random variable with variance 0.5 per dimension. Signal to noise ratio per receive antenna:

7. Space-time trellis codes The probability of decoding erroneously the code matrix C and choosing instead another code matrix E, assuming ideal channel state information, is given by: where and the distance between codewords C and E is given by where ci = element of matrix C and ei = element of matrix E after some manipulation we rewrite the distance as: with: and

7. Space-time trellis codes A(l) is an Hermitian matrix, therefore there exists a unitary matrix U and a diagonal matrix D such that where i(l) = eigenvalues of matrix A(l) Let , so Considering the Chernoff bound of the error probability: Now we must distinguish between two cases.

7. Space-time trellis codes Quasi-static fading gij(l) is constant within a frame of length L and changes randomly from one frame to another. where r(A) is the rank of matrix A. Time-varying fading where Ν(C,E) is the set of indexes of the all zero columns of the difference matrix C-E. where D = C-E is the difference matrix r(D) is its rank Ω(D) is the set of column indexes that differ from zero.

Error Probability for fading channels. Single Input/Single Output (SISO) Multi-antenna (MIMO), from the Chernoff bound of the error probability. (coherent binary PSK, Rayleigh fading) (coherent orthogonal, Rayleigh fading) (orthogonal, noncoherent, Rayleigh fading)

Assume that the total energy of a block is limited to Etot = K.E0 The output noise power of the branch K can be written as: Assume that the total energy of a block is limited to Etot = K.E0 (E0 is the transmitting energy for each source symbol). Using the orthogonal martix H to transmit the sequence (x1,x2,…..,xK) Where Signal-to-Noise (SNR)

The total energy can also be expressed as: Assume the constellation at the receiver satisfies ER = .dR2 , where dR is the minimum distance of the constellation, and  is a constant depending on the different constellations. Using minimum distance sphere bound, the instant symbol error rate bound is: Thus

We assume i,j are independent samples of zero mean complex Gaussian random variables having a variance of 0.5 per dimension. Thus are independent Rayleigh distributed with a PDF of: Thus, the average symbol error rate bound is given by Where or

Probability of error for different numbers of Tx & Rx antennas

Probability of error for large number of Tx antennas (n)

8. Code design Diversity advantage (DA) is the exponent in the error probability bound. In order to improve the performance of ST codes, the diversity advantage must be maximised by maximising the rank of the difference matrix. 1st Design criteria: the minimum of the ranks of all possible matrices D = C-E must be maximised. To achieve the full rank n all matrices D must have full rank. Coding gain (CG) is the term independent of SNR in the upper error bound. It is the product of eigenvalues of the difference matrix or of euclidean distances. 2nd Design criteria: in order to maximise the coding gain, the minimum of the products of euclidean distance (or equivalently the eigenvalues) taken over all pairs of codes C and E must be maximised. for quasi-static fading for time-varying fading

9. System block diagram I. Encoder … c1 x c2 b0 binary cn Sn(t) S2(t) Inf Map Space-Time Ring Encoder b0 binary x Interleaver Frame Building Modulator c2 S1(t) c1 P1 Pn cn …

Space-Time Viterbi Decoder 9. System block diagram II. Decoder Space-Time Viterbi Decoder (Sequence Detection) Detection & Demodulation Channel Estimation Deinterleaver Inv Map rj binary output r1(k) r2(k) rm(k) r1(t) r2(t) rm(t) In case of iterative channel estimation

10. Example Delay diversity code Encoder structure: QPSK modulation code rate ½ 2 Tx antennas Encoder structure: Example: x = 1 3 2 0 1 c1 = 1 3 2 0 1 c2 = 0 1 3 2 0 x ∈ ℤ4 output antenna 1 antenna 2 input D c1 c2 1 symbol delay

10. Example The space-time decoding branch metric calculation is performed using the following equation: For QPSK, received signal consists of I and Q components which are produced separately by the demodulator, so the magnitude of this signal must be taken. Making the above equation:

10. Example For two transmit antennas (n=2), the equation becomes: For two receive antennas (m=2), the equation becomes:

10. Example The trellis structure for the Modulo 4, 4-state 21/3 ST-Ring TCM code is:

10. Example From a computer simulation, with 3dB S/N and a Rayleigh fading variance set to 0.5 per dimension, the following metric is calculated:   r1I = 0.998 g11 = 1.983 c1 = 0.707 g21 = 0.554c2 = -0.707 r1Q = -2.027 g11 = 1.983 c1 = 0.707 g21 = 0.554 c2 = 0.707 r2I = -0.734 g12 = 0.552 c1 = 0.707 g22 = 1.412 c2 = -0.707 r2Q = -1.586 g12 = 0.552 c1 = 0.707 g22 = 1.412 c2 = 0.707 = 0 = 14.597 = 0.016 = 8.848 0 + 14.579 + 0.016 + 8.848 = 23.461

10. Example This value (23.461) is then used as a branch metric value (BMV) and assigned to the following trellis branch: All other branch metric values are calculated in the same way (16 values per codespace pair for a 4-state code). This procedure is then repeated for each received symbol pair through the trellis (from left to right) until the whole frame has been calculated.

10. Example The path metric values (one value per code state) are then calculated in the following manner:   At the start of the trellis, the PMV values are the same as the BMV values leading into that node: For the repetitive trellis sections, there are four (in the case of a Modulo-4 code) paths leading into each node, three competitor paths and one survivor path.

10. Example The second and further sets (deeper into the trellis) are calculated by adding the BMV value of a path entering a node to the PMV value from the previous node connected to that path to form a competitor path, this procedure is then repeated for all four paths and the smallest of these four calculated values is used to form the new node PMV value, if there is more than one value that is the smallest of the four, then one is chosen arbitrarily.   Competitor 1: 28.6 + 2.9 = 31.5 Competitor 2: 0.1 + 16.0 = 16.1 = Survivor Competitor 3: 11.7 + 21.4 = 33.1 Competitor 4: 18.4 + 8.3 = 26.7 This procedure is then repeated for each node throughout the trellis, working from left to right.

10. Example Once these values have been calculated, the traceback operation is performed which consists of identifying the most likely path through the trellis based on low PMV values. This operation is performed from the end of the trellis to the start (i.e. right to left) and so can only be performed when an entire frame has been received. The operation begins with the selection of the smallest PMV for the end (deepest) set. The path backwards (the overall survivor path) to the start of the trellis is calculated based on the selection of the smallest of the four PMVs joining the current node, if there exist more than one PMV with the smallest value then one is chosen arbitrarily. This path is then stored as the modulo-4 value of the systematic MCE output corresponding to the selected path. The survivor path can be seen in the trellis diagram above (in red). This stored encoder output sequence is then reversed in order (symbol by symbol) and becomes the decoded data.

11. BER performance on time-varying Rayleigh fading channels , (Tx=2), QPSK, ideal CSI

12. Conclusions The use of space-time diversity techniques for transmission over fading channels offers a promising increase in the capacity. The capacity of MIMO channels can even increase linearly with the number of transmit antennas. In particular, space-time codes provide a significantly better performance than single antenna systems. The design criteria for space-time codes has been presented. These takes into account two factors: the diversity advantage and the coding gain. Maximum-likelihood detection techniques can be applied in the decoding process without an increase on the decoder complexity. Computer simulations avail these statements by showing a smaller BER at a fixed SNR.

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