Successive Integer-Forcing and its Sum-Rate Optimality Or Ordentlich Joint work with Uri Erez and Bobak Nazer October 2nd, 2013 Allerton conference Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Outline Review of standard successive cancelation decoding (through noise prediction) Review of integer-forcing equalization Successive integer-forcing Optimality of Korkin-Zolotarev reduction Asymmetric rates and sum-rate optimality Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
The MIMO channel Transmitter Channel Receiver z 1 y 1 x 1 . . . w ˆ Encoder . Decoder w H . . z N x M y N y = Hx + z H ∈ R N × M , x ∈ R M × 1 and z ∼ N (0 , I N × N ) Power constraint is E � x m � 2 ≤ SNR for m = 1 , . . . , M Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
The MIMO channel Transmitter Channel Receiver z 1 y 1 x 1 w 1 Enc 1 . . . . w 1 , . . . , ˆ ˆ w M . H . Decoder z N Enc M x M w M y N We only consider BLAST schemes = ⇒ All results are also valid for multiple access channels Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Sum rate optimality of SIC (via noise prediction) Assume each encoder uses an i.i.d. Gaussian codebook, such that x looks like N ( 0 , SNR I ) Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Sum rate optimality of SIC (via noise prediction) Assume each encoder uses an i.i.d. Gaussian codebook, such that x looks like N ( 0 , SNR I ) The receiver first performs linear MMSE estimation of x from SNR I + HH T � − 1 . 1 y = Hx + z . The LMMSE filter is B = H T � Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Sum rate optimality of SIC (via noise prediction) Assume each encoder uses an i.i.d. Gaussian codebook, such that x looks like N ( 0 , SNR I ) The receiver first performs linear MMSE estimation of x from SNR I + HH T � − 1 . 1 y = Hx + z . The LMMSE filter is B = H T � Resulting effective channel is y eff = By = x + e , where e = By − x = ( BH − I ) x + Bz is a Gaussian vector with K ee = SNR( I + SNR H T H ) − 1 = SNR GG T Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Sum rate optimality of SIC (via noise prediction) Assume each encoder uses an i.i.d. Gaussian codebook, such that x looks like N ( 0 , SNR I ) The receiver first performs linear MMSE estimation of x from SNR I + HH T � − 1 . 1 y = Hx + z . The LMMSE filter is B = H T � Resulting effective channel is y eff = By = x + e , where e = By − x = ( BH − I ) x + Bz is a Gaussian vector with K ee = SNR( I + SNR H T H ) − 1 = SNR GG T √ SNR Gw where w ∼ N ( 0 , I ) and G is lower e can be written as e = triangular matrix satisfying ( I + SNR H T H ) − 1 = GG T Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Successive cancelation decoding via noise prediction Equivalent channel after LMMSE estimation is 0 · · · 0 g 11 x 1 w 1 . . √ x 2 w 2 g 21 g 22 0 . y eff = + SNR . . . . . ... . . . . . . . 0 x M w M · · · g M 1 g M 2 g MM Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Successive cancelation decoding via noise prediction Equivalent channel after LMMSE estimation is 0 · · · 0 g 11 x 1 w 1 . . √ x 2 w 2 g 21 g 22 0 . y eff = + SNR . . . . . ... . . . . . . . 0 x M w M · · · g M 1 g M 2 g MM √ Decoding first stream from y eff , 1 = x 1 + SNR g 11 w 1 is possible if R 1 < 1 � SNR � = − 1 2 log( g 2 2 log 1 + − 1 11 ) SNR g 2 11 Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Successive cancelation decoding via noise prediction After decoding first stream, w 1 is also known and can be canceled from remaining streams 0 · · · 0 g 11 x 1 w 1 . . √ x 2 w 2 0 0 . g 22 y (2) eff = + SNR . . . . ... . . . . . . . . 0 x M w M 0 g M 2 · · · g MM Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Successive cancelation decoding via noise prediction After decoding first stream, w 1 is also known and can be canceled from remaining streams 0 · · · 0 g 11 x 1 w 1 . . √ x 2 w 2 0 0 . g 22 y (2) eff = + SNR . . . . ... . . . . . . . . 0 x M w M 0 g M 2 · · · g MM √ Decoding second stream from y (2) eff , 2 = x 2 + SNR g 22 w 2 is possible if R 2 < 1 � SNR � = − 1 2 log( g 2 − 1 2 log 1 + 22 ) SNR g 2 22 Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Successive cancelation decoding via noise prediction Continuing in the same manner, each stream can be decoded if R m < − 1 2 log( g 2 mm ) , m = 1 , . . . , M Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Successive cancelation decoding via noise prediction Continuing in the same manner, each stream can be decoded if R m < − 1 2 log( g 2 mm ) , m = 1 , . . . , M Achievable sum-rate is M M R m = − 1 � � g 2 � � log mm 2 m =1 m =1 � M � = − 1 � g 2 2 log mm m =1 = − 1 � GG T � 2 log det = 1 � � I + SNR H T H 2 log det Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Integer-forcing - background Transmitter Channel Receiver z 1 y 1 x 1 v 1 ˆ w 1 w 1 Enc Dec . . . . . . . . A − 1 . H . B . . z N x M ˆ v M w M w M Enc Dec y N Proposed by Zhan et al . ISIT2010 Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Integer-forcing - background Transmitter Channel Receiver z 1 y 1 x 1 v 1 ˆ w 1 w 1 Enc Dec . . . . . . . . A − 1 . H . B . . z N x M ˆ v M w M w M Enc Dec y N Antennas transmit independent streams (BLAST). All streams are codewords from the same linear code with rate R . Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Integer-forcing - background Transmitter Channel Receiver z 1 y 1 x 1 ˆ v 1 w 1 w 1 Enc Dec . . . . . . . . A − 1 . H . B . . z N x M ˆ v M w M w M Enc Dec y N Rather than estimating x from y as in standard linear equalizers, in IF Ax is estimated for some full-rank A ∈ Z M × M . LMMSE filter is SNR − 1 I + HH T � − 1 B = AH T � Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Integer-forcing - background e 1 v 1 ∈ C y eff , 1 = � M x 1 ∈ C ˜ m =1 a 1 m x m + e 1 . . . . . A . e M v M ∈ C y eff , M = � M x M ∈ C ˜ m =1 a Mm x m + e M Effective channel is ˜ y eff = Ax + e A linear combination of codewords with integer coefficients is a codeword = ⇒ Can decode the linear combinations - remove noise = ⇒ Can solve noiseless linear combinations for transmitted streams Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Integer-forcing - background Transmitter Channel Receiver z 1 y 1 x 1 v 1 ˆ w 1 w 1 Enc Dec . . . . . . . . A − 1 . H . B . . z N x M ˆ v M w M w M Enc Dec y N Effective channel is ˜ y eff = Ax + e A linear combination of codewords with integer coefficients is a codeword = ⇒ Can decode the linear combinations - remove noise = ⇒ Can solve noiseless linear combinations for transmitted streams Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Integer-forcing - background e 1 v 1 ∈ C y eff , 1 = � M x 1 ∈ C ˜ m =1 a 1 m x m + e 1 . . . . . A . e M v M ∈ C y eff , M = � M x M ∈ C ˜ m =1 a Mm x m + e M Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Integer-forcing - background e 1 v 1 ∈ C y eff , 1 = � M x 1 ∈ C ˜ m =1 a 1 m x m + e 1 . . . . . A . e M v M ∈ C y eff , M = � M x M ∈ C ˜ m =1 a Mm x m + e M For capacity achieving codebooks, the estimation errors behave like i.i.d. (in time) Gaussian RVs. The spatial covariance matrix is K ee = SNR A ( I + SNR H T H ) − 1 A T Or Ordentlich, Uri Erez and Bobak Nazer Successive Integer-Forcing and its Sum-Rate Optimality
Recommend
More recommend
