Simultaneous SDR Optimality via a Joint Matrix Decomp.

Transcription

1 Simultaneous SDR Optimality via a Joint Matrix Decomposition Joint work with: Yuval Kochman, MIT Uri Erez, Tel Aviv Uni. May 26, 2011

2 Model: Source Multicasting over MIMO Channels z 1 H 1 y 1 Rx1 ŝ 1 s Tx x z 2 H 2 y 2 Rx2 ŝ 2 s Scalar white Gaussian source of power P S. x Channel input vector of length N t of power P x. y 1, y 2 Output vectors of lengths N (1) r, N (2) r. z 1, z 2 AWGN vectors of lengths N (i) r and entries of power 1. H 1, H 2 Channel matrices of dimensions N (i) r N t.

3 Model: Source Multicasting over MIMO Channels Signal-to-distortion ratio: SDR i Var(S i )/Var(Ŝ i S i ). Point-to-point (single user) Digital scheme (Shannon 48) Quantize source. Send index digitally using a channel code. Optimal: Minimum distortion Maximum SDR. Goal Find optimal SDR trade-off for two users.

4 Digital Scheme Analog Scheme Special Case: Source Multicasting over SISO Channels Scalar Case: y 1 = h 1 x +z 1 y 2 = h 2 x +z 2. Signal-to-noise ratio: SNR i h i 2 P x. Optimal individual performance: SDR i = 1+SNR i.

5 Digital Scheme Analog Scheme Special Case: Source Multicasting over SISO Channels s 1,s 2,... bits x 1,x 2,x 3,... Quantizer Channel Encoder Digital Transmission Quantize source. Send (multicast) digital index to both users. Cannot be simultaneously optimal for both users!

6 Digital Scheme Analog Scheme Special Case: Source Multicasting over SISO Channels Analog transmission (Goblick 61) Adjust power: Multiply source samples by P x /P s. Transmit (analog) source samples. Perform MMSE estimation at Rx. Achieves optimum for both channels simultaneously: SDR 1 = 1+SNR 1 SDR 2 = 1+SNR 2

7 Back to MIMO...

8 Upper Bound on Signal-to-Distortion Ratios Upper bound An outer bound on the achievable SDR-pairs is given by the union over all covariance matrices Cx, satisfying the power constraint on: SDR 1 I +H 1 CxH1 T, SDR 2 I +H 2 CxH T 2. Is this bound achievable for certain cases?

9 Constant-Diagonal Channel Matrices h h h 1 0 H 1 =....., H 0 h = h h 2 Simultaneous optimality: SDR i = (1+SNR i ) N, i = 1,2. Single Antenna: N = 1 Optimum achieved by analog transmission. Digital solution is suboptimal! Multiple Antennas: N > 1 Analog solution cannot exploit all degrees of freedom. Simultaneous optimality impossible (Reznic et al. 06). Some hybrid digital-analog (HDA) schemes offer tradeoffs.

10 2 2 Diagonal Channel Matrices UB Lucky Cases Transformation Opt. Region For simplicity, concentrate on 2 2 diagonal high-snr case. (2 2 Diagonal Matrices / Two parallel channels) ( α1 0 H 1 = 0 β 1 ) ( α2 0, H 2 = 0 β 2 ) Upper bound: The union over power allocation parameter Upper bound : 0 γ 1 of ) SDR i (1+αi 2 γp x )(1+βi 2 (1 γ)p x, i = 1,2. High-SNR limit (P x 1): SDR i α2 i P x 2 β2 i P x 2 = ( ) 2 Px 2 H i, i = 1,2.

11 Lucky Cases UB Lucky Cases Transformation Opt. Region Equal capacity - α 2 1 β2 1 = α2 2 β2 2 Digital solution: Quantize source. Use channel code to convey quantization index to both users. Optimal since channels of same quality (capacity). One gain equal (e.g. β 1 = β 2 ) Hybrid Digital-Analog (HDA) solution (Mittal & Phamdo): Quantize source. Use channel code to convey quantization index over channel of same gain. Transmit quantization error over other band in an analog manner.

12 Lucky Cases UB Lucky Cases Transformation Opt. Region Digital log β 1 /β 2 HDA HDA log α 1 /α 2

13 Solution by Transformation (1) UB Lucky Cases Transformation Opt. Region Equivalent parallel channels representation: [ ] [ ][ ] yi;1 αi 0 x1 = y i;2 0 β i x 2 }{{}}{{}}{{} y i H i x [ zi;1 + z i;2 }{{} z i, where x 1 and x 2 are i.i.d. of power P/2 each. Orthogonal transformation Apply an orthogonal matrix V at Tx: x = Vx. ], i = 1,2 Apply orthogonal matrices U i at Rx-i: ỹ i = U T i y i. upper bound and power stay the same!

14 Solution by Transformation (2) UB Lucky Cases Transformation Opt. Region Equivalent channel adequate for HDA transmission: Need one of the diagonal gains equal for the digital element. Do not need a diagonal matrix: Triangular suffices. Theorem: existence of transformation Iff α1 2 α2 2 and β2 1 β2 2, or vice versa, then: [ ] α1 0 0 β 1 [ ] α2 0 0 β 2 = U 1 [ a1 c 1 0 b = U 2 [ a2 c 2 0 b where U 1, U 2,V are orthogonal matrices. ] V T U 1 R 1 V T ] V T U 2 R 2 V T

15 Solution by Transformation (3) UB Lucky Cases Transformation Opt. Region By applying V at Tx and U T i at Rx, equivalent BC channel: ỹ 1 = [ ỹ1;1 ỹ 1;2 ] = [ a1 c 1 0 b ][ x1 x 2 ] [ z1;1 + z 1;2 ] ỹ 2 = [ ỹ2;1 ỹ 2;2 ] = [ a2 c 2 0 b ][ x1 x 2 ] [ z2;1 + z 2;2 ] Due to orthogonality: a 2 i b2 = α 2 i β2 i. z i have the same statistics as z i.

16 Solution by Transformation (4) UB Lucky Cases Transformation Opt. Region ỹ i;1 = a i x 1 + Interference {}}{ c i x 2 + z i;1 ỹ i;2 = 0 x 1 + b x 2 + z i;2 Optimal transmission over equivalent channel Quantize and send digital channel code over second channel. Send analog quantization error over first channel. Interference cancelation: ỹ i;1 c i x 2 = a i x 1 + z i;1. Optimal reconstruction as in HDA for parallel channels with one equal band.

17 Optimality Region UB Lucky Cases Transformation Opt. Region Digital log β 1 /β 2 HDA HDA log α 1 /α 2

18 Optimality Region UB Lucky Cases Transformation Opt. Region Digital log β 1 /β 2 HDA New HDA log α 1 /α 2 New

19 Generalizations Optimality for general SNR under the same condition. MIMO (non-diagonal) channel: optimality under the same condition, applied to the generalized singular values. Higher dimension (N 1) digital sub-channels, one analog. Explicit condition on singular values - a generalized form of Weyl s condition used in the GTD (Jiang et al.). Results carry over to colored/isi channels (diagonal matrix filter in frequency domain) Special case of Network Modulation : Joint decomposition of channel matrices for MIMO network problems.