Background ρ = 1 Colored Problem Extensions Rematch and Forward: Joint Source-Channel Coding for Communications Anatoly Khina Joint work with: Yuval Kochman, Uri Erez, Ram Zamir Dept. EE - Systems, Tel Aviv University, Tel Aviv, Israel February 21st, 2011
Background ρ = 1 Colored Problem Extensions Model Definitions UB DF CF AF The Parallel Relay Network Z 1 Y 1 Relay 1 X 1 W Z 2 X BC Y Encoder 2 Power P 1 X 2 Relay 2 Z MAC Y MAC Ŵ Decoder Power P BC Power P 2 Z M Y M X M Relay M Power P M Equal bandwidth (ρ = 1) - Schein and Gallager 2000 Bandwidth expansion/compression factor: ρ BW BC BW MAC.
Background ρ = 1 Colored Problem Extensions Model Definitions UB DF CF AF Definitions Symmetric Case Definitions P 1 = P 2 = = P M P MAC S MAC M m=1 P m σ 2 Z MAC = MP MAC σ 2 Z MAC S BC P BC σ 2 Z BC C(S) 1/2 log(1 + S) For now: Assume equal bandwidths (ρ = 1).
Background ρ = 1 Colored Problem Extensions Model Definitions UB DF CF AF Upper Bounds on Capacity Simple Upper Bounds Noiseless BC: C C (MS MAC ) Noiseless MAC: C C (MS BC ) Improved Upper Bounds Schein (Ph.D.) Other cuts. Niesen-Diggavi Consider several different cuts, simultaneously.
Background ρ = 1 Colored Problem Extensions Model Definitions UB DF CF AF Decode-and-Forward Decode and Forward Encode the message at the relays and decode it again for the MAC. C DF = min {C ( ) ( ) } MS MAC, C SBC Remark Rate must be low enough, such that each relay can decode reliably.
Background ρ = 1 Colored Problem Extensions Model Definitions UB DF CF AF Compress-and-Forward Compress and Forward Relays digitally compress their analog inputs and transmit them over the MAC. Optimal Compression = CEO Approach. ( ) C CF C S BC C(S MAC ) (see Gastpar & Vetterli, 2005) Remark Fails to achieve the coherence gain, due to separation.
Background ρ = 1 Colored Problem Extensions Model Definitions UB DF CF AF Amplify-and-Forward Amplify and Forward Send the relay inputs up to proper amplification. ( ) MSMAC S BC C AF = C S MAC + S BC + 1 Remarks Accumulates the noise. Gains coherence gains in both sections! Outperforms CF for all SNRs (ρ = 1).
Background ρ = 1 Colored Problem Extensions Strategies Comparsion of Different Strategies Interpretation of Different Strategies Decode & Forward: Channel coding approach. Compress & Forward: Source coding approach. Amplify & Forward: Joint source-channel coding approach. Strategy A & F D & F C & F BC coherence X X MAC coherence X avoid noise accumulation X X
Background ρ = 1 Colored Problem Extensions Sol. JSC for P2P R&F Max. Analog HDA AM BW Expa Colored Problem General Problem: Noises with general color Symmetric Case: Noises in each section have the same spectrum Interesting Case: Unequal Bandwidths Bandwidth Expansion/Compression
Background ρ = 1 Colored Problem Extensions Sol. JSC for P2P R&F Max. Analog HDA AM BW Expa Possible Solutions for Bandwidth Mismatch Case Possible Solutions C&F and D&F do not exploit the coherence gains. A&F does not exploit full bandwidth. Can we exploit both gains simultaneously? Yes we can! Rematch & Forward
Background ρ = 1 Colored Problem Extensions Sol. JSC for P2P R&F Max. Analog HDA AM BW Expa Joint Source-Channel Coding for Point-to-Point White Colored / W JSCC JSCC Ŵ White Codebook BW Mismatched ENC DEC DEC ENC Gauss. Channel Use white channel codebook of arbitrary BW. Treat W as a source signal. Use joint source-channel coding to transmit W. Treat the reconstruction Ŵ as output of white channel. C = R(D) for MMSE distortion Capacity is Achieved BW mismatch: Equivalent SNR SNR ρ
Background ρ = 1 Colored Problem Extensions Sol. JSC for P2P R&F Max. Analog HDA AM BW Expa Rematch & Forward - Idea Joint Source-Channel Coding Usage White codebook of BW = BW MAC. The codebook is not matched to the BC section. Use optimal JSCC for the first channel section (R(D) = C BC ). Reconstruction = Output of white channel with BW MAC. Apply A&F to reconstructions. Conclusion JSCC exploits coherence gains for BW BC BW MAC.
Background ρ = 1 Colored Problem Extensions Sol. JSC for P2P R&F Max. Analog HDA AM BW Expa Rematch & Forward - Scheme replacements Scheme: W P2P V Encoder Z 1 Y 1 JSCC ˆV 1 X 1 γ decoder Z MAC 1 Mγ JSCC X BC Y MAC ˆV P2P G Encoder BC G MAC Z Decoder M Y M JSCC ˆV M X γ M decoder Transmitter BC Section Relays MAC Section Receiver Ŵ Equivalent scheme: Z 1 ˆV 1 γ W P2P V Encoder ZM Z MAC 1 Mγ ˆV P2P Ŵ Decoder ˆV M γ
Background ρ = 1 Colored Problem Extensions Sol. JSC for P2P R&F Max. Analog HDA AM BW Expa Maximally Analog Reconstruction Error Problem Not every JSSC scheme achieves full possible coherence. Errors should be summed non-coherently. Need analog (codeword independent) JSCC scheme Definition (Maximally Analog Reconstruction Error JSCC Scheme) A JSCC scheme for source with BW SC and channel with BW CH, where the unbiased reconstruction error is independent of the source for all f < min{bw SC, BW CH }.
Background ρ = 1 Colored Problem Extensions Sol. JSC for P2P R&F Max. Analog HDA AM BW Expa Maximally Analog Reconstruction Error JSCC Shemes BW Mismatch Mittal & Phamdo (2002). Reznic et al.(2006). General Colored Case Prabhakaran et al. Kochman and Zamir.
Background ρ = 1 Colored Problem Extensions Sol. JSC for P2P R&F Max. Analog HDA AM BW Expa Hybrid Analog-Digital Schemes (Mittal & Phamdo, 2002) BW Expansion (ρ > 1) Use excess BW to digitally transmit quantized source. Use source BW to analogically transmit quantization error. Reconstruction error = Channel noise in source BW. BW Compression (ρ < 1) Quantize excess BW component of the source. Transmit by superposition: Digital code of quantized component + Channel BW component
Background ρ = 1 Colored Problem Extensions Sol. JSC for P2P R&F Max. Analog HDA AM BW Expa Analog Matching General Colored (Symmetric) Case What can we do when noises have arbitrary spectra? Analog Matching Can match any BW ratio and noise color. Uses time-domain processing. Transmits an analog signal modulo-lattice. Achieves maximally analog estimation error.
Background ρ = 1 Colored Problem Extensions Sol. JSC for P2P R&F Max. Analog HDA AM BW Expa Performance Example: BW Expansion For high SNRs: ( ( ) ) C CF C S ρ C(SMAC ) ρ BC ρ ( C DF = C min ( MS MAC, S ρ ) ) BC ) C AF = C (M (S MAC S BC ) C RF = C (M ( S MAC S ρ ) ) BC where a b ab a + b R&F outperforms all other strategies for large enough M.
Background ρ = 1 Colored Problem Extensions Sol. JSC for P2P R&F Max. Analog HDA AM BW Expa Performance Example: BW Expansion (M=1) ρ = 3, S BC = 10dB 7 6 Rematch and Forward Amplify and Forward Decode and Forward 5 Capacity[bits] 4 3 2 1 0 0 10 20 30 40 50 60 MAC SNR [db]
Background ρ = 1 Colored Problem Extensions Sol. JSC for P2P R&F Max. Analog HDA AM BW Expa Performance Example: BW Expansion (M=2) ρ = 3, S BC = 10dB 7 6 5 Outer bound Rematch and Forward Amplify and Forward Decode and Forward Capacity[bits] 4 3 2 1 0 0 10 20 30 40 50 60 MAC SNR [db]
Background ρ = 1 Colored Problem Extensions Sol. JSC for P2P R&F Max. Analog HDA AM BW Expa Performance Example: BW Expansion (M=8) ρ = 3, S BC = 10dB 7 6 5 Outer bound Rematch and Forward Amplify and Forward Decode and Forward Capacity[bits] 4 3 2 1 0 0 10 20 30 40 50 60 MAC SNR [db]
Background ρ = 1 Colored Problem Extensions Improving ρ = 1 Layered Networks Further Research Improvement over A&F for ρ = 1 PSfrag replacements λ MAC 1 λ MAC Analog Digital Codebook λ BC 1 λ BC Analog Digital BC RF DF λ MAC Analog 1 λ MAC Digital MAC Yellow bands - used for R&F; Cyan bands - used for D&F. For R&F: ρ = λ BC λ MAC.
Background ρ = 1 Colored Problem Extensions Improving ρ = 1 Layered Networks Further Research Improvement over A&F for ρ = 1 3.55 Capacity [bits] 3.5 3.45 3.4 3.35 RF DF AF DF AF DF 3.3 3.25 16 17 18 19 20 21 22 23 24 S MAC [db] R&F-D&F timesharing outperforms any known strategy.
Background ρ = 1 Colored Problem Extensions Improving ρ = 1 Layered Networks Further Research Layered Networks Layered Network
Background ρ = 1 Colored Problem Extensions Improving ρ = 1 Layered Networks Further Research Layered Networks Not a Layered Network
Background ρ = 1 Colored Problem Extensions Improving ρ = 1 Layered Networks Further Research Layered Networks Rematch and Forward can be applied to Layered Networks. Layer 1 Layer 2 Layer 3 Layer 4
Background ρ = 1 Colored Problem Extensions Improving ρ = 1 Layered Networks Further Research Further Research Non-symmetric (different noise spectra) case. Extension to MIMO channels. Usage of R&F for more complex networks. Constructing good JSCC schemes for the MAC section.
Background ρ = 1 Colored Problem Extensions Improving ρ = 1 Layered Networks Further Research HAPPY BIRTHDAY!