Power Preserving 2:1 Bandwidth Reduction Mappings Amir Ingber 1 and Meir Feder 2 In this work we consider dimension reducing mappings that can be used for joint source-channel coding (JSCC) systems. In such systems, the source coding and the channel coding is performed as a single operation. Although it is known by Shannon’s separation theorem that asymptotically JSCC is not required for attaining the optimal performance, utilizing such schemes is beneficial for practical reasons such as delay and implementation simplicity. We specifically focus on the bandwidth reduction case, where the bandwidth of the data is greater than the bandwidth of the channel. More specifically, we focus on bandwidth reduction mappings, where the JSCC operation is performed using a single nonlinear operation. For the case of a source that is a series of iid Gaussian random variables, and the additive white Gaussian noise (AWGN) channel, it was shown that using Archimedes’ spiral attains performance that is close to optimal (see [1] and references within). One property of this mapping is that the power at its output is proportional to the square of the input power. This is not ideal for practical reasons, such as dynamic range, peak to average power ratio, sensitivity to model mismatch etc. Here we present a modification of the spiral mapping, so the power at the output is proportional to that of the input. We do that by taking the square root of the output of the original mapping, and preserving its sign. We prove that our modification indeed results in a power-preserving mapping, and calculate the pdf of the output of our new mapping. Showing that the new mapping indeed preserves the power is not enough for making this mapping desirable. The signal-to-distortion ratio (SDR) is of our interest, and we verify by simulations that the SDR remains similar to that of the original mapping. Sim- ulation results show that the power preserving mapping loses only about 1dB, compared to the original power-squaring spiral mapping. For further research, it would be interesting to see the idea presented here implemented for more general systems such as 3:1 mappings. It would also be interesting to see the principle of power preserving mappings with the bandwidth expansion case (for example, 1:2 mappings). All these topics and more are now under investigation. REFERENCES [1] Fredrik Hekland, Geir E. Oien, and Tor A. Ramstad. Using 2: 1 shannon mapping for joint source-channel coding. In DCC ’05: Proceedings of the Data Compression Conference, pages 223–232, Washington, DC, USA, 2005. IEEE Computer Society. 1 Amimon, Israel; email: [email protected] 2 Department of EE-Systems, Tel-Aviv University, and Amimon, Israel; email: [email protected] 2007 Data Compression Conference (DCC'07) 0-7695-2791-4/07 $20.00 © 2007

[IEEE 2007 Data Compression Conference (DCC'07) - Snowbird, UT, USA (2007.03.27-2007.03.29)] 2007 Data Compression Conference (DCC'07) - Power Preserving 2:1 Bandwidth Reduction Mappings

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Page 1: [IEEE 2007 Data Compression Conference (DCC'07) - Snowbird, UT, USA (2007.03.27-2007.03.29)] 2007 Data Compression Conference (DCC'07) - Power Preserving 2:1 Bandwidth Reduction Mappings

Power Preserving 2:1 Bandwidth ReductionMappings

Amir Ingber1 and Meir Feder2

In this work we consider dimension reducing mappings that can be used for jointsource-channel coding (JSCC) systems. In such systems, the source coding and thechannel coding is performed as a single operation. Although it is known by Shannon’sseparation theorem that asymptotically JSCC is not required for attaining the optimalperformance, utilizing such schemes is beneficial for practical reasons such as delay andimplementation simplicity. We specifically focus on the bandwidth reduction case, wherethe bandwidth of the data is greater than the bandwidth of the channel. More specifically,we focus on bandwidth reduction mappings, where the JSCC operation is performed usinga single nonlinear operation.

For the case of a source that is a series of iid Gaussian random variables, and theadditive white Gaussian noise (AWGN) channel, it was shown that using Archimedes’spiral attains performance that is close to optimal (see [1] and references within). Oneproperty of this mapping is that the power at its output is proportional to the square ofthe input power. This is not ideal for practical reasons, such as dynamic range, peak toaverage power ratio, sensitivity to model mismatch etc.

Here we present a modification of the spiral mapping, so the power at the output isproportional to that of the input. We do that by taking the square root of the output of theoriginal mapping, and preserving its sign. We prove that our modification indeed resultsin a power-preserving mapping, and calculate the pdf of the output of our new mapping.

Showing that the new mapping indeed preserves the power is not enough for makingthis mapping desirable. The signal-to-distortion ratio (SDR) is of our interest, and weverify by simulations that the SDR remains similar to that of the original mapping. Sim-ulation results show that the power preserving mapping loses only about 1dB, comparedto the original power-squaring spiral mapping.

For further research, it would be interesting to see the idea presented here implementedfor more general systems such as 3:1 mappings. It would also be interesting to see theprinciple of power preserving mappings with the bandwidth expansion case (for example,1:2 mappings). All these topics and more are now under investigation.

REFERENCES

[1] Fredrik Hekland, Geir E. Oien, and Tor A. Ramstad. Using 2: 1 shannon mapping for joint source-channel coding.In DCC ’05: Proceedings of the Data Compression Conference, pages 223–232, Washington, DC, USA, 2005.IEEE Computer Society.

1Amimon, Israel; email: [email protected] of EE-Systems, Tel-Aviv University, and Amimon, Israel; email: [email protected]