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Description:
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We study the problem of communicating a discrete time analog source over
a channel such that the resulting distortion is minimized . For ergodic channels ,
Shannon showed that separate source and channel coding is optimal . In this work we
study this problem for non -ergodic channels .
Although not much can be said about the general problem of transmitting any
analog sources over any non -ergodic channels with any distortion metric , for many
practical problems like video broadcast and voice transmission , we can gain insights
by studying the transmission of a Gaussian source over a wireless channel with mean
square error as the distortion measure . Motivated by different applications , we consider three different non -ergodic channel models - (1 ) Additive white Gaussian noise
(AWGN ) channel whose signal -to -noise ratio (SNR ) is unknown at the transmitter ; (2 )
Rayleigh fading multiple -input multiple -output MIMO channel whose SNR is known
at the transmitter ; and (3 ) Rayleigh fading MIMO channel whose SNR is unknown
at the transmitter .
The traditional approach to study these problems has been to fix certain SNRs
of interest and study the corresponding achievable distortion regions . However , the
problems formulated this way have not been solved even for simple setups like 2
SNRs for the AWGN channel . We are interested in performance over a wide range
of SNR and hence we use the distortion SNR exponent metric to study this problem .
Distortion SNR exponent is defined as the rate of decay of distortion with SNR in the high SNR limit .
We study several layered transmissions schemes where the source is first compressed in layers and then the layers are transmitted using channel codes that provide
variable error protection . Results show that in several cases such layered transmission
schemes are optimal in terms of the distortion SNR exponent . Specifically , if the band -
width expansion (number of channel uses per source sample ) is b , we show that the
optimal distortion SNR exponent for the AWGN channel is b and it is achievable using
a superposition based layered scheme . For the L -block Rayleigh fading M x N MIMO
channel the optimal exponent is characterized for b < (|N - M|+1 )= min (M ;N ) and
b > MNL2 . This corresponds to the entire range of b when min (M ;N ) = 1 and
L = 1 . The results also show that the exponents obtained using layered schemes
which are a small subclass of joint source channel coding (JSCC ) schemes are , surprisingly , as good as and better in some cases than achievable exponent of all other
JSCC schemes reported so far . |