BOUNDS ON THE CHANNEL DISTORTION OF VECTOR QUANTIZERS Gal Ben-David and David Malah Technion - Israel Institute of Technology Dept. of Electrical Eng. Haifa 32000, Israel e-mail: galb@netvision.net.il , malah@ee.technion.ac.il ABSTRACT Vector-Quantization (VQ) is a widely implemented method for low-bit-rate signal coding. A common assumption in the design of VQ systems is that the digital information is transmitted through a perfect channel. Under this assumption, the assignment of channel symbols to the VQ Reconstruction Vectors (RV) is of no importance. However, under physical channels, the effect of channel errors on the VQ system performance depends on the index assignment of the RV. For a VQ of size N, there are ! N possible assignments, meaning that an exhaustive search over all possible assignments is practically impossible. In this paper, lower and upper bounds on the performance of VQ systems under channel errors, over all possible assignments, are presented. A related expression for the average performance is also discussed. Numerical examples are given in which the bounds and average performance are compared with index assignments obtained by the index- switching algorithm. 1. INTRODUCTION A typical Vector Quantization (VQ) based communication system is shown in Fig. 1. Source VQ Encoder () t x IA () t y Channel () t z () t z ˆ VQ Decoder () t y ˆ Destination () t x ˆ IA -1 Fig.1 - Vector Quantization based Communication system A discrete-time Source emits signal samples over an infinite (or densely finite) alphabet. The VQ Encoder translates source output vectors into Channel digital sequences. The VQ decoder’s goal is to reconstruct source samples from this digital information and deliver then to the Destination. Since the analog information cannot be perfectly represented by the digital information some Quantization Distortion must be tolerated. In each channel transmission the VQ encodes a K-dimensional vector of source samples - () t x into a Reconstruction Vector index () t y , where the discrete variable t represents the time instant or a channel-use counter. The index is taken from a finite alphabet, () { } 1 , , 1 , 0 - ∈ N t y K , where N is the VQ size (number of reconstruction vectors and number of possible channel symbols). The Index Assignment (IA) is represented in Fig. 1 by a permutation operator, () { } () { } 1 , , 1 , 0 1 , , 1 , 0 : - ∈ → - ∈ Π N t z N t y K K (1) The number of possible permutations, ! N , increases very fast with N, e.g., for just 4-bits indices there are 13 10 2 ! 16 ⋅ ≈ possible permutations. For typical values of N , examination of all possible permutations is therefore impractical. The channel index () () { } t y t z Π = is sent through the channel. For Memoryless Channels, The channel output () t z ˆ is a random mapping of its input () t z , characterized by the Channel Probability Matrix Q , defined by: {} () () { } i n z j n z Q ij = = = ˆ Prob (2) Throughout we shall assume that Q is symmetric (i.e., Symmetric Memoryless Channels). For the special case of the Binary-Symmetric-Channel (BSC): {} () () { } ( ) ( ) ( ) j i H L j i H ij q q i n z j n z Q , , 1 ˆ Prob - - = = = = (3) where L is the number of bits ( L N 2 = ) per channel use, q is the Bit-Error-Rate (BER) and ( ) j i H , is the Hamming Distance between the binary representations of i and j. At the receiver, after inverse-permutation, the VQ Decoder converts the channel output symbols into one of N possible reconstruction vectors - () t x ˆ . The fidelity of the transmission is defined by a distortion measure between the input and the output of the VQ system ( ) x x d ˆ , . Knowledge of the source statistics () x p or a representing Training Sequence is assumed. The performance of the overall system is measured in terms of the average distortion ( ) [ ] x x d E ˆ , . In “classic” discussions of VQ applications, the channel is assumed to be noiseless ( I Q = , where I is the unity matrix), [1], so that no errors occur during transmission and () () t y t y ˆ = for every t. This assumption is based upon using a channel encoder-decoder pair to correct channel errors, causing the distortion due to channel-errors to be negligible. The permutation Π has no effect in this case. Upon knowledge of the source statistics, Lloyd’s algorithm [1] may be used to design the VQ. In Practice, a training sequence is used and the LBG algorithm [1] is implemented. Both methods are iterative and alternately apply the Nearest-Neighbor Condition and the Centroid condition.