Abstract:
A method is provided for transmitting information in a data communication system. The method includes: receiving a codeword having a plurality of bits; mapping more significant bits of the codeword to bit locations of a symbol in a constellation with lower error probabilities, where the constellation represents a modulation scheme; modulating the symbol in accordance with the modulation scheme; and transmitting the symbol from a transmitter in the data communication system.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]     This application claims the benefit of U.S. Provisional Application No. 60/819,004, filed on Jul. 6, 2006. The disclosure of the above application is incorporated herein by reference. 
     
    
     FIELD  
       [0002]     The present disclosure relates to a source-aware information transmission scheme that simultaneously minimizes bit error rate and distortion.  
       BACKGROUND  
       [0003]     Given a source with rate R bits/second and a channel with capacity C bits/second, Shannon&#39;s well known channel capacity theorem, says that if R&lt;C, then there exists a combination of source and channel coders such that the source can be communicated over the channel with fidelity arbitrarily close to perfect. This theorem essentially implies that the source coding and channel coding are fundamentally separable without loss of performance for the overall system.  
         [0004]     Following the “separation principle”, in most modern communication systems, source coding and channel coding are treated independently. In other words, source representation is designed disjointly from information transmission. After A/D conversion of analog source signals, the bit streams are then uniformly encoded and mapped to symbols prior to transmission. Uniform bit-error-rate (BER) has been serving as one of the most commonly used performance measures. However, for systems with analog inputs, the ultimate goal of the communication system is to minimize the overall input-output distortion. While BER plays the dominant role in distortion minimization, a communication system that minimizes the BER does not necessarily minimize the overall input-output distortion.  
         [0005]     Therefore, it is desirable to provide an information transmission scheme that can achieve simultaneous BER and distortion minimization. The statements in this section merely provide background information related to the present disclosure and may not constitute prior art.  
       SUMMARY  
       [0006]     A method is provided for transmitting information in a data communication system. The method includes: receiving a codeword having a plurality of bits; mapping more significant bits of the codeword to bit locations of a symbol in a constellation with lower error probabilities, where the constellation represents a modulation scheme; modulating the symbol in accordance with the modulation scheme; and transmitting the symbol from a transmitter in the data communication system.  
         [0007]     Further areas of applicability will become apparent from the description provided herein. It should be understood that the description and specific examples are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure.  
     
    
     DRAWINGS  
       [0008]      FIG. 1  is a block diagram of an exemplary digital communication system;  
         [0009]      FIG. 2  is a graph of a mapping of quantized values to the 16-QAM constellation;  
         [0010]      FIG. 3  illustrates a one-dimensional 16-AM constellation;  
         [0011]      FIG. 4  illustrates a method for transmitting information in a data communication system;  
         [0012]      FIG. 5A  is a graph of an asymmetric 16-QAM constellation;  
         [0013]      FIG. 5B  is a graph of a symmetric 16-QAM constellation with Gray codes;  
         [0014]      FIGS. 6A and 6B  are graphs illustrating performance results for different transmission schemes using coded and uncoded systems, respectively;  
         [0015]      FIG. 7A  illustrates a source consisting of two Gaussian distributed random processes; and  
         [0016]      FIG. 7B  is a graph depicting normalized MSE under different SNR levels for a uniform 4-AM versus the proposed source-aware 4-AM. 
     
    
       [0017]     The drawings described herein are for illustration purposes only and are not intended to limit the scope of the present disclosure in any way.  
       DETAILED DESCRIPTION  
       [0018]      FIG. 1  illustrates an exemplary digital communication system with analog input. Let x k  be the discrete-time analog input vector resulted from uniform sampling of a continuous signal x(t). x k  is first fed into a quantizer (Q)  12 , which is a mapping of n-dimensional Euclidean space R n  to a finite set P ⊂ R n , given by Q:R n →P, where P={P 0 , P 1 , . . . , P M-1 } is the quantization codebook with P i  ε R n  for 0≦i≦M-1. Assume that the size of P is |P|=M=2 m , where m&gt;0 is an integer. Let y k =Q(x k ) denote the quantization value of x k , y k  is coded into a binary sequence through an index assignment function (π)  14 , and is then fed into a source-aware digital channel encoder  16  and a modulator  18 , i.e., the most significant bits (MSB) and least significant bits (LSB) may be treated distinctly. Let ŷ k  denote the receiver output, which is an estimate of the quantization value y k , the averaged input-output distortion is then given by 
   D   0   =E{d ( x   k   , ŷ   k )},   (1)  
 where d:R n ×R n →R is a non-negative function that measures the distance between two vectors in R n . 
 
         [0019]     Consider the widely used mean-square distortion function d(x, y) ∥x−y∥ 2 . In this case, the optimal quantizer satisfies the well-known nearest neighbor and centroid conditions. The overall distortion D 0  can then be decomposed into two parts, namely, the distortion due to quantization noise, and the distortion due to channel noise, denoted as n q  and n c , respectively. That is,  
           x   k     -       y   ^     k       =         (       x   k     -     y   k       )       ︸     n   q         +       (       y   k     -       y   ^     k       )       ︸     n   c               
 
 When the quantizer satisfies the centroid condition, E{n q }=0. Note that the quantization noise and the channel noise are independent, we have E{n q n c   H }=E{n c n q   H }=0. It then follows that  
                     D   0     =       E   ⁢     {            n   q          2     }       +     E   ⁢     {            n   c          2     }                     =       E   ⁢     {              x   k     -     y   k            2     }       +     E   ⁢     {              y   k     -       y   ^     k            2     }                       (   2   )             
 
 When the quantizer is optimal, the distortion due to quantization error is minimized. Minimization of D o  is thus reduced to minimizing the distortion only due to the channel noise 
 
 D=E{d ( y   k   , ŷ   k )}  (3) 
 
 Joint source index assignment and constellation codeword design for minimum distortion is considered further. 
 
         [0020]     First, we consider to minimize the distortion D=E{∥y k −ŷ k ∥ 2 } through joint design of source index assignment and index mapping. Write y k  as y k =ŷ k +e k  where y k , ŷ k  ε P={P 0 , P 1 , . . . P M-1 }, and e k  is the estimation error. For 0≦i≦M-1, define E i ={P i  P j , 0≦j≦M-1}, it then follows that  
             D   =     E   ⁢     {              y   k     -       y   ^     k            2     }                   =       ∑     i   =   0       M   -   1       ⁢       ∑     j   =   0       M   -   1       ⁢                P   i     -     P   j            2     ⁢     p   ⁡     (         y   ^     k     =         P   j     |     y   k       =     P   i         )       ⁢     p   ⁡     (       y   k     =     P   i       )                         =       ∑     i   =   0       M   -   1       ⁢       p   ⁡     (       y   k     =     P   i       )       ⁢       ∑       e   k     ∈     E   i         ⁢              e   k          2     ⁢       p   ⁡     (     e   k     )       .                       
 
 (4) 
 
 Here p(x) denotes the probability that x occurs. 
 
         [0021]     For efficient transmission, each quantizer output y k  is first coded to a binary sequence then mapped to a symbol in a constellation Ω. When the signal-to-noise ratio (SNR) is reasonably high, as it is for most useful communication systems, each transmitted symbol is more likely to be mistaken for one of its neighbors than for far more distant symbols. Therefore, to minimize the distortion D, the optimal index assignment and constellation codeword design should map the neighboring quantization vectors from the quantization codebook P to neighboring symbols in constellation Ω. More specifically, the optimal 1-1 mapping S:P→Ω should satisfy the following condition: 
 
Let P i ,P j ,{tilde over (P)} i ,{tilde over (P)} j  ε P, then d(P i ,P j )≦d({tilde over (P)} i ,{tilde over (P)} j ) if and only if d(S(P i ),S(P j ))≦d(S({tilde over (P)} i ),S({tilde over (P)} j )).   C1 
 
 That is, ideally, an isomorphic mapping that reserves the geometric structure should exist between the quantization codebook P and the constellation Ω. When the quantizer is optimal, and the constellation is Gray coded, condition (C1) ensures the equivalence between minimizing the BER and minimizing the average distortion. 
 
         [0022]     Next we look at the necessary conditions for the existence of S that satisfies (C1). First, assume that the size of the constellation |Ω|=|P|=M, and then look at the case when |Ω|&lt;|P|. Start with systems equipped with scalar quantizers and two-dimensional constellations. Consider a system with a 4-bit uniform scalar quantizer and a 16-QAM constellation as shown in  FIG. 2 . Since P={P 0 , P 1 , . . . P 15 } ⊂ R, without loss of generality, assume P 0 &lt;P 1 &lt; . . . &lt;P 15 . As can be seen, each P i  has at most two nearest neighbors, but a symbol in a 16-QAM constellation can have as many as 4 nearest neighbors. It is then impossible to find an S:P→Ω that satisfies (C1).  
         [0023]     In fact, assuming there is an S:P→Ω that satisfies (C1), then we should have  
                 d   ⁡     (       S   ⁡     (     P   0     )       ,     S   ⁡     (     P   15     )         )       =       max       x   1     ,       x   2     ∈   Ω         ⁢     d   ⁡     (       x   1     ,     x   2       )           ,           (   5   )                   d   ⁡     (       S   ⁡     (     P   0     )       ,     S   ⁡     (     P   1     )         )       =       d   ⁡     (       S   ⁡     (     P   14     )       ,     S   ⁡     (     P   15     )         )       =       min       x   1     ,       x   3     ∈   Ω         ⁢     d   ⁡     (       x   1     ,     x   2       )             ,           (   6   )                   d   ⁡     (       S   ⁡     (     P   1     )       ,     S   ⁡     (     P   14     )         )       ≥       max       P   i     ,       P   j     ∈   P         ⁢     {     d   ⁡     (       S   ⁡     (     P   i     )       ,     S   ⁡     (     P   j     )         )       }         ,   i   ,     j   ≠   0     ,   1   ,   14   ,   15           (   7   )             
 
 Without loss of generality, assume S(P 0 )=A 41 , S(P 15 )=A 14 , as illustrated in  FIG. 2 . Now consider the pair P 1  and P 14 . For (6) to be satisfied, P 1  and P 14  should be mapped to the nearest neighbors of P 0  and P 15 , respectively. Without loss of generality, assume S(P 1 )=A 42 . Since d(A 42 , A 13 )&gt;d(A 42 , A 24 ), according to (C1), we should have S(P 14 )=A 13  However, this violates (7), since d(A 42 , A 13 )&lt;(A 11  A 44 ) but A 11 , A 44  will correspond to points from {P i , i≠0, 1, 14, 15}. This implies that, to satisfy (7), P 1  and P 14  should be mapped to the pair A II  and A 44 . Clearly, this violates (6). Therefore, an S that satisfies (C1) does not exist. More generally, for systems utilizing scalar quantizer with codebook P and a symmetric (two-dimensional) rectangular or square constellation Ω with |Ω|=|P|=2 m ,m&gt;1, there is no 1-1 mapping S:P→Ω that satisfies (C1). 
 
         [0024]     In contrast, for the 4-bit scalar quantizer discussed above, instead of 16-QAM, consider the one-dimensional constellation 16-AM with Gray code as shown in  FIG. 3 . Still denoting the constellation with Ω, clearly the 1-1 mapping S:P→Ω defined by S(P i )=A i , 0≦i≦15, satisfies (C1), and it minimizes the BER and the distortion D simultaneously. This one-dimensional case implies that simultaneous minimization of BER and distortion D essentially requires that there exists a 1-1 mapping S:P→Ω 0  which satisfies, condition (C1), where Ω 0   ⊂ Ω is a real subset of Ω or Ω itself. This result provides another demonstration to the well known fact that: the essence to obtaining larger coding gain is to design codes in a subspace of signal space with higher dimensionality, as a larger minimum distance can be obtained with the same signal power. For example, two-dimensional constellation such as QAM would be a natural choice for two-dimensional vector quantizers. For multidimensional vector quantizers, multidimensional constellations would fit best. In the case when |Ω|&lt;|P|, more than one constellation symbols are needed to represent one, quantization value. Again, the multidimensional signal constellation obtained from the Cartesian product, Ω N  should be exploited.  
         [0025]      FIG. 4  illustrates this proposed method for transmitting information in a data communication system. Briefly, an incoming analog signal may be sampled and quantized into discrete signal values. Each quantized signal value is then coded into a codeword formed by a binary sequence. Codewords are in turn mapped to symbols of a constellation, where the constellation represent a modulation scheme for the codewords. Simultaneous minimization of bit error rate and distortion could be through constellation-aware source index assignment. Lastly, the modulation symbols may be transmitted in the data communication system. Although reference has been made to quadrature amplitude modulation, it is readily understood that the broader aspect of this disclosure are not limited to a particular modulation scheme.  
         [0026]     Due to lack of a priori statistical information of the input signal, non-entropy coding is widely used for various sources in practice. That is, in each quantization codeword, some bits are more significant than other bits, and an error in a significant bit will result in larger distortion than that in a less significant bit. This universal existence of non-uniformity in source coding calls for non-uniform information transmission, also known as unequal error protection, in which the most significant bits have lower bit error-rates than other bits. In the following, we consider source-aware non-uniform transmission design along the line of joint source index assignment and constellation design.  
         [0027]     With the exception of BPSk and QPSK, non-uniformity exists in most constellations. Asymmetric constellations were originally developed for multiresolution (MR) broadcast in Digital HDTV. The asymmetric constellations were designed to provide more protection to the more significant bits by grouping bits into clouds leading by the most significant bits, and the minimum distance between the clouds is larger than the minimum distance between symbols within a cloud as shown in  FIG. 5A .  
         [0028]     For symmetric constellations, an unequal error protection scheme based on block partitioning is provided in “Multilevel coded modulation for unequal error protection and multistage decoding” by R. H. Morelos-Zaragoza et al., IEEE Trans. Commun., Vol 48, pp 204-213, February 2000, which is the generalization of the Ungerboeck&#39;s mapping by set partitioning. With block partitioning, the number of nearest neighbors is minimized for each bit level b i . It turns out that the resulted codeword design coincide with that of the MR scheme with d 1 =d 2 . As can be seen, constellations resulted from either block partitioning or the MR scheme may no longer be Gray-coded.  
         [0029]     Gray codes are developed to minimize the bit-error-rate, in which the nearest neighbors correspond to bit groups that differ by only one position. Here, we revisit the non-uniformity in constellations with Gray codes, and introduce a non-uniform transmission scheme based on Gray-coded constellations. In the following, we illustrate the idea through Gray coded 16-QAM constellation shown in  FIG. 5B .  
         [0030]     In 16-QAM, each codeword has the form b 0 b 1 b 2 b 3  If we go through the 16 symbols in  FIG. 5B , there are altogether 24 nearest neighbor bit changes, among which b 0  and b 2  each changes 4 times, and b 1  and b 3  each changes 8 times. Note that when channel probability error is sufficiently small, a bit error corresponding to each bit location b i  is most likely to occur when the nearest neighbor has a different value in that specific bit location, i.e., among neighboring pairs where a change occurs. Let P e  denote the error probability, then this implies that when SNR is reasonably high,  
           P   e     ⁡     (     b   0     )       =         P   e     ⁡     (     b   2     )       =         1   2     ⁢       P   e     ⁡     (     b   1     )         =       1   2     ⁢         P   e     ⁡     (     b   3     )       .               
 
 Accordingly, we propose to minimize the average distortion by exploiting the inherent non-uniformity in Gray-coded constellations, that is, to map the more significant bits from the source encoder to bit locations with lower error probability in constellations with Gray codes. For example, consider a 4-bit quantizer and a 16-QAM constellation as in  FIG. 5B , the two MSBs will be mapped to b 0  and b 2 , while the two LSBs be mapped to b 1  and b 3 . This mapping function is preferably performed by the channel encoder  16  shown in  FIG. 1 . 
 
         [0031]     This proposed approach may be summarized as follows. For a non-uniform source, a Grey-coded constellation is defined for the designated modulation scheme. Within the constellation, bit locations having a lower error probability are noted. The more significant bits are identified in the bit sequence received from the quantizer. Given a binary sequence from the source, the more significant bits in the binary sequence are then mapped to the bit locations having the lower error probability in the constellation. Lastly, the binary sequence is modulated in accordance with the constellation. While the above description has been provided with reference to a quadrature amplitude modulation scheme, it is readily understood that this approach is extendible to other types of the modulation schemes.  
         [0032]     The proposed approach can be applied to both symmetric and asymmetric constellations. To illustrate the performance, we compare the proposed Gray-code based non-uniform transmission scheme with the block partitioning based approaches for both coded and uncoded systems (note that the MR scheme is only for asymmetric constellations and coincides with the block partitioning based method in the asymmetric case).  
         [0033]     First, the source is assumed to be analog with the amplitude uniformly distributed within [0,100], quantized using a 12-bit uniform quantizer. We consider various 16-QAM constellations, both symmetric and asymmetric. First, each 12-bit quantization output b 0 b 1 . . .  b 11  is partitioned into three 4-bit strings: b 0 b 1 b 6 b 7 , b 2 b 3 b 8 b 9 , b 4 b 5 b 10 b 11,  then mapped to both symmetric and asymmetric 16-QAM constellations based on the block partitioning (BP) scheme or the proposed Gray-code based non-uniform transmission scheme. By random index assignment, we mean that no distinction is made on MSBs and LSBs, and the strings are mapped to the Gray coded constellation based on their original bit arrangements b 0 b 1 b 2 b 3,  b 4 b 5 b 6 b 7 , b 8 b 9 b 10 b 11.  The result is shown in  FIG. 6A .  
         [0034]     In another example, impact of channel coding is investigated for both systematic and non-systematic coding schemes. Using the same source as in the example above, a 10-bit uniform quantizer is connected with a source-aware channel encoder, for which the first 4 MSBs are fed to a rate 1/3 convolutional (or Turbo) encoder and the rest 6 bits are fed to a rate 1/2 convolutional (or Turbo) encoder, respectively. The channel coding output is then mapped to 16-QAM constellations non-uniformly based on the block partitioning approach and the proposed mapping scheme. The result is shown in  FIG. 6B .  
         [0035]     As demonstrated by the simulation results, while the proposed approach has comparable performance with existing unequal error protection methods for uncoded systems (i.e. when there is no channel coding), the Gray-code based non-uniform transmission outperforms the non-Gray coded methods (i.e., the MR method and the block partitioning based approach) with big margins when channel coding is involved. The underlying arguments are: (i) channel coding may change the geometric structure of the uncoded symbols; and (ii) when SNR is reasonably high, BER of the more significant bits vanishes, and BER of the less significant bits dominates the overall distortion, and hence Gray coded constellations result in much better performance.  
         [0036]     Constellation design has largely been separated from quantizer design in the past. However, we further consider joint quantizer-constellation design for minimum distortion. Following our discussions, we propose to incorporate the source information reflected in optimal quantizer design into constellation design. Note that the optimal quantizers minimize the average distortion between the original sampled values and the quantization values, when considering memoryless AWGN channels, optimal quantizers can be exploited directly for constellation design. We illustrate this idea through the following example.  
         [0037]     Consider a non-uniform scalar quantizer with four possible quantization values. Assuming the quantization code book is P={P 1 , . . . P 4 }, where P i &lt;P i + 1  for i=1, 2, 3 and each P i  occurs with probability p(P i )=P i  for i=1, . . . , 4. Define D ij =|P-P j | 2 . Along the lines of Lemma 1, we consider the design of a 4-AM constellation Ω={A 1 , . . . , A     4   } and assume that the 1-1 map S:Ω→P is designed to satisfy condition (C1). We further assume that the quantizer is optimal, that is, it satisfies the nearest neighbor rule and the centroid criterion. Define d ij =|P i -P j | for i, j=1, . . . ,4, d i =|A i -A i + 1 | for i=1, 2, 3, and let D i  be the average distortion corresponding to symbol A i  for i=1, . . . ,4. Consider a memoryless AWGN channel, for which the noise is zero mean and with variance σ 2 , then we have  
         D   1     =         d   12   2     ⁡     [       Q   ⁡     (       d   1       2   ⁢           ⁢   σ       )       -     Q   ⁡     (         2   ⁢           ⁢     d   1       +     d   2         2   ⁢           ⁢   σ       )         ]       +       d   13   2     ⁡     [       Q   ⁡     (         2   ⁢           ⁢     d   1       +     d   2         2   ⁢           ⁢   σ       )       -     Q   ⁡     (         2   ⁢           ⁢     d   1       +     2   ⁢           ⁢     d   2       +     d   3         2   ⁢           ⁢   σ       )         ]       +       d   14   2     ⁢     Q   ⁡     (         2   ⁢           ⁢     d   1       +     2   ⁢           ⁢     d   2       +     d   3         2   ⁢           ⁢   σ       )               
         D   2     =         d   12   2     ⁢     Q   ⁡     (       d   1       2   ⁢           ⁢   σ       )         +       d   23   2     ⁡     [       Q   ⁡     (       d   2       2   ⁢           ⁢   σ       )       -     Q   ⁡     (         2   ⁢           ⁢     d   2       +     d   3         2   ⁢           ⁢   σ       )         ]       +       d   24   2     ⁢     Q   ⁡     (         2   ⁢           ⁢     d   2       +     d   3         2   ⁢           ⁢   σ       )               
         D   3     =         d   13   2     ⁢     Q   ⁡     (         d   1     +     2   ⁢           ⁢     d   2           2   ⁢           ⁢   σ       )         +       d   23   2     ⁡     [       Q   ⁡     (       d   2       2   ⁢           ⁢   σ       )       -     Q   ⁡     (         2   ⁢           ⁢     d   2       +     d   1         2   ⁢           ⁢   σ       )         ]       +       d   34   2     ⁢     Q   ⁡     (       d   3       2   ⁢           ⁢   σ       )               
         D   4     =         d   34   2     ⁡     [       Q   ⁡     (       d   3       2   ⁢           ⁢   σ       )       -     Q   ⁡     (         d   2     +     2   ⁢     d   3           2   ⁢           ⁢   σ       )         ]       +       d   24   2     ⁡     [       Q   ⁡     (         2   ⁢           ⁢     d   2       +     2   ⁢     d   3           2   ⁢           ⁢   σ       )       -     Q   ⁡     (         d   1     +     2   ⁢           ⁢     d   2       +     2   ⁢     d   3           2   ⁢           ⁢   σ       )         ]       +       d   14   2     ⁢     Q   ⁡     (         d   1     +     2   ⁢           ⁢     d   2       +     d   3         2   ⁢           ⁢   σ       )               
 
 where  
         Q   ⁡     (   x   )       =       1       2   ⁢           ⁢   π         ⁢       ∫   x   ∞     ⁢       ⅇ       -     t   2       /   2       ⁢       ⅆ   t     .               
 
 The overall average distortion can be written as  
             D   =       ∑     i   =   1     4     ⁢       p   i     ⁢     D   i                 (   8   )             
 
 Define  
           γ   1     =       d   1     σ       ,       γ   2     =       d   2       d   1         ,       γ     3   ′       =       d   3       d   1         ,       
 
 the problem of optimal constellation design for, minimum average distortion is reduced to finding γ 1 , γ 2 , γ 3  such that D is minimized, subjected to a power constraint, that is  
                 min       γ   1     ,     γ   2     ,     γ   3         ⁢     D   ⁢           ⁢   subjected   ⁢           ⁢   to   ⁢           ⁢     P   s         ≤   C           (   9   )             
 
 where P s  is the average symbol power and C is a constant. The method used in this example can be extended directly to more general cases. We illustrate the proposed approach through an example. 
 
         [0038]     Consider a source consists of two Gaussian distributed random processes centered at ±5 with variance σ 2 =(5/3) 2 , as shown in  FIG. 7A . Using a 4-level optimal quantizer with codebook P={−6.40, −3.86, 3.69, 6.2}, normalized MSE under different SNR levels is shown in  FIG. 7B  for both uniform 4-AM and the proposed source-aware 4-AM.  
         [0039]     From the simulation result, it can be seen that while symmetric constellations are optimal for uniformly distributed sources, asymmetric constellations reduce the average distortion significantly for sources that require non-uniform quantization. Compared with the asymmetric constellations originally designed for multiresolution broadcasting, the proposed joint quantizer-constellation design scheme generalizes the concept of non-uniform constellation design from the perspective of joint source-channel coding.  
         [0040]     In this disclosure, we studied joint optimization of source index assignment and modulation design for overall input-output distortion minimizations in communication systems. Taking a joint source-channel coding perspective, distortion minimization was carried out through Gray-code based non-uniform mapping and joint quantizer-constellation design. More specifically, our contributions can be summarized as: 
        We focused on distortion minimization for any wireless systems with analog input. Our discussion on simultaneous minimization of BER and average input-output distortion provides an interface between the optimal system design for minimum distortion and the traditional system design focused on BER minimization.     We proposed a novel non-uniform transmission scheme based on Gray-coded constellations. This design makes it possible for simultaneous minimization of distortion and BER. At the same time, the proposed approach outperforms existing unequal error protection approaches with big margins when channel coding is involved. Channel coding is widely used in almost all communication systems. Therefore, this approach can be applied to improve the power and spectral efficiency of virtually any the digital communication systems with analog inputs, particularly for systems with tight power constraints such as wireless sensor networks and space communications.     We also proposed a novel method on optimal constellation design for minimum distortion, by incorporating the source information reflected in optimal quantizer design into constellation design. This scheme generalized the concept of non-uniform constellation design and is particularly attractive for systems with non-uniform sources.        
 
         [0044]     The description in this disclosure is exemplary in nature and is not intended to limit the present disclosure, application, or uses.