Table 1 Summary of mathematical notation
From: Adaptive error protection coding for wireless transmission of motion JPEG 2000 video
Symbol | Meaning | Symbol | Meaning |
|---|---|---|---|
B | Size of the compressed bitstream, in bits | m i | \({m_{i}} = \Sigma _{j = i}^{{N_{k}} - 1}{M_{j}}\), complementary cumulative distortion (CCD) |
N k | Number of pieces the bitstream is split into | \(\hat m\) | \(\hat m = {(\Pi _{i = 0}^{{N_{k}} - 1}{m_{i}})^{1/{N_{k}}}}\), geometric mean of the CCD |
k | Size of each piece (with 2-byte CRC) before channel coding, in bits | \(\bar {r}_{\text {min}}\), \(\bar {n}_{\text {min}}\) | Minimum average channel code rate and piece length after channel coding, in bits or bytes |
n i | Size of the i-th piece after channel coding, in bits | σ 2 | Variance of the compressed frame coefficients |
p i | p i (n 0,n 1,…,n i ), probability that no transmission errors occur up to piece i | Δ ρ | MSE profile sampling step width, in bits/pixel |
M i | MSE distortion of reconstructed frame using the first i pieces | H | Differential entropy of the frame |
Q | Number of available transmission alphabet symbols in the symmetric channel | \(\tilde N\) | Number of mother R-S codes |
P S | Symbol error rate of the Q-ary symmetric channel | \(\tilde {r}_{l}\) | \(\tilde {k}_{l} / \tilde {n}_{l}\), channel code rate of the l-th R-S code |
\(\tilde {P}_{S}\) | Q-ary symbol error rate estimated at the receiver | w i | Index of the R-S code used in the i-th piece |
P b | Effective bit error rate of the Q-ary symmetric channel | α i , β i | Fractional number of codewords, in piece i, with R-S channel code rate \(\tilde {r}_{w_{i}}\) and \(\tilde {r}_{w_{i} + 1}\), respectively |
γ b | Average signal-to-noise ratio of the AWGN or Rayleigh fading channels | α i′, β i′ | Integer number of codewords, in piece i, with R-S channel code rate \(\tilde {r}_{w_{i}}\) and \(\tilde {r}_{w_{i} + 1}\), respectively |
h(n l ) | Probability that a piece of n l bits is channel-decoded without residual errors | \(\delta _{\alpha '_{i}}\) | Amount of rate \(\tilde {r}_{w_{i}}\) codewords normalized to the total number of codewords |
C, s, d | Amplitude, decay, and offset constants of the log-linear model for the combined transmission channel/channel code performance | N row, N col | Dimensions (rows and columns) of the block interleaver |
f D | Doppler spread of the Rayleigh fading channel | λ | Lagrange multiplier used for the optimization step |
r s | Source code rate, in bits/pixel | \(\bar {r}\) | Channel code rate |
R | Combined source and channel rate, in bits/pixel | N F | Number of frames in the YUV video sequence |
R b | Transmitted bit rate, after source and channel coding | \(\epsilon _{i}, \bar {\epsilon }\) | Per-frame and average MSE of the received video sequence |
\(\bar {n}\) | Average piece length after channel coding, in bits or bytes | \(\Gamma _{i}, \bar {\Gamma }\) | Per-frame and average PSNR (in dB) of the received video sequence |
\(\iota _{i}, \bar {\iota }\) | Per-frame and average MSSIM of the received video sequence | F E (ε) | a posteriori cumulative distribution function of the per-frame MSE ε i |