Exercise 1.16: Block Error Probability Bounds for AWGN
We assume the following constellation:
- a linear block code with code rate $R = k/n$ and distance spectrum $\{W_i\}, \ i = 1, \ \text{...} \ , n$,
- an AWGN channel characterized by $E_{\rm B}/N_{0}$ ⇒ convertible to noise power $\sigma^2$,
- a receiver based on soft decision as well as the maximum likelihood criterion.
Under the assumption valid for the entire exercise that always the null word $\underline{x}_{1} = (0, 0, \text{... } \ , 0)$ is sent, the "pairwise error probability" with a different codeword $\underline{x}_{l} (l = 2,\ \text{...} \ , 2^k)$:
- $$ {\rm Pr}[\hspace{0.05cm}\underline{x}_{\hspace{0.02cm}1} \hspace{-0.02cm}\mapsto \hspace{-0.02cm}\underline{x}_{\hspace{0.02cm}l}\hspace{0.05cm}] = {\rm Q}\left ( \sqrt{w_{\rm H}(\underline{x}_{\hspace{0.02cm}l})/\sigma^2} \right ) \hspace{0.05cm}.$$
The derivation of this relation can be found in [Liv10]. Used in this equation are:
- the "complementary Gaussian error function" ${\rm Q}(x)$,
- the "Hamming weight" $w_{\rm H}(\underline{x}_{l})$ of the codeword $\underline{x}_{l}$,
- the "AWGN noise power" $\sigma^2 = (2 - R - E_{\rm B}/N_{0})^{-1}.$
Damit lassen sich verschiedene Schranken für die Blockfehlerwahrscheinlichkeit angeben:
- the so called "Union Bound" (UB):
- $$p_1 = \sum_{l = 2}^{2^k}\hspace{0.05cm}{\rm Pr}[\hspace{0.05cm}\underline{x}_{\hspace{0.02cm}1} \hspace{-0.02cm}\mapsto \hspace{-0.02cm}\underline{x}_{\hspace{0.02cm}l}\hspace{0.05cm}] = \sum_{l \hspace{0.05cm}= \hspace{0.05cm}2}^{2^k}\hspace{0.05cm}{\rm Q}\left ( \sqrt{w_{\rm H}(\underline{x}_{\hspace{0.02cm}l})/\sigma^2} \right ) \hspace{0.05cm},$$
- the so called "Truncated Union Bound" (TUB):
- $$p_2 = W_{d_{\rm min}} \cdot {\rm Q}\left ( \sqrt{d_{\rm min}/\sigma^2} \right ) \hspace{0.05cm},$$
- $$p_3 = W(\beta) - 1\hspace{0.05cm},\hspace{0.2cm} {\rm mit}\hspace{0.15cm} \beta = {\rm e}^{ - 1/(2\sigma^2) } \hspace{0.05cm}.$$
- In this case, replace the distance spectrum $\{W_i\}$ with the weight enumerator function:
- $$\left \{ \hspace{0.05cm} W_i \hspace{0.05cm} \right \} \hspace{0.3cm} \Leftrightarrow \hspace{0.3cm} W(X) = \sum_{i=0 }^{n} W_i \cdot X^{i} = W_0 + W_1 \cdot X + W_2 \cdot X^{2} + ... \hspace{0.05cm} + W_n \cdot X^{n}\hspace{0.05cm}.$$
In the transition from the Union Bound $p_{1}$ to the more imprecise bound $p_{3}$ among others the function ${\rm Q}(x)$ is replaced by the Chernoff-Rubin bound ${\rm Q}_{\rm CR}(x)$ . Both functions are shown in the above graph (red and green curve, respectively).
In the "Exercise 1.16Z" the relationship between these functions is evaluated numerically and referenced to the bounds ${\rm Q}_{o}(x)$ and ${\rm Q}_{u}(x)$ which are also drawn in the above graph.
Hints:
- This exercise belongs to the chapter "Block error probability bounds".
- The above cited reference [Liv10] refers to the lecture manuscript "Liva, G.: Channel Coding. Chair of Communications Engineering, TU Munich and DLR Oberpfaffenhofen, 2010."
- Further we refer to the interactive applet "Complementary Gaussian error functions".
Questions
Solution
- $W_{1}$ indicates how often the Hamming weight $w_{\rm H}(\underline{x}_{i}) = 1$ occurs.
- $W_{n}$ indicates how often the Hamming weight $w_{\rm H}(\underline{x}_{i}) = n$ occurs.
With that, the Union Bound is:
- $$p_1 = {\rm Pr(Union \hspace{0.15cm}Bound)}= \sum_{i = 1}^{n}\hspace{0.05cm}W_i \cdot {\rm Q}\left ( \sqrt{i/\sigma^2} \right ) \hspace{0.05cm}.$$
(2) The distance spectrum of the $(8, 4, 4)$ code was given as $W_{0} = 1 , \ W_{4} = 14, \ W_{8} = 1$. Thus, one obtains for $\sigma = 1$:
- $$p_1 = W_4 \cdot {\rm Q}\left ( 2 \right ) + W_8 \cdot {\rm Q}\left ( 2 \cdot \sqrt{2} \right ) = 14 \cdot 2.28 \cdot 10^{-2}+ 1 \cdot 0.23 \cdot 10^{-2} \hspace{0.15cm}\underline{\approx 32.15\%}\hspace{0.05cm},$$
or for $\sigma = 0.5$:
- $$p_1 = 14 \cdot {\rm Q}\left ( 4 \right ) + {\rm Q}\left ( 4 \cdot \sqrt{2} \right ) = 14 \cdot 3.17 \cdot 10^{-5}+ 1.1 \cdot 10^{-8} \hspace{0.15cm}\underline{\approx 0.0444 \%}\hspace{0.05cm}.$$
(3) With the minimum distance $d_{\rm min} = 4$ we get:
- $$\sigma = 1.0\text{:} \hspace{0.4cm} p_2 \hspace{-0.15cm}\ = \ \hspace{-0.15cm} W_4 \cdot {\rm Q}\left ( 2 \right ) \hspace{0.15cm}\underline{= 31.92\%}\hspace{0.05cm},$$
- $$\sigma = 0.5\text{:} \hspace{0.4cm} p_2 \hspace{-0.15cm}\ = \ \hspace{-0.15cm}W_4 \cdot {\rm Q}\left ( 4 \right ) \approx p_1 \hspace{0.15cm}\underline{ = 0.0444 \%}\hspace{0.05cm}.$$
(4) The correct solution is suggestion 1:
- The Union Bound - denoted here by $p_{1}$ - is an upper bound on the block error probability in all cases.
- For the bound $p_{2}$ (Truncated Union Bound) this is not always true.
- For example, in the $(7, 4, 3)$ Hamming code ⇒ $W_{3} = W_{4} = 7, \ W_{7} = 1$ is obtained with standard deviation $\sigma = 1$:
- $$p_2 \hspace{-0.15cm}\ = \ \hspace{-0.15cm} 7 \cdot {\rm Q}\left ( \sqrt{3} \right ) = 7 \cdot 4.18 \cdot 10^{-2} \approx 0.293\hspace{0.05cm},$$
- $$p_1 \hspace{-0.15cm}\ = \ \hspace{-0.15cm} p_2 + 7 \cdot {\rm Q}\left ( \sqrt{4} \right )+ 1 \cdot {\rm Q}\left ( \sqrt{7} \right ) \approx 0.455 \hspace{0.05cm}.$$
The actual block error probability is likely to be between $p_{2} = 29.3\%$ and $p_{1} = 45.5\%$ (but has not been verified).
That is, $p_{2}$ is not an upper bound.
(5) Correct are suggested solutions 1 and 3, as the following calculation for the $(8, 4, 4)$ code shows:
- It holds ${\rm Q}(x) ≤ {\rm Q_{CR}}(x) = {\rm e}^{-x^2/2}$. Thus, for the Union Bound
- $$p_1 = W_4 \cdot {\rm Q}\left ( \sqrt{4/\sigma^2} \right ) +W_8 \cdot {\rm Q}\left ( \sqrt{8/\sigma^2} \right )$$
- another upper bound can be specified:
- $$p_1 \le W_4 \cdot {\rm e}^{ - {4}/(2 \sigma^2) } +W_8 \cdot {\rm e}^{ - {8}/(2 \sigma^2) } \hspace{0.05cm}.$$
- With $\beta = {\rm e}^{-1/(2\sigma^2)}$ can be written for this also (so the given $\beta = 1/\sigma$ is wrong):
- $$p_1 \le W_4 \cdot \beta^4 + W_8 \cdot \beta^8 \hspace{0.05cm}.$$
- The weight function of the $(8, 4, 4)$ code is:
- $$W(X) = 1 + W_4 \cdot X^4 + W_8 \cdot X^8 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} W(\beta) - 1 = W_4 \cdot \beta^4 + W_8 \cdot \beta^8\hspace{0.3cm} \Rightarrow \hspace{0.3cm} p_3 = W(\beta) - 1 \ge p_1\hspace{0.05cm}.$$
(6) With $\sigma = 1$, the Bhattacharyya parameter is $\beta = {\rm e}^{-0.5} = 0.6065$, and thus one obtains for the Bhattacharyya bound:
- $$p_3 = 14 \cdot \beta^4 + \beta^8 = 14 \cdot 0.135 + 0.018= 1.913 \hspace{0.15cm}\underline{= 191.3%}\hspace{0.05cm}.$$
- Considering that $p_{3}$ is a bound for a probability, $p_{3} = 1.913$ is only a trivial bound.
- For $\sigma = 0.5$, on the other hand, $\beta = {\rm e}^{-2} \approx 0.135.$ Then holds:
- $$p_3 = 14 \cdot \beta^4 + \beta^8 = 14 \cdot 3.35 \cdot 10^{-4} + 1.1 \cdot 10^{-7} \hspace{0.15cm}\underline{= 0.47 \%}\hspace{0.05cm}.$$
A comparison with subtask (2) shows that in the present example the Bhattacharyya bound $p_{3}$ is above the union bound $p_{1}$ by a factor $(0.47 - 10^{-2})/(0.044 - 10^{-2}) > 10$.
- The reason for this large deviation is the Chernoff-Rubin bound, which is well above the ${\rm Q}$ function.
- In "Exercise 1.16Z", the deviation between ${\rm Q}_{\rm CR}$ and ${\rm Q}(x)$ is also calculated quantitatively:
- $${{\rm Q_{CR}}( x )}/{{\rm Q}( x )} \approx 2.5 \cdot x \hspace{0.3cm} \Rightarrow \hspace{0.3cm} {{\rm Q_{CR}}( x = 4 )}/{{\rm Q}( x = 4)} \approx 10 \hspace{0.05cm}.$$
