3.3 Implications of the converse of Shannon's capacity theorem

The converse of Shannon's channel coding theorem applied to the communications system in Fig. 1 implies that any error correcting code with code rate $R_c$ information bits per transmitted bit satisfies

$$R_c (\log_2 M) (1-{\cal H}_b(P_b)) \le C(M,\bar n_s,\bar n_b,T_s, ) ,$$ (12)

where ${\cal H}_b(x) \triangleq -x\log_2 x - (1-x) \log_2(1-x)$ is the binary entropy function, and where $P_b$ is the coded bit error rate. Here, $R_c \log_2 M$ is the rate in bits per channel use. Note that capacity is expressed in bits per channel use, which removes its dependence on $T_d$. We may rewrite Eq. (12) as

$$P_b \ge {\cal H}_b^{-1}\left[1-{C(M,\bar n_s,\bar n_b,T_s,\mbox{detector}) \over R_c\log_2 M}\right].$$ (13)

For a given code rate $R_c$ and fixed $(M,\bar n_s,\bar n_b,T_s,$detector$)$, Eq. (13) gives the minimum BER $P_b$ that any rate $R_c$ code can achieve on the channel. Alternatively, we may write

$$R_c \le {C(M,\bar n_s,\bar n_b,T_s,\mbox{detector}) \over (\log_2 M) (1-{\cal H}_b(P_b))}.$$ (14)

For a given desired error rate, say $P_b = 10^{-6}$, Eq. (14) gives an upper bound on the code rate, i.e., the percentage of the transmission bits that carry information. Since the data rate $R_d = (R_c \log_2 M)/(MT_s + T_d)$ this translates directly into a bound on the data rate as well,

$$R_d \le {C(M,\bar n_s,\bar n_b,T_s,\mbox{detector}) \over (MT_s + T_d)(1-{\cal H}_b(P_b))} \quad \mbox{bits/sec.}$$ (15)