6.30 Capacity of a channel

In addition to the Noisy Channel Coding Theorem and its converse , Shannon also derived the capacity for a bandlimited (to $W$ Hz) additive white noise channel. For this case, the signal set isunrestricted, even to the point that more than one bit can be transmitted each "bit interval." Instead of constrainingchannel code efficiency, the revised Noisy Channel Coding Theorem states that some error-correcting code exists suchthat as the block length increases, error-free transmission is possible if the source coder's datarate, $\langle B(A)\rangle R$ , is less than capacity.

Note that the data rate allowed by the capacity can exceed the bandwidth when the signal-to-noise ratio exceeds 0 dB. Our results for BPSK and FSK indicated the bandwidth they requireexceeds $\frac{1}{T}$ . What kind of signal sets might be used to achieve capacity? Modem signal sets send more than one bit/transmissionusing a number, one of the most popular of which is multi-level signaling . Here, we can transmit several bits during one transmission interval by representingbit by some signal's amplitude. For example, two bits can be sent with a signal set comprised of a sinusoid with amplitudesof $\pm (A)$ and $\pm (\frac{A}{2})$ .