 small spheres should be nonin- tersecting or disjoint. Here comes the key question: how many such disjoint spheres can be fitted inside the hypersphere of radius n(P+N)? 14.1 Gaussian channel The answer comes as follows. The volume Vn of a hypersphere with dimension n and radius r is given by the formula Vn = Anrn, with the coefficient An defined by: (14.14) where T(m) is the gamma function. 3 The maximum number M of disjoint spheres that can be fitted inside the volume Vn is, thus, (14.15) The result in Eq. (14.15) shows that there exist M = 2nR possible messages with non- confusable inputs, corresponding to the channel or code rate (14.16) Since R is the maximum rate for which the messages have nonconfusable inputs (mean- ing that receiving and decoding errors can be made arbitrarily small), the value found in Eq. (14.16) represents an upper bound, which corresponds to the channel capacity. Thus, the demonstration confirms the fundamental result found in Eq. (14.13). The unit of channel capacity is bits or, more accurately, bits per channel use.4 4 Then how does channel capacity relate to channel bandwidth? We shall now address this question as follows. Define B as the signal bit rate. To recover bits from signal waveforms, a basic engineering principle states that the waveform must be sampled at the Nyquist sampling rate of f = 2B per unit time (i.e., twice the bit rate). A continuous communication channel with a capacity Cbit, which is used at the rate f = 2B, gives us the capacity per unit time Cbit/s = 2BChit. Substituting the above result into Eq. (14.13), we obtain (14.17) By dividing both sides in Eq. (14.17) by B, we obtain another definition of channel capacity, which now has the dimensionless unit of (bit/s)/Hz: Communicate (14.18) The results in either Eq. (14.17) or Eq. (14.18) are known as the Shannon-Hartley the- orem (SHT), based on an earlier contribution from Hartley to Shannon's analysis (see previously). The SHT is also referred to as the Shannon-Hartley law, the information The gamma functi