B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

7.9 MAry Digital Carrier Modulation 381
design a frequency increment 8f and let
fm =Ji + (m - 1)8f
m = I, 2, ... , M
For this FSK with equal frequency separation, the frequency deviation (in analyzing the FM
signal) is
!::,.f = fM - f1 = ( M - 1)8f
2 2
It is therefore clear that the selection of the frequency set {Ii} determines the performance and
the bandwidth of the FSK modulation. If 8f is chosen too large, then the M-ary FSK will use
too much bandwidth. On the other hand, if 8f is chosen too small, then over the time interval
of T b second, different FSK symbols will show virtually no difference and the receiver will be
unable to distinguish the different symbols reliably. Thus large 8f leads to bandwidth waste,
whereas small 8f is prone to detection error due to transmission noise and interference.
The task of M-ary FSK design is to determine a small enough 8f that each FSK symbol
A cos Wit is highly distinct from all other FSK symbols. One solution to this problem
of FSK signal design actually can be found in the discussion of orthogonal signal space in
Sec. 2.6.2. If we can design FSK symbols to be orthogonal in T b by selecting a small 8f
(or !::,.f), then the FSK signals will be truly distinct over T b , and the bandwidth consumption
will be small.
To find the minimum 8f that leads to an orthogonal set of FSK signals, the orthogonality
condition according to Sec. 2.6.2 requires that
{Tb
J o A cos (2nf mt) A cos (2nfnt) dt = 0 m-:/= n (7.62)
We can use this requirement to find the minimum 8f. First of all,
Since in practical modulations, if m + fn)T b is very large (often no smaller than 10 3 ), the first
term in Eq. (7.63) is effectively zero and negligible. Thus, the orthogonality condition reduces
to the requirement that for any integer m -f= n,
Because fm = !1 + (m - 1)8f, for mutual orthogonality we have
sin [2n(m - n)8fTb] = 0
m -:/= n
From this requirement, it is therefore clear that the smallest 8f to satisfy the mutual
orthogonality condition is
8f = - Hz
2T b