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

To compute P b from Eq. (10.25b), we need E pq
,
10.3 Coherent Receivers for Digital Carrier Modulations 523
tb E pq
= lo p(t)q(t) dt
In practice w e T b » 1, and the second term on the right-hand side can be ignored. Therefore,
Similarly,
( T b
2 2
E b = Er = E q
= lo [p(t)] dt = A T b
The BER analysis of Eq. (10.25b) for equiprobable binary symbols 1 and O becomes
It is therefore clear that to minimize P b , we should select !}.w for the binary FSK such that
sine (!}.wT b ) is minimum. Figure 10.lla shows sine (!}.wT b ) as a function of (!}.wT b ), The
minimum value of E pq
is -0.217A 2 Tb at /}.w • Tb= 1.43n or when
/}.w 0.715
2n
This leads to the minimum binary FSK BER
!}.j = - = -- = 0.715Rb
T b
(P!!fa.217Eb
Pb -
Q )
N
(10.41a)
When E pq
= 0, we have the case of orthogonal signaling. From Fig. 10. lla, it is clear
that E pq
= 0 for !}.j = n/2T b , where n is any integer. Although it appears that binary FSK
can use any integer n when selecting !}.j , larger !1f means wider separation between signaling
frequencies W e - (!}.w/2) and W e + (!}.w/2), and consequently larger transmission bandwidth.
To minimize the bandwidth, !}.j should be as small as possible. Based on Fig. 10.11 a, the
minimum value of !}.j that can be used for orthogonal signaling is 1 /2T b . FSK using this value
of !}.j is known as minimum shift keying (MSK).
Minimum Shift Keying
In MSK, not only are the two frequencies selected to be separated by 1 /2T b , but we should
also take care to preserve phase continuity when switching between! ± !1f at the transmitter.