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

674 DIGITAL COMMUNICATIONS UNDER LINEARLY DISTORTIVE CHANNELS
In general, the autocorrelation between two noise samples in Eq. (12.21) depends on the
receiver filter which is, in this case, p(-t). In Sec. 7.3, the ISi-free pulse design based on
Nyquist's first criterion is of particular interest. Nyquist's first criterion requires that the total
response from the transmitter to the receiver be free of intersymbol interferences. Without
channel distortion, the QAM system in our current study has a total impulse response of
p(t) * p(-t) {=} IP (!) 1 2
For this combined pulse shape to be free of ISI, we can apply the first Nyquist criterion in the
frequency domain
(12.22a)
This is equivalent to the time domain requirement
{ 1 £=0
p(t) * P(-t) I - =
D
t-tT 0 = ±1 , ±2 , ...
(12.22b)
In other words, the Nyquist pulse-shaping filter is equally split between the transmitter and
the receiver. According to Eq. (12.22a), the pulse-shaping frequency response P(f) is the
square root of a pulse shape that satisfies Nyquist's first criterion in the frequency domain.
If the raised-cosine pulse shape of Section 7.3 is adopted, then P(f) would be known as the
root-raised-cosine pulse. For a given roll-off factor r, the root-raised-cosine pulse in the time
domain is
2 , cos [o + r) -o/ ] + (4 r )-I sin (l - r) -o/
Prrc U) = ;rr;
[1 - (4r f f]
(12.23)
Based on the ISi-free conditions of Eq. (12.22b), we can derive from Eq. (12.21) that
R w [l] = N l oo IP(f)l 2 e-i 2 nf a df
2 - (X)
= Np(t) *P(-t) I
2 t=tT
£ =0
£ = ±1, ±2, . ..
(12.24)
This means that the noise samples {w[n]} are uncorrelated. Since the noise samples {w[n]} are
Gaussian, they are also independent. As a result, the conditional joint probability of Eq. ( 12.19)
becomes much simpler
p (. .., z[n - 1], z[n], z[n + 1], ..-1·.., Sn- I , Sn , Sn+!, ...)
= np (z[n - i]1 ..., Sn- I , Sn , Sn+!, ...) (12.25)