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[s(t) + n c (t)]e *iω c t 2RA LO Re{[s(t) + n c (t)]e *i(ω IF + ∆φ) }
Coherent ∑
Rx
LO
n d (t)
A LO e *iω LO + φ LO

Envelope detector
t = T b
Decision (∙) LPF 2 H I (ω)
device r(T b ) r(t) (∙)

Figure 8.17 A heterodyne receiver with an envelope detector and a matched filter H () for OOK.
I

In Eq. (8.188), the second term on the right-hand side corresponds to the Fourier transform of s()s ( + T −
b
1
t) at =±2 . Since the spectral width of s() is much smaller than , the second term can be ignored.
IF
IF
Therefore,
s (t)
F
I (t)= cos [ (t − T )+Δ], (8.189)
b
IF
F
2
⏟⏟⏟
envelope
where
T b
s (t)= s()s (T + − t) d. (8.190)
F ∫ 1 b
0
The output of the matched filter passes through an envelope detector which can be imagined as a cascade of
squarer, low-pass filter, and square-rootor, as shown in Fig. 8.17. When we square I (t), we obtain a term
F
proportional to cos [2 (t − T )+Δ] which is rejected by the low-pass filter. The signal output of the
IF
b
envelope detector is the envelope of I (t) (shown in Eq. (8.189)), which is given by
F
u(t)= s (t)∕2, (8.191)
F
s (T ) 1 T b
F
b
u(T )= = s()s () d, (8.192)
1
b
2 2 ∫ 0
u(T )= E ∕2 when ‘1’ is sent
b 1
= 0 when ‘0’ is sent. (8.193)
Next, consider the noise component before the matched filter given by Eq. (8.121),
n (t)
d
n(t)= n cos ( t +Δ)+ n sin ( t +Δ)+ . (8.194)
cI IF cQ IF
2RA
LO
Here, we have dropped the scaling factor 2RA . Since the ASE is expressed as the modulated noise process,
LO
it is convenient to express the detector noise as the modulated noise process as well, i.e.,
n (t)= n cos ( t +Δ)+ n dQ sin ( t +Δ), (8.195)
dI
IF
d
IF