Consider transmitting independent and equally likely binary data using the signals
s kt 1k 2P c cos2f c t 2P s sin2f c t, k 1, 2. The received signal due to transmitting s kt is
given by rt 1k 2P c cos2f c t 2P s sin2f c t nt, where is independent random
channel phase that is constant over a bit interval, T, and nt is AWGN with variance n
2 N02. Assume that
f c 1T.
This channel is peculiar since the random i.i.d. channel phase is discrete-valued with
Pr 0 Pr 12.
The receiver has two objectives: (i) Estimate the binary data and (ii) estimate the channel phase.
Show all work.
(a.) Provide a complete signal space description of the transmitted signals. Make sure to completely specify
the basis waveforms and the coefficients of expansion over the bit inerval.
1t s1 ,
2t s2 ,
(b.) Provide the coefficients of expansion for the signals
y kt 1k 2P c cos2f c t 2P s sin2f c t over the same time interval, in the signal space
above.
y1 , y2 ,
(c.)The channel phase is estimated using threshold detection in the quadrature (sin) carrier dimension.
Determine the probability of channel phase estimation error, P Pr ̂ .
P
(d.) The binary data are detected using threshold detection in the in-phase (cos) carrier dimension, after
attempting to remove the effects of the channel phase using the phase estimate ̂ . Determine the probability of
bit error. (For simplicity, use the variable P in your answer.)
P b
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