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Consider transmitting independent and equally likely binary data using the signals s kt  1k 2P c cos2f c t  2P s sin2f c t, k  1, 2. The received signal due to transmitting s kt is given by rt  1k 2P c cos2f c t    2P s sin2f c t    nt, where  is independent random channel phase that is constant over a bit interval, T, and nt is AWGN with variance n 2  N02. Assume that f c  1T. This channel is peculiar since the random i.i.d. channel phase is discrete-valued with Pr  0  Pr    12. 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. 1t  s1  , 2t  s2  , (b.) Provide the coefficients of expansion for the signals y kt  1k 2P c cos2f c t    2P s sin2f c t   over the same time interval, in the signal space above. y1 , y2 , (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  3 of 5

Name: 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 47335
Uploaded: 2023-12-07T22:30:49-08:00
Duration: 2 min 31 s
Channel: Dominador Tan
Description: 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 47335

          Consider transmitting independent and equally likely binary data using the signals
s kt  1k 2P c cos2f c t  2P s sin2f c t, k  1, 2. The received signal due to transmitting s kt is
given by rt  1k 2P c cos2f c t    2P s sin2f c t    nt, where  is independent random
channel phase that is constant over a bit interval, T, and nt is AWGN with variance n
2  N02. Assume that
f c  1T.
This channel is peculiar since the random i.i.d. channel phase is discrete-valued with
Pr  0  Pr    12.
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.
1t  s1  ,
2t  s2  ,
(b.) Provide the coefficients of expansion for the signals
y kt  1k 2P c cos2f c t    2P s sin2f c t   over the same time interval, in the signal space
above.
y1 , y2 ,
(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 
3 of 5

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