Assume a biphase modulated signal in white Gaussian noise of the form y(t) = 2P sin (Ï

Assume a biphase modulated signal in white Gaussian noise of the form 

y(t) = ˆš2P sin (ω_ct ± cos^-1 m + θ) + n(t), 0 ‰¤ t ‰¤ T_s 

where the ± signs are equally probable and θ is to be estimated by a maximum-likelihood procedure. In the preceding equation,

T_s = signaling interval

P = average signal power

ω_c = carrier frequency (rad/s)

m = modulation constant

θ = RF phase (rad)

Let the double-sided power spectral density of n(t) be 1/2 N₀. 

(a) Show that the signal portion of y(t) can be written as

S(t)= ˆš2Pm sin(ω_ct + θ) ± ˆš2P ˆš1- m² cos(ω_ct + θ)

Write in terms of the orthonormal functions Ï•₁ and Ï•₂, given by (11.192) and (11.193). 

(b) Show that the likelihood function can be written as (c) Draw a block diagram of the ML estimator for θ and compare with the block diagram shown in Figure 11.15.

Transcribed Image Text:

2/2P (1–m²) -т, 2m/2P y (t) sin (@̟t + 0) dt +ln cosh y (1) cos (@̟t+0) dt L (0) = No No ,т. (Odt K, tanh ( ) K2 cos (@t + ÔML) VCO У() sin (@t +OML) т, K2 ()dt