Demodulation 2.13 2001/06/27 2005/09/19 14:13:06.456 GMT-5 Behnaam Aazhang aaz@ece.rice.edu Dinesh Rajan dinesh@ece.rice.edu Mohammad Jaber Borran mohammad@ece.rice.edu Roy Ha rha@rice.edu Shawn Stewart mrshawn@alumni.rice.edu Behnaam Aazhang aaz@ece.rice.edu AWGN demodulation detection Gaussian modulation This module serves as an introduction to the problem of demodulation and detection: the decision procedure for deciding which signal of a set of signals was transmitted despite noise and attenuation.

Demodulation Convert the continuous time received signal into a vector without loss of information (or performance). r t s m t N t r t n 1 N s m n ψ n t n 1 N η n ψ n t N t ˜ r t n 1 N s m n η n ψ n t N t ˜ r t n 1 N r n ψ n t N t ˜ The noise projection coefficients η n 's are zero mean, Gaussian random variables and are mutually independent if N t is a white Gaussian process. μ η n η n t 0 T N t ψ n t μ η n t 0 T N t ψ n t 0 η k η n t 0 T N t ψ k t t ′ 0 T N t ′ ψ k t ′ t ′ 0 T t 0 T N t N t ′ ψ k t ψ n t ′ η k η n t ′ 0 T t 0 T R N t t ′ ψ k t ψ n t ′ η k η n N 0 2 t ′ 0 T t 0 T δ t t ′ ψ k t ψ n t ′ η k η n N 0 2 t 0 T ψ k t ψ n t N 0 2 δ k n N 0 2 k n 0 k n η k 's are uncorrelated and since they are Gaussian they are also independent. Therefore, η k Gaussian 0 N 0 2 and R η k n N 0 2 δ k n The r n 's, the projection of the received signal r t onto the orthonormal bases ψ n t 's, are independent from the residual noise process N t ˜ . The residual noise N t ˜ is irrelevant to the decision process on r t . Recall r n s m n η n , given s m t was transmitted. Therefore, μ r n s m n η n s m n Var r n Var η n N 0 2 The correlation between r n and N t ˜ N t ˜ r n N t k 1 N η k ψ k t s m n η n N t ˜ r n N t k 1 N η k ψ k t s m n η k η n k 1 N η k η n ψ k t N t ˜ r n N t t ′ 0 T N t ′ ψ n t ′ k 1 N N 0 2 δ k n ψ k t N t ˜ r n t ′ 0 T N 0 2 δ t t ′ ψ n t ′ N 0 2 ψ n t N t ˜ r n N 0 2 ψ n t N 0 2 ψ n t 0 Since both N t ˜ and r n are Gaussian then N t ˜ and r n are also independent. The conjecture is to ignore N t ˜ and extract information from r 1 r 2 … r N . Knowing the vector r we can reconstruct the relevant part of random process r t for 0 t T r t s m t N t n 1 N r n ψ n t N t ˜

Once the received signal has been converted to a vector, the correct transmitted signal must be detected based upon observations of the input vector. Detection is covered elsewhere.