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4.4 Demodulation

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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 uponobservations of the input vector. Detection is covered elsewhere .
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Source:  OpenStax, Principles of digital communications. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10805/1.1
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