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Demodulation

Module by: Behnaam Aazhang. E-mail the author

Summary: 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 s m t N t
(1)
r t = n =1N s m n ψ n t+ n =1N η n ψ n t+ N t ˜ r t n 1 N s m n ψ n t n 1 N η n ψ n t N t ˜
(2)
r t = n =1N( s m n + η n ) ψ n t+ N t ˜ r t n 1 N s m n η n ψ n t N t ˜
(3)
r t = n =1N r n ψ n t+ N t ˜ r t n 1 N r n ψ n t N t ˜
(4)

Proposition 1

The noise projection coefficients η n η n 's are zero mean, Gaussian random variables and are mutually independent if N t N t is a white Gaussian process.

Proof

μ η n=E η n =E0T N t ψ n td t μ η n η n t 0 T N t ψ n t
(5)
μ η n=0TE N t ψ n td t =0 μ η n t 0 T N t ψ n t 0
(6)
E η k η n ¯=E0T N t ψ k td t 0T N t ¯ ψ k t ¯d t =0T0T N t N t ¯ ψ k t ψ n t dtd t 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
(7)
E η k η n ¯=0T0T R N t t ψ k t ψ n ¯d t d t η k η n t 0 T t 0 T R N t t ψ k t ψ n t
(8)
E η k η n ¯= N 0 20T0Tδt t ψ k t ψ n t ¯d t d t η k η n N 0 2 t 0 T t 0 T δ t t ψ k t ψ n t
(9)
E η k η n ¯= N 0 20T ψ k t ψ n t¯d t = N 0 2 δ k n ={ N 0 2  if  k=n0  if  kn 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
(10)
η k η k 's are uncorrelated and since they are Gaussian they are also independent. Therefore, η k Gaussian0 N 0 2 η k Gaussian 0 N 0 2 and R η kn= N 0 2 δ k n R η k n N 0 2 δ k n

Proposition 2

The r n r n 's, the projection of the received signal r t r t onto the orthonormal bases ψ n t ψ n t 's, are independent from the residual noise process N t ˜ N t ˜ .

The residual noise N t ˜ N t ˜ is irrelevant to the decision process on r t r t .

Proof

Recall r n = s m n + η n r n s m n η n , given s m t s m t was transmitted. Therefore,

μ r n=E s m n + η n = s m n μ r n s m n η n s m n
(11)
Var r n =Var η n = N 0 2 Var r n Var η n N 0 2
(12)
The correlation between r n r n and N t ˜ N t ˜
E N t ˜ r n ¯=E( N t k =1N 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 n
(13)
E N t ˜ r n ¯=E N t k =1N η k ψ k t s m n +E η k η n ¯ k =1NE η k η n ¯ ψ k t 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
(14)
E N t ˜ r n ¯=E N t 0T N t ¯ ψ n t ¯d t k =1N N 0 2 δ k 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
(15)
E N t ˜ r n ¯=0T N 0 2δt t ψ n t d t N 0 2 ψ n t N t ˜ r n t 0 T N 0 2 δ t t ψ n t N 0 2 ψ n t
(16)
E N t ˜ r n ¯= N 0 2 ψ n t N 0 2 ψ n t=0 N t ˜ r n N 0 2 ψ n t N 0 2 ψ n t 0
(17)
Since both N t ˜ N t ˜ and r n r n are Gaussian then N t ˜ N t ˜ and r n r n are also independent.

The conjecture is to ignore N t ˜ N t ˜ and extract information from r 1 r 2 r N r 1 r 2 r N . Knowing the vector r r we can reconstruct the relevant part of random process r t r t for 0tT 0 t T

r t = s m t+ N t = n =1N r n ψ n t+ N t ˜ r t s m t N t n 1 N r n ψ n t N t ˜
(18)

Figure 1
Figure 1 (Figure4-17.png)
Figure 2
Figure 2 (Figure4-18.png)

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.

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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