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Detection by Correlation

Module by: Behnaam Aazhang. E-mail the author

Summary: This module describes a decision procedure for decoding received signals. The implementation and analysis of a correlation receiver is included.

Figure 1
Demodulation and Detection
Demodulation and Detection (Figure4-18.png)

Detection

Decide which s m ⁢t s m t from the set of s 1 ⁢t… s m ⁢t s 1 t … s m t signals was transmitted based on observing r=r1r2⋮rN r r 1 r 2 ⋮ r N , the vector composed of demodulated received signal, that is, the vector of projection of the received signal onto the NN bases.

m ^ =argmax 1≤m≤M Pr s m ⁢t⁢ was transmitted |  r⁢ was observed m ^ 1 m M  r  was observed s m t  was transmitted 
(1)
Note that
Pr s m | r ≜ Pr s m ⁢t⁢was transmitted|  r⁢ was observed = f r | s m ⁢Prsm f r r s m ≜  r  was observed s m t was transmitted f r | s m s m f r
(2)
If Prsm⁢ was transmitted=1M s m  was transmitted 1 M , that is information symbols are equally likely to be transmitted, then
argmax 1≤m≤M Prsm| r =argmax 1≤m≤M f r | s m 1 m M r s m 1 m M f r | s m
(3)
Since r⁢t= s m ⁢t+ N t r t s m t N t for 0≤t≤T 0 t T and for some m=12…M m 1 2 … M then r=sm+η r s m η where η=η1η2⋮ηN η η 1 η 2 ⋮ η N and ηn η n 's are Gaussian and independent.
∀ r n , r n ∈R: f r | s m =12⁢π⁢ N 0 2N2⁢e−∑ n =1Nrn−sm,n22⁢ N 0 2 r n r n f r | s m 1 2 N 0 2 N 2 n 1 N r n s m n 2 2 N 0 2
(4)
m ^ =argmax 1≤m≤M f r | s m =argmax 1≤m≤M ln f r | s m =argmax 1≤m≤M (−(N2⁢lnπ⁢ N 0 ))−1 N 0 ⁢∑ n =1Nrn−sm,n2=argmin 1≤m≤M ∑ n =1Nrn−sm,n2 m ^ 1 m M f r | s m 1 m M f r | s m 1 m M N 2 N 0 1 N 0 n 1 N r n s m n 2 1 m M n 1 N r n s m n 2
(5)
where D⁢rsm D r s m is the l 2 l 2 distance between vectors rr and sm s m defined as D⁢rsm ≜ ∑n=1Nrn−sm,n2 D r s m ≜ n 1 N r n s m n 2
m ^ =argmin 1≤m≤M D⁢rsm=argmin 1≤m≤M ∥r∥2−2⁢〈(r,sm)〉+∥sm∥2 m ^ 1 m M D r s m 1 m M r 2 2 r s m s m 2
(6)
where ∥r∥ r is the l 2 l 2 norm of vector rr defined as ∥r∥ ≜ ∑n=1Nrn2 r ≜ n 1 N r n 2
m ^ =argmax 1≤m≤M 2⁢〈(r,sm)〉−∥sm∥2 m ^ 1 m M 2 r s m s m 2
(7)
This type of receiver system is known as a correlation (or correlator-type) receiver. Examples of the use of such a system are found here. Another type of receiver involves linear, time-invariant filters and is known as a matched filter receiver. An analysis of the performance of a correlator-type receiver using antipodal and orthogonal binary signals can be found in Performance Analysis.

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