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Examples of Correlation Detection

Module by: Behnaam Aazhang

Summary: Examples of the use of a correlator-type receiver

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

The implementation and theory of correlator-type receivers can be found in Detection.

Example 1

Figure 1
Figure 1 (Figure4-20.png)

m ^ =2 m ^ 2 since Drs1>Drs2 D r s 1 D r s 2 or ∥s1∥2=∥s2∥2 s 1 2 s 2 2 and <r,s2>><r,s1> r s 2 r s 1 .

Figure 2
Figure 2 (Figure4-21_1.png)

Example 2

Data symbols "0" or "1" with equal probability. Modulator s 1 t=st s 1 t s t for 0≤t≤T 0 t T and s 2 t=-st s 2 t s t for 0≤t≤T 0 t T .

Figure 3
Figure 3 (Figure4-21_2.png)

ψ 1 t=stA2T ψ 1 t s t A 2 T , s11=AT s 11 A T , and s21=-AT s 21 A T

∀m,m=12:rt= s m t+ N t m m 1 2 r t s m t N t (1)

Figure 4
Figure 4 (Figure4-22_1.png)

r1=AT+ η 1 r 1 A T η 1 (2)
or
r1=-AT+ η 1 r 1 A T η 1 (3)
η 1 η 1 is Gaussian with zero mean and variance N 0 2 N 0 2 .

Figure 5
Figure 5 (Figure4-22_2.png)

m ^ =argmaxATr1-ATr1 m ^ A T r 1 A T r 1 , since AT>0 A T 0 and Pr s 1 =Pr s 1 s 1 s 1 then the MAP decision rule decides.

s 1 t s 1 t was transmitted if r1≥0 r 1 0

s 2 t s 2 t was transmitted if r1<0 r 1 0

An alternate demodulator:

rt= s m t+ N t ⇒r=sm+η r t s m t N t r s m η (4)

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