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Performance Analysis of Orthogonal Binary Signals with Matched Filters

Module by: Tuan Do-Hong. E-mail the author

r t ⇒(Y= Y 1 ⁢T Y 2 ⁢T) r t Y Y 1 T Y 2 T (1)

If s 1 ⁢t s 1 t is transmitted

Y 1 ⁢T=∫−∞∞ s 1 ⁢τ⁢ h 1 opt ⁢T−τd τ + ν 1 ⁢T=∫−∞∞ s 1 ⁢τ⁢ s 1 * ⁢τd τ + ν 1 ⁢T= E s + ν 1 ⁢T Y 1 T τ s 1 τ h 1 opt T τ ν 1 T τ s 1 τ s 1 * τ ν 1 T E s ν 1 T (2)
Y 2 ⁢T=∫−∞∞ s 1 ⁢τ⁢ s 2 * ⁢τd τ + ν 2 ⁢T= ν 2 ⁢T Y 2 T τ s 1 τ s 2 * τ ν 2 T ν 2 T (3)

If s 2 ⁢t s 2 t is transmitted, Y 1 ⁢T= ν 1 ⁢T Y 1 T ν 1 T and Y 2 ⁢T= E s + ν 2 ⁢T Y 2 T E s ν 2 T .

Figure 1
Figure 1 (Figure4-39.png)

H0

Y= E s 0+ ν 1 ν 2 Y E s 0 ν 1 ν 2 (4)

H1

Y=0 E s + ν 1 ν 2 Y 0 E s ν 1 ν 2 (5)
where ν 1 ν 1 and ν 2 ν 2 are independent are Gaussian with zero mean and variance N 0 2⁢ E s N 0 2 E s . The analysis is identical to the correlator example.
P e =Q⁢ E s N 0 P e Q E s N 0 (6)
Note that the maximum likelihood detector decides based on comparing Y 1 Y 1 and Y 2 Y 2 . If Y 1 ≥ Y 2 Y 1 Y 2 then s 1 s 1 was sent; otherwise s 2 s 2 was transmitted. See Figure 2 or Figure 3.
Figure 2
Figure 2 (Figure4-40_1.png)
Figure 3
Figure 3 (Figure4-40_2.png)

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