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

# Performance Analysis of Orthogonal Binary Signals with Matched Filters

Module by: Behnaam Aazhang. E-mail the author

Summary: Bit-error analysis for an orthogonal binary signal set by using a matched filter receiver.

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 Tτd 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 .

## 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. For a similar analysis for binary antipodal signals, refer here. See Figure 2 or Figure 3.

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