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Examples with Matched Filters

Module by: Behnaam Aazhang

Summary: Examples with the use of matched filters.

The theory and rationale behind matched filter receivers can be found in Matched Filters.
Example 1 
Figure4-27_1.png
Figure 1
s 1 t=t s 1 t t for 0≤t≤T 0 t T
s 2 t=-t s 2 t t for 0≤t≤T 0 t T
h 1 t=T-t h 1 t T t for 0≤t≤T 0 t T
h 2 t=-T+t h 2 t T t for 0≤t≤T 0 t T
Figure4-27_2.png
Figure 2
∀t,0≤t≤2T: s 1 ˜ t=∫-∞∞ s 1 τ h 1 t-τdτ t 0 t 2 T s 1 ˜ t τ s 1 τ h 1 t τ (1)
s 1 ˜ t=∫0tτT-t+τdτ=12T-tτ2|0t+13τ3|0t=t22T-t3 s 1 ˜ t τ 0 t τ T t τ 0 t 1 2 T t τ 2 0 t 1 3 τ 3 t 2 2 T t 3 (2)
s 1 ˜ T=T33 s 1 ˜ T T 3 3 (3)
Compared to the correlator-type demodulation
ψ 1 t= s 1 t E s ψ 1 t s 1 t E s (4)
s 11 =∫0T s 1 τ ψ 1 τdτ s 11 τ 0 T s 1 τ ψ 1 τ (5)
∫0t s 1 τ ψ 1 τdτ=1 E s ∫0tττdτ=1 E s 13t3 τ 0 t s 1 τ ψ 1 τ 1 E s τ 0 t τ τ 1 E s 1 3 t 3 (6)
Figure4-28.png
Figure 3
Example 2 
Assume binary data is transmitted at the rate of 1T 1 T Hertz.
0⇒b=1⇒ s 1 t=st 0 b 1 s 1 t s t for 0≤t≤T 0 t T
1⇒b=-1⇒ s 2 t=-st 1 b -1 s 2 t s t   for   0≤t≤T 0 t T
X t =∑i=-PP b i st-iT X t i P P b i s t i T (7)
Figure4-29.png
Figure 4

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