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Examples with Matched Filters

Module by: Behnaam Aazhang. E-mail the author

Summary: Examples with the use of matched filters.

The theory and rationale behind matched filter receivers can be found in Matched Filters.

Example 1

Figure 1
Figure 1 (Figure4-27_1.png)

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

Figure 2
Figure 2 (Figure4-27_2.png)

∀t,0≤t≤2⁢T: s 1 ˜ ⁢t=∫−∞∞ s 1 ⁢τ⁢ h 1 ⁢t−τd τ t 0 t 2 T s 1 ˜ t τ s 1 τ h 1 t τ
(1)
s 1 ˜ ⁢t=∫0tτ⁢(T−t+τ)d τ =12⁢(T−t)⁢τ2|0t+13⁢τ3|0t=t22⁢(T−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 ⁢τd τ s 11 τ 0 T s 1 τ ψ 1 τ
(5)
∫0t s 1 ⁢τ⁢ ψ 1 ⁢τd τ =1 E s ⁢∫0tτ⁢τd τ =1 E s ⁢13⁢t3 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)

Figure 3
Figure 3 (Figure4-28.png)

Example 2

Assume binary data is transmitted at the rate of 1T 1 T Hertz.

0⇒(b=1)⇒( s 1 ⁢t=s⁢t) 0 b 1 s 1 t s t for 0≤t≤T 0 t T

1⇒(b=-1)⇒( s 2 ⁢t=−s⁢t) 1 b -1 s 2 t s t   for   0≤t≤T 0 t T

X t =∑ i =−PP b i ⁢s⁢t−i⁢T X t i P P b i s t i T
(7)

Figure 4
Figure 4 (Figure4-29.png)

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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