Skip to content Skip to navigation

Connexions

You are here: Home » Content » Matched Filters

Navigation

Lenses

What is a lens?

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.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • VOCW

    This module is included inLens: Vietnam OpenCourseWare's Lens
    By: Vietnam OpenCourseWare

    Click the "VOCW" link to see all content affiliated with them.

Recently Viewed

This feature requires Javascript to be enabled.
 

Matched Filters

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

Signal to Noise Ratio (SNR) at the output of the demodulator is a measure of the quality of the demodulator.

SNR=signal energynoise energy SNR signal energy noise energy (1)
In the correlator described earlier, E s =| s m |2 E s s m 2 and σ η n 2= N 0 2 σ η n 2 N 0 2 . Is it possible to design a demodulator based on linear time-invariant filters with maximum signal-to-noise ratio?

Figure 1
Figure 1 (Figure4-24.png)

If s m ⁢t s m t is the transmitted signal, then the output of the kth kth filter is given as

y k ⁢t=∫−∞∞ r τ ⁢ h k ⁢t−τd τ =∫−∞∞( s m ⁢τ+ N τ )⁢ h k ⁢t−τd τ =∫−∞∞ s m ⁢τ⁢ h k ⁢t−τd τ +∫−∞∞ N τ ⁢ h k ⁢t−τd τ y k t τ r τ h k t τ τ s m τ N τ h k t τ τ s m τ h k t τ τ N τ h k t τ (2)
Sampling the output at time TT yields
y k ⁢T=∫−∞∞ s m ⁢τ⁢ h k ⁢T−τd τ +∫−∞∞ N τ ⁢ h k ⁢T−τd τ y k T τ s m τ h k T τ τ N τ h k T τ (3)
The noise contribution:
ν k =∫−∞∞ N τ ⁢ h k ⁢T−τd τ ν k τ N τ h k T τ (4)
The expected value of the noise component is
E ν k =E∫−∞∞ N τ ⁢ h k ⁢T−τd τ =0 ν k τ N τ h k T τ 0 (5)
The variance of the noise component is the second moment since the mean is zero and is given as
σ( ν k )2=E ν k 2=E∫−∞∞ N τ ⁢ h k ⁢T−τd τ ⁢∫−∞∞ N τ ' ¯⁢ h k ⁢T− τ ' ¯d τ ' ν k ν k 2 τ N τ h k T τ τ ' N τ ' h k T τ ' (6)
E ν k 2=∫−∞∞∫−∞∞ N 0 2⁢δ⁢τ− τ ' ⁢ h k ⁢T−τ⁢ h k ⁢T− τ ' ¯d τ d τ ' = N 0 2⁢∫−∞∞| h k ⁢T−τ|2d τ ν k 2 τ ' τ N 0 2 δ τ τ ' h k T τ h k T τ ' N 0 2 τ h k T τ 2 (7)

Signal Energy can be written as

∫−∞∞ s m ⁢τ⁢ h k ⁢T−τd τ 2 τ s m τ h k T τ 2 (8)

and the signal-to-noise ratio (SNR) as

SNR=∫−∞∞ s m ⁢τ⁢ h k ⁢T−τd τ 2 N 0 2⁢∫−∞∞| h k ⁢T−τ|2d τ SNR τ s m τ h k T τ 2 N 0 2 τ h k T τ 2 (9)

The signal-to-noise ratio, can be maximized considering the well-known Cauchy-Schwarz Inequality

∫−∞∞ g 1 ⁢x⁢ g 2 ⁢x¯d x 2≤∫−∞∞| g 1 ⁢x|2d x ⁢∫−∞∞| g 2 ⁢x|2d x x g 1 x g 2 x 2 x g 1 x 2 x g 2 x 2 (10)
with equality when g 1 ⁢x=α⁢ g 2 ⁢x g 1 x α g 2 x . Applying the inequality directly yields an upper bound on SNR
∫−∞∞ s m ⁢τ⁢ h k ⁢T−τd τ 2 N 0 2⁢∫−∞∞| h k ⁢T−τ|2d τ ≤2 N 0 ⁢∫−∞∞| s m ⁢τ|2d τ τ s m τ h k T τ 2 N 0 2 τ h k T τ 2 2 N 0 τ s m τ 2 (11)
with equality ∀τ: h k opt ⁢T−τ=α⁢ s m ⁢τ¯ τ h k opt T τ α s m τ . Therefore, the filter to examine signal m m should be

Matched Filter

∀τ: h m opt ⁢τ= s m ⁢T−τ¯ τ τ h m opt τ s m T τ (12)
The constant factor is not relevant when one considers the signal to noise ratio. The maximum SNR is unchanged when both the numerator and denominator are scaled.
2 N 0 ⁢∫−∞∞| s m ⁢τ|2d τ =2⁢ E s N 0 2 N 0 τ s m τ 2 2 E s N 0 (13)
Examples involving matched filter receivers can be found here.

Another type of receiver system is the correlation receiver. A performance analysis of matched filters can be found in Performance Analysis.

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

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.

| External bookmarks