Skip to content Skip to navigation

Connexions

You are here: Home » Content » Non-overlapping Received Signals

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

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

      What is in a lens?

      Lens makers point to Connexions 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 Connexions 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
  • E-mail the author
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    		(0 ratings)

Recently Viewed

This feature requires Javascript to be enabled.

Non-overlapping Received Signals

Module by: Don Johnson

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

If we restrict to observing rt r t over 0T 0 T , the minimum P e P e receiver computes N 0 2ln π i +<si*,r>−∥si*∥22 N 0 2 π i s i * r s i * 2 2 and chooses the largest. Here, <si*,r>=∫0T s i * trtdt s i * r t 0 T s i * t r t ∥si*∥2=∫0T s i * 2tdt s i * 2 t 0 T s i * t 2 When we have a binary signal set, the probability of error is given by P e =Q∥s0*−s1*∥2 N 0 P e Q s 0 * s 1 * 2 N 0

One question of interest is what the "best" signal set at the transmitter to use to maximize the performance. We want to find the s i t s i t that maximizes ∥s0*−s1*∥2 s 0 * s 1 * 2 for a given channel model. As usual, we constrain the transmitter signal energies by ∥si∥2≤E s i 2 E . Now ∥s0*−s1*∥2=∫0T s 0 * 2t−2 s 0 * t s 1 * t+ s 1 * 2tdt s 0 * s 1 * 2 t 0 T s 0 * t 2 2 s 0 * t s 1 * t s 1 * t 2 As s i * = s i * h CH s i * s i h CH , we have ∫ s i * t s j * tdt=∫∫ h CH tα s i αdα∫ h CH tβ s j βdβdt t s i * t s j * t t α h CH t α s i α β h CH t β s j β Interchanging the order of integration, this integral can be written as ∫∫∫ h CH tα h CH tβdt s i α s j βdαdβ β α t h CH t α h CH t β s i α s j β Defining the innermost integral as Q CH αβ Q CH α β , we can write the squared distance between the received signals as

∥s0*−s1*∥2=∫∫ Q CH αβ s 0 α s 0 β− s 0 α s 1 β− s 0 β s 1 α+ s 1 α s 1 βdαdβ=∫∫ Q CH αβ s 0 α− s 1 α s 0 β− s 1 βdαdβ=∥s0−s1∥ Q CH 2 s 0 * s 1 * 2 β α Q CH α β s 0 α s 0 β s 0 α s 1 β s 0 β s 1 α s 1 α s 1 β β α Q CH α β s 0 α s 1 α s 0 β s 1 β Q CH s 0 s 1 2 (1)
where ∥x∥ Q CH Q CH x denotes the norm induced by Q CH αβ Q CH α β . ∥x∥ Q CH 2=∫∫ Q CH αβxαxβdαdβ Q CH x 2 β α Q CH α β x α x β This quantity is a valid inner product because Q CH αβ Q CH α β is a positive-definite, symmetric function of αα and ββ. The relevant inner product can be written as ∥s0−s1∥ Q CH 2=∥ s 0 ∥ Q CH 2−2< Q CH , s 0 >+∥ s 1 ∥ Q CH 2 Q CH s 0 s 1 2 Q CH s 0 2 2 Q CH s 0 s 1 Q CH s 1 2 Therefore, if we require ∥ s 0 ∥ Q CH 2=∥ s 1 ∥ Q CH 2 Q CH s 0 2 Q CH s 1 2 , the maximum value of this norm occurs when < Q CH , s 0 >=-∥ s 1 ∥ Q CH 2 Q CH s 0 s 1 Q CH s 1 2 . In other words, s 0 =- s 1 s 0 s 1 . Consequently, antipodal signals still have the best performance. Because of the weighting function Q CH ·· Q CH · · in the inner product, certain signals are better than others. We seek to find those signals. According to Mercer's theorem, Q CH αβ Q CH α β may be represented in an orthogonal expansion. Q CH αβ=∑j=1∞ λ j φ j α φ j β Q CH α β j 1 λ j φ j α φ j β where φ j t φ j t and λ j λ j are the eigenfunctions and eigenvalues of Q CH αβ Q CH α β . As s 0 t=∑j=1∞ s 0 j φ j t s 0 t j 1 s 0 j φ j t , the squared norm of a signal can be written ∥ s 0 ∥ Q CH 2=∫∫ Q CH αβ s 0 α s 0 βdαdβ=∑j=1∞ λ j s 0 j 2 Q CH s 0 2 β α Q CH α β s 0 α s 0 β j 1 λ j s 0 j 2 When we have an antipodal signal set, ∥s0−s1∥ Q CH 2=4∑ λ j s 0 j 2 Q CH s 0 s 1 2 4 λ j s 0 j 2 . To constrain the energy of s 0 t s 0 t means that ∑ s 0 j 2≤ E 0 s 0 j 2 E 0 . If h CH tτ=δt−τ h CH t τ δ t τ , Q CH αβ=δα−β⇒ λ j =1 Q CH α β δ α β λ j 1 for all jj. Consequently, the distance with respect to Q CH αβ Q CH α β is maximized for any signal having energy E 0 E 0 . On the other hand, if the λ j λ j are not identical, the values of the representation s 0 j s 0 j affect the distance. To maximize this distance, set s 0 j =0 s 0 j 0 except for that component corresponding to the largest eigenvalue λ max λ max . Best results are obtained, therefore, when s 0 t=E φ m t s 0 t E φ m t , where λ max =maxj{ λ j } λ max j λ j . In this case, ∥s0−s1∥ Q CH 2=4 λ max E Q CH s 0 s 1 2 4 λ max E . With this optimum signal selection, P e =Q2 λ max E N 0 P e Q 2 λ max E N 0

Example 1

Consider a multipath channel that has the impulse response h CH tτ=δt−τ+aδt−τ−Δ h CH t τ δ t τ a δ t τ Δ where ΔΔ is the delay in the multipath and aa is the gain (usually less than one). The kernal Q CH αβ Q CH α β becomes in this case Q CH αβ=1+a2δα−β+aδα−β−Δ+aδα−β+Δ Q CH α β 1 a 2 δ α β a δ α β Δ a δ α β Δ The eigenfunctions of this kernal are determined by the equation λφt=1+a2φt+aφt−Δ+aφt+Δ λ φ t 1 a 2 φ t a φ t Δ a φ t Δ The solutions of this equation are sinusoids having frequencies harmonically related to ΔΔ. φ k t=Asin2πkΔt+Bcos2πkΔt φ k t A 2 k Δ t B 2 k Δ t λ k =1+a2 λ k 1 a 2 In principal, any signal represented by a Fourier series having a period of ΔΔ would yield the smallest probability of error over this multipath channel.

When this solution is considered in detail, no time-limited solution to the eigenequation exists. Thus, this analysis does not yield a realistic signal set for transmitting digital information over a multipath channel. However, this result does suggest that a time-limited version of this signal set would result in a receiver insensitive to the multipath. Consider the antipodal signal set s 0 t=ET−Δif0≤t<T−Δ0ifT−Δ≤t<T s 0 t E T Δ 0 t T Δ 0 T Δ t T s 1 t=- s 0 t s 1 t s 0 t The signal s 0 * t s 0 * t is given by s 0 * t=ET−Δif0≤t<Δ1+aET−ΔifΔ≤t<T−ΔaET−ΔifT−Δ≤t<T s 0 * t E T Δ 0 t Δ 1 a E T Δ Δ t T Δ a E T Δ T Δ t T The probability of error obtained when a receiver uses this signal in its matched filter is P e =Q2E N 0 1+a2−2aΔT−Δ P e Q 2 E N 0 1 a 2 2 a Δ T Δ The probability of error of the optimum (unobtainable) receiver is given by P e =Q21+a2E N 0 P e Q 2 1 a 2 E N 0 Thus, the suboptimum receiver can perform almost as well as the optimum one under certain conditions (i.e., ΔΔ small compared to T−Δ T Δ ).

Comments, questions, feedback, criticisms?

Send feedback