This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 7.3 Signal Space Concepts We have seen in the previous section that the statistical relation between the hypotheses { H i } and the observation Y are expressed in terms of the conditional densities p ( y  i ). We are now interested in applying this framework for derive optimal decision rules (and the receiver structures required to implement them) for the problem of Mary signaling in AWGN. In the language of hypothesis testing, the observation here is the received signal y ( t ) modeled as follows: H i : y ( t ) = s i ( t ) + n ( t ) , i = 0 , 1 , ..., M 1 (7.32) where s i ( t ) is the transmitted signal corresponding to hypothesis H i , and n ( t ) is WGN with PSD 2 = N / 2. Before we can apply the framework of the previous section, however, we must figure out how to define conditional densities when the observation is a continuoustime signal. Here is how we do it: We first observe that, while the signals s i ( t ) live in an infinitedimensional, continuoustime space, if we are only interested in the M signals that could be transmitted under each of the M hypotheses, then we can limit attention to a finitedimensional subspace of dimension at most M . We call this the signal space. We can then express the signals as vectors corresponding to an expansion with respect to an orthonormal basis for the subspace. The projection of WGN onto the signal space gives us a noise vector whose components are i.i.d. Gaussian. Furthermore, we show that the component of the received signal orthogonal to the signal space is irrelevant: that is, we can throw it away without compromising performance. We can therefore restrict attention to projection of the received signal onto the signal space without loss of performance. This projection can be expressed as a finitedimensional vector which is modeled as a discrete time analogue of (7.32). We can now apply the hypothesis testing framework of Section 7.2 to infer the optimal (ML and MPE) decision rules. We then translate the optimal decision rules back to continuous time to infer the structure of the optimal receiver. 7.3.1 Representing signals as vectors Let us begin with an example illustrating how continuoustime signals can be represented as finitedimensional vectors by projecting onto the signal space. Example 7.3.1 (Signal space for twodimensional modulation): Consider a single complex valued symbol b = b c + jb s (assume that there is no intersymbol interference) sent using two dimensional passband linear modulation. The set of possible transmitted signals are given by s b c ,b s ( t ) = b c p ( t ) cos 2 f c t b s p ( t ) sin 2 f c t where ( b c , b s ) takes M possible values for an Mary constellation (e.g., M = 4 for QPSK, M = 16 for 16QAM), and where p ( t ) is a baseband pulse of bandwidth smaller than the carrier frequency f c . Setting c ( t ) = p ( t ) cos 2 f c t and s ( t ) = p ( t ) sin 2 f c t , we see that we can write the set of transmitted signals as a linear combination of these signals as follows:...
View
Full
Document
This note was uploaded on 12/28/2011 for the course ECE 146b taught by Professor Staff during the Fall '08 term at UCSB.
 Fall '08
 Staff

Click to edit the document details