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ch7_part3

# ch7_part3 - 7.3 Signal Space Concepts We have seen in the...

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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 M -ary 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 σ 2 = N 0 / 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 continuous-time signal. Here is how we do it: We first observe that, while the signals s i ( t ) live in an infinite-dimensional, continuous-time 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 finite-dimensional 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 finite-dimensional 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 continuous-time signals can be represented as finite-dimensional vectors by projecting onto the signal space. Example 7.3.1 (Signal space for two-dimensional 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 π f c t - b s p ( t ) sin 2 π f c t where ( b c , b s ) takes M possible values for an M -ary 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 φ c ( t ) = p ( t ) cos 2 π f c t and φ s ( t ) = - p ( t ) sin 2 π f c t , we see that we can write the set of transmitted signals as a linear combination of these signals as follows: s b c ,b s ( t ) = b c φ c ( t ) + b s φ s ( t ) so that the signal space has dimension at most 2. From Chapter 2, we know that φ c and φ s are orthogonal (I-Q orthogonality), and hence linearly independent. Thus, the signal space has

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ch7_part3 - 7.3 Signal Space Concepts We have seen in the...

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