A Detection and estimation in additive Gaussian noise in .NET Draw 39 barcode in .NET A Detection and estimation in additive Gaussian noise

Appendix A Detection and estimation in additive Gaussian noise use visual studio .net 39 barcode encoder todeploy barcode 3/9 with .net OneCode The signal is i Code 39 for .NET n the direction v = uA uB / uA uB (A.47).

Projection of t he received vector y onto v provides a (complex) scalar sufficient statistic: y = v y 1 u + uB 2 A = x uA uB + w (A.48). where w 0 N0 Code 39 Extended for .NET . Note that since x is real ( 1/2), we can further extract a sufficient statistic by looking only at the real component of y: y = x uA uB + where w (A.

49). w N 0 N0 /2 . The error probability is exactly as in (A.44): Q uA uB 2 N0 /2 (A.

50). Note that altho Code 3 of 9 for .NET ugh uA and uB are complex vectors, the transmit vectors x u A uB + 1 u + uB 2 A x = 1 (A.51).

lie in a subspa ce of one real dimension and hence we can extract a real sufficient statistic. If there are more than two possible transmit vectors and they are of the form hxi , where xi is complex valued, h y is still a sufficient statistic but h y is sufficient only if x is real (for example, when we are transmitting a PAM constellation). The main results of our discussion are summarized below.

. Summary A.2 Vector detection in complex Gaussian noise Binary signals The transmit vector u is either uA or uB and we wish to detect u from received vector y = u+w (A.52). where w 0 N0 .net vs 2010 3 of 9 I . The ML detector picks the transmit vector closest to y and the error probability is Q u A uB 2 N0 /2 (A.

53). A.3 Estimation in Gaussian noise Collinear signa .net vs 2010 Code 39 ls The transmit symbol x is equally likely to take one of a finite set of values in (the constellation points) and the received vector is y = hx + w where h is a fixed vector. Projecting y onto the unit vector v = h/ h yields a scalar sufficient statistic: v y = h x + w Here w 0 N0 .

(A.55) (A.54).

If further the constellation is real-valued, then v y = h x + is sufficient. Here w 0 N0 /2 . w (A.

56). With antipodal Code 3 of 9 for .NET signalling, x = a, the ML error probability is simply Q a h N0 /2 (A.57).

Via a translati on, the binary signal detection problem in the first part of the summary can be reduced to this antipodal signalling scenario.. A.3 Estimation in Gaussian noise A.3.1 Scalar estimation Consider a zero Code39 for .NET -mean real signal x embedded in independent additive real Gaussian noise (w 0 N0 /2 ): y = x+w (A.58).

Suppose we wish to come up with an estimate x of x and we use the mean squared error (MSE) to evaluate the performance: MSE = x x . (A.59). Appendix A Detection and estimation in additive Gaussian noise where the avera bar code 39 for .NET ging is over the randomness of both the signal x and the noise w. This problem is quite different from the detection problem studied in Section A.

2. The estimate that yields the smallest mean squared error is the classical conditional mean: x= xy (A.60).

which has the i mportant orthogonality property: the error is independent of the observation. In particular, this implies that x x y = 0 (A.61).

The orthogonali barcode 39 for .NET ty principle is a classical result and all standard textbooks dealing with probability theory and random variables treat this material. In general, the conditional mean x y is some complicated non-linear function of y.

To simplify the analysis, one studies the restricted class of linear estimates that minimize the MSE. This restriction is without loss of generality in the important case when x is a Gaussian random variable because, in this case, the conditional mean operator is actually linear. Since x is zero mean, linear estimates are of the form x = cy for some real number c.

What is the best coefficient c This can be derived directly or via using the orthogonality principle (cf. (A.61)): c= x2 x2 + N0 /2 (A.

62). Intuitively, we are weighting the received signal y by the transmitted signal energy as a fraction of the received signal energy. The corresponding minimum mean squared error (MMSE) is MMSE = x2 N0 /2 x2 + N0 /2 (A.63).

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