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The corresponding minimum mean squared error (MMSE) is: MMSE = x 2 N0 x h 2 + N0 in .NET Print 39 barcode in .NET The corresponding minimum mean squared error (MMSE) is: MMSE = x 2 N0 x h 2 + N0

The corresponding minimum mean squared error (MMSE) is: MMSE = x 2 N0 x h 2 + N0 use .net framework code-39 generating toencode uss code 39 in .net Code 9/3 (A.85). 2 In the special case w hen x , this estimator yields the minimum mean squared error among all estimators, linear or non-linear.. A.4 Exercises Exercise A.1 Consider t he n-dimensional standard Gaussian random vector w 0 In and its squared magnitude w 2 . 1.

With n = 1, show that the density of w 2 is 1 a exp f1 a = 2 2 a a 0 (A.86). Appendix A Detection and estimation in additive Gaussian noise 2. For any n, show that the density of w relation: fn+2 a = (denoted by fn ) satisfies the recursive a f a n n (A.87). 3. Using the formulas f or the densities for n = 1 and 2 ((A.86) and (A.

9), respectively) and the recurisve relation in (A.87) determine the density of w 2 for n 3. Exercise A.

2 Let w t be white Gaussian noise with power spectral density N0 /2. Let s1 sM be a set of finite orthonormal waveforms (i.e.

, orthogonal and unit energy), and define zi = w t si t dt. Find the joint distribution of z. Hint: Recall the isotropic property of the normalized Gaussian random vector (see (A.

8)). Exercise A.3 Consider a complex random vector x.

1. Verify that the second-order statistics of x (i.e.

, the covariance matrix of the real representation x x t ) can be completely specified by the covariance and pseudo-covariance matrices of x, defined in (A.15) and (A.16) respectively.

2. In the case where x is circular symmetric, express the covariance matrix x x t in terms of the covariance matrix of the complex vector x only. Exercise A.

4 Consider a complex Gaussian random vector x. 1. Show that a necessary and sufficient condition for x to be circular symmetric is that the mean and the pseudo-covariance matrix J are zero.

2. Now suppose the relationship between the covariance matrix of x x t and the covariance matrix of x in part (2) of Exercise A.3 holds.

Can we conclude that x is circular symmetric Exercise A.5 Show that a circular symmetric complex Gaussian random variable must have i.i.

d. real and imaginary components. Exercise A.

6 Let x be an n-dimensional i.i.d.

complex Gaussian random vector, with the real and imaginary parts distributed as 0 Kx where Kx is a 2 2 covariance matrix. Suppose U is a unitary matrix (i.e.

, U U = I). Identify the conditions on Kx under which Ux has the same distribution as x. Exercise A.

7 Let z be an n-dimensional i.i.d.

complex Gaussian random vector, with the real and imaginary parts distributed as 0 Kx where Kx is a 2 2 covariance matrix. We wish to detect a scalar x, equally likely to be 1 from y = hx + z (A.88).

where x and z are indep barcode 3 of 9 for .NET endent and h is a fixed vector in n . Identify the conditions on Kx under which the scalar h y is a sufficient statistic to detect x from y.

Exercise A.8 Consider estimating the real zero-mean scalar x from: y = hx + w where w 0 N0 /2I is uncorrelated with x and h is a fixed vector in. (A.89) ..

A.4 Exercises 1. Consider the scaled linear estimate ct y (with the normalization c = 1): x = act y = act h x + act z Show that the constant a that minimizes the mean square error ( equal to x2 c t h 2 ct h 2 + N0 /2 (A.90) x x .

) is (A.91). 2. Calculate the minima visual .net Code 3 of 9 l mean square error (denoted by MMSE) of the linear estimate in (A.

90) (by using the value of a in (A.91). Show that x2 = 1 + SNR = 1 + MMSE x2 c t h 2 N0 /2 (A.

92). For every fixed linear estimator c, this shows the relationship between the corresponding SNR and MMSE (of an appropriately scaled estimate). In particular, this relation holds when we optimize over all c leading to the best linear estimator..

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