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1. The generalized Marcum Q-function may be defined as 2 OM dr where is the modified Bessel function of the first kind of order M - -1. (a) Find OM (a,0) and OM (0,b) (b) Let X = (X1,X2 XN ) be a sequence of i.i.d. observations, each with a Rice N distribution with specular component a. If = , then it can be shown j=] that the pdf of Y is given by 20 where =N². Derive the Neyman-Pearsor test for the hypothesis testing problem H0 : B =0 versus H1 : B>0 (c) Derive the expressions for Pp = p(y>21B>0) and P1 = p(Y>21B=0) = 3. Let X1, X, X, n be a sequence of i.i.d. observations, each with the exponential pdf Consider the hypothesis testing problem Hb : 0 = o versus H0 : 0=0, = The following test statistic (called Fisher's Detector) is used for the test where pi = P(X,>x) is the significance level of the observation X, (a) Find the pdf of T11 FD (X) when Ho is true. (b) Find the pdf of TFD (X) when H1 is true. 4. Let X = (X1,X2 x,) be a sequence of independent and identically distributed observations, each with pdf f F(x) and corresponding cdf F(x). Let the linear detector be defined as Ty LD (x)=x, Find the asymptotic relative efficiency (ARE) of the following detectors relative to the linear detector (LD). (a) The sign detector (SD): Tso(x)=[X(x) i=] where u(.) is the unit step function. (b) The Fisher detector (SD): where pi = 1 - F(x,) is the significance level of X, i.e., pi =Pr(X,> xi). 5. Consider the generalized Gaussian pdf defined as C exp (a) Show that if a is fixed, f (x) converges as c 00 to the uniform pdf on the interval [-a.a]. . (b) Define the sign correlator (SC) detector as (X) Find the limit to which the ARE of the sign correlator relative to the linear detector converges as c 00, (c) Verify your result in part (b) by direct calculation of the ARE for the uniform distribution.

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