Exercises on Systems and Density 2.6 2002/01/11 2004/01/07 09:58:10.516 US/Central Behnaam Aazhang aaz@ece.rice.edu Dinesh Rajan dinesh@ece.rice.edu Mohammad Jaber Borran mohammad@ece.rice.edu Behnaam Aazhang aaz@ece.rice.edu Shawn Stewart mrshawn@alumni.rice.edu Roy Ha rha@rice.edu homework Elec 430 Exercises Consider the following system

Assume that N τ is a white Gaussian process with zero mean and spectral height N 0 2 . If b is "0" then X τ A p T τ and if b is "1" then X τ A p T τ where p T τ 1 0 τ T 0 . Suppose b 1 b 0 12 . Find the probability density function Z T when bit "0" is transmitted and also when bit "1" is transmitted. Refer to these two densities as f Z T H 0 z and f Z T H 1 z , where H 0 denotes the hypothesis that bit "0" is transmitted and H 1 denotes the hypothesis that bit "1" is transmitted. Consider the ratio of the above two densities; i.e., Λ z f Z T H 0 z f Z T H 1 z and its natural log Λ z . A reasonable scheme to decide which bit was actually transmitted is to compare Λ z to a fixed threshold γ. ( Λ z is often referred to as the likelihood function and Λ z as the log likelihood function). Given threshold γ is used to decide b ^ 0 when Λ z γ then find b ^ b (note that we will say b ^ 1 when Λ z γ ). Find a γ that minimizes b ^ b . Proakis and Salehi, problems 7.7, 7.17, and 7.19 Proakis and Salehi, problem 7.20, 7.28, and 7.23