ps12-Prob1-union

ps12-Prob1-union - 6.450 Principles of Digital...

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Unformatted text preview: 6.450 Principles of Digital Communication Wednesday, December 4, 2000 MIT, Fall 2001 Handout #35 Due: Not to be passed in Problem Set 12 Problem 12.1 (Orthogonal signal sets, continued) Consider the set of m orthogonal signals from problem set 10.5. Each signal has energy E A and the WGN has spectral density N / 2. View the signals in a coordinate system where each vector is collinear with one coordinate. (a) Use the union bound to show that Pr { e } is bounded by Pr { e } ≤ ( m- 1) Q ( p E A /N ) . (b) Let m → ∞ with E b = E A / log m held constant. Using the upper bound for Q ( x ) in Problem 10.4 part (d), show that Pr( e ) < 1 2 exp- b E b 2 N- ln2 Show that this approaches 0 exponentially in b for fixed E b /N > 2ln2. (c) Compare this result with that in the last section of lecture 19. Draw a sketch of the bound on- ln( P e ) /b as given there and as given above as a function of E b /N . This problem shows that the above union bound is good for small E b /N but not for large E b /N . See the end of lecture 19 for more discussion of this point. Problem 12.2 : More on the union bound: Let E 1 ,E 2 ,...,E k be independent events each with probability p . (a) Show that Pr( ∪ k j =1 E j ) = 1- (1- p ) k . (b) Show that pk- ( pk ) 2 / 2 ≤ Pr( ∪ k j =1 E j ) ≤ pk . Problem 12.3 : Consider a wireless channel with two paths, both of equal strength, operating at a carrier frequency f c . Assume that the baseband equivalent system function is given by ˆ g ( f,t ) = 1 + e iφ exp[- 2 πi ( f + f c ) τ 2 ( t )] (1) (a) Assume that the length of path 1 is a fixed value r and the length of path 2 is r + Δ r + vt . Show (using (1)) that ˆ g ( f,t ) ≈ 1 + e iψ exp- 2 πi f Δ r c + f c vt c (2) Explain what the parameter...
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ps12-Prob1-union - 6.450 Principles of Digital...

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