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hw7_soln - ECE 562 Fall 2011 December 6 2011 HOMEWORK...

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ECE 562 Fall 2011 December 6, 2011 HOMEWORK ASSIGNMENT 7 SOLUTIONS 1. True or False. Determine if the following statements are True or False. You need to provide a brief justification for your answer to get credit. (a) If the mobile speed is 72 km/hr, the carrier frequency is 900 MHz, and symbol rate for communication is 10 symbols a second, then the mobile experiences slow fading. False: f max = 60 Hz, and T c = 0 . 1 /f max 0 . 1 = T s . (b) In wireless communication, if the speed of the mobile is doubled the delay spread is also doubled. False: The mobile speed has little effect on the delay spread. It is the Doppler speed that gets doubled. (c) A wireless channel with τ ds = 10 - 4 seconds and f max = 100 Hz operating with a bandwidth of 1 . 25 MHz is a flat fading channel. False: W τ ds (d) For a fixed γ b , P b for binary DPSK signaling over a slow flat fading channel is always larger than P b without fading, irrespective of the distribution of the fading. True: By Jensen’s inequality since exp is convex. 2. Useful Result. Show that 0 Q ( x ) e - x/γ γ dx = 1 2 1 - γ 2 + γ . Hint: Start with: x =0 Q ( x ) e - x/γ γ dx = x =0 t = x e - t 2 / 2 2 π e - x/γ γ dt dx = t =0 t 2 x =0 e - t 2 / 2 2 π e - x/γ γ dx dt and use the fact that the Gaussian pdf integrates to 1 to conclude the result. Ans: Starting with the hint, x =0 Q ( x ) e - x/γ γ dx = 0 1 2 π e - 1 2 t 2 - 1 2 π e - 1 2 ( γ +2 γ ) t 2 dt = 1 2 - 0 1 2 π e - 1 2 ( γ +2 γ ) t 2 dt = 1 2 - γ 2 + γ 0 1 2 π γ 2+ γ e - 1 2 ( γ +2 γ ) t 2 dt = 1 2 1 - γ 2 + γ c V.V. Veeravalli, 2011 1
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3. For MPSK signaling,
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