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EE422-Spring 2002 Quiz 3 Dr. Dickerson 1. A binary symmetric channel has an error probability of PE. The probability of transmitting 1 is Q and the probability of transmitting 0 is 1 – Q. If the receiver detects the incoming digit as 0 what is the probability that the corresponding transmitted digit was a 1? This question is looking for the probability that a 1 was sent given that a 0 was detected. This is opposite of how we usually think of conditional probabilities: Pr(1 transmitted/0 detected). However, we can still apply Bayes rule (theorem) to solve for it: Prd etected transmitted 0 1 Prtransmitted 1 Prtrans det 1 0 Prdetected 0 PE Q Prtransmitted 1 Prd etected transmitted 0 1 Prtransmitted 0 Prd etected transmitted 0 0 PE Q QPE 1 Q 1 PE PE Q 1 PE Q 1 2 PE 2. Two random processes x(t) and y(t) are: x t A cos 0t and y t B cos n0t n where n is an integer not equal to 1 and A and B are constants. is a uniformly distributed RV in the range (0, 2). Show that the two processes are incoherent. Solution: Definition of Incoherent: Rxy 0 Starting from the definition of cross correlation: ______________________________ Rxy AB cos 0t cos n0 t n AB _____________________________ ____________________________ cos 0t n0 t n 1 cos n0 t 0t n 1 2 Calculating the statistical averages gives: _____________________________ 1 2 cos 0t n0 t n 1 cos 0t n0 t n 1 d 2 0 0 Similarly _____________________________ cos 0t n0 t n 1 0 and the cross correlation is Rxy 0 (incoherent) EE422-Spring 2002 Quiz 3 Dr. Dickerson 3. The input to a linear system is a deterministic sinusoid ( s t Asin 0t ) and a white Gaussian noise process with rms value N/2. a. Find the input signal to noise ratio b. Find the output signal to noise ratio of the filter with the following impulse response: h t e j0t e . Use the attached integral and Fourier transform tables to calculate the result as a t far as you are able. Do not be alarmed if the result is not in the table. Show all work. Solution: A2 a. Input power of a sinusoid is Ps (from 421 and previously this semester) 2 Noise power is: ____ N n Sn f df i 2 df 2 The input SNR is zero since any number over infinity is zero. b. The Output signal to noise ratio is calculated as in Lecture 14: Output is y(t)=so(t)+no(t) (superposition of linear systems) b. Output Signal Power: Calculate output signal using convolution so t si t h t A H f o sin 0t ph H f o 2 A so t H f0 2 2 2 From the Fourier Transform table given at the end of the quiz: a t 2a e a0 a 2 2 and the complex exponential in the front is a frequency shift, so 2a h t e j0 t e H 2 a t a 0 2 Substituting into the above equation gives: 2 A2 2 2 A2 s t 2 o 2 2 a a Output Noise Power Spectral Density 2 2a N Sn0 H Sn 2 a 2 0 2 2 Power of the random process is: 1 1 Pn Rn 0 Sno d 0 Sno d o o 2 EE422-Spring 2002 Quiz 3 Dr. Dickerson 2 A2 SNRout a2 2 N 2a 2 d 0 a 0 2 2

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posted: | 7/9/2011 |

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