homework5f_solution

homework5f_solution - ECE 5325/6325 Fall 2009 Homework 5(f...

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Unformatted text preview: ECE 5325/6325 Fall 2009: Homework 5(f) Solution From the Lecture 11 notes: Frequency Shift Keying : Assume Ts >> 1/fc , and show that these two are orthogonal. φ1 (t) = φ2 (t) = cos (2πfc t), 0 ≤ t ≤ Ts 0, o.w. cos 2π fc + 0, 1 Ts t , 0 ≤ t ≤ Ts o.w. Solution: The integral of the product of the two must be zero. Checking, and using the identity for the product of two cosines, Ts cos (2πfc t) cos 2π fc + 0 1 t dt Ts = = 1 2 Ts Ts cos (2πt/Ts ) dt + 0 Ts 0 0 cos (4πfc t + 2πfc t/Ts ) dt Ts 0 1 Ts sin (2πt/Ts ) 2 2π 1 sin (2π (2fc + 1/Ts )t) + 2π (2fc + 1/Ts ) The second term has a 2π(2fc1 /Ts ) constant out front. Because fc is very high (we’re given +1 fc >> 1/Ts ), this term will be very very low. The sine term is limited to between -1 and +1 so it will not cause the second term to be large. So we will approximate this second term as zero. ≈ 1 Ts [sin(2π ) − sin(0)] = 0 2 2π Thus the two frequency waveforms are orthogonal. ...
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