homework5f_solution

homework5f_solution - ECE 5325/6325 Fall 2009: Homework...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

Ask a homework question - tutors are online