# problems5 - pare it with the signal constellation of BPAM...

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Digital Modulation 1 - Problem Set 5 Problem 1 (BFSK) The BFSK transmitter maps the information bits into harmonic waveforms with dif- ferent frequencies: 0 m→ s 1 ( t ) = 2 P cos(2 πf 1 t + φ 1 ) , 1 m→ s 2 ( t ) = 2 P cos(2 πf 2 t + φ 2 ) , where 0 t < T . This modulation scheme is called binary frequency-shift keying (BFSK). We assume that the receiver knows φ 1 and φ 2 and thus can use this knowl- edge for the demodulation. This kind of demodulation for BFSK is called coherent demodulation. 1. Plot the waveforms s i ( t ), i = 1 , 2 (You may use Matlab). 2. Show that the minimum frequency separation | f 2 f 1 | such that these functions are orthogonal equals 1 2 T for the case where φ 1 = φ 2 , and 1 T for the case where φ 1 n = φ 2 . Assume that f 1 + f 2 = r T with r 1. (The calculations for this problem may take some time.) 3. Consider two waveforms with su±cient frequency separation and a receiver with ²lters matched to these waveforms. Plot the signals at the output of the matched ²lters. (You might also use Matlab). 4. Plot the signal constellation of BFSK with the above frequency separation. Com-

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Unformatted text preview: pare it with the signal constellation of BPAM and BPPM. 5. What can you infer from the above comparison about the BEP of BFSK (with coherent demodulation), BPAM, and BPPM? BFl, TPe. Digital Modulation 1 NavCom Digital Modulation 1 - Problem Set 5 Problem 2 (Transformation of a WGN vector) Let W = ( W 1 , W 2 ) denote a 2-dimensional white Gaussian noise vector (WGNV) with component variance σ 2 , i.e. W 1 ∼ N (0 , σ 2 ) and W 2 ∼ N (0 , σ 2 ). The components ˜ W 1 and ˜ W 2 of the WGNV with respect to a rotated system of coordinates (see the Fgure below) can be obtained from W 1 and W 2 by the linear transformation ˜ W 1 = W 1 cos α + W 2 sin α, ˜ W 2 = − W 1 sin α + W 2 cos α. Show that ˜ W = ( ˜ W 1 , ˜ W 2 ) is also a WGNV with component variance σ 2 . α ˜ ξ 2 ξ 2 ˜ W 1 α W ˜ W 2 ˜ ξ 1 W 2 ξ 1 W 1 B±l, TPe. Digital Modulation 1 NavCom...
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problems5 - pare it with the signal constellation of BPAM...

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