tailieunhanh - Proakis J. (2002) Communication Systems Engineering - Solutions Manual (299s) Episode 12

Tham khảo tài liệu 'proakis j. (2002) communication systems engineering - solutions manual (299s) episode 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | b It is apparent from S f that the terms Sk f 2 decay proportionally as f__fk 2. also note that S fm2_ TEb sin nTb f - fk 2 sin nTb f fk 2 Sk f 2 I n f - fk Ml n f fk because the product sin nTb f - fk X sin nTb f fk _ 0 n f - fk n f fk due to the relation that the carrier frequency fc T. Problem 1 The autocorrelation function of the information symbols an is Ra k E ana n k 4 X an 2ỗ k ỗ k Thus the power spectral density of v t is Sv f T Sa f G f 2 T G f 2 where G f 5 g t . If g t An R we obtain G f 2 A2T2sinc2 fT and therefore Sv f A2Tsinc2 fT In the next figure we plot SV f for T A 1. 2 If g t A sin n n T then G f A 24S f - 4 - 1 S f 1 1 Tsinc fE Ĩ -2J 4 2j 4 - Af ỗ f - 1 - ỗ f 1 sinc fT e-j 2nf T 2 2 4 4 ATe-jn f-1 T 11 sinc f - 1 T - sinc f - 1 T e-j cy J 1 VVv A Thus A2T2 r 1 1 G f 2 4- sinc2 f 1 T sinc2 f - 1 T -2sinc f 4 t sinc f - 4 t cos L 218 and the power spectral of the transmitted signal is Sv f 4 AT rsinc2 f 4 T sinc2 f - 1 T -2sinc f 4 t sinc f - 4 t cos 2- In the next figure we plot SV f for two special values of the time interval T. The amplitude of the signal A was set to 1 for both cases. 3 The first spectral null of the power spectrum density in part 1 is at position Wnull T The 3-dB bandwidth is specified by solving the equation SV W3dB 2 SV 0 Since sinc2 0 1 we obtain W3dBT 2 sm nW3dBT nW3dBT Solving the latter equation numerically we find that 3dB nT T To find the first spectral null and the 3-dB bandwidth for the signal with power spectral density in part 2 we assume that T 1. In this case A2 r 1 1 1 Sv f sinc2 f 1 sinc2 f - 1 1 1 4 4 and as it is observed there is no value of f that makes Sv f equal to zero. Thus Wnull c. To find the 3-dB bandwidth note that . A2 1 A2 Sv 0 2sinc 4 Solving numerically the equation 1 A2 Sv Wxffi 1 I I 62 12 we find that W3dB . As it is observed the 3-dB bandwidth is more robust as a measure for the bandwidth of the signal. 219 Problem The transition probability matrix P is .

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