Quiz1_solution - Solutions to EE3008 Quiz 1 Problems...

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Solutions to EE3008 Quiz 1 Problems Problem 1: (a) H ( f ) = 1. (b) Given the Fourier transform pair s ( t ) = ( A, | t | ≤ τ/ 2 0 , otherwise S ( f ) = sinc( ) and using the duality property, we have s ( t ) = sinc( ) S ( f ) = ( A, | f | ≤ τ/ 2 0 , otherwise. Then, letting τ = 2 and A = 0 . 5, we obtain s ( t ) = sinc(2 t ) S ( f ) = ( 0 . 5 , | f | ≤ 1 0 , otherwise. (c) As H ( f ) = 1, we have Y ( f ) = S ( f ) and hence y ( t ) = sinc(2 t ). Using the Parseval’s theorem, the signal energy of y ( t ) is obtained as E y = Z -∞ | S ( f ) | 2 df = Z 1 - 1 0 . 25 df = 0 . 5 . Since E y < , y ( t ) is an energy-type signal. 1
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Problem 2: (a) (i) We observe from Figure 2 that the bandwidth of the modulating signal s ( t ) is B s = 10 Hz. The bandwidth of the modulated signal s AM - DSB - SC ( t ) is B m = 2 B s . Therefore, B m = 20 Hz. (ii) No. Since the carrier frequency is 200 Hz, the required frequency range of the channel is [190 Hz, 210 Hz]. (b) Recall in Problem 1(b) that s ( t ) = sinc( ) S ( f ) = ( A, | f | ≤ τ/ 2 0 , otherwise. Then, letting τ = 100 and A = 1, we obtain s ( t ) = 100sinc(100 t ) S ( f ) = ( 1 , | f | ≤ 50 0 , otherwise.
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