assignment2_w11_soln

assignment2_w11_soln - 1 (f) + 0.25 [ X 1 (f- 2 f c ) + X 1...

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SYSC 3501 W11 Assignment 2 Solution Set
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Problem 2 [10 Marks] Consider the system shown below, where the input signals, x 1 ( t ) and x 2 ( t ), both have a bandwidth of W Hz. If the lowpass filter also has a bandwidth of W Hz, and f c >>W , express z ( t ) in terms of x 1 ( t ) and x 2 ( t ) in the simplest form possible. [Hint: Solving in the frequency domain using Fourier transforms may be simpler] Modulation of x 1 ( t ): Let y 1 ( t ) = x 1 ( t ) cos(2 π f c t ), Æ Y 1 (f) = 0.5 X 1 (f- f c ) + 0.5 X 1 (f + f c ) Modulation of x 2 ( t ): Let y 2 ( t ) = x 2 ( t ) sin(2 f c t ), Æ Y 2 (f) = 0.5 j X 2 (f- f c ) - 0.5 j X 2 (f + f c ) Multiplexing or adding: Let y 3 ( t ) = y 1 ( t )+ y 2 ( t ) Æ Y 3 (f) = Y 1 (f) + Y 2 (f) Demodulating y 3 ( t ): Let y 4 ( t ) = y 3 ( t ) cos(2 f c t ), Æ Y 4 (f) = 0.5 Y 3 (f- f c ) + 0.5 Y 3 (f + f c ) Y 4 (f) = 0.5[Y 1 (f- f c ) + Y 2 (f- f c ) + Y 1 (f + f c ) + Y 2 (f + f c )] = 0.5[0.5 X 1 (f- 2 f c ) + 0.5 X 1 (f) + 0.5 j X 2 (f- 2 f c ) - 0.5 j X 2 (f) + 0.5 X 1 (f) + 0.5 X 1 (f + 2 f c ) +0.5 j X 2 (f) - 0.5 j X 2 (f + 2 f c )] = 0.5 [0.5 X 1 (f- 2 f c ) + X 1 (f) + 0.5 j X 2 (f- 2 f c ) + 0.5 X 1 (f + 2 f c ) + - 0.5 j X 2 (f + 2 f c ) ]
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= 0.5 X
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Unformatted text preview: 1 (f) + 0.25 [ X 1 (f- 2 f c ) + X 1 (f + 2 f c )] + 0.25 j [X 2 (f- 2 f c ) + X 2 (f + 2 f c ) ]] Applying the LPF to y 4 ( t ): Z(f) = 0.5 X 1 (f) all the higher frequency components will be cut off z(t) = 0.5 x 1 (t) Problem 3 [10 Marks] The signal, x ( t ) = 1 + cos(2 t ) + 2sin( 4 t ) Volt, is passed through an LTI filter, h ( t ) , whose transfer function is shown in the figure below. The output signal of the filter is y ( t ) . a) Find the Fourier transform of x ( t ) and sketch its magnitude spectrum, | X ( f ) |. The magnitude spectrum is sketched by plotting weights of the delta functions. b) What is the fundamental frequency f 0 of x ( t ) ? c) Find the input PSD, P x ( f ) and sketch it. d) Find the output PSD, P y ( f ) and sketch it. P y (f) = |H(f)| 2 P x (f) 1 1 16...
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assignment2_w11_soln - 1 (f) + 0.25 [ X 1 (f- 2 f c ) + X 1...

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