{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

assignment2_w11_soln

# assignment2_w11_soln - 1(f 0.25 X 1(f 2 f c X 1(f 2 f c...

This preview shows pages 1–5. Sign up to view the full content.

SYSC 3501 W11 Assignment 2 Solution Set

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 π 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 π 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 ) ]
= 0.5 X 1 (f ) + 0.25 [

This preview has intentionally blurred sections. Sign up to view the full version.

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

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 π 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...
View Full Document

{[ snackBarMessage ]}