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hw6 - R 12 τ = G 1 w G 2 w 6 We know that e-| t | FT ↔ 2...

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EE210 Signal Systems (March 18, 2009) Home Work Set 6 1. (a) Find the Fourier Transform(FT) of y ( t ) = (1 + μx ( t )) cos ( ω c t ) This signal is the AM (Amplitude modulated) signal and μ is referred to as the modulation index. (b) Find a scheme (with mathematical justification) for recovering x ( t ) from y ( t ). 2. Show that e - πt 2 FT e - w 2 4 π . 3. Show that FT[ x 1 ( t ) x 2 ( t )] = 1 2 ( X 1 ( ω ) X 2 ( ω )) 4. Let x(t) ←→ X(w). Moreover, let x ( t ) be real for all t . Then show that | X ( w ) | = | X ( - w ) | and X ( w ) = - X ( - w ) i.e., Magnitude spectrum is an even function and the phase spectrum is an odd function. 5. Let R 12 ( τ ) = -∞ g 1 ( t ) g * 2 ( t - τ ) . Show that (a) R 12 ( τ ) = R 12 ( - τ ) (b) FT[
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Unformatted text preview: R 12 ( τ )] = G 1 ( w ) G * 2 ( w ) 6. We know that e-| t | FT ↔ 2 1 + ω 2 Now use an appropriate property of FT to find the FT of te-| t | . 7. Let y ( t ) = ( x ( t ) cos 2 ( t )) ⊗ sin ( t ) πt Assume that X( ω ) = 0 for | ω |≥ 1. Show that ∃ a linear time invariant system h ( t ), such that y ( t ) = x ( t ) ⊗ h ( t ). Find h ( t ). (Here ⊗ stands for convolution operator). 8. Let x(t) FT ↔ X( ω ). Show that Z t-∞ x ( τ ) dτ FT ↔ X ( ω ) jω + X (0) δ ( ω ) ....
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