\u03c9 4 \u03c0 2 \u03c0 2 \u03c0 4 \u03c0 X f \u03c9 1 140 The signal x t is used as the input to a

Ω 4 π 2 π 2 π 4 π x f ω 1 140 the signal x t is

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ω - 4 π - 2 π 0 2 π 4 π X f ( ω ) 1 140
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The signal x ( t ) is used as the input to a continuous-time LTI system having the frequency response H f ( ω ) shown. ω - 4 π - 2 π 0 2 π 4 π H f ( ω ) 1 1 Accurately sketch the spectrum Y f ( ω ) of the output signal. 3.1.16 Two continuous-time LTI system are used in cascade. Their impulse responses are h 1 ( t ) = sinc(3 t ) h 2 ( t ) = sinc(5 t ) . - h 1 ( t ) - h 2 ( t ) - Find the impulse response and sketch the frequency response of the total system. 3.1.17 Consider a continuous-time LTI system with the impulse response h ( t ) = 3 sinc(3 t ) . (a) Accurately sketch the frequency response H f ( ω ). (b) Find the output signal y ( t ) produced by the input signal x ( t ) = 2 + 5 cos( πt ) + 7 cos(4 πt ) . (c) Consider a second continuous-time LTI system with impulse response g ( t ) = h ( t - 2) where h ( t ) is as above. For this second system, find the output signal y ( t ) produced by the input signal x ( t ) = 2 + 5 cos( πt ) + 7 cos(4 πt ) . 3.1.18 The signal x ( t ) is given the product of two sine functions, x ( t ) = sin( π t ) · sin(2 π t ) . Find the Fourier transform X f ( ω ). 3.1.19 A continuous-time signal x ( t ) has the spectrum X f ( ω ), ω - 4 π - 2 π 0 2 π 4 π X f ( ω ) 1 (a) The signal g ( t ) is defined as g ( t ) = x ( t ) cos(4 πt ) . Accurately sketch the Fourier transform of g ( t ). (b) The signal f ( t ) is defined as f ( t ) = x ( t ) cos( πt ) . Accurately sketch the Fourier transform of f ( t ). 141
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3.1.20 The signal x ( t ): -6 -4 -2 0 2 4 6 0 0.5 1 1.5 2 t x(t) has the Fourier transform X f ( ω ): -3π -2π 0 π 0 1 2 3 4 5 ω X f ( ω ) Accurately sketch the signal g ( t ) that has the spectrum G f ( ω ): -3π -2π 0 π 0 1 2 3 4 5 ω G f ( ω ) Note that the spectrum G f ( ω ) is a sum of left- and right-shifted copies of X f ( ω ). Specifically, G f ( ω ) = X f ( ω - 2 π ) + X f ( ω + 2 π ) . In your sketch of the signal g ( t ) indicate its zero-crossings. Show and explain your work. 3.1.21 The left-hand column below shows four continuous-time signals. The Fourier transform of each signal appears in the right-hand column in mixed-up order. Match the signal to its Fourier transform. Signal Fourier transform 1 2 3 4 142
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-5 0 5 0 0.5 1 1.5 t x(t) #1 -5 0 5 -1 0 1 2 ω / π X f ( ω ) C -5 0 5 0 0.5 1 1.5 t x(t) #2 -5 0 5 0 2 4 ω / π X f ( ω ) D -5 0 5 -0.5 0 0.5 1 1.5 t x(t) #3 -5 0 5 0 0.5 1 1.5 ω / π X f ( ω ) B -5 0 5 -1 0 1 2 t x(t) #4 -5 0 5 0 0.5 1 1.5 ω / π X f ( ω ) A 143
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3.1.22 The ideal continuous-time high-pass filter has the frequency response H f ( ω ) = 0 , | ω | ≤ ω c 1 , | ω | > ω c . Find the impulse response h ( t ). 3.1.23 The ideal continuous-time band-stop filter has the frequency response H f ( ω ) = 1 | ω | ≤ ω 1 0 ω 1 < | ω | < ω 2 1 | ω | ≥ ω 2 . (a) Sketch H f ( ω ). (b) What is the impulse response h ( t ) of the ideal band-stop filter? (c) Describe how to implement the ideal band-stop filter using only lowpass and highpass filters. 3.1.24 The ideal continuous-time band-pass filter has the frequency response H f ( ω ) = 0 | ω | ≤ ω 1 1 ω 1 < | ω | < ω 2 0 | ω | ≥ ω 2 .
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