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Question

# A continuous time signal x(t) is defined as x(t)=2+cos(50πt) is sampled with sampling interval Ts=0.025sec and passed through an ideal low pass filter whose frequency response is as shown in figure below The spectrum of output will be

A
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B
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C
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D
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Solution

## The correct option is D Given, x(t)=2+cos(50πt) Frequency of signal ωsig=50π Ts = 0.025 sec ∴ sampling frequency ωs = 2πTs = 80πrad/sec Then, X(jω)=4πδ(ω)+π[δ(ω+50π)+δ(ω−50π)] Let the sampled signal be represented as Xs(jω) , where Xs(jω) is given as Xs(jω)=1Ts∞∑n=−∞X(j(ω−nωs)) Xs(jω)=40∞∑n=−∞[4πδ(ω−80π)]+πδ(ω−50π−80πn)−πδ(ω+50π−80πn) Now, the sampled input Xs(jω) is passed through a low passed filter having cut-off frequency at ω=40π Therefore the output Y(jω) will contain only the components which are less than ω=40π Now by putting Ts=0.025, we will get,

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