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Question 6.9: CTFT of a modulated sinusoid Find the CTFT of x(t) = 24cos(1......

CTFT of a modulated sinusoid

Find the CTFT of x(t) = 24cos(100πt)sin(10,000πt).

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This is the product of two functions. Therefore, using the multiplication–convolution duality property, the CTFT will be the convolution of their individual CTFTs. Using

cos(2πf_0t)\overset{\mathcal{F}}{\longleftrightarrow }(1/2)[δ( f − f_0 ) + δ( f + f_0 )]

and

sin(2πf_0t)\xleftrightarrow{\mathcal{F}}(j/2)[δ( f + f_0 ) – δ( f – f_0 )]

we get

24cos(100πt)\overset{\mathcal{F}}{\longleftrightarrow }12[δ( f − 50 ) + δ( f + 50 )]

and

sin(10,000πt)\overset{\mathcal{F}}{\longleftrightarrow }(j/2)[δ( f + 5000 ) – δ( f – 5000 )]

Then the overall CTFT is

24cos(100πt)sin(10,000πt)\overset{\mathcal{F}}{\longleftrightarrow }12[δ( f − 50 ) + δ( f + 50 )]∗( j/2)[δ[ f + 5000] − δ( f − 5000)]

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