ASN4 -Fourier Transform property

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Find the Fourier transform of cos(2πf0t)g(t)


[cos(2πf0t)g(t)]

Applying the forward Fourier transform

=cos(2πf0t)g(t)ej2πftdt

Applying Euler's cosine identity

=[12ej2πf0t+12ej2πf0t]g(t)ej2πftdt

=12ej2πf0tej2πftdt+12ej2πf0tg(t)ej2πftdt

=12ej2π(ff0)tg(t)dt+12ej2π(f+f0)tg(t)dt

Note the Inverse Fourier Transform expressions in the above equation. With substitution the result is

[cos(2πf0t)g(t)]=12G(ff0)+12[G(f+f0)