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Using the convolution property, find the spectrum for

$$w(t)= \sin(2\pi f_1 t) \cos(2\pi f_2 t).$$

I'm confused on how to solve this question. Can you give me any aproach?

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2 Answers

The low-tech approach is to use the identity $\sin x\cos y =\frac12 (\sin(x+y)-\sin(x-y))$: $$w(t)=\frac12 \sin(2\pi(f_1+f_2)t)- \frac12\sin(2\pi(f_1-f_2)t)$$ The spectrum consists of just two frequencies, $f_1\pm f_2$.

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The Fourier Transform of the product of two signals (functions) is the convolution of their spectra, i.e. the convolution of their individual Fourier Transforms. The spectra of $sin(2\pi f_1t), cos(2 \pi f_2t)$ are well known (dirac functions at appropriate locations and with appropriate coefficients) and convolving them is also easy, using the property of the delta function $\delta(t-t_0) * f(t)=f(t_0)$ where $*$ denotes convolution.

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