Final answer:
The modulation property of a Fourier transform pair is proven using the Fourier Transform properties, expressing the cosine as a sum of complex exponentials, and acknowledging that multiplication in time corresponds to a shift in frequency domain.
Step-by-step explanation:
The question asks to prove the modulation property of a Fourier transform pair, stating that the product of a function g(t) and a cosine function representing a sinusoidal modulation results in a combination of the Fourier-transformed function G(f) shifted by ± the frequency of the cosine. The mathematical proof of the modulation property involves using the Fourier Transform properties of linearity and time-shifting.
Modulation Property Proof
To prove g(t)cos(2πf0t) = 1/2[G(f-f0)+G(f+f0)], we acknowledge that cos(2πf0t) can be expressed using Euler's formula as (ei2πf0t + e-i2πf0t)/2. Under Fourier transform, multiplication in the time domain corresponds to convolution in the frequency domain, which in this case simplifies to a shift due to the sinusoidal nature of the modulating function.
Thus, the transformation of the modulated function produces the original Fourier-transformed function G(f), shifted by the modulation frequency f0 in both positive and negative directions (due to the cosine being an even function), confirming the property to be proven.