Frequency translation by multiplication: If v(t) = m(t) cos(2π f_c t) and the Fourier transform of m(t) is M(f), what is the Fourier transform of v(t)?

Introduction / Context:This question checks your understanding of frequency translation (modulation) in signals and systems. Multiplying a baseband signal m(t) by a cosine of frequency f_c produces a double sideband spectrum centered at ±f_c. Recognizing how the Fourier transform behaves under multiplication by a sinusoid is fundamental in analog modulation and spectral analysis.

Given Data / Assumptions:

• v(t) = m(t) cos(2π f_c t).
• Fourier transform pair: m(t) ⇄ M(f).
• Real-valued signals and standard engineering Fourier transform convention are assumed.

Concept / Approach:

Use the modulation property of the Fourier transform and the identity cos(2π f_c t) = (1/2)(e^{j 2π f_c t} + e^{-j 2π f_c t}). Multiplication in time corresponds to convolution in frequency; however, when multiplying by exponentials, the effect is simply to shift the spectrum.

Step-by-Step Solution:

Verification / Alternative check:

Check symmetry: if M(f) is bandlimited around DC, the result produces two mirrored lobes around ±f_c. This is exactly the spectrum of double-sideband suppressed-carrier modulation, confirming the formula.

Why Other Options Are Wrong:

• Only 0.5 M(f + f_c) or only 0.5 M(f - f_c): misses one of the sidebands.
• Duplicated shift term 0.5 M(f - f_c) + 0.5 M(f - f_c): ignores the +f_c sideband.
• 'Shifted by 2 f_c only': incorrect shift amount and ignores the two-sided nature.

Common Pitfalls:

• Confusing multiplication by cos with a single frequency shift; cos always creates two symmetric shifts.
• Forgetting the factor 0.5 from the exponential identity.

Final Answer:

0.5 M(f + f_c) + 0.5 M(f - f_c)