Frequency Shifting property of the Fourier Transform


Frequency Shifting (or Amplitude Modulation)

This property states that if $ F(\omega)=\mathcal{F}[f(t)] $, then
$$\mathcal{F}\left[f(t) e^{j \omega_{0} t}\right]=F\left(\omega-\omega_{0}\right) \tag{1} $$
meaning, a frequency shift in the frequency domain adds a phase shift to the time function. By definition,
$$\begin{aligned}\mathcal{F}\left[f(t) e^{j \omega o t}\right] &=\int_{-\infty}^{\infty} f(t) e^{j \omega_{0} t} e^{-j \omega t} d t \\&=\int_{-\infty}^{\infty} f(t) e^{-j\left(\omega-\omega_{0}\right) t} d t=F\left(\omega-\omega_{0}\right)\end{aligned}$$
For example, $ \cos \omega_{0} t=\frac{1}{2}\left(e^{j \omega_{0} t}+e^{-j \omega_{0} t}\right) $. Using the property in Eq. (1),
$$\begin{aligned}\mathcal{F}\left[f(t) \cos \omega_{0} t\right] &=\frac{1}{2} \mathcal{F}\left[f(t) e^{j \omega_{0} t}\right]+\frac{1}{2} \mathcal{F}\left[f(t) e^{-j \omega_{0} t}\right] \\&=\frac{1}{2} F\left(\omega-\omega_{0}\right)+\frac{1}{2} F\left(\omega+\omega_{0}\right) \end{aligned}$$
This is an important result in modulation where frequency components of a signal are shifted.
Fig. 1: Amplitude spectra of: (a) signal f (t), (b) modulated signal $f (t) =\cos ωt$.
If, for example, the amplitude spectrum of $ f(t) $ is as shown in Fig. 1(a), then the amplitude spectrum of $ f(t) \cos \omega_{0} t $ will be as shown in Fig. 1(b).

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