Shift Theorem

The shift theorem states that shifting a function in one domain multiplies its Fourier transform by a complex exponential.

Statement

If $f(x)$ has Fourier transform $F(f_x)$, then:

$$ \mathcal{F}\{f(x - a)\} = F(f_x) \cdot e^{-j2\pi a f_x} $$

A shift by $a$ in the spatial domain introduces a linear phase $e^{-j2\pi a f_x}$ in the frequency domain.

Derivation

From the definition of the Fourier transform:

$$ \mathcal{F}\{f(x - a)\} = \int_{-\infty}^{\infty} f(x - a) \, e^{-j2\pi f_x x} \, dx $$

Substituting $u = x - a$ (so $x = u + a$, $dx = du$):

$$ = \int_{-\infty}^{\infty} f(u) \, e^{-j2\pi f_x (u + a)} \, du = e^{-j2\pi a f_x} \int_{-\infty}^{\infty} f(u) \, e^{-j2\pi f_x u} \, du = e^{-j2\pi a f_x} \, F(f_x) $$

Special Case: Shifted Delta Function

Since $\mathcal{F}\{\delta(x)\} = 1$, applying the shift theorem gives:

$$ \mathcal{F}\{\delta(x - a)\} = e^{-j2\pi a f_x} $$

This result is used frequently in Fourier optics when computing diffraction patterns of multi-element apertures (e.g., double slits, gratings).

See Also

• Fourier transform
• Convolution theorem