Fourier transform duality principle

Figure 10.12 Figure a) is the real part of a spectrum, and figure b) is its inverse Fourier transform. Along with Figure 10.11, these figures demonstrate the duality principle. 

This is shown in figme 10.12. When we compare equation 10.25 to equation 10.22 and figure 10.11 to figure 10.12 we see that the Fourier transform of a rectangular pulse is a sine spectrum, and the inverse Fourier transform of a rectangular spectrum is a sine waveform. This demonstrates another special property of the Fourier transform known as the duality principle. These and other general properties of the Fourier transform are discussed further in section 10.3. 

As expected from the scaling property, the Fourier transform of an impulse is a function that is infinitely stretched , that is, the Fourier Transform is 1 at all frequencies. Using the duality principle, a signal x(t) = 1 for all t will have a Fourier transform of 6( ), that is, an impulse at time 00 = 0. This is to be expected - a constant signal (a d.c. signal in electrical terms) has no variation and hence no information at frequencies other than 0. 

---

See also in sourсe #XX -- [ ]

See also in sourсe #XX -- [ ]

---