Fourier transform pairs and properties

Pick a pulse, then shift it, squeeze it, reverse it, modulate it or split it into even and odd parts. The spectrum changes beside it, magnitude and phase or real and imaginary, with the property that explains the change and the symmetry shortcut GATE uses to eliminate options.

Signal

Operation

Pair:

\text{rect}(t/T)\ \longleftrightarrow\ T\,\text{sinc}(fT)

Property:

x(t-t_0)\ \longleftrightarrow\ X(f)\,e^{-j2\pi f t_0}

Time domain x(t)

before the operationx(t)

Spectrum X(f)

|X(f)|

before|X(f)|

Phase ∠X(f), in units of π (drawn only where |X| is not negligible)

∠X / π

Signal width T1.00

t_0 \text{ (delay)}

1.00

x(t) isneither even nor odd

so X(f) iscomplex (|X| even, phase odd)

X(0) = ∫ x(t) dt1.0000 = 1.0000

x(0) = ∫ X(f) df0.0000 = 0.0002

Energy ∫|x|²dt vs ∫|X|²df0.9995 vs 0.9997

The dashed grey curves are the signal and spectrum before the operation, so you can see exactly what it changed. Both sides of each equation are computed separately (the right-hand ones by numerical integration), so they agree only because the theorem holds. At a jump, x(0) takes the midpoint value.

What to notice

A delay never changes $|X(f)|$ — only the phase, which gains $-2\pi f t_0$ : a straight line of slope $-2\pi t_0 = -2.00\pi$ per hertz. It wraps every $1/|t_0| = 1.00$ Hz.
That is why a delay is called linear phase, and why the group delay $-\frac{1}{2\pi}\frac{d\phi}{df}$ equals $t_0$ at every frequency.
Symmetry shortcut: a real signal that is neither even nor odd has a complex spectrum, but always with $|X(-f)| = |X(f)|$ and $\angle X(-f) = -\angle X(f)$ .
Two GATE favourites, read off the boxes on the right: $X(0)$ is the area under $x(t)$ , and $x(0)$ is the area under $X(f)$ . Parseval: the energy is the same in both domains.

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