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3.4 Fourier transforms 119

• Reversal: The Fourier transform of a reversed signal is the complex conjugate of the
signal’s transform.
• Convolution: The Fourier transform of a pair of convolved signals is the product of
their transforms.

• Correlation: The Fourier transform of a correlation is the product of the first transform
times the complex conjugate of the second one.

• Multiplication: The Fourier transform of the product of two signals is the convolution
of their transforms.

• Differentiation: The Fourier transform of the derivative of a signal is that signal’s
transform multiplied by the frequency. In other words, differentiation linearly empha-
sizes (magnifies) higher frequencies.

• Domain scaling: The Fourier transform of a stretched signal is the equivalently com-
pressed (and scaled) version of the original transform and vice versa.
• Real images: The Fourier transform of a real-valued signal is symmetric around the
origin. This fact can be used to save space and to double the speed of image FFTs
by packing alternating scanlines into the real and imaginary parts of the signal being
transformed.

• Parseval’s Theorem: The energy (sum of squared values) of a signal is the same as
the energy of its Fourier transform.

All of these properties are relatively straightforward to prove (see Exercise 3.15) and they will
come in handy later in the book, e.g., when designing optimum Wiener filters (Section 3.4.3)
or performing fast image correlations (Section 8.1.2).

3.4.1 Fourier transform pairs

Now that we have these properties in place, let us look at the Fourier transform pairs of some
commonly occurring filters and signals, as listed in Table 3.2. In more detail, these pairs are
as follows:

• Impulse: The impulse response has a constant (all frequency) transform.

• Shifted impulse: The shifted impulse has unit magnitude and linear phase.
• Box filter: The box (moving average) filter

1 if |x|≤ 1
box(x)= (3.55)
0 else
has a sinc Fourier transform,
sin ω
sinc(ω)= , (3.56)
ω
which has an infinite number of side lobes. Conversely, the sinc filter is an ideal low-
pass filter. For a non-unit box, the width of the box a and the spacing of the zero
crossings in the sinc 1/a are inversely proportional.