Fourier Transform—Infinite Extension Functions - Vision with Direction - page 75

= 2 π

μ m ,f

Ω f ( t m )

(6.33)

which uses no integrals, this result is a fairly simple tool to find projection coeffi-
cients.

Example 6.2. We illustrate sampling and periodization by Fig. 6.4.

• In the top, left graph, we have drawn a band-limited signal f ( t ). The very def-

inition of f as band-limited implies that F ( ω ) has a limited extension in the
spectrum, which is drawn in the top, right graph. The width of F is Ω .

• In the middle-left graph, we have drawn f after discretization. In the middle

right graph, we have drawn the ω -domain after discretization. The discretization
in one domain is equivalent to a periodization in the other domain. Also, the
discretization step is inversely proportional to the period of the other domain.

• In the bottom, left graph we have shown the sampled f , when we periodize F ,

withalargerperiodthantheextensionof F .Thisperiodizationof F ,usingtwice
as large an Ω as compared to the extension of F , is shown in the bottom, right
graph.

We summarize the results of this section by the Nyquist theorem .

Theorem 6.3 (Nyquist). Let f and F be a Fourier transform pair. Then sampling of
either of the functions is equivalent to periodization of the other function. To be more
precise, the following two statements are valid:

T
2

, T
2

• (Time-limited signals) If f is nonzero only in the interval [ ] , then F can

be sampled without loss of information provided that the sampling period is 2 T
or less.

−

Ω
2

, Ω
2

• (Band-limited signals) If F is nonzero only in the interval [ ] then f can

be sampled without loss of information provided that the sampling period is 2 Ω
or less.

−

When working with band-limited signals, the distance between the samples in
the t -domain can be assumed to be equal to 1, for convenience. Then we obtain the
critical frequency domain parameter Ω =2 π , so that the critical frequency in the
theorem yields:

Ω
2

= π ⇔− π<ω<π (6.34)

Ω
2

In signal processing literature, is often referred to as the Nyquist frequency. Even

its normalized version, π , is called the Nyquist frequency, because a scaling of ω
and t -domains is always achievable. The basic interval − pi<ω<π is sometimes

called the Nyquist period ,orthe Nyquist block .