Frequencies in the ``Cracks''

The DFT is defined only for frequencies . If we are analyzing one or more periods of an exactly periodic signal, where the period is exactly samples (or some integer divisor of ), then these really are the only frequencies present in the signal, and the spectrum is actually zero everywhere but at , . However, we use the DFT to analyze arbitrary signals from nature. What happens when a frequency is present in a signal that is not one of the DFT-sinusoid frequencies ?

To find out, let's project a length segment of a sinusoid at an arbitrary frequency onto the th DFT sinusoid:

The coefficient of projection is proportional to

using the closed-form expression for a geometric series sum once
again. As shown in §6.3-§6.4 above,
the sum is if
and zero at , for . However,
the sum is *nonzero at all other frequencies* .

Since we are only looking at samples, any sinusoidal segment can be
projected onto the DFT sinusoids and be reconstructed exactly by a
linear combination of them. Another way to say this is that the DFT
sinusoids form a *basis* for , so that any length signal
whatsoever can be expressed as linear combination of them. Therefore, when
analyzing segments of recorded signals, we must interpret what we see
accordingly.

The typical way to think about this in practice is to consider the DFT
operation as a *digital filter* for each , whose input is
and whose output is
at time .^{6.4} The
*frequency response* of this filter is what we just
computed,^{6.5} and its magnitude is

We see that
is sensitive to *all* frequencies between dc
and the sampling rate *except* the other DFT-sinusoid frequencies
for . This is sometimes called *spectral leakage*
or *cross-talk* in the spectrum analysis. Again, there is *no
leakage* when the signal being analyzed is truly periodic and we can choose
to be exactly a period, or some multiple of a period. Normally,
however, this cannot be easily arranged, and spectral leakage can
be a problem.

Note that peak spectral leakage is not reduced by increasing
.^{6.7} It can be thought of as being caused by abruptly
*truncating* a sinusoid at the beginning and/or end of the
-sample time window. Only the DFT sinusoids are not cut off at the
window boundaries. All other frequencies will suffer some truncation
distortion, and the spectral content of the abrupt cut-off or turn-on
transient can be viewed as the source of the sidelobes. Remember
that, as far as the DFT is concerned, the input signal is the
same as its
*periodic extension* (more about this in
§7.1.2). If we repeat samples of a sinusoid at frequency
(for any
), there will be a ``glitch''
every samples since the signal is not periodic in samples.
This glitch can be considered a source of new energy over the entire
spectrum. See
Fig.8.3 for an example waveform.

To reduce spectral leakage (cross-talk from far-away
frequencies), we typically use a
*window*
function, such as a
``raised cosine'' window, to *taper* the data record gracefully
to zero at both endpoints of the window. As a result of the smooth
tapering, the *main lobe widens* and the *sidelobes
decrease* in the DFT response. Using no window is better viewed as
using a *rectangular window* of length , unless the signal is
exactly periodic in samples. These topics are considered further
in Chapter 8.

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