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

,
![$ k\in[0,N-1]$](http://www.dsprelated.com/josimages_new/mdft/img1046.png)
.
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 a 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
(shown in Fig.
6.3a for

). At all other integer values of

, the frequency response is the same but shifted (circularly) left or right so
that the peak is centered on

. The secondary peaks away from

are called
sidelobes of the DFT response, while the main
peak may be called the
main lobe of the response. Since we are
normally most interested in
spectra from an audio perspective, the same
plot is repeated using a
decibel vertical scale in
Fig.
6.3b
6.6(clipped at
dB). We see that the sidelobes are really quite high
from an audio perspective. Sinusoids with frequencies near

, for example, are only attenuated approximately
dB in the DFT
output

.
Figure:
Magnitude frequency response of a particular DFT ``bin'' (where ``bin'' is defined in §6.8).
The solid curve shows the relative contribution of arbitrary frequency
components to the spectral bin at one-fourth the sampling rate.
![\includegraphics[width=\twidth]{eps/dftfilter}](http://www.dsprelated.com/josimages_new/mdft/img1059.png) |
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.
Next Section: Spectral Bin NumbersPrevious Section: The Discrete Fourier Transform (DFT)