Filling the FFT Input Buffer (Step 2)
The FFT size
is normally chosen to be the first power of two that
is at least twice the window length
, with the difference
filled with zeros (``zero-padded''). The reason for increasing the FFT
size and filling in with zeros is that zero-padding in the time domain
corresponds to interpolation in the frequency domain, and
interpolating the spectrum is useful in various ways. First, the
problem of finding spectral peaks which are not exact bin frequencies
is made easier when the spectrum is more densely sampled. Second,
plots of the magnitude of the more smoothly sampled spectrum are less
likely to confuse the untrained eye. (Only signals truly periodic in
samples should not be zero-padded. They should also be windowed
only by the Rectangular window.) Third, for overlap-add synthesis from
spectral modifications, the zero-padding allows for multiplicative
modification in the frequency domain (convolutional modification in
the time domain) without time aliasing in the inverse FFT. The length
of the allowed convolution in the time domain (the impulse response of
the effective digital filter) equals the number of extra zeros (plus
one) in the zero padding.
If
is the number of samples in the main lobe when the
zero-padding factor is 1 (
), then a zero-padding factor of
gives
samples for the same main lobe (and same main-lobe
bandwidth). The zero-padding (interpolation) factor
should
be large enough to enable accurate estimation of the true maximum of
the main lobe after it has been frequency shifted by some arbitrary
amount equal to the frequency of a sinusoidal component in the input
signal. We have determined by computational search that, for a
rectangularly windowed sinusoid (of any frequency), quadratic
frequency interpolation (using the three highest bins) yields at
least
(of the distance from the sinc peak to the first
zero-crossing) accuracy if the zero-padding factor
is 5 or
higher.
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As mentioned in the previous section, we facilitate phase detection
by using a zero-phase window, i.e., the windowed data (using an odd
length window) is centered about the time origin. A zero-centered,
length
data frame appears in the length
FFT input buffer as
shown in Fig.H.1c. The first
samples of the windowed
data, the ``negative-time'' portion, will be stored at the end of the
buffer, from sample
to
, and the remaining
samples, the zero- and ``positive-time'' portion, will be stored
starting at the beginning of the buffer, from sample 0
to
. Thus, all zero padding occurs in the middle of the
FFT input buffer.
Next Section:
Peak Detection (Steps 3 and 4)
Previous Section:
Choice of Hop Size