Causal Zero Padding

In practice, a signal
is often an
-sample *frame* of
data taken from some longer signal, and its true starting time can be
anything. In such cases, it is common to treat the start-time of the
frame as zero, with no negative-time samples. In other words,
represents an
-sample signal-segment that is translated in time to
start at time 0
. In this case (no negative-time samples in the
frame), it is proper to zero-pad by simply appending zeros at the end
of the frame. Thus, we define
*e.g.*,

Causal zero-padding should not be used on a spectrum of a real signal because, as we will see in §7.4.3 below, the magnitude spectrum of every real signal is symmetric about frequency zero. For the same reason, we cannot simply append zeros in the time domain when the signal frame is considered to include negative-time samples, as in ``zero-centered FFT processing'' (discussed in Book IV [73]). Nevertheless, in practice, appending zeros is perhaps the most common form of zero-padding. It is implemented automatically, for example, by the matlab function

In summary, we have defined two types of zero-padding that arise in
practice, which we may term ``causal'' and ``zero-centered'' (or
``zero-phase'', or even ``periodic''). The zero-centered case is the
more natural with respect to the mathematics of the DFT, so it is
taken as the ``official'' definition of ZEROPAD(). In both cases,
however, when properly used, we will have the basic Fourier theorem
(§7.4.12 below) stating that *zero-padding in the time domain
corresponds to ideal bandlimited interpolation in the frequency
domain*, and vice versa.

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