### 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.*,

`fft(x,N)`when the FFT size

`N`exceeds the length of the signal vector

`x`.

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.

**Next Section:**

Zero Padding Applications

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Causal (Periodic) Signals