### 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

**Previous Section:**

Causal (Periodic) Signals