### Periodic Interpolation (Spectral Zero Padding)

The dual of the zero-padding theorem states formally that zero padding in the frequency domain corresponds to periodic interpolation in the time domain:

Definition: For all and any integer ,

 (7.7)

where zero padding is defined in §7.2.7 and illustrated in Figure 7.7. In other words, zero-padding a DFT by the factor in the frequency domain (by inserting zeros at bin number corresponding to the folding frequency7.21) gives rise to periodic interpolation'' by the factor in the time domain. It is straightforward to show that the interpolation kernel used in periodic interpolation is an aliased sinc function, that is, a sinc function that has been time-aliased on a block of length . Such an aliased sinc function is of course periodic with period samples. See Appendix D for a discussion of ideal bandlimited interpolation, in which the interpolating sinc function is not aliased.

Periodic interpolation is ideal for signals that are periodic in samples, where is the DFT length. For non-periodic signals, which is almost always the case in practice, bandlimited interpolation should be used instead (Appendix D).

#### Relation to Stretch Theorem

It is instructive to interpret the periodic interpolation theorem in terms of the stretch theorem, . To do this, it is convenient to define a zero-centered rectangular window'' operator:

Definition: For any and any odd integer we define the length even rectangular windowing operation by

Thus, this zero-phase rectangular window,'' when applied to a spectrum , sets the spectrum to zero everywhere outside a zero-centered interval of samples. Note that is the ideal lowpass filtering operation in the frequency domain. The cut-off frequency'' is radians per sample. For even , we allow to be passed'' by the window, but in our usage (below), this sample should always be zero anyway. With this notation defined we can efficiently restate periodic interpolation in terms of the operator:

Theorem: When consists of one or more periods from a periodic signal ,

In other words, ideal periodic interpolation of one period of by the integer factor may be carried out by first stretching by the factor (inserting zeros between adjacent samples of ), taking the DFT, applying the ideal lowpass filter as an -point rectangular window in the frequency domain, and performing the inverse DFT.

Proof: First, recall that . That is, stretching a signal by the factor gives a new signal which has a spectrum consisting of copies of repeated around the unit circle. The baseband copy'' of in can be defined as the -sample sequence centered about frequency zero. Therefore, we can use an ideal filter'' to pass'' the baseband spectral copy and zero out all others, thereby converting to . I.e.,

The last step is provided by the zero-padding theorem7.4.12).

#### Bandlimited Interpolation of Time-Limited Signals

The previous result can be extended toward bandlimited interpolation of which includes all nonzero samples from an arbitrary time-limited signal (i.e., going beyond the interpolation of only periodic bandlimited signals given one or more periods ) by

1. replacing the rectangular window with a smoother spectral window , and
2. using extra zero-padding in the time domain to convert the cyclic convolution between and into an acyclic convolution between them (recall §7.2.4).
The smoother spectral window can be thought of as the frequency response of the FIR7.22 filter used as the bandlimited interpolation kernel in the time domain. The number of zeros needed in the zero-padding of in the time domain is simply length of minus 1, and the number of zeros to be appended to is the length of minus 1. With this much zero-padding, the cyclic convolution of and implemented using the DFT becomes equivalent to acyclic convolution, as desired for the time-limited signals and . Thus, if denotes the nonzero length of , then the nonzero length of is , and we require the DFT length to be , where is the filter length. In operator notation, we can express bandlimited sampling-rate up-conversion by the factor for time-limited signals by

 (7.8)

The approximation symbol ' approaches equality as the spectral window approaches (the frequency response of the ideal lowpass filter passing only the original spectrum ), while at the same time allowing no time aliasing (convolution remains acyclic in the time domain).

Equation (7.8) can provide the basis for a high-quality sampling-rate conversion algorithm. Arbitrarily long signals can be accommodated by breaking them into segments of length , applying the above algorithm to each block, and summing the up-sampled blocks using overlap-add. That is, the lowpass filter rings'' into the next block and possibly beyond (or even into both adjacent time blocks when is not causal), and this ringing must be summed into all affected adjacent blocks. Finally, the filter can `window away'' more than the top copies of in , thereby preparing the time-domain signal for downsampling, say by :

where now the lowpass filter frequency response must be close to zero for all . While such a sampling-rate conversion algorithm can be made more efficient by using an FFT in place of the DFT (see Appendix A), it is not necessarily the most efficient algorithm possible. This is because (1) out of output samples from the IDFT need not be computed at all, and (2) has many zeros in it which do not need explicit handling. For an introduction to time-domain sampling-rate conversion (bandlimited interpolation) algorithms which take advantage of points (1) and (2) in this paragraph, see, e.g., Appendix D and [72].

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