Relation to Stretch Theorem

It is instructive to interpret the periodic interpolation theorem in terms of the stretch theorem, $\hbox{\sc Stretch}_L(x) \;\longleftrightarrow\;\hbox{\sc Repeat}_L(X)$ . To do this, it is convenient to define a ``zero-centered rectangular window'' operator:

Definition: For any $X\in\mathbb{C}^N$ and any odd integer $M<N$ we define the length $M$ even rectangular windowing operation by

$$\hbox{\sc Chop}_{M,k}(X) \isdef \left\{\begin{array}{ll} X(k), & -\frac{M-1}{2}\leq k \leq \frac{M-1}{2} \\ [5pt] 0, & \frac{M+1}{2} \leq \left\vert k\right\vert \leq \frac{N}{2}. \\ \end{array} \right.$$

Thus, this ``zero-phase rectangular window,'' when applied to a spectrum $X$ , sets the spectrum to zero everywhere outside a zero-centered interval of $M$ samples. Note that $\hbox{\sc Chop}_M(X)$ is the ideal lowpass filtering operation in the frequency domain. The ``cut-off frequency'' is $\omega_c = 2\pi[(M-1)/2]/N$ radians per sample. For even $M$ , we allow $X(-M/2)$ 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 $\hbox{\sc Stretch}()$ operator:

Theorem: When $x\in\mathbb{C}^N$ consists of one or more periods from a periodic signal $x^\prime\in \mathbb{C}^\infty$ ,

$$\zbox {\hbox{\sc PerInterp}_L(x) = \hbox{\sc IDFT}(\hbox{\sc Chop}_N(\hbox{\sc DFT}(\hbox{\sc Stretch}_L(x)))).}$$

In other words, ideal periodic interpolation of one period of $x$ by the integer factor $L$ may be carried out by first stretching $x$ by the factor $L$ (inserting $L-1$ zeros between adjacent samples of $x$ ), taking the DFT, applying the ideal lowpass filter as an $N$ -point rectangular window in the frequency domain, and performing the inverse DFT.

Proof: First, recall that $\hbox{\sc Stretch}_L(x)\leftrightarrow \hbox{\sc Repeat}_L(X)$ . That is, stretching a signal by the factor $L$ gives a new signal $y=\hbox{\sc Stretch}_L(x)$ which has a spectrum $Y$ consisting of $L$ copies of $X$ repeated around the unit circle. The ``baseband copy'' of $X$ in $Y$ can be defined as the $N$ -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 $\hbox{\sc Repeat}_L(X)$ to $\hbox{\sc ZeroPad}_{LN}(X)$ . I.e.,

$$\hbox{\sc Chop}_N(\hbox{\sc Repeat}_L(X)) = \hbox{\sc ZeroPad}_{LN}(X) \;\longleftrightarrow\;\hbox{\sc Interp}_L(x).$$

The last step is provided by the zero-padding theorem (§7.4.12).