Interpolation Operator

The interpolation operator $\hbox{\sc Interp}_L()$ interpolates a signal by an integer factor $L$ using bandlimited interpolation. For frequency-domain signals $X(\omega_k)$ , $k=0,1,2,\ldots,N-1$ , we may write spectral interpolation as follows:

$$\begin{eqnarray*} \hbox{\sc Interp}_{L,k^\prime }(X) &\isdef & X(\omega_{k^\prime }), \mbox{ where}\\ \omega_{k^\prime }&=& 2\pi k^\prime /M,\; k^\prime =0,1,2,\dots,M-1,\;\\ M&\isdef & LN. \end{eqnarray*}$$

Since $X(\omega_k )\isdeftext \hbox{\sc DFT}_{N,k}(x)$ is initially only defined over the $N$ roots of unity in the $z$ plane, while $X(\omega_{k^\prime })$ is defined over $M=LN$ roots of unity, we define $X(\omega_{k^\prime })$ for $\omega_{k^\prime }\neq\omega_k$ by ideal bandlimited interpolation (specifically time-limited spectral interpolation in this case).

For time-domain signals $x(n)$ , exact interpolation is similarly bandlimited interpolation, as derived in Appendix D.