Interpolation Operator

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

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

Since $X(\omega_k )\isdeftext \hbox{\sc DFT}_{N,k}(x)$ is initially only defined over the $N$ roots of unity, 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.

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