Ideal Spectral Interpolation

Let $x\in{\bf C}^N$ include all nonzero samples from a time-limited (as opposed to periodic) signal $x^\prime\in{\bf C}^\infty$, and define $y =$ZEROPAD$_M(x)$. Then $y\in{\bf C}^M$ with $M\geq N$. Denote the original frequency index by $k$, where $\omega_k \isdeftext 2\pi k/N$, and the new frequency index by $k^\prime$, where $\omega_{k^\prime }\isdeftext 2\pi k^\prime /M$.

Definition: Given a sampled spectrum $X(\omega_k )\isdeftext$   DFT$_{N,k}(x)$, for $k\in[0,N-1]$, its ideal bandlimited interpolation at frequency $\omega\in[-\pi,\pi)$ is defined as

$$X(\omega) \isdef \sum_{n=0}^{N-1} x(n) e^{-j\omega n} \protect$$ (7.4)

$$= \lim_{M\to\infty}\sum_{n=0}^{N-1} y(n) e^{-j\omega_{k^\prime(\omega)} n}. \protect$$ (7.5)

where we may define $k^\prime(\omega)\isdeftext \lfloor\omega M/(2\pi)\rfloor$. Note that this is just the definition of the DFT with $\omega_k$ replaced by $\omega$. That is, the spectrum is interpolated by projecting onto new sinusoids at arbitrary frequencies $\omega$ exactly as if they were DFT sinusoids (see Chapter 6). Since the interval $n\in [0,N-1]$ spans all nonzero samples from the time-limited signal $x^\prime$, the inner product between $x$ and any sampled sinusoid reduces to exactly Eq. (7.4) above. Thus, for time limited signals, this kind of spectral interpolation is ideal.