Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Sponsor

NEW! TMS320C6474: 3x the performance. 1/3 the cost. Three 1 GHz cores on 1 chip.

Chapters

Physical Audio Signal Processing
    Higher Order Delay Line Interpolation
       Bandlimited Interpolation
          Theory of Ideal Bandlimited Interpolation

Chapter Contents:

Search Physical Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

Theory of Ideal Bandlimited Interpolation

We review briefly the ``analog interpretation'' of sampling rate conversion [97] on which the present method is based. Suppose we have samples $ x(nT)$ of a continuous absolutely integrable signal $ x(t)$, where $ t$ is time in seconds (real), $ n$ ranges over the integers, and $ T$ is the sampling period. We assume $ x(t)$ is bandlimited to $ \pm f_s/2$, where $ f_s=1/T$ is the sampling rate. If $ X(\omega)$ denotes the Fourier transform of $ x(t)$, i.e., $ X(\omega)=\int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt$, then we assume $ X(\omega)=0$ for $ \vert\omega\vert\geq\pi f_s$. Consequently, Shannon's sampling theorem gives us that $ x(t)$ can be uniquely reconstructed from the samples $ x(nT)$ via

$\displaystyle {\hat x}(t) \isdef \sum_{n=-\infty}^{\infty} x(nT) h_s(t-nT) \equiv x(t),$ (K.4)

where

$\displaystyle h_s(t) \isdef$   sinc$\displaystyle (f_st) \isdef \frac{\sin(\pi f_st)}{\pi f_st}. $

To