Reconstruction from Samples--The Math

Let $x_d(n) \isdef x(nT)$ denote the th sample of the original sound , where is time in seconds. Thus, ranges over the integers, and is the sampling interval in seconds. The sampling rate in Hertz (Hz) is just the reciprocal of the sampling period, i.e.,

$\displaystyle f_s\isdef \frac{1}{T}.$

To avoid losing any information as a result of sampling, we must assume is bandlimited to less than half the sampling rate. This means there can be no energy in at frequency or above. We will prove this mathematically when we prove the sampling theorem in §D.3 below.

Let $X(\omega)$ denote the Fourier transform of , i.e.,

$\displaystyle X(\omega)\isdef \int_{-\infty}^\infty x(t) e^{-j\omega t} dt .$

Then we can say

is bandlimited to less than half the sampling rate if and only if $X(\omega)=0$ for all $\vert\omega\vert\geq\pi f_s$ . In this case, the sampling theorem gives us that

can be uniquely reconstructed from the samples

by summing up shifted, scaled, sinc functions:

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

where

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

The sinc function is the impulse response of the ideal lowpass filter. This means its Fourier transform is a rectangular window in the frequency domain. The particular sinc function used here corresponds to the ideal lowpass filter which cuts off at half the sampling rate. In other words, it has a gain of 1 between frequencies 0 and

, and a gain of zero at all higher frequencies.

The reconstruction of a sound from its samples can thus be interpreted as follows: convert the sample stream into a weighted impulse train, and pass that signal through an ideal lowpass filter which cuts off at half the sampling rate. These are the fundamental steps of digital to analog conversion (DAC). In practice, neither the impulses nor the lowpass filter are ideal, but they are usually close enough to ideal that one cannot hear any difference. Practical lowpass-filter design is discussed in the context of bandlimited interpolation [69].

Order a Hardcopy of Mathematics of the DFT

Previous: The Sinc Function
Next: Aliasing of Sampled Signals

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

Sign in

Search Online Books

Free Online Books

Ads

Chapters

Mathematics of the DFT
    Sampling Theory
       Introduction to Sampling
          Reconstruction from Samples--The Math