Reconstruction of Non-Uniformly Sampled Audio Signals

Abstract

Present-day digital audio systems are based on the well-known Nyquist theorem, which states that a signal may be completely reconstructed from reguarly-spaced samples of that signal as long as the highest frequency in the original signal is less than one-half of the sampling frequency. This paper will show a unique decoding algorithm that can completely reconstruct a signal based on non-uniformly spaced samples of that signal, where the non-uniformity consists of reguarly-spaced missing or incorrect samples. We will show that for this case, the signal may be completely reconstructed if the highest frequency present in the original signal is less than one-half of the "average" sample rate. This algorithm has several potential applications in digital audio systems, such as error concealment and adding a low bit-rate side channel to existing digital recorders or transmission devices. To develop this theory, we start with the following assumption. If a signal that is bandlimited to a frequency wl is applied to a FIR linear-phase lowpass filter with a cutoff frequency of w2 where w2 > wl, then the output signal equals the input signal (with delay) with an accuracy determined by the passband ripple of the low-pass filter. The response of the filter between w l and w2 does not affect the input signal, since the input signal has no energy in this frequency range. Fig. 1 shows this theory graphically. Fig. 2 shows the basic block diagram of the proposed scheme. We start with a sampling operation that is non-uniform in a regular pattern. In this example, we use a sampler that samples for 3 consecutive periods and then skips a sample. This example will be used throughout this paper, and the reader will appreziate that extending the technique to other sampling patterns is straightforward. We will assume that the input signal is bandlimited to < 3/4*(Fs/2), where Fs = l/r and T is the spacing in time between the three consecutive samples. In practice, some gaurd-band is needed to allow for filter transition bands. This non-uniformly sampled signal is then applied to a digital FIR low-pass filter. This filter is a linear-phase filter with passband ripple R and delay D. Note that the input to this filter is a continuously-sampled signal at Fs, where the missing sample has been replaced by a sample of arbitrary value or zero. The decoded output will be derived by a switching between the filtered signal …