. In fact, for arbitrary-length signals, the difference between oversampled peak estimates and actual peak levels can’t be bounded. Consider the pathological case:
x(n) = -(-1)n n>0
x(0) = 1
x(n) = -x(-n) n<0
As shown in Figure 5, this signal alternates between +1 and -1, with two consecutive +1 samples at n=0,1. The interpolated peak signal value at the location n=0.5 grows logarithmically with the length of x, so that if x is infinitely long, the peak value is also infinite. For a signal x, which would barely fit on an audio CD, the peak signal value is about 22dB above the largest sample value! The situation is further complicated by the fact that the calculated oversampled peak value depends on the length of window used in the oversampling filter.
Figure 5 shows the calculated interpolated signal for an oversampler with a length of 32. This gives an interpolated peak about 11dB above the largest sample value. The calculated peak value grows logarithmically with the length of oversampling window used.
One possible alternative method to choose levels for digitally encoded signals would be to oversample by a high factor using an interpolation filter that simulates the behavior of a typical D/A reconstruction filter. Such an approach would not estimate the “ideal” peak signal levels; rather, it would create an estimate of the headroom needed for D/A conversion. In any event, it is clear that care should be taken when dealing with digitally encoded signals, as peak signal levels can fall significantly above maximum sample values.