Minimum Zero-Padding for Low-Frequency Peaks

Sharper bounds on the zero-padding factor needed for low-frequency peaks (below roughly 1 kHZ) may be obtained based on the measured Just-Noticeable-Difference (JND) in frequency and/or amplitude [259]. In particular, a % relative-error spec is good above 1 kHz (being conservative by approximately a factor of 2), but overly conservative at lower frequencies where the JND flattens out. Below 1 kHz, a fixed 1 Hz spec satisfies perceptual requirements and gives smaller minimum zero-padding factors than the % relative-error spec.

The following data, extracted from [259, Table I, p. 89] gives frequency JNDs at a presentation level of 60 dB SPL (the most sensitive case measured):

  f =    [    62,    125,    250,    500,   1000,   2000,   4000];
  dfof = [0.0346, 0.0269, 0.0098, 0.0035, 0.0034, 0.0018, 0.0020];

Thus, the frequency JND at 4 kHz was measured to be two tenths of a percent. (These measurements were made by averaging experimental results for five men between the ages of 20 and 30.) Converting relative frequency to absolute frequency in Hz yields (in matlab syntax):

  df = dfof .* f; % = [2.15, 3.36, 2.45, 1.75, 3.40, 3.60, 8.00];

For purposes of computing the minimum zero-padding factor required, we see that the absolute tuning error due to bias can be limited to 1 Hz, based on measurements at 500 Hz (at 60 dB). Doing this for frequencies below 1 kHz yields the results shown in Table 4.4. Note that the Blackman window needs no zero padding below 125 Hz, and the Hamming/Hann window requires no zero padding below 62.5 Hz.

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About the Author: 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.