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 . 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 [276, 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 5.4. 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.