#### 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 . 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.

Table: Minimum zero-padding factors for keeping peak-frequency bias below approximately 1 Hz (well under 1.75 Hz), assuming the window length to span one period of the fundamental frequency.
 Window Type (Hz) Rectangular 1000 4.1 500 3.3 250 2.6 125 2.1 62.5 1.7 Gen. Hamming 1000 2.4 500 1.9 250 1.5 125 1.2 62.5 1 Blackman 1000 1.8 500 1.5 250 1.2 125 1 62.5 1

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Matlab for Computing Minimum Zero-Padding Factors
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