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 [276]. In particular, a $0.1$ % 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 $0.1$ % 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 $L_{\hbox{min}}=N_{\hbox{min}}/M$ for keeping peak-frequency bias below approximately 1 Hz (well under 1.75 Hz), assuming the window length $M$ to span one period of the fundamental frequency.

Window Type	$f$ (Hz)	$\mathbf{L_{\hbox{min}}}$
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