Bias of Parabolic Peak Interpolation
Since the true window transform is not a parabola (except for the conceptual case of a Gaussian window transform expressed in dB), there is generally some error in the interpolated peak due to this mismatch. Such a systematic error in an estimated quantity (due to modeling error, not noise), is often called a bias. Parabolic interpolation is unbiased when the peak occurs at a spectral sample (FFT bin frequency), and also when the peak is exactly halfway between spectral samples (due to symmetry of the window transform about its midpoint). For other peak frequencies, quadratic interpolation yields a biased estimate of both peak frequency and peak amplitude. (Phase is essentially unbiased [1].)
Since zeropadding in the time domain gives ideal interpolation in the frequency domain, there is no bias introduced by this type of interpolation. Thus, if enough zeropadding is used so that a spectral sample appears at the peak frequency, simply finding the largestmagnitude spectral sample will give an unbiased peakfrequency estimator. (We will learn in §5.7.2 that this is also the maximum likelihood estimator for the frequency of a sinusoid in additive white Gaussian noise.)
While we could choose our zeropadding factor large enough to yield any desired degree of accuracy in peak frequency measurements, it is more efficient in practice to combine zeropadding with parabolic interpolation (or some other simple, loworder interpolator). In such hybrid schemes, the zeropadding is simply chosen large enough so that the bias due to parabolic interpolation is negligible. In §5.7 below, the Quadratically Interpolated FFT (QIFFT) method is described as one such hybrid scheme.
Minimum ZeroPadding for HighFrequency Peaks

Table 5.3 gives zeropadding factors sufficient for keeping the bias below Hz, where denotes the sampling rate in Hz, and is the window length in samples. For fundamental frequency estimation, can be interpreted as the relative frequency error `` '' when the window length is one period. In this case, is the fundamental frequency in Hz. More generally, is the bandwidth of each sidelobe in the DTFT of a length rectangular, generalized Hamming, or Blackman window (any member of the BlackmanHarris window family, as elaborated in Chapter 3).
Note from Table 5.3 that the Blackman window requires no zeropadding at all when only % accuracy is required in peakfrequency measurement. It should also be understood that a frequency error of % is inaudible in most audio applications.^{6.10}
Minimum ZeroPadding for LowFrequency Peaks
Sharper bounds on the zeropadding factor needed for lowfrequency peaks (below roughly 1 kHz) may be obtained based on the measured JustNoticeableDifference (JND) in frequency and/or amplitude [276]. In particular, a % relativeerror 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 zeropadding factors than the % relativeerror 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 zeropadding 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.

Matlab for Computing Minimum ZeroPadding Factors
The minimum zeropadding factors in the previous two subsections were computed using the matlab function zpfmin listed in §F.2.4. For example, both tables above are included in the output of the following matlab program:
windows={'rect','hann','hamming','blackman'}; freqs=[1000,500,250,125,62.5]; for i=1:length(windows) w = sprintf("%s",windows(i)) for j=1:length(freqs) f = freqs(j); zpfmin(w,1/f,0.01*f) % 1 percent spec (large for audio) zpfmin(w,1/f,0.001*f) % 0.1 percent spec (good > 1 kHz) zpfmin(w,1/f,1) % 1 Hz spec (good below 1 kHz) end end
In addition to ``perceptually exact'' detection of spectral peaks, there are times when we need to find spectral parameters as accurately as possible, irrespective of perception. For example, one can estimate the stiffness of a piano string by measuring the stretched overtonefrequencies in the spectrum of that string's vibration. Additionally, we may have measurement noise, in which case we want our measurements to be minimally influenced by this noise. The following sections discuss optimal estimation of spectralpeak parameters due to sinusoids in the presence of noise.
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Least Squares Sinusoidal Parameter Estimation
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Quadratic Interpolation of Spectral Peaks