A Quadrature Signals Tutorial: Complex, But Not Complicated

Understanding the 'Phasing Method' of Single Sideband Demodulation

Complex Digital Signal Processing in Telecommunications

Introduction to Sound Processing

Introduction of C Programming for DSP Applications

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Due to the sampled nature of spectra obtained using the STFT, each peak (location and height) found by finding the maximum-magnitude frequency bin is only accurate to within half a bin. A bin represents a frequency interval of Hz, where is the FFT size. Zero-padding increases the number of FFT bins per Hz and thus increases the accuracy of the simple peak detection. However, to obtain frequency accuracy on the level of of the distance from a sinc maximum to its first zero crossing (in the case of a rectangular window), the zero-padding factor required is . (Note that with no zero padding, the STFT analysis parameters are typically arranged so that the distance from the sinc peak to its first zero-crossing is equal to the fundamental frequency of a harmonic sound. Under these conditions, of this interval is equal to the relative accuracy in the fundamental frequency measurement. Thus, this is a realistic specification in view of pitch discrimination accuracy.) Since we would nominally take two periods into the data frame (for a Rectangular window), a Hz sinusoid at a sampling rate of KHz would have a period of samples, so that the FFT size would have to exceed one million. A more efficient spectral interpolation scheme is to zero-pad only enough so that quadratic (or other simple) spectral interpolation, using only bins immediately surrounding the maximum-magnitude bin, suffices to refine the estimate to accuracy. PARSHL uses a parabolic interpolator which fits a parabola through the highest three samples of a peak to estimate the true peak location and height (cf. Fig.I.2).

We have seen that each sinusoid appears as a shifted window transform which is a sinc-like function. A robust method for estimating peak frequency with very high accuracy would be to fit a window transform to the sampled spectral peaks by cross-correlating the whole window transform with the entire spectrum and taking and interpolated peak location in the cross-correlation function as the frequency estimate. This method offers much greater immunity to noise and interference from other signal components.

To describe the parabolic interpolation strategy, let's define a
coordinate system centered at
, where is the bin
number of the spectral magnitude maximum, *i.e.*,
for all
. An example is shown in
Figure 4. We desire a general parabola of the form

(I.2) | |||

(I.3) | |||

(I.4) |

We have found empirically that the frequencies tend to be about twice as accurate when dB magnitude is used rather than just linear magnitude. An interesting open question is what is the optimum nonlinear compression of the magnitude spectrum when quadratically interpolating it to estimate peak locations.

Solving for the parabola peak location , we get

Once an interpolated peak location has been found, the entire local maximum in the spectrum is removed. This allows the same algorithm to be used for the next peak. This peak detection and deletion process is continued until the maximum number of peaks specified by the user is found.

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (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.

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