Spectrum Analysis of an Oboe Tone

In this section we compare three FFT windows applied to an oboe recording. The examples demonstrate that more gracefully tapered windows support a larger spectral dynamic range, at the cost of reduced frequency resolution.

Rectangular-Windowed Oboe Recording

Figure 3.16a shows a segment of two quasi periods from an oboe recording at the pitch C4, and Fig.3.16b shows the corresponding FFT magnitude. The window length was set to the next integer greater than twice the period length in samples. The FFT size was set to the next power of 2 which was at least five times the window length (for a minimum zero-padding factor of 5). The complete Matlab script is listed in §F.2.5.

Figure 3.16: (a) Rectangularly windowed segment of two periods from the steady state portion of an oboe recording of pitch `C4'. (b) Zero-padded FFT magnitude.
\includegraphics[width=\twidth]{eps/oboeboxcar}


Hamming-Windowed Oboe Recording

Figure 3.17a shows a segment of four quasi periods from the same oboe recording as in the previous figure multiplied by a Hamming window, and Fig.3.17b shows the corresponding zero-padded FFT magnitude. Note how the lower side-lobes of the Hamming window significantly improve the visibility of spectral components above 6 kHz or so.

Figure 3.17: (a) Hamming-windowed segment of four periods from the steady state portion of an oboe recording of pitch `C4'. (b) Zero-padded FFT magnitude.
\includegraphics[width=\twidth]{eps/oboehamming}


Blackman-Windowed Oboe Recording

Figure 3.18a shows a segment of six quasi periods from the same oboe recording as in Fig.3.16 multiplied by a Blackman window, and Fig.3.18b shows the corresponding zero-padded FFT magnitude data. The lower side lobes of the Blackman window significantly improve over the Hamming-window results at high frequencies.

Figure 3.18: (a) Blackman-windowed segment of six periods from the steady state portion of an oboe recording of pitch `C4'. (b) Zero-padded FFT magnitude.
\includegraphics[width=\twidth]{eps/oboeblackman}


Conclusions

In summary, only the Blackman window clearly revealed all of the oboe harmonics. This is because the spectral dynamic range of signal exceeded that of the window transform in the case of rectangular and Hamming windows. In other words, the side lobes corresponding to the loudest low-frequency harmonics were comparable to or louder than the signal harmonics at high frequencies.

Note that preemphasis (flattening the spectral envelope using a preemphasis filter) would have helped here by reducing the spectral dynamic range of the signal (see §10.3 for a number of methods). In voice signal processing, approximately $ +6$ dB/octave preemphasis is common because voice spectra generally roll off at $ -6$ dB per octave [162]. If $ X(\omega)$ denotes the original voice spectrum and $ X_p(\omega)$ the preemphasized spectrum, then one method is to use a ``leaky first-order difference''

$\displaystyle X_p(\omega) = (1-0.95\,e^{-j\omega T})X(\omega).$ (4.30)

For voice signals, the preemphasized spectrum $ \vert X_p(\omega)\vert$ tends to have a relatively ``flat'' magnitude envelope compared to $ \vert X(\omega)\vert$ . This preemphasis can be taken out (inverted) by the simple one-pole filter $ 1/(1-0.95z^{-1})$ .


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Bartlett (``Triangular'') Window
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Blackman-Harris Window Family