Analyze signals octave by octave
Although it gives less detail than spectrum analysis, octave analysis better emulates how we perceive sound.
Sam Shearman, National Instruments, Austin, TX -- Test & Measurement World, 4/15/2001
Power-spectral plots generated by frequency sweeps or fast Fourier transforms (FFTs) can show you a signal’s frequency content, but sometimes an FFT provides more detail than you need. If you design and test audio equipment or electromechanical devices and aren’t interested in detail as much as you are in representing a person’s hearing, octave analysis (or fractional octave analysis) may be a better option.
Octave analysis displays a signal’s frequency characteristics in frequency bands where each frequency band covers an octave—a band from frequency f to frequency 2f such as from 250 Hz to 500 Hz. Each band, therefore, occupies a bandwidth that’s twice as wide as the previous band and half as wide as the next band. Think of octave analysis as passing a signal through a series of bandpass filters, each covering one octave. Figure 1 shows how octave analysis divides the audio-frequency band into octaves.
Octave analysis provides insight into how people perceive sounds because it lets you compare signal levels across broad frequency ranges. Breaking frequencies into octaves helps you measure the subjective qualities of sound. For example, you can use octave analysis to find the frequencies and levels at which a sound becomes uncomfortable to the human ear. Or, you might use octave analysis during the manufacturing test of a computer hard drive, automobile engine, or consumer device to test for objectionable sound levels and frequency ranges.
Octave analysis shares some similarities with power-spectral analysis, but in spectral analysis all frequency bands (bins) occupy equal bandwidth. For example, in a 1000-point FFT that covers DC to 10 kHz, each bin occupies 10 Hz. The frequency resolution is linear over the entire range. Because of its log scale, octave analysis can show results over multiple octaves, allowing you to see and compare signal levels over a broad frequency range.
Each octave band’s frequency range falls between fl and fh, where fh=2fl. The octave’s 2:1 ratio sets a geometric spacing between octave bands, meaning that the bandwidth of each octave maintains a constant ratio with its geometric mean. To calculate the geometric mean of the octave’s end frequencies, use the equation below:
For greater resolution, you can divide the spectrum into geometrically equal subdivisions of each band. For instance, 1/3 octave analysis sets the band’s highest frequency (fh) and lowest frequency (fl) so that fh = fl · 21/3. Other common fractional octave analyses include 1/6, 1/12, and 1/24 of an octave. You can choose even more resolution by setting an octave bandwidth of 1/n so that
f h = f1· 21/n. By convention, the first full octave chosen for signal analysis has a center frequency of 1 kHz. The other octave exists at center frequencies above and below 1 kHz such as at 250 Hz, 500 Hz, 2 kHz, 4 kHz, and so on.
Figure 1. Octave analysis passes a time-domain signal through a series of octave (or a fraction of an octave) band-pass filters, producing an octave plot.
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| Figure 2. Fractional octaves provide more resolution than full octaves, yet they still retain the shape that's lost in spectral analysis. |
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| Figure 3. Octave analysis shows that people will hear some, but not all, frequencies from a race car (red) over traffic noise (yellow). |
Figure 2 compares a spectral plot (red trace) to full-octave analysis (green trace) and to 1/3 octave analysis (yellow trace). Note that the octave analysis produces bar graphs while the FFT spectrum shows a single spike at 1 kHz. Octave analysis, though, shows a series of bar graphs where each bar represents the power in one octave or in 1/3 octave. Common octave analyses will contain around 10 of these so-called octave bands.
At first glance, you might expect octave analysis to produce a plot of a single bar surrounding the signal’s frequency. Yet, Figure 2 shows the signal containing energy across several octaves. How can an octave-analysis plot produce bars in more than one octave when the power spectrum shows a single frequency? ANSI and IEC-compliant octave analysis standards specify digital filters that have some roll-off—they emulate analog filters. The simulated analog filters spread the signal’s spectrum over several octaves.Applying the analysis
You can apply octave analysis in psychoacoustic measurements (Ref. 1). For example, you could characterize the signal components that comprise office sounds, traffic, aircraft noise, machinery, and other noise sources. This characterization lets you compare these sounds to criteria set by noise standards. ISO 7779, for example, specifies the measurement of airborne noise emitted by computers and business equipment (Ref. 2). For measurement uniformity, this standard and others commonly reference standards such as ANSI S1.11-1986 and IEC 61260:1995 that define how to perform octave and fractional octave analysis (Refs. 3, 4).
When applying octave analysis for noise measurement, tone perception, and other psychoacoustic measures, you are often trying to estimate a perception. In such circumstances, you aren’t necessarily interested in the detail of an FFT spectrum but in judging subjective qualities such as loudness, harshness, or annoyance. For these jobs, octave analysis gives you the opportunity to employ visual inspection and comparison.
To appreciate the usefulness of this visual inspection, consider Figure 3, which compares the sound of a race car (red trace) to the sound of everyday traffic noise (yellow trace). The everyday traffic produces more of a drone, with more low-frequency components and less deviation than does the race car. T&MW
References
1. Zwicker, Eberhard, H. Fastl, and H. Frater, Psychoacoustics Facts and Models, 2nd ed., Springer-Verlag, Berlin, 1999.
2. ISO 7779:1999, Acoustics - Measurement of airborne noise emitted by information technology and telecommunications equipment, International Standards Organization (ISO), Geneva, Switzerland. www.iso.ch.
3. ANSI S1.11-1986 (Revised 1998), American National Standard Specification for Octave-Band and Fractional-Octave-Band Analog and Digital Filters, Acoustical Society of America, New York, NY. asa.aip.org/standards/s1.html.
4. IEC 61260(1995-08), Electroacoustics - Octave-band and fractional-octave-band filters, International Electrotechnical Commission (IEC), Geneva, Switzerland. www.iec.ch .
For more information
Crocker, Malcolm J., Handbook of Acoustics, John Wiley & Sons, New York, NY, ed., 1998.
Syreeni, Sampo, “Sound, synthesis and audio reproduction,” an online document found at
www.iki.fi/~decoy/dsound/dsound.
Sam Shearman is the signal processing and analysis software product manager at National Instruments. He has a BSEE from Georgia Institute of Technology.
