Raven's Spectrum Analysis: Understanding Time-Varying Signals with Fourier (80 characters), Study notes of Biology

Download Raven's Spectrum Analysis: Understanding Time-Varying Signals with Fourier (80 characters) and more Study notes Biology in PDF only on Docsity! Appendix B A Biologist’s Introduction to Spectrum Analysis About this appendix This appendix provides some conceptual background for making and interpreting spectrogram and spectrogram slice views with Raven. It introduces the short-time Fourier transform (STFT), the mathematical technique used by Raven for making spectrograms. We do not discuss the mathematics of the STFT, but instead treat it here as a black box. This black box has controls on its outside that affect its operation in important ways. One aim of this appendix is to convey enough qualitative understanding of the behavior of this box to allow intelligent use of its controls, without delving into the box’s internal mechanism. Specific details of the controls are covered in Chapter 3, “Spectrographic Analysis”. A second aim of this appendix is to explain some of the limitations and tradeoffs intrinsic to spectrum analysis of time-varying signals. More rigorous mathematical treatments of spectral analysis, at several levels of sophistication, can be found in the references listed at the end of the appendix. Several approaches can be taken to explaining the fundamentals of digital spectrum analysis. The approach taken in this appendix is geared specifi- cally to spectrum analysis with Raven; thus some of the terms and con- cepts used here may not appear in other, more general discussions of spectrum analysis, such as those listed at the end of the appendix. The discussions in this appendix assume a basic understanding of how sound is recorded and represented digitally. If you are not already acquainted with concepts such as sampling rate and sample size, you should read Appendix A, “Digital Representation of Sound” before pro- ceeding. What sound is Sound consists of traveling waves of alternating compression and rarefac- tion in an elastic medium (such as air or water), generated by some vibrat- ing object (a sound source). Sound pressure is the (usually small) alternating incremental change in pressure from ambient pressure that results from a sound. When no sound is present in a medium (i.e., there is no propagating pressure change), we say that sound pressure is zero, even though the medium does exert some static ambient pressure. The dimensions of pressure are force per unit Raven 1.2 User’s Manual 137 Appendix B: Spectrum Analysisarea. The usual unit of sound pressure is the pascal (abbreviated Pa); one pascal equals one newton per square meter. Since the smallest audible sound pressures in air are on the order of 10-6 Pa, sound pressures are usu- ally expressed in µPa. To measure or record sound at a particular location in space, we use a device such as a microphone that responds to sound pressure. A micro- phone produces a time-varying electrical voltage that is proportional to the increase or decrease in local pressure that constitutes sound. This con- tinuous time-varying voltage is an electric analog of the acoustic signal. The continuous electric signal can be converted to a digital representation suitable for manipulation by a computer as discussed in Appendix A, “Digital Representation of Sound”. Time domain and frequency domain representations of sound Any acoustic signal can be graphically or mathematically depicted in either of two forms, called the time domain and frequency domain represen- tations. In the time domain, instantaneous pressure is represented as a function of time. Figure B.1a shows the time domain representation of the simplest type of acoustic signal, a pure tone. Such a signal is called a sinu- soid because its amplitude is a sine function of time, characterized by some frequency, which is measured in cycles per second, or Hertz (Hz). The fre- quency of a sinusoid is most easily determined by measuring the length of one period, which is the reciprocal of the frequency. The amplitude of the signal in the time domain is measured in pressure units. (Once an acoustic signal has been converted by a microphone into an electrical signal, ampli- tude is measured as voltage, which is directly proportional to the sound pressure.) In the frequency domain, the amplitude of a signal is repre- sented as a function of frequency. The frequency domain representation of a pure tone is a vertical line (Figure B.1b).138 Raven 1.2 User’s Manual Appendix B: Spectrum AnalysisFigure B.3. DFT schematic Figure B.3. Schematic representation of the discrete Fourier transform (DFT) as a black box. The input to the DFT is a sequence of digitized amplitude values (x0, x1, x2, ... xN-1) at N discrete points in time. The number of input values N is called the DFT size. The output is a sequence of amplitude values (A0, A1, A2, ... A(N/2)) at N/2 discrete fre- quencies. The highest frequency, f(N/2)-1, is equal to half the sampling rate (= 1/(2T), where T is the sampling period, as shown in the figure). The output can be plotted as a magnitude spectrum. In practice, a spectrum is always made over some finite time interval. This interval may encompass the full length of a signal, or it may consist of some shorter part of a signal. Spectral analysis of time-varying signals: spectrograms and STFT analysis Most signals of biological interest change over time in frequency (spectral) composition. Indeed the changes in spectrum over time are often among the most interesting aspects of such signals. But in order to create a spec- trum, we must examine an interval of time— there is no way to measure a signal’s “instantaneous” spectrum. An individual magnitude spectrum of a signal provides no information about temporal changes in frequency composition during the interval over which the spectrum is made. If we were to make a single magnitude spectrum over the entire duration of a spectrally varying signal such as a typical bird song, we would have a rep- resentation of the relative intensities of the various frequency components of the signal, but we would have no information about how the intensities of different frequencies varied over time during the signal. To see how the frequency composition of a signal changes over time, we can examine a sound spectrogram.1 The spectrograms produced by Raven 1. Sound spectrograms are sometimes called sonagrams. Strictly speaking, how- ever, the term sonagram is a trademark for a sound spectrogram produced by a particular type of spectrum analysis machine called a Sonagraph, produced by the Kay Elemetrics Co.Raven 1.2 User’s Manual 141 Appendix B: Spectrum Analysisplot frequency on the vertical axis versus time on the horizontal; the amplitude of a given frequency component at a given time is represented by a color (by default, grayscale) value (Figure B.4). Figure B.4. Spectrogram example. Figure B.4. Smoothed sound spectrogram of part of a song of a chest- nut-sided warbler, digitized at 44.1 kHz. Spectrograms are produced by a procedure known as the short-time Fourier transform (STFT). The STFT divides the entire signal into a series of succes- sive short time segments, called records (or frames). Each record is used as the input to a DFT, generating a series of spectra (one for each record). To display a spectrogram, the spectra of successive records are plotted side by side with frequency running vertically and amplitude at each fre- quency represented by a color (by default, grayscale) value. Raven’s spec- trogram slice view displays the spectrum of one record at a time as a line graph, with frequency on the horizontal axis, and amplitude on the verti- cal axis. A spectrogram can be characterized by its DFT size, expressed as the number of digitized amplitude samples that are processed to create each individual spectrum. The STFT can be considered as equivalent in function to a bank of N/2 + 1 bandpass filters, where N is the DFT size. Each filter is centered at a slightly different analysis frequency. The output amplitude of each filter is proportional to the amplitude of the signal in a discrete frequency band or bin, centered on the analysis frequency of the filter. In this “filterbank” model of STFT analysis, the spectrogram is considered as representing the time-varying output amplitudes of filters at successive analysis frequen- cies plotted above each other, with amplitude again represented by color (by default, grayscale) values. A spectrogram can be characterized by its bandwidth, the range of input frequencies around the central analysis fre- quency that are passed by each filter. All of the filters in a spectrogram have the same bandwidth, irrespective of analysis frequency. Record length, bandwidth, and the time-frequency uncertainty principle The record length of a STFT determines the time analysis resolution (∆t) of the spectrogram. Changes in the signal that occur within one record (e.g., the end of one sound and the beginning of another, or changes in fre-142 Raven 1.2 User’s Manual Appendix B: Spectrum Analysisquency) cannot be resolved as separate events. Thus, shorter record lengths allow better time analysis resolution. Similarly, the bandwidth of a STFT determines the frequency analysis resolu- tion (∆f) of the spectrogram: frequency components that differ by less than one filter-bandwidth cannot be distinguished from each other in the out- put of the filterbank. Thus a STFT with a relatively wide bandwidth will have poorer frequency analysis resolution than one with a narrower band- width. Ideally we might like to have very fine time and frequency analysis resolu- tion in a spectrogram. These two demands are intrinsically incompatible, however: the record length and filter bandwidth of a STFT are inversely proportional to each other, and cannot be varied independently. Although a short record length yields a spectrogram with finer time analysis resolu- tion, it also results in wide bandwidth filters and correspondingly poor frequency analysis resolution. Thus a tradeoff exists between how pre- cisely a spectrogram can specify the spectral (frequency) composition of a signal and how precisely it can specify the time at which the signal exhib- ited that particular spectrum. The relationship between record length and filter bandwidth applies to each of the individual spectra that collectively constitute a spectrogram. Figure B.5 illustrates the relationship between record length and filter bandwidth in individual spectra. The two spectra, of a 2000 Hz pure tone digitized at 22.05 kHz, were made with different record lengths and thus different bandwidths. Spectrum (a), with a record length of 1024 points (46.0 mS) , shows a fairly sharp peak at 2000 Hz because of its relatively narrow bandwidth (35.3 Hz) filter; spectrum (b), with a record length of 256 points (11.5 mS), corresponding to a wider bandwidth (141 Hz) filter, has poorer frequency resolution.Raven 1.2 User’s Manual 143 Appendix B: Spectrum Analysisrecord length of 64 points (= 2.9 mS; bandwidth = 496 Hz), the beginning and end of each tone can be clearly distinguished and are well-aligned with the corresponding features of the waveform. However, the frequency analysis resolution is poor: each tone appears as a bar that is nearly 1200 Hz in thickness. In spectrogram (b), the record length is 512 points, or 23 mS (filter bandwidth = 61.9 Hz), or about as long as each tone in the signal. Most of the records therefore span more than one tone, in some cases including a tone and a silent interval, in other cases including two tones and an interval. The result is poor time resolution: the beginning and end of the bars representing the tones are fuzzy and poorly aligned with fea- tures of the waveform (compare, for example, the beginning time of the first pulse in the waveform with the corresponding bar in the spectro- gram). However, this spectrogram has much better frequency resolution than spectrogram (a): the bar representing each tone is only about 100 Hz in thickness. Figure B.7. Time vs freq resolution. Figure B.7. Effect of record length and filter bandwidth on time and fre- quency resolution. The signal consists of a sequence of four tones with frequencies of 1, 2, 3, and 4 kHz, at a sampling rate of 22.05 kHz. Each tone is 20 mS in duration. The interval between tones is 10 mS. Both spectrograms have the same time grid spacing = 1.45 mS, and window function = Hann. The selection boundaries show the start and end of the second tone. (a) Wide-band spectrogram: record length = 64 points ( = 2.90 mS), 3 dB bandwidth = 496 Hz. (b) Waveform, showing timing of the tones. (c) Narrow-band spectrogram: record length = 512 points ( = 23.2 mS), 3 dB bandwidth = 61.9 Hz. The waveform between the spectrograms shows the timing of the pulses.146 Raven 1.2 User’s Manual Appendix B: Spectrum AnalysisWhat is the “best” window size to choose? The answer depends on how rapidly the signal’s frequency spectrum changes, and on what type of information is most important to show in the spectrogram, given your particular application. For many applications, Raven’s default window size (512 samples) provides a reasonable balance between time and fre- quency resolution. If you need to observe very short events or rapid changes in the signal, a shorter window may be better; if precise frequency representation is more important, a longer window may be better1. If you need better time and frequency resolution than you can achieve in one spectrogram, you may need to make two spectrograms: a wide-band spec- trogram with a small window for making precise time measurements, and a narrow-band spectrogram with a larger window for precise frequency measurements. Time grid spacing and window overlap Time grid spacing is the time between the beginnings of successive records. In an unsmoothed spectrogram, this interval is visible as the width of the individual boxes (Figure B.6). Successive records that are ana- lyzed may be overlapping (positive overlap), contiguous (zero overlap), or discontiguous (negative overlap). Overlap between records is usually expressed as a percentage of the record length. Figure B.8 illustrates the different effects of changes to record length and time grid spacing. The signal is a frequency-modulated tone that sweeps upward in frequency from 4 to 6 kHz, sampled at 22.05 kHz. Spectrograms (a) and (c) both have a record length of 512 points (= 23.2 mS; 3 dB band- width = 61.9 Hz). (a) was made with 0% overlap (time grid spacing = 23.2 mS), whereas (c) was made with an overlap of 93.8% (time grid spacing = 1.45 mS). In the low-resolution spectrogram (a), each box is as wide as one data record, which in turn is one quarter of the length of the tone. The result is a spectrogram that gives an extremely misleading picture of the signal. Spectrogram (c), with a greater record overlap, is much “smoother” than the one with less overlap, and it more accurately portrays the contin- uous frequency modulation of the signal. It still provides poor time analy- sis resolution, however, because of its large record length— notice the fuzzy beginning and end of the spectrogram image of the tone and the poor alignment with the beginning and end of the tone in the waveform. Comparison of the spectrograms in Figure B.8 demonstrates that improved time grid spacing is not a substitute for finer time analysis reso- lution, which can be obtained only by using a shorter record. 1. If the features that you’re interested in are distinguishable in the waveform (e.g., the beginning or end of a sound, or some other rapid change in ampli- tude), you’ll achieve better precision and accuracy by making time measure- ments on the waveform rather than the spectrogram.Raven 1.2 User’s Manual 147 Appendix B: Spectrum AnalysisFigure B.8. Window size window size - overlap Figure B.8. Different effects on spectrograms of changing record length (= window size, or time analysis resolution) and time grid spac- ing. The signal is a frequency-modulated tone, 100 mS long, sampled at 22.05 kHz. The tone sweeps upward in frequency from 4 to 6 kHz. Spectrograms (a) and (c) have the same window size, but (c) has finer time grid spacing (higher record overlap). (b) and (c) have the same time grid spacing, but (c) has a shorter record length (finer time analysis resolution). (a) Record length = 512 points = 23.2 mS (3 dB bandwidth = 61.9 Hz); Time grid spacing = 23.2 mS (overlap = 0%). (b) Waveform view, with duration of tone highlighted. (c) Record length = 512 points = 23.2 mS (3 dB bandwidth = 61.9 Hz); Time grid spacing = 1.45 mS (overlap = 93.8%). (d) Record length = 64 points = 2.9 mS (3 dB bandwidth = 448 Hz); Time grid spacing = 1.45 mS (overlap = 50%). Frequency grid spacing and DFT size Frequency grid spacing is the difference (in Hz) between the central analy- sis frequencies of adjacent filters in the filterbank modeled by a STFT, and thus the size of the frequency bins in a spectrogram. In an unsmoothed spectrogram, this spacing is visible as the height of the individual boxes (Figure B.6). Frequency grid spacing depends on the sample rate (which is fixed for a given digitized signal) and DFT size. The relationship is frequency grid spacing = (sample rate) / DFT size where frequency grid spacing and sample rate are measured in Hz, and DFT size is measured in samples. Thus a larger DFT size draws the spec-148 Raven 1.2 User’s Manual Appendix B: Spectrum AnalysisThird, each filter does not completely block the passage of all frequencies outside of its nominal passband. For each filter there is an infinite series of diminishing sidelobes in the filter’s response to frequencies above and below the passband (Figure B.10). These sidelobes arise because of the onset and termination of the portion of the signal that appears in a single record. Since a spectrum of a pure tone made by passing the tone through a set of bandpass filters resembles the frequency response of a single filter (Figure B.9), a STFT spectrum of any signal (even a pure tone) contains sidelobes. Figure B.10. Filter sidelobes Figure B.10. Frequency response of a hypothetical bandpass filter from a set of filters simulated by a short-time Fourier transform, show- ing sidelobes above and below the central lobe, or passband. The mag- nitude of the sidelobes relative to the central lobe can be reduced by use of a window function (see text). Note that a spectrum produced by passing a pure tone through a set of overlapping filters is shaped like the frequency response of a single one of the filters (see Figure B.9). Window functions The magnitude of the sidelobes (relative to the magnitude of the central lobe) in a spectrogram or spectrum of a pure tone is related to how abruptly the windowed signal’s amplitude changes at the beginning and end of a record. A sinusoidal tone that instantly rises to its full amplitude at the beginning of a record, and then instantly falls to zero at the end, has higher sidelobes than a tone that rises and falls gradually in amplitude (Figure B.11).Raven 1.2 User’s Manual 151 Appendix B: Spectrum AnalysisFigure B.11. Windowing Figure B.11. Relationship between abruptness of onset and termina- tion of signal in one record and spectral sidelobes. Each panel shows a signal on the left, and its spectrum on the right. (a) A single record of an untapered sinusoidal signal has a spectrum that contains a band of energy around the central frequency, flanked by sidelobes, as if the signal had been passed through a bank of bandpass filters like the one shown in Figure B.10. (b) A single record of a sinusoidal signal multiplied by a “taper” or win- dow function, has smaller sidelobes. The magnitude of the sidelobes in a spectrum or spectrogram can be reduced by multiplying the record by a window function that tapers the waveform as shown in Figure B.11. Tapering the waveform in the record is equivalent to changing the shape of the analysis filter (in particular, lower- ing it sidelobes). Each window function reduces the height of the highest sidelobe to some particular proportion of the height of the central peak; this reduction in sidelobe magnitude is termed the sidelobe rejection, and is expressed in decibels (Table B.1). Given a particular record length, the choice of window function thus determines the sidelobe rejection, and also the width of the center lobe. The width of the center lobe in the spectrum of a pure tone is the filter bandwidth. Table B.1. Sidelobe rejection for Raven’s five window types. The sidelobe rejection for each type is expressed as the height of the highest sidelobe relative to the peak of the main lobe. Window type Sidelobe rejection (dB) Blackman -57 Hamming -41 Hann -31 Rectangular -13 Triangular -25152 Raven 1.2 User’s Manual Appendix B: Spectrum AnalysisFor further reading The books and articles listed below can provide entry at several levels into the vast literature on spectrum analysis and digital signal processing. Beecher, M. D. 1988. Spectrographic analysis of animal vocalizations: Implications of the “uncertainty principle.” Bioacoustics 1:(1): 187- 207. Includes a discussion of choosing an “optimum” filter bandwidth for the analysis of frequency-modulated bioacoustic signals. Bradbury, J. and S. Vehrencamp. 1998. Principles of Animal Communica- tion. Sinauer Associates, Sunderland, MA. 882 pp. Chapter 3 provides an excellent introduction for non-specialist readers to the principles of spectrum analysis, and also discusses spectral properties of the basic types of animal acoustic signals. Cohen, L. 1995. Time-frequency analysis. Prentice-Hall, Englewood Cliffs, NJ. Hlawatsch, F. and G.F. Boudreaux-Bartels. 1992. Linear and quadratic time-frequency signal representations. IEEE Signal Processing Magazine, 9(2): 21-67. A technical overview and comparison of the properties of a variety of time-frequency representations (including spectrograms), writ- ten for engineers. Jaffe, D. A. 1987. Spectrum analysis tutorial. Part 1: The Discrete Fourier Transform; Part 2: Properties and applications of the Discrete Fou- rier Transform. Computer Music Journal, 11(3): 9-35. An excellent introduction to the foundations of digital spectrum analysis. These tutorials assume no mathematics beyond high school algebra, trigonometry, and geometry. More advanced math- ematical tools (e.g., vector and complex number manipulations) are developed as needed in these articles. Marler, P. 1969. Tonal quality of bird sounds. In: Bird Vocalizations: Their Relation to Current Problems in Biology and Psychology (ed. R. A. Hinde), pp. 5-18. Cambridge University Press. Includes an excellent qualitative discussion of how the time and frequency analysis resolution of a spectrum analyzer interact with signal characteristics to affect the “appearance” of a sound either as a spectrogram or as an acoustic sensation. Oppenheim, A.V. and Schafer, R.W. 1975. Digital Signal Processing. Pren- tice-Hall, Englewood Cliffs, NJ. xiv + 585 p. A classic reference, written principally for engineers.Raven 1.2 User’s Manual 153