Chirp design for acoustical system identification John C. Burgessal 1 Deprimeni o f Mechanical Engineedng, Uniwrsiiy o f Haunii. 2540 Doie Street, Hdulu. Hamii 96822

(Received 15 March 1991; acceptcd for publication 28 Octobcr 199 1 )

A r e ~ t e d linear chirp (choip) is a broadband waveform with the lowest possiblt peak to rms amplitude ratio ( p u k fuctor) also having the property that the frcqumcy band fi>r its power concentration @wer band) can be specified. A low peak factor is dsirable to maximi7e signal power while avoiding clipping or saturation effects. The capability to concentrate power is desirable to maximize the signal to noise ratio. Choips, having both properties, makc reduced measurement time possible in practical system identification relative to impulse and maximum lengtli sequence waveforms. This is particularly useful with time-varying systems Choip for use with andog-digital data acquisition systems can be daigncd casily and quickly. They caii bc designed to minimizz leakage and aliasing. For application to a typicnl acoustical system. over 99% of a choip's power can be concent rated in a specified power band.

PACS numbcrs: 43.60 - c, 43.85.Kr

INTRODUCTION 4

The p k factor of a waveform (also called a crest fa& tor) is its peak to rms amplitude ratio. Thepo~rbundof a wavefonn is the frequmcy band within which its power is conmtrated, and band limits refer to the lower and uppcr frequency limits ofa power band. A quick, simpIe, and effec- tive procedure is proposed for designing waveforms having both a 1ow peak factor and a desired power band. The proce- dure is intended for use with analog-digital data aquisition (input/output, or I/O) systemzi where data are proc.es.. by the fart Fourier transfonn (FTT). The procedure involves interaction among band limits, frequency resolution, FFT block size, I/O sarnpling rate. leakage, and aliasing. The ba- sic wavefvrm is a linear chirp that is repeated with a regular period. This wavcform is called a "choip" in the following discussion (the "o" is a reminder that the waveform is circu- lar). A major expectd application is to system identification by physical measurement, w h m a system's impulse re- sponse, or equivalently its Fourier transform, the frequency response. is a desired outcome.

A linear chirp is a waveform w h m instantaneous fre- quency i n c r e í í lincirly with time between two specified frequencies. Single FM chirps are uscd in sonar and radar applications, whcre they have well-known properties for tar- gct dctection and for estimating target rangc and ve1ocity.'-' Birds, whales. and bats are known to use chirplike acoustic signals for ranging and/or cominunication. While single chirps have received much attention, choips, despite their importante to practica1 system identifimtion, appear less well known. In the latter context, modified forms of repeat- ed, stepped-frequency waveforms have been proposed. but discussion of the features and advantages of the direct and more simple choip havc been omitted.' While a commercial software package includes a broadband choip as an excita- tion w a ~ d o t m . ~ ihe docurnentation does no1 identify the

'' lhls uiork wac conductcd in the AcowIics ind Signai Pnxrnins Sectmn d the National Rexarch Council ol Canada whik thc author w ~ s on b v e frmi thc Univcnity of H a d i .

d h o i p properties that are ihe focus here, and there is no pro-

-, vision for specifying a power band.

A low peak factor waveform with ik pwer concentrat- ed in a dcsired power band ha. thc propcrty that the highest feasible signal to noise leve1 can be achieved in the desired band. When such a waveform is used for systeni identifica- tion, the required measuremmt time can bequite short rela- tive to the time required for repetitive measurernents with other waveforms. For ernmple, impulse excitation may re- quire repetition to reduce the effccts of noise, random noisc excitation requires repetiiion to achievestatistical reliability, and sequeniial sinusoidal excitation requires many measurt- ments. A short nieasureinent time is pariicuIarly valuable with time-varying systems, such as the atmosphere, the ocmn,' and biological apecimens.

Waveforms used for system identification muqt be gen- erated by and must transit cquipment that limits the peak power transrnitted without distortion. While an impulse is theoretically the ideal broadband wavefom with which to identify a systern. its high peak factor and the inability to specify a power band are well-known practica1 disadvan- tages. The high peak factor disadvantage ha.. led to consider- able effort to identify broadband low peak factor waverorms. This effort has been concentrated in two arcas: waveform speciíication by hannonic magnitudes and phases in the f r e quency dornain, and as maximum length sequences in ihe time domain.

One way to obtain a low peak factor wavefonn is to specify it in the frequency domain as a sum of equal arnpli- tude harmonics with frequencies limited to a desired power band and with phases selected appropnately." Unfortu- nately, theoptimization process required toselect the phases can take a significant amount of computer time. Pumplina hxs reporid up to 30 days machine time requird on an IBM AT-class computer. Furthennore, while the results appear to be locally optimum sets of ph-, there appean to be no eficient methd to assure a global optimum.

Maximurn length sequcnces (MLSs) provide another way to obtain low peak íactor wavef~rrns?'~ MLSs are peri-

1525 J. A w t Coc. Am. 91 (3). March 1892 üüül-4966/92/031525~.80 @ 1882 Acdusüeel Soeiety al America 1525

odic binary sequcnces. They are special forms of pscudoran- dom noise, hriving only two values, such as + 1 and - 1, where timc durations at a c h valuc vary so that the wave- form has a broud power band. While MLS waveforms are simple to generate, it is difficuli to conlrol thcir power bands.

In contrast to waveform specification in the frequency domain or as MLSs, specificntion by either chirp or choip provides a waveform guamiiteed to have the optimai peak factor, 2"'. that is consisteni with the property that its pow- cr band can be spccificd. An attractive additionai feature of choips is thnt power bands can be specifid so that the wave- form has little power near thc Nyquist frequency (Sec 11). This d u c s the effccts of aliasing, and may make pcrisible some mcasuremcnts without the need for antialiasing Alters.

l. BASIC EQUATIONS The arnplitudc of a discrctc-time chirp waveform isl'

A cos &n), wlierc the general quadratic form of the time- dcpendcnt signal phase is +(t i ) = a + h + cn2, n is an in- tegcr time index (La, norrnalized diwreie time), A is a con- stant waveforrn maxirnurn arnplitudc, and a. 6, and c are constmts. A choip is formed by concalcnating chirps,

x(n) = A w 4 ( n ) mod N, (la) whcrcNis the number ofdata valucrr

xc*) = A a5 Ld (a

Timo Indox. n

(b)

O. 1

FIG. 1. (a) Wavdorm (sampled), choip wiUi s broad power band. O.OSc,fa 4.45 [m pcriod, O<n < N, N = 128). (b) Spcctmm (FFT) for wivcform in Fw 1 (a).

qudratic phase can be exprcssed in a form uscful for choip design,

whcre a is a mnstant phase factor and/, and y",, the nor- rnaiizcdkwer and ~ppe r band lirnits, are lirnited to the range O(fn (0.5. (Subscript n means "normalked;" otlicrwise, n mprescnts time index.) Timc indcx n = O corresponds to f, =/;, , and n = N corresponds to f, = f,, . For al1 exam- ples discussed here, A = 1, and N is an FFT block sím.

Normalized discrete frequency is a frequcncy ratio; ¡.e.,

A" =Al/; ( 2 ) is the normalized lower band limit, wtiere/, and/; are tlie lower band limit and m p l i n g frequency in Hz [also see Q. (611.

The instaritaneuus normalized frequency is deftried in continuous time os f, (t) = (d4/dt)nr- Applying this de- finition to Eq. ( lb) givcs

Figure 1 shows one period of a clioip [h. ( 1 ) ] and iis linearspectrum wlierc the band limits are selecte. to cover u broad power b d , from/, = 0.05 to f,, = 0.45. The appar- ent pcak amplilude variation shown in Fig. 1 (a) is a sam- pling artifact. Sarnpled vdues are seldorn precisely at wave- form p k s . Eyuation (1) assures that thc true pcak values are al1 tlie samc. Thc magnitude spcctrurn sliown >g. 1 (b) has ii linear ordinate wale to crnphasizc details ~,:ts varintion with frcqucncy. The maxirnurn pcak-to-pcak rip ple in the power band, approximately 4 iiB, is iypical of choip spectra.

11. IMPROVED EQUATIONS

The nonzcro spectral magnitudes scen in Fig. 1 (b) out- side thepowcr band mult from leakage,'2 thespreading of a spectmm thnt can occor when a Fourier transfonn is taken of u finite-leiigth seciiori of a waveforni. Leakage results

FIG. 2. Spcctmm (FFT), choip wilh a nanow power bnd, 0.1u. 50.2. Example of a "but" case (a = O, m c 154). wiih I he m h and Imt dc- rivativcs amiinuais at tlu wivdorm * ' e W (N = 1024).

1526 J. Aawst. Soc. Am., Vol. 01, No. 3, March 1982

when there are discontinuous derivatives at the "ends" of the finite-lcngth section." Since an FFT trcats a finitc-length waveform as if it were periodically extended, discontinuous derivatives normally occur at the periodic waveform "ends." It is well known that leakage can be reduced by using a data window that sets to zem one or more derivatives o ía wave- form at its "ends."i2.LJ

Reasoning used to control leakage can be applied with profit to design an improved choip. By imposing thc condi- tion that the wavcform amplitude must be + 1 (or - 1) at the "ends" ofthe waveform, both the zeroth and first deriva- tives of the waveform are guaranteed to be continuous. This condition is achieved for the lower band lirnit at n = O by setting a = O [Eq. (lb)]. The condition is achieved for the upper band limit at n = N by making thc phasc +(N) a multiple of 2lr. With a = 0, m an integer, and &N) = 2mn, Eq. ( 1 b ) can be rearranged to find

The procedure to design an irnproved clioip is toselect Narid the daired band limib, solve Eq. (4) for m, set m Lo the nearcst integcr, solve a rearranged Eq. (4) for eitherh or A'". e-g.9

and use the md i f id value in place of the dcsired valuc. A choip is formed by calcuiating values for a single chirp, plac- ing copies nase to tail in computer memory, and reading the entire sequmce into an I/O system. Unless explicitly slated othenvisc, the improvcd proccdure ha, becn used with al1 examples presented here.

Asan example oían irnprovcd choip with a narrow pow- er band, let N = 1024. l e t h = 0.1 and findf,,, 10.2. Equa- tion (4) gives m = 153.6; Eq. (5) with m roundcd to 154 results in f,, = 0.200 781. The spectra for this "best" case are shown in Fig. 2.

The effiects on leakage duc to mismatching derivatives can bescen by comparing Figs. 2 and 3. Figure 3 is an exam- ple of a "worst" case. where a = - n/2 [¡.e., x ( n ) = A - sin & ( a ) ] , and the fractional part of m is 0.5 (rn = 153.5, f, = 0.199 805). Some power band values are given in Ta- ble T.

For the special case where the desired power band is the entire frequency range, O u , (0.5, several choices of choip are sumrnarized in Table 11. Magnitude spectra range frorn nearly flat (a = - r/2: note srnall ripple) to zcro at both

F l u r m a 1 i zed Frsquency

FIG. 3. Spectrum (FFT), choip with a narrow powcr band, O.I(j, 50.2. Exarnplcda"worst"care (a = - n/2. m = 153.5). wiih a sirong d i m n - tinuity in ihe first cknvatinni ihc wnveíorrn "'ends" (N-. 1024).

ends of the power band (a = + n/4). While there are more useful choices for most practica1 applications in acoustics (see, e.g., Sec. IV), broadband choips can be designed luiv- ing signidcant pwer down to dc or with no pawer at dc

It mighi be reasaned that leakage can be eliminated by making al1 derivatives continuous at waveform "ends." i'his appears to be possible if a waverorm is fomed from an up- chirp (increiising frequency) followed by a minor image downchirp. Unfartunately, the s trum of such a wave- form. cornputed by FFT, shows v8">e rnagnitude varia- tions in the power bnd , making it unattractive for applica- tion to system identification.

The ripple displaycd in the magnitude spectrum of a chirp within its pwerband [Fig. I (b)] is reminisccnt ofthe Gibbs phenomcnon. But the Gibbs phenomenon conccrns ripple near discontinuities in a wavcform when an attempt is made to appmximate tlie wavefonn by a finite number of terms in its Fourier series expansion. The npple seen in Fig. 1 (b) is in the spectrum, not the waveform.

Small changes in waveforrn can rcsult in marked changes in p k spectra (see, e.g., Figs. 2 md 3). Thissensi- tivity illustrates the dificulty encountered in finding suit- able phase sets when a low peak factor waveform is specifieá in the frequency domain.'

TARLE 1. Gain at powcrhand lirnits, in-band pnwcr, m d in-band ripplc, í w a n b a t " ~ n d ~wonl"cl;irnplcoín n s r . w band chaip.QI u. Sü.2 (N a 1024).

Gain at Powcr Ri~ple

A 1". I n band Peak 10 peak fig. dB dD 96 Mna. dB

1527 J. Aco~~st. Soc. Am.. Vol. 91. No. 3. Mar& 1892 John C. Burgess 1527

TADLE 11. Giin ai p o m band limiis and h-band ripple f a four choips wiih pomr band q u a l to rntin nonnaüd frcquaicy range, O d v <0.5 ( .Va 1014).

Gain at Ripple

f. =O f = 0.5 Ptnktopak a dB dB Max. OB

While a choip has a line spectrum, a (single) chirp has a úequencyconiinuous spectrum. Spectra of ihe "best" and "worst" cases diussed abovc amount to sampling the cor- responding chirp spectra near the smailest and greatmt val- ursin tlieir lobestructures. Ari easy way tosee tliis is to pada chirp with a sufficient nurnbcrof zero valucs and examine its FFTspecirum. While values within ihe power band are little affccted, variations outside the powcr band (leakage) are signiricant.

111. DESIGN CRlTERlA AND PROCEDURE

The following steps outline the design procedure for an effective choip.

(1) Concentrate waveform power within the passband of the meaiurement systcm, including reconstmction and antidiasing filters. This passband sels the outside limiis for thc power band, (A),,, U<(/, ),,,, and assures that Fourier components needcd io maintain the low peak factor of the waveform are passed by the mcasurement system without significant distortion.

(2) Concentrate waveform power within the power band for which the target systcm frequency respnsc in Hz is desired,/,(fdm. This maximizes wavcform eíñciency (sig- nal to noisc ratio).

(3) Selecta value Iess than 0.5 for he normaiized upper band limit f,,. This will decrease waveform power ncar the Nyquist frcquency, íhus decreasing the effccts of aiirising.

(4) Spccify a rnaximum frequency resolution, (Af ),,. If a relative frcquency resolution is speciíied (cg., Af /f< l/lOoct), acommm pmctice in ncoustics, (4f 1, is the frequency resolution at the lower band limit,/,.

(5) Choose a combination of sampling me/, and FFT block size N to accommodale thc upper band limit fu, and the disired maximum frcquency resolution (u),,. This may require iteration íind/or multiple power bands. Tlie governing relationships are

L f l . - - ..-.. . .. ... ... * . . .* "" : .- _. . : ...-.,.-1:'.-.-y: A -. . -.-- . . . . . t..*.: . ..-.*-

: . .. . . .. as.-.S-. s..: ! -, : -..- -. .-.e. -- - - - .t.. . --..::; . . . . -. .-S .-e .- . ,- .

Normalizad Frequoncy

FIG. 4. Spctrum (m), choip wiih power band for the cntire audio fre- quency rnnge, 20 IIzC/<20 W z wiih 1 =50 kHz (N= 1024, 0.0004C/, 50.4).

is sufficienl for the specific application. This rnay be ajudge- ment cail. Revise N and iterate if nccessary.

(8) The maximum timeduration ofa choip may be lim- ited by timevarying propcrties of a target systcm or by ex- perimental system design. The time duration of a single chirp is a measure of the time ovcr which the target system is assumed to remnin essentially stationary. For measurements in threcdimensional space, the time between pnmary trans- mitted signal amval at the target system output transducer and the first echo or multipath signnl may further limit us- able choip time duration. The time duration of a single chirp is

T, =N//,. (9)

(6) adjustments eitherhi orLn t0 re- PIG. 5. Spairurn (m). two choip d*ncd Tor 4f/f<0.1. havins thc duce leakagc [%S. (4) and ( 5 ) 1. lame normalid power band. 0.0 1 C/, 5; 0.4. and tlic wcrlappingadio frc-

(7) Gheck that thc number of harmonic ~0tnpOlientS q- powcr bnds 20 HrCjc800 & w i i h l - 2 k H + d MO HrU<ZO (lines) in the power band, kHz withA= 50 kHz (N = 1024).

1528 J. Acoust Soc. Am., Vol. 01, No. 3. March 1892 John C. Rurgess 1528

TABLE 111. Gain at power band limits, in-band powcr, and in-band ripplc for thc choips dcsigncd for thc frqucncy range 20 H z u < 2 0 kHz with Af/f<O.I ( N = 1024).

Frquency Gain ai

Normalized Power Ripple A Rangc A f A. Ln In band Peak lo peak

Fig. kHz Hz-kHz Hz A fum dB dB % Max. dB

IV. APPLlCATlON TO AN ACOUSTICAL EXAMPLE

The design procedure can be illustrated by application to an idealized acoustical example. A more detailed version is given in the Appendix. Assume the following specifica- tions for adesired measurement: ( a ) The objective is to mea- sure the frequency responsc of an acoustical system over the entire audio frequency range, 20 Hz(/á20 kHz; (b) the measurernent system passband (O dB>gain > - 3 dB) is also 20 Hz(f62O kHz; (c) the maximum acceptable fre- quency resolution is 10% of the frequency (¡.e., Af/f<O. I ); ( d ) the maximum usable FFT block size is 1024; and (e) the I/O system has sampling rate capability at 1 kHz intervals up to 50 kHz.

Design criteria ( 1 ) and (2) can be met by specifying I, = 20 Hz and f, = 20 kHz as the desird minimum and maximum power band lirnits.

Criteria (2)-(5) require some calculation and judge- ment. The 20 kHz upper band limit can be met only by sam- pling ratesl,>40 kHz. Selecting f,, = 0.4 results inl, = 50 kHz [Eq. (6) l. Selecting the maximum allowable block size N = 1024 then results in Af = 48.8 Hz [Eq. (7) 1. These values forl, and Nmeet thespecification for 10% maxirnum frequency resolution only for f 2488 Hz. This means that (at least) two separate measurements, called here a "lower part" and a "higher part," are required to cover thc entire audio frequency range while meeting the specification for maximum frequency resolution. The higher part rneasure- ment coven the frequency range 488 Hz to 20 kHz. The lower part must cover the frequency range 20 Hz to at least 488 Hz.

The lower part measurement must meet the specifica-

quency range (Fig. 4) 20 Hz(f620 kHz (/, = 50 kHz: 0.0004(/1, S; 0.4). 'The other measurement can be made over the lower part of the audio frequency range 20 Hz(/á800 Hz ( I , = 2 kHz: 0.01 (/,, S 0.4). These specifications can be refined slightly to use the sume normalized power band, 0.01 (/,, S 0.4 (Fig. 5 ). With this refinement, one measure- ment (lower part) would be made over the frquency range 20 HzQá800 Hz with I; = 2 kHz, and the other (upper part) would be made over the frequency range 500 H z G á 2 0 kHz withl, = 50 kHz. The 3 W H z overlap allows checking for interna1 consistency. The values chosen f o r h andj;,,, assure little power near zero and Nyquist frequen- cies. Increasingh from 0.0004 to 0.01 has the desirable ef- fect of decreasing waveform power near dc, which is appar- ent by inspection of the magnitude spectra in Figs. 4 and 5. Power band parameters are summarized in Table 111.

Choosing a power band well within the minimum and maximum normalized frequency limits, I, = 0 and f,,, = 0.5, has two major advantages. At high frequencies, it assures little aliasing. At both low and high frequencies, it assures that magnitude and phase distortion by the measure- ment system does not significantly modify the low peak fac- torof the waveform. Phascdistortion near measurement sys- tem passband limits, such as by reconstruction filters, may degrade a waveform's peak factor by causing disproportion- ate time shifts of some Fourier components.

At l, = 50 kHz, the time duration of a single chirp is 20.48 ms [Eq. (9) 1. At l, = 2 kHz, it is 5 12 ms. These are measures of the minimum time duration over which the tar- get systern must be stationary, and the environrnent at the target system response transducer free of unwanted echos of the excitation.

tion for 10% maximum frequency rewiution at the Iower V. suMMARY AND CONCLUS~ONS band limit,I, = 20 Hz. This gives A/C2 Hz. Equation (7) with N = 1024 then specifies a maximum 2048-Hz sampling A choip is an optimum low peak factor waveform for

u s i n g ~ = 2 k~~ ves ~ f = 1-95 H~ acceptable which the power band can also be specified. It has been

value. The upper band limit for the lower part can be set at shown that choips with little leakage and aliasing can be

any frequency up to the Nyquist frequency, 1 kHz. Again designed eas i l~ and quickly. A choip desi@ methodolog~

using f,, = 0.4 gives an upper band limit f, = 800 Hz, has been demonstrated.

which meets both the requirement f, >488 Hz and the need to minimize aliasing. Modifying e i therh or/,, to minimize leakage [Eqs. (4) and ( 5 ) 1 has a minimal effect on power band values.

These calculations show that the specifications can be met by selecting N = 1024 and rnaking two measurements. One measurement can be made over the entire audio fre-

1528 J. Acoust. Soc. Arn., Vol. 81. No. 3. March 1992

ACKNOWLEDGMENTS

My thanks go to Yetclr Schuck for providing pertinent references and to David Havelock for helpful discussions and for his careful review of the manuscript. This work was supported in part by the Lanikai Foundation and the Na- tional Research Council of Canada.

John C. Burgess 1528