presto-chirps: using flexible discrete chirp-lets based stimuli to measure acoustic impulse...

ICSV22, Florence (Italy) 12-16 July 2015 1

PRESTO-CHIRPS: USING FLEXIBLE DISCRETE CHIRP-LETS BASED STIMULI TO MEASURE ACOUSTIC IMPULSE RESPONSES

Massimo Serafini

RAI – Radiotelevisione Italiana, radio department, Via Asiago, 10 - 00195 Rome, Italy

e-mail: [email protected]

This paper acknowledges the possibility of measuring the linear impulse response of acoustic

systems by means of a set of discrete narrowband chirplets (dubbed presto-chirps), whose

amplitude and durations can be varied arbitrarily within purposely composed stimuli. Presto-

chirps are linear frequency varying chirps, and although there is no constraint in terms of

which frequency they can be centred around and their bandwidth, in this presented research

they were centred around musical notes with a bandwidth of a musical semitone (1/12th

oc-

tave band). The method employs a set of 89 presto-chirps, which match the 88 keys of a

grand piano, plus an 89th

one. This latter extends the measured frequency range up to 4.7 kHz

permitting the analysis of the third-octave band centred at 4 kHz. The use of presto-chirps al-

lows composing music-like stimuli, thereby facilitating acoustic measurements in presence of

listeners. Time windowing and frequency shifting have been used to mitigate frequency-

domain ripples and discontinuities in the excited spectrum. Nonetheless, a comb-like bias is

introduced in the frequency response of measured systems, which is of the order of ±1 dB on

average. The results from an in situ measurement are presented along with the details of the

used stimulus.

1. Introduction

Measuring impulse responses (IRs) is the preferred solution to characterise the acoustical behav-

iour of spaces and systems. Moreover, although occupied measurements are acknowledged to give

the actual acoustic profile of spaces/systems when in-use, such measurements are rarely carried out

due to yet unsolved issues (obtrusiveness of testing stimuli, time variance, etc...).

In fact, common used methods such as the exponential sine sweep (ESS) and the maximum

length sequence (MLS) utilises specific broadband stimuli and deconvolution techniques to perform

IR measurements. For example, the ESS method developed by [1], employs an exponential chirp

and an inverse filtering deconvolution technique, and as outlined and demonstrated in [2], it allows

to separately measure the linear component and each other order of non-linearity of an IR. As a

consequence distortion free IRs can be obtained even when amplifiers and loudspeakers are pushed

over their linear response to increase the signal-to-noise-ratio (SNR). These features have made the

ESS method preferred over the MLS, which however, conversely to the ESS, can be used to per-

form measurements within very low initial SNR. This feature theoretically allows carrying out in-

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International Congress on Sound and Vibration

ICSV22, Florence, Italy, 12-16 July 2015 2

audible measurements [3], but the MLS time variance susceptibility hampers measurements accu-

racy when over-prolonged averages are taken [4].

Machine learning based technique can successfully extract some acoustical parameters (like RT,

STI, etc...) from played music [5], but cannot measure impulse responses. On the other hand, the

dual channel FFT technique [6] although it can make use of live music to analyse the frequency

response of rooms, the source signal itself may lack frequency fullness or prolonged averaging may

be needed, which makes measurements susceptible to time variance.

Conversely, the presto-chirps method works around such limitations by using narrowband chirps

signals, dubbed “presto-chirps”, to compose frequency-continuous music-like stimuli, which can be

used to measure impulse responses in presence of audience, in a reasonable time.

2. The Method

The prefix “presto” is used in music notations to refer to a movement that is executed quickly,

which, in this context, symbolically prelude the use of short duration chirps. The idea is to match

the frequencies of each presto-chirp with the frequencies of musical notes (equal tempered scale A

= 440 Hz), and to use them to compose musical sounding stimuli. To facilitate compositions presto-

chirps are associated to the keys of an 88-keys piano. Thus, the lower measured frequency corre-

sponds to 27.5 Hz (note A0, the first key of a grand piano). An additional 89th

note is also used to

extend the measured frequency range up to a 4.698 kHz (which corresponds to the note D8), to

permit measuring the third-octave band centred at 4 kHz.

2.1 Presto-chirps

Presto-chirps are Hanning windowed linear sine sweeps. Thus

(1)

Where ( ) is a Hanning window, and is a linear sine sweep, thus

(2)

(3)

Where A is the amplitude and N is the number of samples of both the presto-chirp and the win-

dow. The terms and are the final and the initial frequencies of the chirp, respectively. The

index i relate a MIDI number to the frequency of musical notes, as for

(4)

The presto-chirps realise, using a musical term, “semitone glissando” notes, thus notes whose

frequency is shift upward, or downward, of a semitone. The Hanning window is needed to mitigate

the performed measurements. Because of the windowing, the presto-chirps’ bandwidths are reduced.

Then to avoid frequency gaps between presto-chirps their start and stop frequencies, and , are

shifted by an amount equal to 5/6 of their bandwidths B ( ). Thus

(5)

The term 5/6, found by trial and error, causes the spectra of adjoined presto-chirps to overlap at

about -3.01 dB at the exact frequencies of musical notes. As an example, the magnitude responses

of the presto-chirps and before and after the frequencies shift are depicted in Figure 1.

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Figure 1 – Spectra of the presto-chirps – and – . The

dashed lines and the solid lines represent the presto-chirps before and after the frequencies shift.

The set of 89 presto-chirps covers the frequency range from 27.5 Hz to 4698 Hz and divide it

into non-uniformly spaced sub bands. The bandwidths of presto-chirps’ doubles at each octave,

which implies a magnitude drop of 3 dB/Octave (for a set of presto-chirps having fixed duration and

amplitude), like a logarithmic sine sweep or pink noise. Their magnitude responses are superim-

posed in Figure 2.

Figure 2 – Overlaid magnitude responses of the set of 89 presto-chirps.

2.2 Rationale

The presto-chirps method exploits the superposition property of linear systems, which guarantees

that distinctively measured narrowband parts of an IR can be synthesised together to form a broad-

band IR. Hence, M partial impulse responses are first measured, through the use of presto-

chirps, and then summed to form . Mathematically,

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(6)

The measurement consists on feeding a stimulus ( ), composed of M presto-chirps each

played at the time interval ,

(7)

into the room under test, which gives the output

(8)

Where is the convolution operator. The impulse response of the system is obtained

through the following steps:

The output is sliced into M chunks ( ), each starting at the time position and of

lengths , where is the length of and is an analysis window that is needed to

capture the room’s reverberation tail.

Each is convolved by the inverse filter , which match the presto-chirp used in that

specific chunk, giving the partial impulse response The M partial impulse responses are (opportunely weighted and) summed together.

From a signal processing point of view, each could be regarded as a band-pass filtered ver-

sion of obtained through a forward-backward filtering, thus

(9)

where is the autocorrelation function of , since for real

signals, and can be thought as the impulse responses of zero-phase pass band filters with order of

twice minus one the length of the signals. The sum of all the gives the autocorrelation function

of the stimulus, thus

(10)

The Fourier transform of Eq. (10), thus is the auto spectrum of

the stimulus, which is represented for an “ideal” stimulus formed by the 89 presto-chirps in Figure 3.

Figure 3 – Frequency response of .

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The magnitude differences between presto-chirps, having equal duration and amplitude, are only

due to their different bandwidths (which increases as frequency increase). To compensate for such

difference, each inverse filter ( is multiplied by a weighting factor obtained as the ratio of a

presto-chirps’ bandwidth and the minimum bandwidth among the whole presto-chirps. Thus where is the weighting factor, is the bandwidth, and . The spectra of

the 89 weighted inverse-filters, , are overlaid in Figure 4.

Figure 4 – Overlaid magnitude responses of the 89 weighted inverse-filters.

The spectra of the so weighted autocorrelation functions are shown

overlaid in Figure 5, which when synthesised result in the spectrum shown in Figure 6. Its slightly

oscillating magnitude response is due to the “gaps” between the autocorrelation functions (which

overlap at -6 dB). This introduces a bias in the measurements with magnitude of 1 dB on average

and of about out 2 dB in the lower part of the spectrum (from around 100 Hz).

Figure 5 - Overlaid magnitude responses of the 89 autocorrelation functions

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Figure 6 - Synthesised magnitude responses of the 89 autocorrelation functions

3. Validation

Codes were developed in MatLab to extract from MIDI files, the amplitude (A), the pitch ( ), and the duration (N in samples) of each note composing a song. Those parameters used in Eq. (1)

define the presto-chirps. Such approach facilitates stimuli composition, which, however, is sub-

jected to three main limitations. The first related to the way notes can be arranged and used without

creating artefacts (not discussed in this paper). The second related to the writing of compositions

that use all the 89 notes. The last, but of primary importance, is related to the composing of “enter-

taining” stimuli. So far, only few usable compositions have been found, whose authors kindly gave

consent to use them for this research. In this paper, the song ‘Via del Corso’ (a famous road in

Rome) written by the Finish composer Tero-Pekka Henell has been adapted and used to perform

measurements. The time-frequency representation of such composition played using presto-chirps is

reported in Figure 7. Note that each note is used only once, and that the presto-chirps are either as-

cending or descending. This latter choice, for this research, was made to emphasise such possibility.

Figure 7 – Presto-chirps stimulus realised from the song "Via del Corso" by Tero-Pekka Henell.

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4. Measurement

A listening room (in accordance with the ITU-R BS 1116-1 standard) at university of Salford

(Manchester – UK) was used to test the reliability of the method. The room’s impulse response was

acquired using an ESS signal, and was subsequently bounded in the range from 27 Hz to 4.7 kHz

using a 1000-taps linear phase FIR filter, and used as reference. The stimulus presented in Figure 7

was fed into the listening room, the room’s response acquired, and then partitioned and decon-

volved as discussed in 2.2. The partial impulse responses were summed as

(11)

where is the weighting factor (note that ); is the

peak value of a presto-chirp autocorrelation function, which is used to compensate the different

amplitude and durations of presto-chirps; the term represents the number of times a partial im-

pulse response has been measured, which in this specific case is always equal to one as presto-

chirps are used only once – for a total of partial IRs. The analysis window, thus the length

of the partial impulse responses, was of one second. For sake of argument, it should be said that the

used stimulus fails to measure IRs longer than one second (the reasons are not discussed in this pa-

per). The early reflections of the listening room’s IR calculated using the presto-chirps stimulus

(dashed line) and the one calculated using ESS (solid line), are shown for comparison in Figure 8.

The differences between the two IRs are due to a slightly time variance of the used room, rather

than to the method itself.

Figure 8 – Early reflections of the listening room measured using the presto-chirp method (dashed red

line), and the reference early reflections (solid black line).

The acoustical parameters: clarity (C80), definition (D50), early decay time (EDT), and the re-

verberation times RT30 and RT60, were also calculated (using the plug-in “Aurora” for “Audacity”

[7] for the octave bands from 125 Hz up to 4 kHz, for both the reference and the measured IRs and

then compared. They are reported in Table 1. All the values are within the difference limen of the

specific acoustic parameter (for small rooms).

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Table 1 – Acoustic parameters in octave bands, from 125 Hz to 4 kHz, extracted from the impulser

response measured by mean of the presto-chirps stimulus.

Parameter Bands [Hz]

125 250 500 1k 2k 4k

C80 [dB] 13.26 (13.52) 19.7 (19.8) 19.6 (19.8) 20.5 (20.78) 23.6 (23.1) 24.6 (24.45)

D50 [%] 79.8 (79.9) 91.6 (92.56) 94.4 (19.8) 94.2 (94) 97.1 (97.04) 97.5 (97.74)

EDT [sec] 0.39 (0.39) 0.32 (0.32) 0.27 (0.27) 0.23 (0.23) 0.19 (0.18) 0.18 (0.18)

RT30 [sec] 0.32 (0.29) 0.22 (0.23) 0.22 (0.22) 0.23 (0.22) 0.23 (0.24) 0.21 (0.21)

RT60 [sec] 0.32 (0.3) 0.29 (0.28) 0.25 (0.24) 0.24 (0.24) 0.23 (0.24) 0.21 (0.23)

5. Conclusions and Further Works

The use of pesto-chirps to compose music-like stimuli, supplies an unobtrusive alternative to the

annoying sound of broadband chirps. Indeed, the presented method opens up a new horizon of re-

search into the use of deterministic synthetic musical stimuli to measure the impulse responses.

Nonetheless, further work is needed to create more artistically and entertaining musical stimuli.

Because the method uses narrowband linear chirps as stimuli to measure impulse responses,

some operations have to be performed to mitigate discontinuities in the excited spectrum, like win-

dowing and frequency shifting. However, the use of segmented stimuli produces a slight bias in

measured IRs, of about 1 dB on average, with a peak of 2 dB at frequencies below about 100 Hz.

However, an in situ measurement has proved that the method, under the tested condition, is reliable

and accurate, as the measured impulse response had negligible error.

Besides room acoustics application, a set of presto-chirps covering the whole audio spectrum

may be used in an electroacoustics context to measure loudspeakers and microphones, since simi-

larly to chirp based measurements, they can separate the non-linear part from the linear part in an

impulse response and therefore handle systems with non-linearity. In addition, presto-chirps stimuli

would be representative to what is played through these transducers in real uses. They would also

avoid technical workers being exposed to boring testing signals (i.e. broadband sine sweeps) for

prolonged periods over their working hours. Moreover, it may be used to measure the impulse re-

sponse of ears canals for hearing aids equalisation.

An additional feature of presto-chirps is that they could be arranged accordingly to the back-

ground noise, thus a greater number of presto-chirps around the frequency range of the noise can be

used and results averaged to achieve a generally equal SNR across the whole spectrum, or similarly,

their amplitude can be increased to achieve greater SNR in the noisy sub bands.

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