9 Acoustic Impedance Measurement Methodssymphony.arch.rpi.edu/~xiangn/Teach/SRL_II_Spring2020_chap08.pdf · ment the two-microphone transfer function method. will lead to large measurement

9

Acoustic Impedance Measurement Methods

One of practical measurement techniques of acoustic reflection coefficient,surface impedance and characteristic impedance is conducted in narrow, andlong tubes. In long, and narrow tubes a plane sound waves can be conve-niently generated. There existing a number of measurement techniques, fornormal incidence measurement of surface reflection coefficient, characteristicimpedance of porous materials, and sound transmission loss of a materialunder test. Standing wave ratio method discussed in Sec. 2.3 represents aclassical technique to determine acoustic reflection properties of a materialunder test in a so-called Kunt’s tube. One of its drawbacks is that the mea-surement can only be conducted under single-tone excitations. Each time, onereflection/absorbtion value can be obtained at one harmonic frequency. Thischapter introduces some advanced methods.

9.1 Two-Microphone Transfer-Function Method

Figure 9.1 schematically illustrates a segment near one end of long and thintube where a material under test is properly mounted at the tube termina-tion. Two microphones are placed at positions labeled by ’1’ and ’2’, separatedby distance, s, also termed separation in short.1 They sense sound pressuresignals, expressed in the frequency domain as P 1 and P 2. When acousticwavelength is much larger than the diameter of the tube, λ� D , the acous-tic wavefront is the same across each individual cross-section labeled by ’1’,’2’ and ’m’, respectively. The distance between position ’1’ and ’m’, namelybetween the microphone ’1’ and the surface of the material is known to be’L’.

Both incident and reflected components are conceivably existing at po-sition ’1’, ’2’, and ’m’, with ’m’ denoting the front surface of the material

1 The method was first introduced by Chung and Blaser [4, 5] using a time-domainexplanation. The method introduced in this Chapter explains the theory in fre-quency domain.

112 9 Acoustic Impedance Measurement Methods

Fig. 9.1. Normal incidence plane wave for acoustic reflection coefficient measure-ment using two microphones [4, 5]. A transfer function, H 12 is defined betweenmicrophone position 1 and 2, separated by distance, s.

under test. The medium is considered lossless. If an incident wave componenttravels from position ’1’ to position ’2’ (forward-propagating) as illustratedin Figure 9.1, the sound pressure in frequency domain at position ’2’ can beexpressed as

P 2,i = P 1,i e−jβ s, (9.1)

where P 1,i and P 2,i represent the incident components of sound pressuresin the frequency domain, and s represents separation distance of the twomicrophones. When the reflected wave component travels from position ’2’ toposition ’1’, it can be written as

P 2,r = P 1,r e jβ s. (9.2)

where P 1,r and P 2,r represent the reflected components of sound pressuresat the two positions in the frequency domain. In similar fashion, the soundpressure components between position ’1’ and position ’m’ as illustrated byFigure 9.1 are expressed by

P m,i = P 1,i e−jβ L, (9.3)

P m,r = P 1,r e jβ L. (9.4)

Between two microphones at position ’1’ and ’2’, the two sound pressuresignals, P 1, P 2, in the frequency domain build a transfer function, introducedin Sec. 1.6 as

H 12 =P 2

P 1

=P 2,i + P 2,r

P 1,i + P 1,r

=P 1,i e−jβ s + P 1,r e jβ s

P 1,r + P 1,i

, (9.5)

where Equations (9.1) and (9.2) come to use here in the nominator on theright-hand side. Rearranging

H 12(P 1,r + P 1,i) = P 1,i e−jβ s + P 1,r e jβ s, (9.6)

(e jβ s −H 12)P 1,r = (H 12 − e−jβ s)P 1,i, (9.7)

so thatP 1,r

P 1,i

=H 12 − e−jβ s

e jβ s −H 12

. (9.8)

9.1 Two-Microphone Transfer-Function Method 113

Surface reflection coefficient

Using Equations (9.3) and (9.4), the surface reflection coefficient at the frontsurface of the material ’m’ is defined as the reflected component vs. the inci-dent component, which can be expressed through P 1,r/P 1,i,

R =P m,r

P m,i

=P 1,r

P 1,i

e j 2β L, (9.9)

it is also termed surface reflectance [1]. Substituting Equation (9.8) into Equa-tion (9.9) results in

R(f) =H 12(f)− e−jβ s

e jβ s −H 12(f)e j 2β L, (9.10)

where the reflectance (reflection coefficient), R(f), and the transfer function,H 12(f), are explicitly denoted as functions of frequency. Particularly,

H 12(f) =P 2(f)

P 1(f). (9.11)

The normal incidence surface impedance can be derived from the reflectioncoefficient,

Z s = % c1 +R

1−R, (9.12)

with % c being the characteristic resistance of air, % is air density, and c issound speed. The normal incidence absorption coefficient is determined by

α = 1− |R |2. (9.13)

Valid frequency range

Equation (9.10) expresses the reflection coefficient as a function of frequency,determined through the transfer function, H 12(f). Notably, Equation (9.8)and the fraction in Equation (9.10) will become

H 12 − e−jβ s

e jβ s −H 12

= −1, at e−jβ s = e jβ s, (9.14)

which represents singular conditions for this method [4]

β s = nπ with n = 1, 2, 3, . . . . (9.15)

With n = 1 and β = 2π f/c the microphone separation, s, can be determinedupon a limit frequency for the tube measurement

s <c

2 f=λ

2, (9.16)


where f represents the limit of the valid frequency, and λ is the wavelengthat frequency f .

In addition, the normal incidence measurement requires a plane wave con-dition, which is

fH<K · cd

, (9.17)

with d being the diameter of the circular cross-section tube, where K = 1. Forsquare tubes with d being their diagonal length, K = 0.5. Equation (9.17) isdecisive to determine the higher limit frequency, f

H, assuming that the tube is

long enough. Given the higher limit frequency dictated by the tube diameter,a lower bound of the microphone separation, s, can be determined. In orderto ensure the separation being sufficiently smaller than half wavelength givenin Equation (9.16), the recommended rule of thumb [7] is to take 4/5 of thatin the singular condition,

ssmall≤ 4

5

λH

2=

2λH

5, (9.18)

while the tube should be valid for the frequency range towards lower endat which, the separation has to be selected [7] at least 1 % of the longestwavelength in order to be able to rely on the measured differences betweenthe two microphone positions.

slarge

> 0.01λL. (9.19)

The lower limit frequency is often dictated by the length of the tube. It needssufficient long in order to ensure sound waves becoming plane wave whenreaching the microphones and the material under test.

Absorbing wedges are often inserted at the opposite end near the soundsource. Considering the frequency range near the lower frequency limit, evencareful selection of absorbing materials may not meet the need of total ab-sorption reaching as high as α > 0.98, and the separation has to follow thecondition in Equation (9.19), there will still be some reflection residuals neg-atively impacting the measurement accuracy. For this reason, the lower fre-quency limit, fL = c/λL should be dictated by capability of eliminating thesereflection residuals. Equations (9.18) and (9.19) indicate that there exists arange of the separation distance

ssmall≤ s ≤ s

large(9.20)

in order to cover a sufficient bandwidth which fulfills at the same time thesingular condition given by Equation (9.16). To this end a combination ofdifferent separations is often exercised.

Two-microphone transfer function method discussed so far, requires twosimultaneous microphone channels. This requires highly accurate phase-matchof the two microphones even the magnitude responses of microphone sensitiv-ities are flat enough. Small mismatches in phases or in sensitivity responses

9.2 Three-Microphone Transfer-Function Method 115

Fig. 9.2. Possible implementation of the transfer function method using one micro-phone. Sequential measurements at two microphone positions synthetically imple-ment the two-microphone transfer function method.

will lead to large measurement errors and uncertainties. With the adventcorrelation technique (as introduced in Chapter 6), transfer functions and im-pulse responses of acoustic devices / systems can be experimentally measuredwith high accuracy of phase-locking (synchronization), this enables sequen-tial measurements using single microphone in order to avoid highly accuratephase-matching of two or multiple simultaneous microphone channels [3]. Fig-ure 9.2 illustrates one possible implementation using a single microphone interms of sequential measurement at two or more microphone positions, wherethe advanced measurement techniques discussed in Chapter 6 have to be used.In particular, impulse responses between the sound source at one end of themeasuring tube and the microphone at position ’1’ and ’2’ can be sequentiallymeasured. The transfer function in Equation (9.11) is then determined usingfast Fourier transforms. In order to cover both low frequency and high fre-quency ranges, some additional microphone positions to ensure small or largeseparations need to be measured.

9.2 Three-Microphone Transfer-Function Method

The two-microphone method discussed previously yields measurement resultsof normal incidence surface reflection, absorption coefficient or the normal in-cidence surface impedance, which cannot directly characterize the materials.This Section introduces one of advanced methods suitable for characteriza-tion of fibreous and porous materials [11], commonly used for a wide varietyof absorbing materials in acoustics applications and noise control engineering.The technique is essentially an extension of the two-microphone method underplane wave assumptions. Figure 9.3 schematically illustrates the tube config-uration for determining the characteristic impedance, Z c, and the complex-valued propagation coefficient, γ, of porous materials or those materials whichcan be considered as equivalent fluid media.

One more microphone is placed at position ’3’ in the rigid backing righton the back surface of the porous material as shown in Figure 9.3. Two micro-phones at position ’1’ and ’2’, separated by a distance, s, are placed in front ofthe material surface with microphone ’1’ being positioned at L distance from


the material surface, similar to that in Figure 9.1. The porous material undertest has a known thickness, d. The measurement technique experimentallydetermines its characteristic impedance, Z c and propagation coefficient, γ.As before, Equation (9.10) yields the reflection coefficient, r, at the front-sidewith respect to the material surface with L, when the microphone separation,s, is known.

Fig. 9.3. Three microphone transfer function method for characterizing porousmaterials [11]. The medium sinde the tube is considered as lossless, with its charac-teristic resistance, % c, and its propagation coefficient, β.

Denoting the sound pressure, P 3, and particle velocity, V 3 at position,’3’, the sound pressure is measured by microphone at position ’3’. The soundpressure, P m, and particle velocity, V m, at the material front surface canthen be expressed [2] as:[

P m

V m

]=

[cos γd jZ c sin γd

j sin γd/Z c cos γd

] [P 3

V 3

]. (9.21)

At the rigid termination, V 3 = 0, so the characteristic impedance, Z c, canbe derived from the above equation in terms of the surface impedance, Z s =P m / V m, and the propagation coefficient, γ, when the material thickness, d,is given,

Z c = jZ s tan γd, (9.22)

where the surface impedance, Z s, is determined by Equation (9.12), via Equa-tion (9.10) through experimentally measured pressure spectra from the twomicrophones at position ’1’ and ’2’, in front-side of the tube as discussed inSec. 9.2. From Equation (9.27), it also follows that

P m = P 3 cos γd, (9.23)

resulting in the complex-valued propagation coefficient

γ = cos−1(H 3m) / d, (9.24)

with

H 3m =P m

P 3

=P m

P 1

P 1

P 3

=P m

P 1

H 31, (9.25)

9.2 Three-Microphone Transfer-Function Method 117

and

H 31(f) =P 1(f)

P 3(f). (9.26)

Similar to P 1 as discussed previously, P 3 represents the sound pressure infrequency domain measured by the microphone at position, ’3’.

The sound pressures, P 1 and P m, represent superpositions of the incidentand reflected components

P 1 = P 1,i + P 1,r = P m,i(ej β L +R e−j β L), (9.27)

P m = P m,i + P m,r = P m,i(1 +R ), (9.28)

Complex division of Equation (9.28) by Equation (9.27) leads to

P m

P 1

=1 +R

e j β L +R e−j β L. (9.29)

A substitution of Equation (9.29) into Equation (9.25), and then into Equa-tion (9.24) leads to the complex-valued wavenumber containing the propaga-tion coefficient

γ d = cos−1

(1 +R

e jβ L +R e−jβ LH 31

), (9.30)

where the transfer function, H 31, can be measured via microphones at posi-tion ’1’ and ’3’, as in Equation (9.26) through the pressure spectra, while Ris determined using Equation (9.10). Note that the complex-valued wavenum-ber, γ d, containing the propagation coefficient of the material and its thick-ness, d, are most important quantity, as used in Equations (9.27)-(9.23). Asubstitution of Equation (9.30) into Equation (9.22) yields the characteristicimpedance, Z c.

Using three microphones, two on up-stream (source) side near the samplematerial, and one right at the back surface, the characteristic impedance andthe propagation coefficient can be experimentally determined which relies alsoon intermediate step of getting the acoustic surface impedance of the mate-rial under test [in Equation (9.12) via Equation (9.22) and Equation (9.30)].The valid frequency range and the singular conditions as presented in Sec. 9.1hold in general for the three-microphone method. To cover a wide frequencyrange, a combination of different microphone separations is a common prac-tice as well, as discussed using Equations (9.18)-(9.20). Similar to the two-microphone method, the phase-accurate correlation technique can be appliedin sequential measurements using one single microphone at three microphonepositions. The three-microphone method is also advantageous over the four-microphone method [12] (each two on both sides of the material under test;two microphone on the up-stream side and two on down-stream side). Not onlyone channel/measurement position less needs to be involved in the measure-ment procedure, but the microphone right on the back surface will measureenhanced signals due to the rigid termination, in comparison with those onthe down-stream side of the four-microphone method.


9.3 Free-Field Methods

The acoustic surface reflectance is defined as

R(f) =P r(f)

P i(f)=

FT[ pr(t)]

FT[ pi(t)], (9.31)

where pi(t), and pr(t), represent incident and reflected pressure signals intime domain at the front surface of the material under consideration, FT( )stands for Fourier transform, while P i(f) and P r(f) are pressure spectra forincident and reflected pressure signals, respectively. Note that the definitionof reflectance in Equation (9.31) implicitly assumes a plane wave condition,restricting to plane wave reflectance (spherical wave reflectance will be clearlydifferent). Equation (9.31) also hints at an experimental approach in free-fieldif the pressure signals, either in time-domain or spectral signals in frequencydomain can be obtained. Ideally the measurement needs to be carried out infree-field, such as in an anechoic chamber in laboratory. In practice, it is alsodesirable to carry out the measurement in quasi-free-field, such as in largespaces with the materials already built into the site or already in use. Thiskind of measurements so-called in-situ, requires careful arrangement near thematerial sample under test, this Section discusses two approaches.

9.3.1 In-situ Windowing Method

Figure 9.4 schematically illustrates one possible configuration [6]. In Fig-ure 9.4 (a) a microphone is placed between a sound source (loudspeaker)and a large material panel under test with a perpendicular distance, d, to thematerial surface. The sound source is placed with a perpendicular distance,D, to the material surface. When measuring an impulse response of this mea-surement set up, Figure 9.4 (b) schematically illustrates a straightforwardwindowing approach to separating the incident impulse response segment,hi(t), and reflected impulse response segment, hr(t), contained in an in-situmeasurement of the impulse response of this set up [6]

hm(t) =

direct sound︷︸︸︷h i(t− t0) +

surface reflection︷︸︸︷h r(t− t0 − τr) +n(t)

+

speaker reflection︷︸︸︷h s(t− t0 − τs) +

other unwanted︷︸︸︷∑u

Ku h r(t− t0 − τu), (9.32)

where

t0 =D − dc

, τr =2 d

c, and τs =

2D

c. (9.33)

Ku represents coefficients of unwanted reflections from irrelevant surfaces, orobjects in the measurement site. n(t) is background noise. In anechoic envi-ronment, the last sum-term on the right-hand side of Equation (9.32) would

9.3 Free-Field Methods 119

Fig. 9.4. Free-field measurement method for normal incidence reflectance measure-ment [6]. (a) Measurement setup with a sound source and a microphone relative tomaterial panel under test. (b) Time trace of idealized echogram, indicating the tim-ing relationship with dash-line windows used to separate the incident and reflectedpulses.

technically disappear. However, during the measurement in-situ, it is possibleto make τu ≥ τs through adjustment of the speaker distance to the materialsample under test and the material panel size, so that a straightforward ap-plication of two window functions as sketched in Figure 9.4 (b) will be ableto extract two impulse response portions at the given microphone positions.Considering the acoustic surface reflection as a linear time-invariant systemwith a plane wave reflection impulse response, r

P(t), it follows [6]

hr(t− t0 − τr) ≈D − dD + d

rP(t) ∗ hi(t− t0), (9.34)

with symbol, ’*’ denoting linear convolution, and the ratio, (D − d)/(D +d), represents a path-length correction factor, τr represents the delay timecaused by a round-trip distance from microphone to the material surface andback, and ’≈’ implies that the background nois,e n(t), is ignored here andthroughout the following discussion.

Fourier transforming the both sides of Equation (9.34) yields


H r(f) e−j 2π f (t0+τr) =D − dD + d

R(f)H i(f) e−j 2π f t0 , (9.35)

leading to the reflectance as a function of frequency

R(f) =D + d

D − dH r(f)

H i(f)e−j 2 β d, (9.36)

with R(f) = FT[ rP(t)], β = 2π f/c, τr in Equation (9.33) comes into use.

Fig. 9.5. Measurement setup for surface reflectance using the subtractionmethod [9]. (a) Measurement setup with a microphone near to material surfaceunder test. (b) Time trace of idealized echogram, indicating the timing relation-ship with one dash-line window used to separate a composition of the incident andreflected pulses.

9.3.2 In-situ Subtraction Method

Figure 9.5 (a) schematically illustrates an alternative method for the sur-face reflectance measurement. One microphone is placed near the surface oftest sample panel, along the perpendicular axis from the sound source to thesample surface. The major difference to the windowing method previouslydiscussed in Sec. 9.3.1 is that the reflected signal from the surface is movedtowards the direct sound signal as schematically shown in Figure 9.5 (c), whilethe direct sound signal and the loudspeaker-face reflected signal are remainedthe same if the loudspeaker distance to the sample surface remains the same.


The method is more suitable for in-situ applications [9, 13], it inherentlyextends the time window width, therefore, extending the valid frequency rangetowards lower limit as will be discussed in the following in Sec. 9.3.3. Themethod is more challenging than the direct windowing method, the in-sitemeasurement yields an impulse response, which can also be expressed in Equa-tion (9.32), but with

t0 =D

c, τr =

2 d

c, and τs =

2D

c. (9.37)

The profound difference lies in an extremely short delay time τr due to asmall distance, d, to the material surface, often on order of a few millimetersas indicated in a magnified sketch in Figure 9.5 (b), as the reflected signalfalls within the time range of the direct sound impulse response. Unlike thewindowing method as discussed in Sec. 9.3.1, a straightforward separationof the two signals, hi(t − t0), and hr(t − t0 − τr), in terms of windowing isnot possible any more. Figure 9.5 (c) illustrates a windowing function withwindow width of approx. 2D/c would only extract a composition of the directsound and the surface reflection impulses

h(w)m (t) = h i(t− t0) + h r(t− t0 − τr), (9.38)

where the two components on the right-hand side are temporally overlappedwith each other.

Mommertz [9] suggested to conduct a reference measurement with thesample material being absence, in addition to the material measurement withthe microphone near the material surface. The intension is to obtain identicaldirect sound impulse response through the reference measurement,

h(w)ref (t) = h i(t− t0), (9.39)

where the reference measurement could also be conducted in-situ, and thesame windowing as used in Equation (9.38) can be applied to the measuredimpulse response to isolate the windowed direct sound impulse as the refer-

ence, h(w)ref (t).

The critical issue is that the relative postion/distance between the loud-speaker and the microphone has to be strictly unchanged over the both mea-surements, so that the measured signal in front of material can be processedso as to subtract the direct sound signal using Equation (9.39) from the com-position of the direct sound and the reflected signal in Equation (9.38) con-tained in the material measurement result. A subtraction of Equation (9.38)by Equation (9.39) yields

h(w)m (t)− h(w)

ref (t) = h r(t− t0 − τr). (9.40)

Particularly for in-situ measurements, a dedicated fixture, stably holdingthe microphone relative to the loudspeaker, is highly required. The reference


measurement needs to be conducted without the material being in front ofthe microphone. This can be performed when the loudspeaker-microphonesetup is facing to empty space, or facing to the space where the unwantedobstacles are far away. Advanced correlation technique of impulse responsemethods [8] needs to be applied so as to conduct sequential measurements ofthe reference and material impulse responses. The sequential measurementsshould be conducted within as a short time interval as possible so that anypossible ambient environment changes will insignificantly impact accuracyof the attempted identical signals, since the time-variance between the twomeasurements will negatively impact the subtraction results [9, 10].

Despite of all careful conduction of both measurements in sequential man-ner, experimental practice has indicated that a straightforward subtraction inEquation (9.40) would rarely yield satisfactory results [9, 10]. Ambient envi-ronment changes, such as, temperature, and medium drafts inevitably make

the two sequential measurements of the reference signal, h(w)ref (t) in Equa-

tion (9.39), and direct sound component, h i(t − t0) in Equation (9.38) con-tained in material signals, not accurately identical. To implement a subtrac-tion successfully, an numerical over-sampling scheme is recommended, unlessthe measurements were conducted already at a sufficiently over-sampled fre-quency, which requires highly demanding measurement resources (such as ata sampling frequency between 315 kHz and 1 MHz).

Fig. 9.6. Flow diagram of over-sampled subtraction for obtaining the acoustic re-flectance, R(f).


Figure 9.6 recommends a numerical over-sampling scheme, summarized asfollows

� Up-sampling: the two impulse responses from both material and referencemeasurements (often at an audio sampling frequency) are up-sampled bya factor ranging from 8 to 32

� Alignment: In the up-sampled domain, a temporal and amplitude align-ment of the material impulse response with respect to up-sampled referenceimpulse response is undertaken if the refined differences of direct soundportion can be recognized

� Subtraction: in the up-sampled domain, the subtraction similar to Equa-tion (9.40) is carried out

� Down-sampling: the subtracted result is still in up-sampled domain, adown-sampling by the same factor is applied to the subtracted result, lead-ing to a reflection impulse response from the material surface in originalsampling domain

� Windowing: the same window function (a right-half window is recommend-able) is applied to both material reflection impulse and followed by possiblezero-padding

� FFT / Division: the zero-padding makes the two signals to be featuredby a FFT-length (in form of 2m points), so as to perform fast Fouriertransforms of the two signals followed by complex-valued division of thereflection spectrum by the incidence spectrum similar to Equation (9.31).

9.3.3 Windowing, Limit Frequency, Source and Receiver Positions

Both the windowing method in Sec. 9.3.1 and the subtraction method inSec. 9.3.2 require windowing. The window width, TW in time, is decisive forthe lower limit frequency of data analysis, in general, and of the reflectancemeasurement, in particular as discussed in this Chapter. The lower limit fre-quency, f

Lis reciprocal to the window width [6]

fL

= 4f =1

TW

. (9.41)

To shade light to this reciprocal relation, Figure 9.7 illustrates the datapoints in discrete frequency domain, assuming the impulse response segment,either hi, and hr as illustrated within the dash-line window frames in Figure 9.4

or h(w)m (t) within the dash-line window frame in Figure 9.5 (c). These win-

dowed impulse response segments carry useful information limited by the win-dow width, expressed in time as T

W. The discrete frequency axis in Figure 9.7

highlights the discrete Fourier transform of the windowed impulse responsesegments, in form of data dots. Each is separated by the frequency interval /resolution,4f = 1/TW . The first two data dots represent the both data pointsat 0-frequency and at the4f -frequency, below the first4f -frequency the data


Fig. 9.7. Relationship between the window width, TW , frequency resolution, 4f ,and the lower limit frequency, fL , of the windowing method and the subtractionmethod.

contain no useful spectral content of the respective impulse response. Sincethe second data point at frequency of the first 4f begins to carry the spectralcontent of the measured data. According to Equation (9.41), the lower limitfrequency of the in-situ windowing method as shown in Figure 9.4 (b) is

fL≥ c

2 d, given D ≥ 2 d, (9.42)

and the lower limit frequency of the in-situ subtraction method in Fig-ure 9.5 (c) is

fL ≥c

2D. (9.43)

Figures 9.4 and 9.5 illustrate that the impulse responses are assumed tobe direct measurement results for decisive processing steps, they need to beextracted from the entire impulse responses by windowing, particularly in-situ,where the entire room impulse responses are primarily the raw data. Thereexist a number of window functions, often in symmetrical forms as exemplifiedin Figure 9.8 (a). For these impulse responses which are often featured withsteep on-sets in responding to the direct sound arrival, the classical windowingfunctions in symmetrical forms are less suitable, rather those in right-halfwindow forms as in Figure 9.8 (b). The top value of the window function isequal to 1.0, it is reasonable to extend the right-half window to feature a ’flat-top’ (plateau). For the methods discussed in this Chapter, window functionswith zero-value at the end, such as Hanning window or Black-Harris windoware recommended, as they favor zero-padding after windowing.

Note that zero-padding after the right-half windowing, may prolong thedata length, however, the zero-padding definitely doesn’t enrich the frequencycontent of the windowed impulse response at all, therefore has no effort onlower limit frequency, f

L. The only change is to fill up additional interpo-

lating points between the original data points as illustrated in Figure 9.7.The changes also include spectral changes on the material reflection impulseand the reference impulse, since the multiplicative manipulations (windowing)of the impulse response segments in time-domain correspond to convolutionof the window spectrum with the spectra of the measured impulse response


Fig. 9.8. Window functions used to separate the useful impulse response portions.(a) Symmetrical Black-Harris and Hanning windows. (b) Right-half Black-Harrisand Hanning windows with a plateau before the start of the half window functions.

segments in frequency-domain. But these changes are less violate than therectangular windows, particularly the complex division as indicated in Equa-tion (9.31) largely removes these spectral changes caused by windowing. Forthis reason, the windowing functions applied to the two different impulseresponse segments need to be the same length and at the same temporallocations.

The speaker distance should not be too far away from the material undertest, it will require the material sample to be even larger, otherwise, the mate-rial edge diffractions cause some residual reflections which may fall in the timerange prior to the speaker-face reflection. Figure 9.4 (b) and Figure 9.5 (c)


indicate temporal locations of the speaker reflection. For the in-situ window-ing method, due to the decisive window length given by the Window 1 as thedash-line frame indicated in Figure 9.4 (b), the minimum speaker distanceto the material surface should be D = 2 d, twice of the microphone distanceto the surface. The minimum speaker distance directly impacts the requiredsample size. The measurement setup as shown in Figure 9.4 (a) should beapplicable to in-situ measurements, however, the speaker distance and themicrophone distance to other obstacles, such as floor, light-fixtures and soon are crucial to the accuracy of the measurement results. The measurementsetup has to be implemented within reflection-free zone, although the speaker-face and sample edge diffractions will inevitably cause reflections within themeasured impulse response. But these unwanted reflections can be pushedaway beyond the window limit by properly adjusting the speaker distance tothe sample under test and the sample size.

For the in-situ subtraction method in addition, the microphone should beplaced as close to the plane surface of the material under test as possible,so as not to increase the low frequency limit. But the microphone shouldnot be placed in the surface, in this case, the reflection is unlikely to bemeasurable. The microphone location at the same time, needs to registerreflection from the surface which should not significantly impact the on-setand the amplitude of the incident impulse response (direct sound), often itdepends on reflective properties of the material under test. In extreme cases,when the material under test is highly reflective, the microphone needs to beplaced more away from the surface than those of highly absorptive ones. If themicrophone distance, d, is substantially large so as to impact the measurementaccuracy, a similar correction strategy as stated in Equation (9.35) should beapplied.

References

1. Blauert, J., Xiang, N.: Acoustics for Engineers – Troy Lectures. Springer-Verlag,Berlin Heidelberg (2009). Chap 7

2. Blauert, J., Xiang, N.: Acoustics for Engineers – Troy Lectures. Springer-Verlag,Berlin Heidelberg (2009). Chap 8

3. Chu, W.T.: Extension of the two-microphone transfer function method forimpedance tube measurements. J. Acoust. Soc. Am. 80, 347–348 (1986)

4. Chung, J.Y., Blaser, D.A.: Transfer function method of measuring in-duct acous-tic properties, I. Theory. J. Acoust. Soc. Am. 68, 907–913 (1980)

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9 Acoustic Impedance Measurement Methodssymphony.arch.rpi.edu/~xiangn/Teach/SRL_II_Spring2020_chap08.pdf · ment the two-microphone transfer function method. will lead to large measurement