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# Engineering Acoustics ME 413

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This 3 page Class Notes was uploaded by Roman Jaskolski on Friday October 23, 2015. The Class Notes belongs to ME 413 at University of Idaho taught by Michael Anderson in Fall. Since its upload, it has received 33 views. For similar materials see /class/227897/me-413-university-of-idaho in Mechanical Engineering at University of Idaho.

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Date Created: 10/23/15

Use of Frequency Decomposition to Computer Sound Pressure Level for Wideband Noise Prepared by Michael Anderson Last Revised March 6 2002 I ommltino RMS Acoustic Pressure from a Sampled quot Voltage Signal A sampling of a microphone voltage signal yields N sampled voltage points vquot Upon computing the DFT coefficients Vk from the sampled voltages vquot the voltage signal can be visualized in the time domain as the reconstructed series NZil A k Va cos27rfkt pk fk I argVk 161 We assume that the signal is more or less deterministic so that the coefficients Vk would not change for successive samples of N voltages The DC component 1NVg is not present because the voltage signal corresponding to acoustic pressure has no mean component The mean component of total pressure is the ambient pressure 2 EV For each frequency the acoustic pressure amplitude will be related to the voltage amplitude using the appropriate frequency dependent sensitivity N271 1 2 pt7 Pkcos27rfktyk Pkim Vk k71N271 kl 1 1 k P V f N2 NZi k MfN2N T Here the phase W is unknown because microphone calibration data does not often include phase information in the sensitivity coefficient M It is understood in the above expression that the appropriate sensitivity free field or pressure response is indicated by Mfk The RMS pressure amplitude Prms can be computed from the pressure amplitudes Pk using Parseval39s theorm as Nil NZil 1 l l 2 Prms F 2 Pg 3 E P E PNz quot0 k1 Often a microphone is chosen such that its sensitivity is a constant over the appropriate frequency range so that M02 is just a constant This would be the case for the measurement of a plane wave whose wavelength is at least 4 times larger than the microphone diameter The above procedure for calculation of Prms can now be used to determine the sound pressure level using the formula SPL 201ogM ref This is often denoted the sound pressure level LP in dB re Pref Weighting Scales A further frequency dependent modi cation of measured pressure amplitudes is required to account for the sensitivity of human hearing to sounds For example it is commonly known that tones at at frequencies exceeding 20 kHz cannot be heard by humans at all The frequency dependent human sensitivity to sound has both physical and perceptual causes Consider first human sensitivity to a harmonic tone at frequency f indicated by ptPcos27tftp In this case the RMS pressure used in the computation of sound pressure level is PrmsP To account for the frequency dependent sensitivity of human hearing the amplitude is multiplied by a frequency dependent weight Wf to obtain pWtW Pcos27t p The subscript Whas been added to the acoustic pressure to denote that it has been weighted Now the RMS pressure is PrmsWWfPrms where the Windicates that it is a weighted RMS pressure There are several weighting scales the most common being the A scale followed by the B and C scales The functional form for the AWeight scale is m 187210 f f2 205989972Jf2 107652652f2 737862232f2 121942172 where Wf is the magnitude of the desired Aweight filter transfer function The A Weight scale is designed to replicate the human sensitivity to acoustic disturbances To report a weighted sound measurement the weighted RMS pressure PrmsA is used in the computation of SPL For the measurement of a single frequency harmonic tone considered above the computation of AWeighted sound pressure level is SPL 2010g W 2010g Pref Pref where Prms is the RMS pressure amplitude computed from the microphone signal using the appropriate microphone sensitivity In this case however the measurement is reported as SPL in dBA re Pref and the level is known as the AWeighted Sound Pressure Level LPA In the case of a multifreqeuncy signal the unweighted pressure amplitudes are computed using the DFT coefficients of the voltage signal and the appropriate microphone sensitivity by Pk 1 ivk fk k 12N2 1 MHZ N T LV NZ MfN2 N NZ 39 Then the pressure amplitudes at each frequency fk are multiplied by the weight Wm evaluated at that frequency and placed into Parseval s theorem 1 71 2 1 2 MM EZWVU QR 35WfkgtPNz The AWeighted RMS acoustic pressure is then used to compute the AWeighted Sound Pressure level with SPL 20 log M ref The units are now dBA re Pref

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