Acoustic Research Institute ARI

Transcription

1 Austrian Academy of Sciences Acoustic Research Institute ARI System Identification in Audio Engineering P. Majdak Institut für Schallforschung, Österreichische Akademie der Wissenschaften; A- Wien, Reichsratsstrasse 7 Tel: +43 / Fax: +43 / piotr@majdak.com

2 System Identification in Audio Engineering Devices Under Test (DUT): Acoustical systems Buildings, concert halls, rooms, chambers Acoustical elements: absorber, reflectors (Sound generators: musical instruments, engines) Electro-acoustic systems Speakers Microphones DUT in general: Linear, Time Invariant Systems (LTI)

3 LTI-Systems Linearity: Asin T T {a x t b y t }=a T { x t } b T { y t } Bsin T In praxis: weakly nonlinear systems: Asin T Bsin T B n sin n T n n

4 LTI-Systems Time Invariance: T { x t }=T { x t T }= y t In praxis: a priori knowledge is essential: estimation of time constants repetition after decay of internal energy For nearly time variant systems: short measurement time slot measurement at steady state Additional problem: Noise internal noise in measurement equipment internal noise in DUT

5 Measurement path Devices: DAC ADC Symbolic: Noise u n DUT Signal x n + Output y i n y n

6 Parameters Transfer Function Spectrum H f, H k Impulse Response h t, h n Finite (FIR), Infinite (IIR) Pole-Zero-Diagram Signal-To-Noise-Ratio: SNR=2 log E signal max y i n u n A RMS A A RMS A/ Minimize the crest factor!

7 Parameters - Total Harmonic Distortion Amp_champ.freq._adc.wav THD: 2 A n n=2 2 n=2 A A signal search for peak range peak found max of noise threshold of peak energy bins above threshold -2 % 2 n THD+N: amp in db THD= E out E in THD N = %-6 E out THD+N -8 db f in Hz THD=.55296% 4 Signal In Noise And Distortion x

8 Parameters - Intermodulation Distortions IMD= A f, f2 A f A f2 %

9 Direct Measurement Method Direct measurement of amplitude and phase: Vanderkooy (986) Simple procedure (Stepped Sine) Measurement in steady state!!! Long duration for high frequency resolution High SNR (crest factor: 2 ) Improvement: Time Delay Spectrometry

10 Time Delay Spectrometry Signal: Linear Sweep x t =cos t Response: y t = H cos [ t ] Vanderkooy (986) y t = H {cos t cos [ ] sin t sin [ ] } Demodulation [ cos 2 t ] cos[ ] 2 and LP-Filtering: y R t = H cos [ ] 2 y I t = H sin [ ] 2

11 2 pass TDS TDS with 2 passes: pass Vanderkooy (986) 2 pass Vanderkooy (986) Vanderkooy (986)

12 Impulse Excitation Signal: unit pulse with amplitude A Impulse response immediately available 2 Little energy in the excitation signal: E signal = A High crest factor: A/ A RMS = A N Averaging necessary: Periodic Impulse Excitation (PIE) doubling the no. of pulses: +3dB SNR low SNR

13 - or 2-channel FFT Signal: white noise: Amplitude spectrum: DC=, the rest= Phase spectrum: random (equal distributed) Signal: random, gaussian distribution Crest factor: Example: fs=48khz, T=.5s 72 samples Averaging necessary Crest Factor Probability 32,% 2 4,8% 3,37% 3,3,% 3,9,% 4 63 ppm 4,4 ppm -channel-fft: for amplitude spectrum only 2-channel-FFT: for total transfer function

14 2-channel FFT Identification of amplitude and phase spectrum: X f Y f Y f H f = X f Müller & Massarani (2)

15 Pseudo Random Sequences From the system theory: r xy n =h n r xx n White noise decorrelated signal: r xx n = n With a decorrelated signal: Substitution: white noise r xy n =h n decorrelated signal Wanted: decorrelated signal determinist crest factor of binary pseudo random sequences

16 Maximum Length Sequence (MLS) 2 Generation: N shift registers feedback with EX-OR Z- Z- Z- x(n) EXOR Length of sequence: N L=2 Autocorrelation: r xx n = n L Unit pulse with a little offset Dunn & Hawksford (993)

17 MLS Calculation of the IR: r xx n = n r xy n =h n r xx n L N r xy n =h n h n = L n= L L L L N N r xy n =h n h n h n L n= L L n= Mittelwert von h(n) = DC DC L r xy n =h n DC [ ] L

18 MLS AC-Coupling: r xy n =h n DC-Coupling: r xy n h n DC Calculation of the cross-correlation: frequency domain: FT (not FFT!) direct method: L r xy n = x [i n] mod L y i L i= Signal as a matrix: create a circular matrix X from x(n) Calculation: r xy = L X y Hadamard Transformation

19 Fast Hadamard Transformation (FHT) Algorithmus ähnlich der DFT: X k = ℱ x n Matrix-Operator: Hadamard-Matrix: [ H 2= H =[] H2 n [ H2 = H2 n n ] ] H2 = H 2 H 2 H 2 n n n H8 n Algorithmus: Butterfly, aber kein Bit-reversal nur Additionen/Subtraktionen H 32

20 MLS MLS ist keine Hadamard-Matrix Umformung Hadamard-Matrix zur MLS-Matrix: [ ] X 2 =P 2 S 2 H 2 S P P, P 2... Permutationsmatrizen P =... Begrenzungsmatrizen (supress) S, S 2 S 2= S = n n [ ] [ ] r xy = P 2 S 2 {H 2 [ S P Y ]} L n

21 System Identification with MLS Length of MLS: at least the length of the expected IR SNR: crest factor: best SNR we can get! log L higher than PIE SNR gain: +45dB fs=48khz, T=.7s: doubling PIE length 5 times measurement time with PIE: 9 hours Averaging of time variations Very sensitive to nonlinear distortions

22 Sensitivity of MLS on distortions System: LP-Filter, fc=khz System output: Dunn & Hawksford (993) IR of filter: With distortion: 3 d { x f n }= db [ x n ] Dunn & Hawksford (993) Dunn & Hawksford (993) Dunn & Hawksford (993)

23 Sensitivity of MLS on distortions Error signal: e n =hd n h n hd n =r xy n =r x hd n =r x y n r dy n e n =r xd n f f d y n Error: [ x n ] 2 5 [ x n ] Energy distribution: Dunn & Hawksford (993) Dunn & Hawksford (993)

24 System identification procedure with MLS Immunity against noise: Noise I n = A Immunity against distortions: L= db Distortions I d = r A Depending on system: optimal excitation amplitude L= 2 db Increase MLS order instead of length doubling L= 6 db OK! Dunn & Hawksford (993)

25 Inverse Repeated Sequence (IRS) Canceling distortions for even orders: x n L = x n Dunn & Hawksford (993) IRS: { m n, n even, n 2 L x n = m n, n odd, n 2 L m(n)... MLS 2 r xy = L x n x n k 2 L k = { r my n, n even = r my n, n odd Dunn & Hawksford (993) n = n n L L n 2 L

26 Comparison: PIE, MLS, IRS Distortion Immunity: Filter: LP f = khz Distortion: -2dB Length: 247 samples Dunn & Hawksford (993) Noise Immunity (normalized to distortion immunity): Dunn & Hawksford (993)

27 Improving Distortion Immunity Problems of MLS: Sensitivity to distortions What we want: Identification of the linear part All harmonics separated Solution: Exponential sweep: x t =sin [ A e t / ] A= T ln 2 / T = ln 2 / T... sweep length with:... start frequency 2... end frequency

28 Exponential Sweep linear sweep Constant distance between all harmonics Farina (2) Weakly nonlinear system as example T=const T=const exponential sweep Farina (2)

29 Sweep Response To Impulse Response Sweep: Y f = X f H f H f =Y f X f Inverse sweep: ℱ { x t } X f = X f 2

30 Separating Harmonics t2 Impulse Response: Farina (2) IR of harmonics: 5th 3rd 2nd Output signal: t2 T ln N tn= ln 2 / Farina (2) linear

31 Simultaneous Measurement of THD and IR Amplitude spectra of separated harmonics: Farina (2) khz

32 Measurement of multiple systems 55 systems á.5 sec. Exp Sweep: 29' MLS: (3') 52' PIE: 38.5 days

33 Measurement of multiple systems

34 Measurement of multiple systems 55 systems with overlapping of 2 systems (5 sec total) 4 Minutes vs. 29 Min.

35 Increasing SNR with Gabor Multiplier Improvement in this example: +4dB

36 Summary - Comparison Direct Method, PIE Maximum Length Sequence: highest possible SNR sensitive to distortions Exponential Sweeps: separation of nonlinear distortions simultaneous measurement of THD measurements of multiple systems high SNR (-3dB compared to MLS) sensitive to transients