PHYSICAL AND APPLIED ACOUSTICS: AN INTRODUCTION

By RICHARD RAMIREZ

CONTENTS ANALYSIS OF ACOUSTIC SIGNALS SOME BASICS—MEASURES OF SOUND ONE-THIRD OCTAVE BAND LEVELS A-WEIGHTED SOUND LEVELS NARROWBAND SIGNAL ANALYSIS POWER SPECTRAL DENSITY BY THE FINITE FOURIER TRANSFORM SPECTRAL ANALYSIS OF MEASURED TIME SERIES SOUND LEVEL CALCULATIONS FROM NARROWBAND POWER SPECTRA THE “SLOW” FOURIER TRANSFORM: LEAST SQUARES EXTRACTION OF A HARMONIC SIGNAL FREQUENCY RESPONSE OF LINEAR SYSTEMS WITH RANDOM INPUT INPUT–OUTPUT RELATIONSHIPS FOR A LINEAR SYSTEM SPECTRAL APPROACH TO EVALUATING THE CONVOLUTION INTEGRAL EXAMPLE: FFT FOR RESPONSE OF A SPRING/MASS/DAMPER EXAMPLE: NUMERICAL DIFFERENTIATION AND INTEGRATION USING THE FFT FFT AND IFFT TO ESTIMATE THE FOURIER TRANSFORM FOR COMPLEX FUNCTIONS PROBLEMS ONE DIMENSIONAL SOUND FIELDS NEWTON’S SECOND LAW CONSERVATION OF MASS EQUATION OF STATE SOLUTIONS FOR SOME SIMPLE FIELDS SOUND INTENSITY AND THE SOUND ABSORPTION COEF?CIENT D’ALEMBERT’S SOLUTION SOUND IN AN IN?NITE TUBE SOUND TRANSMISSION LOSS THE MASS LAW RANDOM INCIDENCE SOUND TRANSMISSION LOSS TRANSMISSION LOSS CALCULATIONS USING TRANSFER MATRICES TRANSFER MATRIX FOR A SOLID ELEMENT TRANSFER MATRIX FOR AN AIR GAP AIR GAP WITH NON-NORMAL INCIDENT SOUND DOUBLE WALLS SIMPLE AIR GAP LOW FREQUENCIES SLIGHTLY HIGHER FREQUENCIES—DOUBLE WALL RESONANCE VERY HIGH FREQUENCIES ANALYSIS OF DOUBLE WALLS WITH MECHANICAL COUPLING REFERENCES FOR CHAPTER 3 ANALYSIS OF MUF?ERS AND DUCTS THE JUNCTION OF TWO PIPES THE EXPANSION MUF?ER THE HELMHOLTZ RESONATOR SIDE BRANCHES GENERAL SIDE BRANCH SIMPLE PIPE SIDE BRANCH SOUND IN TUBES OF VARYING CROSS-SECTIONAL AREA CONICAL HORN ASSEMBLE CONICAL ELEMENTS TO MODEL A TUBE WITH ARBITRARY CROSS-SECTIONAL AREA SOLVING FOR THE PRESSURE AND VELOCITY WITHIN THE DOMAIN THE EXPONENTIAL HORN NUMERICAL EXAMPLE FOR A NON-UNIFORM DUCT REFERENCE FOR CHAPTER 4 SOUND RADIATION IN THREE DIMENSIONS WAVE EQUATION IN THREE DIMENSIONS SOUND INTENSITY OF SPHERICAL WAVES SOUND RADIATION BY A SIMPLE SOURCE SOUND FIELD INSIDE A PULSATING SPHERE SIMPLE MODEL OF SOUND RADIATION FROM A LOUDSPEAKER THE IMAGE SOURCE PERIODIC FREQUENCY RESPONSE DUE TO RE?ECTIONS THE ACOUSTIC DIPOLE THE LINE SOURCE SURFACE SOURCE SOUND RADIATION FROM A PISTON IN A BAF?E SINGLE PISTON PISTON VIBRATION MULTIPLE PISTON RADIATORS ANALYSIS AND DESIGN OF LOUDSPEAKERS IN VENTED BOXES LOUDSPEAKER DIAPHRAGM IN A CLOSED BOX DIAPHRAGM FORCE DUE TO BACK VOLUME OF AIR EFFECT OF THE AIR IN THE VENT RESPONSE DUE TO HARMONIC SOUND FIELDS SOUND RADIATION FROM LOUDSPEAKERS IN VENTED BOXES COMPUTER-AIDED ACOUSTICS THE GREEN’S FUNCTION GREEN’S FUNCTION FOR IN?NITE HALF-SPACE BOUNDED BY AN IN?NITE RIGID PLANE INTEGRAL EQUATION FOR THE GENERAL SOUND RADIATION PROBLEM NUMERICAL SOLUTION EFFECT OF BOUNDARY IMPEDANCE SOUND FIELD DUE TO A POINT SOURCE IN AN ENCLOSURE CALCULATION OF THE PRESSURE AT AN ARBITRARY POINT IN SPACE APPROXIMATE EVALUATION OF INTEGRALS ELEMENTS THAT ARE SUF?CIENTLY SEPARATED: ESTIMATION USING MIDPOINTS ESTIMATION OF BLL ESTIMATION OF CLL COMPUTER-AIDED DESIGN MODEL TO SPECIFY GEOMETRY EVALUATION OF SINGULAR INTEGRALS FOR CLL NUMERICAL RESULTS SOUND INSIDE A RECTANGULAR TUBE SOUND INSIDE A RADIALLY OSCILLATING SPHERE SOUND RADIATION FROM A RADIALLY OSCILLATING CYLINDER REFERENCES FOR CHAPTER 6 MODAL SOLUTIONS FOR THE SOUND IN ENCLOSURES NATURAL FREQUENCIES AND EIGENFUNCTIONS FOR THE RECTANGULAR ENCLOSURE SOLUTION FOR THE PRESSURE FIELD GEOMETRICAL ROOM ACOUSTICS RELATION BETWEEN ENERGY DENSITY AND INTENSITY EFFECT OF SOUND ABSORPTION COEF?CIENT ON STEADY-STATE SOUND FIELDS EFFECT OF SOUND ABSORPTION COEF?CIENT ON DECAYING SOUND FIELDS EFFECTS OF VISCOSITY BASIC EQUATIONS FOR SOUND IN A VISCOUS FLUID VISCOUS FLOW IN ONE DIMENSION SQUEEZE FILM DAMPING IN A COMPRESSIBLE FLUID IN TWO DIMENSIONS LIMITING CASES AT LOW AND HIGH FREQUENCIES VISCOUS FORCE ON AN OSCILLATING CYLINDER VISCOUS ACOUSTIC EXCITATION OF A THIN BEAM HAVING CIRCULAR CROSS SECTION SOLUTIONS FOR THE ACOUSTIC RESPONSE RESPONSE OF AN IN?NITELY LONG UNIFORM FIBER RESPONSE OF A FIBER OF FINITE LENGTH PRESSURE AND FLOW AROUND A PERIODIC ARRAY OF THIN BEAMS INCLUDING VISCOSITY AND COMPRESSIBILITY FOURIER TRANSFORM SOLUTION FOR THE VELOCITIES AND PRESSURE NUMERICAL EVALUATION NUMERICAL EXAMPLE SOUND PRESSURE AND VISCOUS FLOW AROUND A PERIODIC ARRAY OF THIN BEAMS REFERENCES FOR CHAPTER 9 ACOUSTIC SENSING MECHANICAL SENSITIVITY OF A MICROPHONE DIAPHRAGM DIAPHRAGM WITH NO VENT THE MICROPHONE PACKAGE AS A HELMHOLTZ RESONATOR AN IMPORTANT NOTE ON THE IDEAL DIAPHRAGM PROPERTIES DAMPING DUE TO THE VENT ACOUSTIC RADIATION LOADING ON THE DIAPHRAGM NUMERICAL EXAMPLES ANALYSIS OF THERMAL-MECHANICAL NOISE IN A PRESSURE-SENSING MICROPHONE LIMITING CASE OF THERMAL-MECHANICAL NOISE IN AN IDEAL PRESSURE-SENSING MICROPHONE THERMAL RESPONSE OF A VELOCITY DRIVEN SPRING/MASS APPLICATION TO A HAIR FOR SENSING SOUND THERMAL NOISE RESPONSE INCREASES WITH THE NUMBER OF DEGREES OF FREEDOM VISCOUS FLOW-INDUCED RESPONSE TO SOUND AND SIGNAL TO NOISE RATIO, SNR EFFECT OF SOUND ANGLE OF INCIDENCE ON A PRESSURE MICROPHONE ACCELERATION SENSITIVITY BASICS OF DIRECTIONAL MICROPHONES FIRST-ORDER DIRECTIONAL MICROPHONES SECOND-ORDER DIRECTIONAL MICROPHONES THE RIBBON MICROPHONE EFFECT OF GAIN ERRORS ON DIRECTIVITY INDEX EMPIRICAL MICROPHONE COMPENSATION FOR DESIRED DIRECTIVITY NOISE IN DIRECTIONAL MICROPHONES DIRECTIVITY INDEX BY INTEGRATION IN THE WAVE VECTOR DOMAIN ACOUSTIC PARTICLE VELOCITY ESTIMATION USING A PRESSURE DIFFERENTIAL MICROPHONE MEASUREMENT OF ACOUSTIC PARTICLE VELOCITY PRESSURE GRADIENT AND VELOCITY MICROPHONES MEASUREMENT ERROR TWO-MICROPHONE INTENSITY MEASUREMENT REFERENCES FOR CHAPTER 10 ELECTRONIC TRANSDUCTION FOR ACOUSTIC SENSORS ELECTRICAL SENSITIVITY OF A MICROPHONE MICROPHONE SENSITIVITY USING A CONSTANT CHARGE, HIGH IMPEDANCE VOLTAGE AMPLI?ER EFFECT OF BUFFER AMPLI?ER INPUT IMPEDANCE—PARASITIC CAPACITANCE MICROPHONE SENSITIVITY USING A CONSTANT VOLTAGE BIAS EFFECT OF THE ELECTRONIC CIRCUIT ON THE DIAPHRAGM MECHANICAL RESPONSE ELECTRIC FORCE DUE TO CONSTANT CHARGE ELECTRIC FORCE DUE TO CONSTANT VOLTAGE OPTIMUM CAPACITANCE FOR SENSITIVITY AND STABILITY SOUND INPUT-REFERRED NOISE EFFECTS OF ELECTRONIC NOISE AND STRAY CAPACITANCE NOISE ANALYSIS OF A CHARGE AMPLI?ER MICROPHONE EQUIVALENT INPUT NOISE INCLUDING ELECTRONICS ELECTRODYNAMIC MICROPHONES ESTIMATION OF CAPACITANCE COULOMB’S LAW AND THE ELECTRIC POTENTIAL INTEGRAL EQUATION RELATING CHARGE AND POTENTIAL NUMERICAL EVALUATION FOR THREE-DIMENSIONAL DOMAINS POTENTIAL ENERGY FOR A DISTRIBUTION OF CHARGE DENSITIES EQUATION OF MOTION FOR A SYSTEM HAVING A SINGLE DEGREE OF FREEDOM EXAMPLE OF A SOFTENING ELECTRODE: PARALLEL PLATE EXAMPLE OF A STIFFENING ELECTRODE: ELECTROSTATIC PENDULUM EXAMPLE OF AN ADJUSTABLE ELECTRODE: ELECTROSTATIC PENDULUM WITH REDUCED STIFFNESS AT EQUILIBRIUM EXAMPLE OF AN ADJUSTABLE ELECTRODE: A BISTABLE ELECTROSTATIC PENDULUM ELECTRONIC SENSITIVITY COMPLIANT REPULSIVE ACTUATOR REFERENCES FOR CHAPTER 12 PARAMETER IDENTI?CATION OF ACOUSTIC SYSTEMS LEAST SQUARES MODEL OF A COMPLEX TRANSFER FUNCTION WITH REAL UNKNOWNS ACCOUNTING FOR AN UNKNOWN TIME DELAY IN H EXAMPLES REFERENCES FOR CHAPTER 13 CHAPTER REF , (1 1 1) WHERE < P2(T) > IS THE MEAN OF THE SQUARE OF THE ?UCTUATING SOUND PRESSURE AND PREF = 20 × 10?6 PA IS THE STANDARD REFERENCE PRESSURE THE SOUND PRESSURE LEVEL IS DIMENSIONLESS BUT IT IS NORMALLY STATED WITH THE UNITS OF DECIBELS (DB) THE MEAN OF ANALYSIS OF ACOUSTIC SIGNALS THE SQUARE OF THE ?UCTUATING PRESSURE (OR MEAN SQUARE PRESSURE) MAY BE DETERMINED FROM < P2(T) >= LIM T?? T T ?94 DB (1 1 3) HUMANS WITH GOOD HEARING CAN EXPERIENCE (AND DETECT) SOUND LEVELS OVER A RANGE OF ZERO DB UP TO ABOUT 130DB THIS CORRESPONDS TO MEAN SQUARE PRESSURES THAT COVER A RANGE FROM 20 × 10?6 TO ABOUT 63PA IN NEARLY ALL ACOUSTICAL ENVIRONMENTS, SOUND SOURCES TEND TO BE INDEPENDENT OF EACH OTHER SO THAT THEIR OUTPUTS CAN BE CONSIDERED TO BE UNCORRELATED SIGNALS SUPPOSE WE HAVE TWO UNCORRELATED SOUND PRESSURES P1(T) AND P2(T) AND WE’D LIKE TO KNOW THE SOUND PRESSURE LEVEL WHEN BOTH ARE PRESENT SINCE THE TOTAL PRESSURE WILL BE P(T) = P1(T) + P2(T), APPLYING EQ (1 1 2) GIVES < P2(T) >= LIM T ?? P2(T)DT = LIM T ?? (P1(T) + P2(T))2DT = LIM T ?? P1(T)2DT + LIM T ?? P2(T)2DT + 2 LIM T ?? REF (1 1 5) IF THE SOUND PRESSURE LEVELS SPL1 AND SPL2 THAT CORRESPOND TO P1(T) AND P2(T) ARE KNOWN, THEN EQ (1 1 1) CAN BE MANIPULATED TO DETERMINE THE CORRESPONDING MEAN SQUARES THIS RESULT CAN THEN BE COMBINED WITH EQ (1 1 5) TO CALCULATE THE SOUND PRESSURE LEVEL WHEN BOTH SIGNALS ARE PRESENT, 1 3 A-WEIGHTED SOUND LEVELS A-WEIGHTED SOUND LEVELS WHILE THE OVERALL SOUND PRESSURE LEVEL PROVIDES A CONVENIENT SINGLE NUMBER TO REP- RESENT THE AMPLITUDE OF A GIVEN SOUND, IT OFTEN DOESN’T ADEQUATELY CHARACTERIZE SOUND ?ELDS THIS IS BECAUSE DIFFERENT FREQUENCIES CREATE SIGNI?CANTLY DIFFERENT LEVELS OF ANNOYANCE ALONG WITH HEARING DAMAGE TO BETTER REPRESENT THE PERCEPTION AND HEAR- ING DAMAGE DUE TO A GIVEN NOISE ENVIRONMENT, IT CAN BE HELPFUL TO USE A FREQUENCY- WEIGHTED MEASURE OF THE SOUND PRESSURE LEVEL RATHER THAN THE OVERALL SOUND PRESSURE LEVEL (OASPL) THE MOST IMPORTANT (AND COMMONLY USED) FREQUENCY-WEIGHTED MEA- SURE OF SOUND IS THE DBA SCALE TO APPLY THIS WEIGHTING, ONE EFFECTIVELY PASSES THE ?UCTUATING PRESSURE P(T) THROUGH A FREQUENCY-DEPENDENT ?LTER BEFORE CALCULATING SOUND PRESSURE LEVEL THE A-WEIGHTING ?LTER IS DE?NED BY THE STANDARD, ANSI S1 42- 2001, DESIGN RESPONSE OF WEIGHTING NETWORKS FOR ACOUSTICAL MEASUREMENTS ANALYSIS OF ACOUSTIC SIGNALS TABLE 1 1 APPROXIMATE ONE-THIRD OCTAVE BAND CENTER AND CUTOFF FREQUENCIES WITH DBA WEIGHTINGS BAND LOWER FREQ CENTER FREQ UPPER FREQ DBA ?( F ) , (1 3 1) 1 3 A-WEIGHTED SOUND LEVELS NARROWBAND SIGNAL ANALYSIS WHILE OVERALL, A-WEIGHTED AND OCTAVE BAND LEVELS PROVIDE EXCELLENT MEASURES OF LOUDNESS IN MANY SITUATIONS, IT IS ALSO OFTEN NECESSARY TO RESORT TO FREQUENCY DOMAIN ANALYSIS IN NARROW FREQUENCY BANDS THIS IS OFTEN PROVIDED IN THE FORM OF A POWER SPECTRAL DENSITY WITH UNITS OF PASCALS2/HZ OR PASCALS2/(RAD/S) THESE DATA ARE NOR- MALLY PROVIDED FOR FREQUENCY BANDS THAT HAVE THE SAME WIDTH F OVER THE ENTIRE FREQUENCY RANGE RATHER THAN THE LOGARITHMIC SPACING USED IN OCTAVE BANDS THE POWER SPECTRAL DENSITY IS AN EXTREMELY IMPORTANT STATISTIC FOR THE CHARAC- TERIZATION OF COMPLEX, RANDOM SIGNALS AS ARE OFTEN ENCOUNTERED IN ACOUSTICS THE POWER SPECTRAL DENSITY, SPP(?), IS DE?NED TO BE THE FOURIER TRANSFORM OF THE AUTO- CORRELATION FUNCTION FOR A WEAKLY STATIONARY ERGODIC RANDOM PROCESS, RPP(?) = E[P(T)P(T + ?)], WHERE E[] DENOTES THE EXPECTED VALUE WEAK STATIONARITY MEANS THAT CERTAIN ?RST-ORDER STATISTICS OF THE PROCESS ARE INDEPENDENT OF TIME, T WHICH IS WHY THE ARGUMENT OF RPP DEPENDS ONLY ON ? IT IS IMPORTANT TO NOTE THAT THE AUTOCORRELATION FUNCTION IS AN EVEN FUNCTION OF ? THIS IS A RESULT OF THE FACT THAT IT IS INDEPENDENT OF A SHIFT IN THE TIME ORIGIN LET T0 = T + ? SO THAT THE AUTOCORRELATION FUNCTION BECOMES RPP(?) = E[P(T)P(T + ?)] = E[P(T0 ??)P(T0)] = RPP(??) (1 4 1) ANALYSIS OF ACOUSTIC SIGNALS THE FOURIER TRANSFORM OF THIS GIVES THE POWER SPECTRAL DENSITY, SPP(?) = 2? ? ?? RPP(?)E?I??D? (1 4 2) ALTHOUGH THE FOURIER TRANSFORM OF A REAL FUNCTION OFTEN RESULTS IN A COMPLEX QUANTITY, THE POWER SPECTRAL DENSITY IS ALWAYS ENTIRELY REAL THIS IS A RESULT OF THE FACT THAT THE AUTOCORRELATION FUNCTION IS AN EVEN FUNCTION OF ? TO SEE THIS, TAKE THE COMPLEX CONJUGATE OF THE POWER SPECTRAL DENSITY, S? PP(?) = 2? ?? ? RPP(???)E?I?? ?D?? = 2? ? ?? RPP(??)E?I?? ?D?? = SPP(?), (1 4 4) WHERE WE HAVE USED THE FACT THAT RPP(? ?) = RPP(?? ?) EQUATION(1 4 4) ALSO SHOWS THAT THE POWER SPECTRAL DENSITY IS AN EVEN FUNCTION OF ?, SPP(?) = SPP(??) THIS FACT LEADS US TO REFER TO SPP(?) AS THE DOUBLE-SIDED POWER SPECTRAL DENSITY SINCE THE POWER SPECTRAL DENSITY IS OBTAINED VIA THE FOURIER TRANSFORM OF THE AUTOCORRELATION FUNCTION, THESE TWO QUANTITIES ARE ALSO RELATED THROUGH THE INVERSE FOURIER TRANSFORM, RPP(?) = ? ?? SPP(?)EI??D? (1 4 5) EQUATIONS(1 4 2) AND (1 4 5) THUS FORM A FOURIER TRANSFORM PAIR AN EXTREMELY IMPORTANT FEATURE OF THE POWER SPECTRAL DENSITY IS THAT THE INTEGRAL OVER ALL FREQUENCIES IS EQUAL TO THE MEAN SQUARE RESPONSE TO SEE THIS, SET ? = 0 IN EQS (1 4 1) AND (1 4 5), RPP(0) = E[P2(T)] = ? ?? SPP(?)D? (1 4 6) THE INTEGRAL OF THE POWER SPECTRAL DENSITY BETWEEN TWO FREQUENCIES, ?1 AND ?2 WILL THEN GIVE THE CONTRIBUTION TO THE MEAN SQUARE ASSOCIATED WITH THAT FREQUENCY RANGE WHILE THE DOUBLE-SIDED POWER SPECTRAL DENSITY SPP(?) FOLLOWS FROM ITS DE?NITION IN TERMS OF THE TRANSFORM OF THE AUTOCORRELATION FUNCTION, ENGINEERS GENERALLY PREFER THE SINGLE-SIDED DENSITY G PP( F ), WHICH IS EXPRESSED IN UNITS OF PASCALS2/HZ THIS IS DE?NED SO THAT, AGAIN, 1 4 NARROWBAND SIGNAL ANALYSIS < P2(T) >= ? G PP ? POWER SPECTRAL DENSITY BY THE FINITE FOURIER TRANSFORM THUS FAR, WE HAVE COMPUTED THE POWER SPECTRAL DENSITY BY ?RST DETERMINING THE AUTOCORRELATION FUNCTION AND THEN USING THE FOURIER TRANSFORM IN THE ANALYSIS OF MEASURED DATA, IT IS FAR MORE EF?CIENT TO EMPLOY THE FAST FOURIER TRANSFORM (FFT) TO COMPUTE SPECTRA TO DO THIS WE NEED TO ?RST COMPUTE THE POWER SPECTRUM IN TERMS OF A ?NITE FOURIER TRANSFORM GIVEN A STATIONARY RANDOM PROCESS, X(T), WE WILL SHOW THAT THE POWER SPECTRAL DENSITY, SXX(?) MAY BE COMPUTED BY SXX(?) = LIM T ?? ? T E[|X(?, T )|2], (1 5 1) WHERE THE ?NITE FOURIER TRANSFORM OF X(T) IS, X(?, T ) = 2? T ?T X(T)E?I?TDT (1 5 2) MULTIPLYING EQ (1 5 2) BY ITS COMPLEX CONJUGATE AND TAKING THE EXPECTED VALUE GIVES ANALYSIS OF ACOUSTIC SIGNALS FIG 1 1 DOMAIN OF INTEGRATION FOR COMPUTING THE FOURIER TRANSFORM OF THE AUTOCORRELATION FUNCTION IN TERMS OF THE FINITE FOURIER TRANSFORM E[|X(?, T )|2] = (1 5 3) BRINGING THE EXPECTED VALUE OPERATION INSIDE THE INTEGRAL GIVES E[|X(?, T )|2] = T ?T T ?T RXX(T1 ?T2)E?I?(T1?T2)DT1DT2 (1 5 4) TO CARRY OUT THE DOUBLE INTEGRATION IN EQ (1 5 4) MAKE A CHANGE OF VARIABLES FROM T1 TO ?, WHERE ? = T1 ?T2 ELIMINATING T1 IN EQ (1 5 4) GIVES E[|X(?, T )|2] = RXX(?)E?I?(?)D?DT2 (1 5 5) THE INTEGRATION RANGE IS NOW OVER THE DOMAIN SHOWN IN FIG 1 1 IT IS CONVENIENT TO ?RST INTEGRATE OVER T2 AND THEN OVER ? WHEN CHANGING THE ORDER OF THE INTEGRATION WE MUST INTEGRATE OVER SEPARATE TRIANGLES FOR POSITIVE AND NEGATIVE VALUES OF ? FOR THE UPPER TRIANGLE WE WILL INTEGRATE OVER ? FOR 0 ?? ?2T THE RANGE FOR T2 FOR THIS TRIANGLE WILL BE FROM ?T TO THE LINE DESCRIBED BY ? = T ?T2 FOR THE LOWER TRIANGLE, ? WILL BE INTEGRATED FROM ?2T TO 0, OR, ?2T ?? ?0, AND THE RANGE FOR T2 WILL BE FROM THE LINE DESCRIBED BY ? = ?T ?T2 TO T THE INTEGRAL IN EQ (1 5 5) MAY THEN BE WRITTEN AS E[|X(?, T )|2] = T ?? ?T RXX(?)E?I?(?)DT2D? + ?2T T ?T ?? RXX(?)E?I?(?)DT2D? (1 5 6) 1 5 POWER SPECTRAL DENSITY BY THE FINITE FOURIER TRANSFORM SINCE THE INTEGRANDS IN EQ (1 5 6) DO NOT DEPEND ON T2, THE INTEGRALS OVER T2 ARE EASILY EVALUATED WHICH GIVES E[|X(?, T )|2] = RXX(?)E?I?(?)(2T ??)D? + ?2T RXX(?)E?I?(?)(2T + ?)D? (1 5 7) COMBINING THE INTEGRALS IN EQ (1 5 7) GIVES E[|X(?, T )|2] = 2T ?2T RXX(?)E?I?(?)(2T ?|?|)D? (1 5 8) TO EXAMINE THE VALIDITY OF EQ (1 5 1) WE MUST MULTIPLY BY ? T AND TAKE THE LIMIT, LIM T?? ? T E[|X(?, T )|2] = LIM T?? 4?T 2T ?2T RXX(?)E?I?(?)(2T ?|?|)D? = LIM T ?? 4? 2T ?2T RXX(?)E?I?(?) |?| T D? (1 5 9) IT CAN BE SHOWN THAT LIM T ?? SPECTRAL ANALYSIS OF MEASURED TIME SERIES IN THE FOLLOWING, WE WILL USE OUR ABILITY TO COMPUTE THE POWER SPECTRAL DENSITY FROM THE ?NITE FOURIER TRANSFORM SHOWN IN THE PREVIOUS SECTION TO SHOW HOW TO APPROXIMATE THE POWER SPECTRAL DENSITY OF MEASURED SIGNALS BY USING THE FAST FOURIER TRANSFORM (FFT) THISPROVIDESANEXTREMELYIMPORTANTTOOLFORTHEANALYSISOFTIMEDOMAINDATA THE FFT ALGORITHM IS THE BASIS OF VIRTUALLY ALL METHODS OF EXAMINING THE FREQUENCY CONTENT OF SIGNALS ANALYSIS OF ACOUSTIC SIGNALS SUPPOSE A MEASURED TIME SERIES, X(T) IS SAMPLED AT DISCRETE VALUES OF TIME, T, THEN LET XN = X(TN) = X(NT), N = 0, 1, , N ?1, (1 6 1) WHERE THE TIME DURATION BETWEEN THE EQUALLY SPACED SAMPLE TIMES IS T TO SATISFY THE NYQUIST SAMPLING CRITERION, THE TIME SERIES, X(T) MUST BE SAMPLED AT A HIGH ENOUGH RATE TO OBTAIN TWO SAMPLES DURING ONE PERIOD OF THE HIGHEST FREQUENCY COMPONENT IN THE SIGNAL IF ?MAX IS THE HIGHEST FREQUENCY COMPONENT IN THE SIGNAL, THE NYQUIST CRITERION IS T ? 2? ?MAX 2 (1 6 2) IF A TOTAL OF N DATA POINTS ARE OBTAINED, THEN THE TIME DURATION OF THE TIME SERIES IS T = NT IN THE PREVIOUS SECTION WE HAVE SHOWN THAT THE POWER SPECTRAL DENSITY MAY BE COMPUTED FROM SXX(?) = LIM T ?? ? T E[|X(?, T )|2], (1 6 3) WHERE X(?, T ) IS THE ?NITE FOURIER TRANSFORM OF X(T), X(?, T ) = 2? T ?T X(T)E?I?TDT (1 6 4) THE ?NITE FOURIER TRANSFORM IN EQ (1 6 4) IS EXPRESSED AS AN INTEGRAL OVER BOTH NEGA- TIVE AND POSITIVE TIMES, T IN THE ANALYSIS OF MEASURED DATA IT IS CONVENIENT TO CONSIDER THE BEGINNING OF THE TIME RECORD AS BEING AT T = 0 RATHER THAN AT T = ?T SINCE OUR RANDOM PROCESS IS ASSUMED TO BE STATIONARY, THE STATISTICS ARE INDEPENDENT OF A SHIFT IN THE TIME ORIGIN EQUATION (1 6 4) MAY THEN BE REWRITTEN AS X(?, T ) = 2? 2T WHICH ALLOWS US TO WRITE EQ (1 6 3) AS SXX(?) = LIM T/2?? ? T/2E[|X(?, T/2)|2], (1 6 6) AND THE FOURIER TRANSFORM IN EQ (1 6 5) MAY BE WRITTEN AS X(?, T/2) = 2? T X(T)E?I?TDT (1 6 7) 1 6 SPECTRAL ANALYSIS OF MEASURED TIME SERIES XNE?I?TNT (1 6 8) TO EMPLOY THE FFT ALGORITHM TO EVALUATE EQ (1 6 8), SUPPOSE WE WISH TO EVALUATE EQ (1 6 8) ONLY AT DISCRETE VALUES OF ?, ? J = J?, J = 0, 1, , N ?1, WHERE N IS, AS ABOVE, THE TOTAL NUMBER OF TIME POINTS IN THE RECORD XN ? WILL BE THE LOWEST FREQUENCY COMPONENT IN THE SIGNAL IN RADIANS/SECOND, ? = 2? T (1 6 9) AFTER SUBSTITUTING ? J FOR ?, USING EQ (1 6 9), AND USING THE FACT THAT T = T/N, EQ (1 6 8) BECOMES X(? J, T/2) = XNE?I? JTNT = XNE?I J?NTT = XLE?I2? JL/N, (1 6 13) M T 2? M T 2? N 2 |X JL|2 (1 6 18) IT IS IMPORTANT TO NOTE THAT EQ (1 6 18) CAN BE EVALUATED FOR ANY J IT TURNS OUT, HOW- EVER, THAT NOT ALL OF THESE VALUES ARE UNIQUE PLOTTING THE RESULT OF EQ (1 6 18) VERSUS 1 6 SPECTRAL ANALYSIS OF MEASURED TIME SERIES SOUND LEVEL CALCULATIONS FROM NARROWBAND POWER SPECTRA IN THE FOLLOWING, A PROCEDURE IS OUTLINED TO COMPUTE ONE-THIRD OCTAVE BAND LEVELS FROM THE DOUBLE-SIDED POWER SPECTRAL DENSITY OF A SOUND PRESSURE, SPP(?) KNOWING THE ONE-THIRD OCTAVE LEVELS ONE CAN THEN ESTIMATE THE EQUIVALENT DBA LEVEL ETC THE ONE-THIRD OCTAVE BAND LEVEL AT THE FREQUENCY F3 OF A WEAKLY STATIONARY RANDOM SIGNAL, P(T), IS GIVEN BY ? 21/3 (1 7 2) TO DETERMINE < P2 > F3 IT IS HELPFUL TO ?RST CONVERT THE DOUBLE-SIDED POWER SPECTRAL DENSITY, SPP(?), TO A SINGLE-SIDED POWER SPECTRAL DENSITY, G PP( F ), WITH UNITS OF PASCALS2/HZ GIVEN THE DOUBLE-SIDED POWER SPECTRAL DENSITY OF A RANDOM PROCESS, SPP(?) AND THE SINGLE-SIDED SPECTRAL DENSITY G PP( F ), IT IS USEFUL TO RECALL THAT THE TOTAL MEAN SQUARE RESPONSE, E[P2] IS OBTAINED BY INTEGRATING THE SPECTRAL DENSITY OVER ALL FREQUENCIES, E[P2] = ? G PP( F )D F = ? ?? SPP(?)D? = SPP(?)D?, (1 7 3) WHERE THE LAST INTEGRAL RESULTS FROM THE FACT THAT THE DOUBLE-SIDED POWER SPECTRAL DENSITY IS AN EVEN FUNCTION, SPP(?) = SPP(??) SINCE ? = 2? F , SPP(?)D? = SPP(2? F )2?D F = ? 4? SPP(2? F )D F (1 7 6) IN THE SPECIAL CASE WHERE THE POWER SPECTRUM IS INDEPENDENT OF FREQUENCY, THE INTEGRAL CAN BE PERFORMED ANALYTICALLY IF SPP(2? F ) = SPP, CARRYING OUT THE INTEGRATION IN EQ (1 7 6) THEN GIVES 1 7 SOUND LEVEL CALCULATIONS FROM NARROWBAND POWER SPECTRA ? 21/3 ? THE “SLOW” FOURIER TRANSFORM: LEAST SQUARES EXTRACTION OF A HARMONIC SIGNAL THE METHODS DESCRIBED ABOVE ARE USED TO EXAMINE THE FREQUENCY CONTENT OF BROAD- BAND SIGNALS THERE ARE MANY SITUATIONS WHERE WE ARE INTERESTED IN DETERMINING THE AMPLITUDE AND PHASE OF A SIGNAL AT A KNOWN FREQUENCY THAT IS SUPERIMPOSED ON OTHER UNCORRELATED SIGNALS SUCH AS NOISE TO ACCOMPLISH THIS WE MIGHT BE TEMPTED TO EMPLOY A FFT ALGORITHM WHICH WOULD PROVIDE THE AMPLITUDE AND PHASE AT NUMEROUS FREQUENCIES THIS, HOWEVER, PROVIDES A SUBSTANTIAL AMOUNT OF INFORMATION ON THE CON- TRIBUTIONS DUE TO FREQUENCIES THAT ARE IRRELEVANT TO WHAT IS SOUGHT; I E ONE MAY CARE ONLY ABOUT THE CONTRIBUTION AT ONE PARTICULAR FREQUENCY, AND THAT FREQUENCY MAY NOT MATCH UP WELL WITH THE TIME STEP USED IN THE ANALYSIS FORTUNATELY, BECAUSE THE FRE- QUENCY OF INTEREST IS KNOWN, WE CAN EASILY CONSTRUCT A LINEAR LEAST SQUARES APPROACH TO OBTAIN BOTH THE MAGNITUDE AND PHASE OF THE DESIRED SIGNAL SUPPOSE THAT THE MEASURED SIGNAL IS X(T), WHICH IS SAMPLED AT N DISCRETE INSTANTS OF TIME, TN, FOR N = 1, , N UNLIKE THE USE OF THE FFT ALGORITHM, IN THE PRESENT DISCUSSION WE DO NOT REQUIRE THAT THE SAMPLE TIMES TN BE EVENLY SPACED WE MERELY NEED TO KNOW WHAT THEY ARE THE MEASURED DATA ARE THEN STORED IN AN ARRAY, XN = X(TN), N = 1, , N OUR TASK IS TO ESTIMATE THE AMPLITUDE AND PHASE OF A COMPONENT OF THE MEASURED SIGNAL HAVING THE KNOWN FREQUENCY, ? THIS PROBLEM MAY THEN BE EXPRESSED AS ONE OF ESTIMATING THE COEF?CIENTS A, B, AND C IN XN = A COS(?TN) + B SIN(?TN) + C + ?N N = 1, , N, (1 8 1) WHERE THE CONSTANT C ACCOUNTS FOR THE FACT THAT IN REAL SIGNALS THERE IS OFTEN A DC SHIFT CAUSING THE AVERAGE OF THE SIGNAL OVER TIME TO NOT BE EQUAL TO ZERO ?N IS THE ERROR AT TIME TN, ?N = XN ?A COS(?TN) ?B SIN(?TN) ?C (1 8 2) N (1 8 3) MINIMIZING THE TOTAL SQUARED ERROR WITH RESPECT TO THE UNKNOWN COEF?CIENTS IN THIS LINEAR MODEL IS THE TYPICAL LEAST SQUARES PROBLEM THE SOLUTION IS OBTAINED BY ?NDING THE EXTREMUM OF THE ERROR, ?E ?A = ?E ?B = ?E ?C = 0 (1 8 4) EQUATIONS (1 8 2) THROUGH (1 8 4) GIVE ?N COS(?TN) = 0, (1 8 5) ?N SIN(?TN) = 0, (1 8 6) AND (XN ?A COS(?TN) ?B SIN(?TN) ?C) = 0 (1 8 10) 1 8 THE “SLOW” FOURIER TRANSFORM: LEAST SQUARES EXTRACTION OF A HARMONIC SIGNAL EQUATION (1 8 8) THROUGH (1 8 10) MAY BE REARRANGED AS A SET OF EQUATIONS THAT MAY EASILY BE SOLVED FOR THE UNKNOWNS A, B, AND C, ? ?? N N=1 COS2(?TN) N N=1 COS(?TN) SIN(?TN) N N=1 COS(?TN) N N=1 COS(?TN) SIN(?TN) N N=1 SIN2(?TN) N N=1 SIN(?TN) N N=1 COS(?TN) N N=1 SIN(?TN) N N=1 ? ?? ? ? A B C ? ?= ? ?? N N=1 XN COS(?TN) N N=1 XN SIN(?TN) N N=1 XN ? ?? (1 8 11) KNOWING A, B, AND C, THE LEAST SQUARES ESTIMATE, XLS(T), OF THE SIGNAL AT THE FREQUENCY ? MAY BE COMPUTED FROM XLS(T) = A COS(?T) + B SIN(?T) + C (1 8 12) THE AMPLITUDE, X, AND PHASE, ?, OF THE HARMONIC COMPONENT OF THE SIGNAL CAN BE DETERMINED BY A COS(?T) + B SIN(?T) = X SIN(?T + ?) = X SIN(?T) COS(?) + X COS(?T) SIN(?), (1 8 13) SO THAT X = A2 + B2, TAN(?) = A B (1 8 14) THE CALCULATIONS IN EQ (1 8 11) MAY BE EASILY ACCOMPLISHED IN MATLAB BY EXPRESS- ING EQ (1 8 1) AS AN OVERDETERMINED SYSTEM OF EQUATIONS AND NEGLECTING THE ERROR DE?NE A MATRIX HAVING N ROWS AND THREE COLUMNS, = COS(?TN) SIN(?TN) , (1 8 15) WHERE EACH ROW CORRESPONDS TO A VALUE OF THE INDEX N THE MEASURED DATA IS STORED IN THE ARRAY X, WHERE THE NTH ROW CONTAINS THE VALUE XN THE SOLUTION FOR THE UNKNOWN COEF?CIENTS IS THEN EASILY ACCOMPLISHED BY THE BACKSLASH OPERATOR IN MATLAB, ? ? A B C ? ?= \X (1 8 16) THE LEAST SQUARES ESTIMATE OF THE SIGNAL AT THE FREQUENCY ? IS THEN COMPUTED BY XLS = ? ? A B C ? ? (1 8 17) FREQUENCY RESPONSE OF LINEAR SYSTEMS WITH RANDOM INPUT THE NARROWBAND ANALYSIS OF ACOUSTIC SIGNALS DISCUSSED ABOVE HAS BEEN CENTERED ON THE USE OF THE AUTO-POWER SPECTRAL DENSITY, SPP(?) AN EVEN MORE POWERFUL STATISTIC AVAILABLE FOR THE CHARACTERIZATION OF RANDOM SIGNALS IS THE CROSS POWER SPECTRAL DENSITY OF TWO SIGNALS, SAY P1(T) AND P2(T), SP1 P2(?) UNLIKE THE AUTO-POWER SPECTRAL DENSITY, THIS QUANTITY WILL GENERALLY BE COMPLEX TO ILLUSTRATE ONE IMPORTANT APPLICATION OF THE CROSS-SPECTRAL DENSITY, IN THE FOLLOWING WE WILL DERIVE AN EXPRESSION FOR THE FREQUENCY RESPONSE OF A LINEAR SYSTEM IN TERMS OF THE CROSS POWER SPECTRAL DENSITY OF THE INPUT AND OUTPUT SIGNALS AND THE AUTO-POWER SPECTRAL DENSITY OF THE INPUT 1 9 FREQUENCY RESPONSE OF LINEAR SYSTEMS WITH RANDOM INPUT WE WILL ASSUME THAT A LINEAR, TIME-INVARIANT SYSTEM HAS A RANDOM INPUT, F (T) AND AN OUTPUT, P(T) THE SYSTEM IS CHARACTERIZED BY ITS IMPULSE RESPONSE, H(T), WHICH IS ZERO FOR NEGATIVE VALUES OF T IF THE SYSTEM IS INITIALLY AT REST, THE IMPULSE RESPONSE ENABLES THE DETERMINATION OF THE SYSTEM OUTPUT THROUGH THE CONVOLUTION INTEGRAL, P(T) = T H(T ?? ?) F (? ?)D? ? (1 9 1) THE COMPLEX FREQUENCY RESPONSE OF THE SYSTEM IS DE?NED TO BE THE FOURIER TRANSFORM OF THE IMPULSE RESPONSE, H(?) = ? H(?)E?I??D? (1 9 2) THE IMPULSE RESPONSE MAY BE OBTAINED FROM THE FREQUENCY RESPONSE FUNCTION THROUGH THE USE OF THE INVERSE TRANSFORM, H(?) = 2? ? ?? H(?)EI??D? (1 9 3) NOTE THAT EQS (1 9 2) AND (1 9 3) FORM A FOURIER TRANSFORM PAIR WHERE THE FACTOR OF 1/(2?) HAS BEEN INTERCHANGED FROM THE USUAL CONVENTION WE WISH TO SHOW THAT GIVEN THE CROSS POWER SPECTRAL DENSITY, S F P(?) AND THE AUTO- POWER SPECTRAL DENSITY, S F F (?), THE FREQUENCY RESPONSE FUNCTION MAY BE DETERMINED FROM H(?) = S F P(?) S F F (?) (1 9 4) WE WILL ASSUME THAT THE INPUT SIGNAL, F (T) IS A WEAKLY STATIONARY RANDOM PROCESS S F P(?) IS DE?NED TO BE THE FOURIER TRANSFORM OF THE CROSS- CORRELATION FUNCTION, R F P(?) = E[ F (T)P(T + ?)], WHERE E[ ] DENOTES THE EXPECTED VALUE THE WEAK STA- TIONARITY OF THE PROCESSES RESULTS IN THE CROSS-CORRELATION BEING INDEPENDENT OF TIME, T EQUATION (1 9 1) ENABLES US TO WRITE THE CROSS-CORRELATION FUNCTION AS R F P(?) = E[ F (T)P(T + ?)] = E F (T) T+? H(T + ? ?? ?) F (? ?)D? ? (1 9 5) THE CROSS POWER SPECTRAL DENSITY IS THEN OBTAINED BY TAKING THE FOURIER TRANSFORM, S F P(?) = 2? ? ?? R F P(?)E?I??D? = 2? ? ?? E F (T) T+? H(T + ? ?? ?) F (? ?)D? ? E?I??D? (1 9 6) ANALYSIS OF ACOUSTIC SIGNALS THE UPPER LIMIT IN THE INTEGRATION OVER ? ? CAN BE CHANGED TO ?SINCE H(T) = 0 FOR T < 0 THE LOWER LIMIT CAN BE CHANGED TO ??SINCE THE FORCE STARTS AT TIME T = 0, F (T) = 0 FOR T < 0 CHANGING THE LIMITS OF INTEGRATION ENABLES US TO ALSO CHANGE THE ORDER OF THE INTEGRATIONS AND REARRANGE EQ (1 9 6) TO THE FORM S F P(?) = 2? ? ?? ? ?? H(T + ? ?? ?)E[ F (T) F (? ?)]E?I??D?D? ?, (1 9 7) WHERE WE HAVE ALSO USED THE FACT THAT H(T) IS DETERMINISTIC SO IT CAN BE TAKEN OUT- SIDE THE EXPECTED VALUE OPERATOR THE EXPECTED VALUE IN EQ (1 9 7) CONSISTS OF THE AUTOCORRELATION FUNCTION, R F F (T ?? ?) = R F F (? ? ?T) = E[ F (T) F (? ?)], (1 9 8) SINCE R F F (?) IS AN EVEN FUNCTION OF ? IT IS NOW HELPFUL TO MAKE A CHANGE OF VARIABLES BY LETTING ? ?? = T ?? ? SO THAT D? ? = ?D? ?? WHICH ENABLES US TO WRITE EQ (1 9 7) AS S F P(?) = 2? ? ?? ? ?? H(? ?? + ?)R F F (? ??)E?I??E?I?? ??EI?? ??D?D? ??, (1 9 9) WHERE WE HAVE ALSO MULTIPLIED BY UNITY, 1 = E?I?? ??EI?? ?? (1 9 10) MAKE ONE MORE CHANGE OF VARIABLES, ?1 = ? ?? + ? SO THAT EQ (1 9 9) BECOMES S F P(?) = 2? ? ?? ? H(?1)R F F (? ??)E?I??1EI?? ??D?1D? ??, (1 9 11) WHERE WE HAVE REPLACED THE LOWER LIMIT OF INTEGRATION OVER ?1 WITH ZERO SINCE H(?1) = 0 FOR ?1 ?0 THE TWO INTEGRATIONS MAY NOW BE CARRIED OUT SEPARATELY SINCE THE POWER SPECTRAL DENSITY IS DE?NED TO BE THE FOURIER TRANSFORM OF THE AUTOCORRELATION FUNCTION, S F F (?) = 2? ? ?? R F F (? ??)E?I?? ??D? ?? (1 9 12) EQUATIONS (1 9 2), (1 9 11), AND (1 9 12) GIVE S F P(?) = H(?)S F F (?) (1 9 13) IN AGREEMENT WITH EQ (1 9 4) WHICH IS WHAT WE WISHED TO SHOW IT IS IMPORTANT TO NOTE THAT THE RESULTS PRESENTED IN PREVIOUS SECTIONS TO ESTIMATE THE AUTO-POWER SPECTRAL DENSITY FROM DISCRETE TIME DATA CAN BE EXTENDED TO THE ESTIMATION 1 9 FREQUENCY RESPONSE OF LINEAR SYSTEMS WITH RANDOM INPUT INPUT–OUTPUT RELATIONSHIPS FOR A LINEAR SYSTEM IN THE FOLLOWING, WE EXAMINE THE CALCULATION OF THE RESPONSE OF DYNAMIC SYSTEMS THROUGH THE USE OF THE FFT ALGORITHM KNOWING THE COMPLEX TRANSFER FUNCTION, H(?), WHICH RELATES THE INPUT F (T), TO THE OUTPUT SIGNAL, X(T), OF A LINEAR SYSTEM IN THE FREQUENCY DOMAIN, ONE CAN OBTAIN THE RESPONSE OF THE SYSTEM TO ANY INPUT SIGNAL, F (T) THIS ALGORITHM CAN ALSO BE EXTREMELY USEFUL FOR OBTAINING THE NUMERICAL DERIVATIVE OR INTEGRAL OF ARBITRARY TIME DOMAIN SIGNALS CONSIDER A GENERAL LINEAR SYSTEM WHICH HAS AN INPUT, F (T) AND AN OUTPUT RESPONSE X(T) THIS SYSTEM MAY BE COMPOSED OF AN ELECTRONIC CIRCUIT OR ?LTER HAVING ANY NUMBER OF POLES AND ZEROS OR A MECHANICAL STRUCTURE HAVING ANY NUMBER OF RESONANT MODES IT IS CONVENIENT TO REPRESENT THE RESPONSE OF SUCH SYSTEMS BY USE OF THE FOURIER TRANSFORM THE FOURIER TRANSFORM OF THE INPUT AND OUTPUT SIGNALS ARE F(?) = 2? ? ?? E?I?T F (T)DT X(?) = 2? ? ?? E?I?TX(T)DT (1 10 1) THE TRANSFER FUNCTION OF THE SYSTEM IS THE RATIO OF THESE TRANSFORMS, H(?) = X(?) F(?) (1 10 2) IF ONE KNOWS F(?) OR X(?), THE CORRESPONDING TIME DOMAIN SIGNALS MAY BE OBTAINED BY THE INVERSE FOURIER TRANSFORM, F (T) = ? ?? EI?T F(?)D? X(T) = ? ?? EI?T X(?)D? (1 10 3) IF H(?) AND F(?) ARE KNOWN, THE SECOND OF EQ (1 10 3) GIVES X(T) = ? ?? EI?T H(?)F(?)D? (1 10 4) ANALYSIS OF ACOUSTIC SIGNALS IN SOME PROBLEMS, IT CAN BE HELPFUL TO EXPRESS THE RESPONSE X(T) USING THE CONVOLUTION INTEGRAL, X(T) = T H(T ??) F (?)D?, (1 10 5) WHERE H(T) IS THE IMPULSE RESPONSE WHICH IS DE?NED TO BE THE INVERSE FOURIER TRANS- FORM OF THE TRANSFER FUNCTION, H(T) = ? ?? EI?T H(?)D? (1 10 6) THE IMPULSE RESPONSE IS ALSO DE?NED SUCH THAT H(T) = 0, FOR T < 0 (1 10 7) EQUATION (1 10 5) MAY BE SHOWN TO REPRESENT THE SYSTEM RESPONSE BY SUBSTITUTING IT INTO EQ (1 10 1) AND USING EQ (1 10 2), H(?)F(?) = 2? ? ?? E?I?T T H(T ??) F (?)D?DT (1 10 8) WE WILL ASSUME THAT THE FORCE DOES NOT ACT FOR NEGATIVE VALUES OF TIME SO THAT F (?) = FOR ? < 0 IN THIS CASE, THE LOWER LIMIT OF THE INTEGRAL OVER ? MAY BE CHANGED FROM ZERO TO ??WITHOUT CHANGING THE RESULT, H(?)F(?) = 2? ? ?? E?I?T T ?? H(T ??) F (?)D?DT (1 10 9) IN ADDITION, BECAUSE OF EQ (1 10 7), THE UPPER LIMIT OF THE INTEGRAL OVER ? MAY ALSO BE CHANGED TO ?WITHOUT CHANGING THE RESULT, H(?)F(?) = 2? ? ?? E?I?T ? ?? H(T ??) F (?)D?DT (1 10 10) NOW WE WILL CHANGE THE ORDER OF THE INTEGRATION AND MULTIPLY BY E?I??EI?? = 1, H(?)F(?) = 2? ? ?? ? ?? E?I?TH(T ??) F (?)E?I??EI??DTD? = 2? ? ?? ? ?? E?I?(T??)H(T ??)DT F (?)E?I??D? (1 10 11) LET T? = T ?? SO WE CAN CARRY OUT THE INNER INTEGRAL, 1 10 INPUT–OUTPUT RELATIONSHIPS FOR A LINEAR SYSTEM H(?)F(?) = 2? ? ?? ? ?? E?I?T?H(T?)DT? F (?)E?I??D? = SPECTRAL APPROACH TO EVALUATING THE CONVOLUTION INTEGRAL WHEN THE IMPULSE RESPONSE, H(T), AND THE INPUT SIGNAL F (T) ARE GIVEN AS SPECI?C FUNC- TIONS, THERE ARE A FEW SITUATIONS WHERE THE INTEGRAL IN EQ (1 10 5) CAN BE EXPRESSED IN TERMS OF A REASONABLY SIMPLE CLOSED FORM EXPRESSION MORE OFTEN, HOWEVER, THE FUNC- TIONS F (T) AND H(T) DO NOT PERMIT THE INTEGRAL TO BE READILY EVALUATED WE MUST OFTEN RESORT TO NUMERICAL METHODS NUMEROUS TECHNIQUES ARE AVAILABLE IN THE FOLLOWING