THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

194 9. Sound-Measuring Instrumentation9.13 Fast Fourier Transform AnalysisThe Fast Fourier Transform (FFT) technique, rendered feasible by the advent ofmicrocomputers, employs both digital sampling and digitization. Instead relyingon bandpass filters to measure the analog amplitudes that formulate a signal’sspectrum, the FFT analyzer executes an efficient transformation of the signal fromthe time domain to the frequency domain. As it is capable of executing nearly anyanalysis function on the signal at high speed, the FFT technique is an extremelypowerful analytical method. The FFT analyzer captures a block of sampled dataof finite length (generally 1024 or 2048 samples) for a processing interval. Intransforming from the time domain to the frequency domain, the Fourier transformrelates a function of time g(t) to a function of frequency F(ω) in the followingmanner:F(ω) =∫ +∞−∞g(t)e −iωt dt (9.13)In measuring noise, a microphone assembly generates a voltage proportional tosound pressure. A time series is formed when the voltage is sampled at equal intervals,as shown in Figure 9.16. In order to transform this series into the frequencydomain, Equation (9.13) must be reformulated into the discrete Fourier transform(DFT), given byF(k) = 1 Nn=N−1 ∑n=0g(n)e −2πikn/N (9.14)The matrix format of Equation (9.14) (Randall, 1977) isF = 1 N[A] g (9.15)Figure 9.16. Sound pressure sampling at discrete intervals.