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

9.16 Measurement Error 199whereβ = the resolution which is the frequency increment between lines in aspectrum, Hzf s = sampling frequency, the reciprocal of sampling period T sN = number of samples in the original time seriesf R = frequency range, normally from 0 to the Nyquist frequencySome versions of FFT analyzers incorporate a “zoom” capability which displaysselected portions of a spectrum with finer reduction. Normally, in order to increasethe resolution (which means that β must decrease) the number of samplesmust be larger. This is not always feasible since either the processing time willincrease or the block size is limited by machine memory capacity. The zoomcapability overcomes this limitation by permitting the spectrum analyzer to concentrateits entire resolution, whether it be 200 or 800 or 1000 lines, on a smallfrequency interval selected by the user. While arbitrarily fine resolution is achievable,a compromise must be effected between resolution and the required samplingtime, as attested by Equation (9.18). Note that according to this equation, if thefrequency resolution for a signal is 10 Hz, the sampling time is 0.1 s, but if thefrequency resolution is changed to 0.1 Hz, the corresponding sampling time will be10 s.A number of FFT analyzers incorporate large digital processing capacitiesthat can present measurement results in the format of three-dimensional plots(or “waterfall” or “cascade” plots) with the vertical coordinate representing theamplitudes of the spectra as functions of the other two coordinates, one representingthe frequency f and the other time t. The scan presented on a displaycan show a “running” plot that moves in the direction of increasing t. This typeof display is most useful in observing the behavioral characteristics of transientsounds.9.16 Measurement ErrorNoise is usually random in nature, i.e., its sound pressure level cannot be predictedfor any instant. But statistical means can be used to describe random noise. If thenoise is relatively constant in level and frequency content then it may be deemeda stationary random process, one in which statistical parameters are invariantwith respect to time. A machine operating in a constant cyclic manner may emitdifferent levels of sound, with corresponding changes in frequency content, foreach successive instant of the cycle, but a measurement interval over a group ofcycles will yield a consistent spectral distribution over time. Analysis data basedon a very short interval that is less than the length of a single cycle is certain toyield misleading results.Let us consider noise with an idealized Gaussian or normal probability distribution.The standard deviation ε, which is the uncertainty in the rms signal dividedby the long-term average rms signal, relates to the ideal filter bandwidth bw and