Engineering Acoustics. Malcolm J. Crocker

band filter.

      Solution

      1/12-octave filters are obtained by dividing each one-octave band into 12 geometrically equal sub‐sections, i.e. f_U/f_L = 2^1/12. By the same procedure as for octave and one‐third‐octave filters, we get the result that for 1/12‐octave bands the cutoff frequencies, f_L and f_U, are f_C × 2^−1/24 and f_C × 2^1/24, respectively. Then, the bandwidth is given by Δf = f_U − f_L = f_C(2^1/24 − 2^−1/24), so Δf ≈ 6% (f_C).

There are two main types of constant percentage filters in common use: (i) those with a fixed center frequency and bandwidth which is a certain percentage of the center frequency, and (ii) those with a variable (or tunable) center frequency and a bandwidth which can be set to certain selected percentages of the center frequency. The first type of filter is perhaps in most common use. In practical instruments, many different parallel filters each with a different center frequency are assembled in one unit. The instrument is provided with a root mean square detector and a display.

      On the other hand, instruments for constant bandwidth filter analysis are normally constructed so that the center frequency of a single filter can be tracked effectively throughout the frequency range of interest. Often different bandwidth settings are available on the same instrument (e.g. 1, 5, 10, 20 Hz). The narrower the bandwidth chosen, the slower the tracking rate should be to obtain reliable results. A rule which should be used in spectrum analysis is that the duration, T, of the noise sample length (or of the analysis time) must be at least as long as the reciprocal of the bandwidth Δf,

      (1.16)

      This fundamental principle, also known as the uncertainty principle, puts a limit on the corresponding resolutions in the time and frequency domain, meaning that narrow resolution in one domain means wide resolution in the other domain [10].

      If fine frequency resolution is not needed or if the signal is broadband in nature, then octave band readings are sufficient. One‐third octave band readings (or narrower) should be used if the signal spectrum is not smooth or if pure tones are present. For diagnostic work on machinery, it may be necessary to use constant bandwidth filters (e.g. if the fan blade passing frequency and its higher harmonics must be separated).

When changes in level and frequency of a signal occur in a short period of time, real‐time frequency analysis is required to observe rapid variations in the signal and showing the results on a continuously updated display. Real‐time analysis can be performed using a frequency analyzer made up of a set of parallel filters and a detector (see Figure 1.12). The input signal is previously conditioned in terms of level (gain/attenuation) and high‐ and/or low‐pass filtering. A digital analyzer will require an anti‐aliasing filter at the input before the ADC. The conditioner is then connected to a large number of parallel band‐pass filter channels (usually between 30 and 40 for a standard one‐third octave band model). The detector detects the power in the transmitted signal in terms of its mean square or rms value. In Figure 1.12, only one detector is shown, and this is supposed to work as detector for all the filters in the situation of a parallel filter bank [10].

Figure 1.11 Comparison between bandwidths of (a) constant percentage and (b) constant bandwidth filters at the same frequency.

Figure 1.12 Simplified block diagram of a parallel‐filter real‐time analyzer [10].

1.5 Fast Fourier Transform Analysis

      Although equipment for analyzing noise and vibration signals with constant frequency bandwidth filters and constant percentage bandwidth (CPB) filters are still available, these instruments have largely been replaced by Fast Fourier Transform (FFT) Digital Fourier analyzers. These types of analyzers give similar results in a fraction of the time and at a lower cost. These Fourier analyzers are manufactured by several companies and make use of the FFT algorithm which was published by Cooley and Tukey in l965 [18]. This algorithm is much faster than conventional digital Fourier transform algorithms and has made the digital FFT analyzer a useful, efficient means of signal analysis [19].

      In digital FFT analyzers, the analog signal needs to be converted to digital. The signal is not only sampled discretely in time but quantized as well with discrete amplitude values. Because of sampling, the digital system is limited in frequency, which may cause some problems, such as aliasing. During the sampling process, the amplitude of each sample is represented and stored as finite binary numbers in a computer. Rounding errors produce a signal‐correlated noise, called quantization noise, which is another disadvantage of the digital system. So the sampling rate and resolution are two basic considerations of the ADC [20].

      FFT analyzers operate on discrete blocks of data where each sample block is captured then analyzed while the next block is being captured, and so on. Unlike equipment which works on purely analog or hybrid analog/digital principles, the digital FFT analyzer makes all the analysis digitally. Since all the analysis results are in digital form, numerous calculations can be performed by such analyzers. The Fourier transform X(f) of a time signal x(t) may be calculated (see Eq. (1.6)). In addition, the auto‐power spectral density G_x(f) (Eq. (1.15)) may be calculated. A dual‐channel FFT analyzer is able to sample two input signals simultaneously and compute several joint functions. This type of analyzer is widely used in modal testing, electroacoustics, and vibroacoustics applications. If two signals x₁(t), x₂(t) are fed into the computer at once, then the cross‐spectral density G₁₂(f) can be calculated [12, 13],

      (1.17)

where the asterisk denotes the complex conjugate.

      In general G₁₂(f) is a complex quantity having both an amplitude and a phase. The phase is the relative phase between the two signals. The real and imaginary parts of the cross‐spectrum are referred to as the co‐spectrum and quad‐spectrum, respectively. The auto‐power spectral densities

      (1.18)

      (1.19)

      are real quantities.

      If x₁(t) were an input (for example, a measured force) and x₂(t) an output (for example, a measured displacement), then the transfer function H₁₂(f) can be computed [5, 13],

      (1.20)