called the energy density function or equivalently the energy spectral density, S(f), is defined:
The energy spectral density S(f) is the “energy” of the sound or vibration signal in a bandwidth of 1 Hz. Note that S(ω) = 2πS(f) where S(ω) is the “energy” in a 1 rad/s bandwidth. We use the term “energy” because if x(t) were converted into a voltage signal, S(ω) would have the units of energy if the voltage were applied across a 1 Ω resistor. In the case of the pure tone, if x(t) is assumed to be a voltage, then the mean square value in Eq. (1.8) represents the power in watts.
In the case of random sound or vibration signal we define a power spectral density Gx(f). This may be derived through the filtering – squaring – averaging approach or the finite Fourier transform approach. We will consider both approaches in turn.
Suppose we filter the time signal through a filter of bandwidth Δf, then the mean square value
(1.13)
where x(t,f,Δf) is the filtered frequency component of the signal after it is passed through a filter of bandwidth Δf centered on frequency f. In the practical case, the filter bandwidth, Δf, could be, for example, a one‐third octave or smaller. The power spectral density is defined as:
(1.14)
The power spectral density may also be defined via the finite Fourier transform [12, 13]
Note the difference between Eqs. (1.15) and (1.12). We notice that Gx must be a power spectral density because of the division by time. The unit of power spectral density is U2/Hz, where U is the unit of the measured signal. The square root of the power spectral density, often called the rms spectral density, has a unit of
Figure 1.8 Power spectral density of random noise.
1.4 Frequency Analysis Using Filters
Sound and vibration signals can be combined, but they can also be broken down into frequency components as shown by Fourier over 200 years ago. The ear seems to work as a frequency analyzer. We also can make instruments to analyze sound signals into frequency components.
In order to determine experimentally the contribution of the overall signal in some particular frequency band we filter the signal. Most of the important points concerning filters can be observed from a simple R–C passive circuit. By passive we mean that no electrical power is supplied. However, practical filters are usually made from more complicated R–C passive filters or active filters, involving R–C components, inductances, amplifiers, and a power supply. Modern digital signal processing can also be used to filter a signal. Digital filters are implemented through a series of digital operations (addition, multiplication, and time delay) on a digitized signal [3, 5, 14].
In a high‐pass filter, only high‐frequency signals are passed without attenuation. An ideal high‐pass filter would pass no signal below the cutoff frequency and would have a vertical “skirt,” which is impossible to achieve by use of a simple RC passive filter. Practical high‐pass filters have more complicated circuits and normally incorporate inductances and/or active components. High‐pass filters are often used when the displacement signal from a transducer is analyzed. This is because frequently the high‐frequency displacement is of interest and large amplitude low‐frequency signals must be filtered out to prevent the dynamic range of the instrumentation from being exceeded.
In a low‐pass filter only low‐frequency signals are passed without attenuation. In practice, the skirt of a simple RC filter is usually not steep enough and a more complicated circuit is also needed. In modern noise and vibration instrumentation, low‐pass filters are often used as anti‐aliasing filters at the input of an analog‐to‐digital converter (ADC) to suppress unwanted high‐frequency signals. Aliasing errors occur when signals at high frequency (above the upper frequency limit) are interpreted as lower frequency signals below the upper frequency limit.
In a band‐stop (or band‐rejection) filter most frequencies are passed without attenuation; but those frequencies in a specific range are attenuated to very low levels. It is the opposite of a band‐pass filter. Band‐stop filters are commonly used as anti‐hum filters and to remove specific interference frequencies in a complex signal.
By combining the simple high‐pass and low‐pass filters, a simple band‐pass filter is produced. In a band‐pass filter all simple harmonic frequency signals within a band are passed with little, if any, attenuation. All signals outside this band are attenuated. In practice it is usually desirable to make the skirts of such a filter steeper. This requires the use of additional electrical circuit elements and/or the use of active filters. Figure 1.9 illustrates the filtering process and shows only the magnitude of the signals in the frequency domain [10].
Figure 1.9 Different types of filter. Low‐pass, high‐pass, band‐pass, and band‐stop [10].
Band‐pass filters are frequently used in acoustics and vibration. A frequency weighting network such as A‐weighting is a special kind of band‐pass filter. Frequency analysis using band‐pass filters is commonly carried out using (i) constant frequency band filters and (ii) constant percentage filters. The following symbol notation is used in the characterization of a band‐pass filter: fL and fU are the lower and upper cutoff frequencies, and fC and Δf are the band center frequency and the frequency bandwidth, respectively. Thus Δf = fU − fL. See
