Frequency Span (F max ) Selection and Relation to Sampling Fre-

One of the main parameters selected during the setup of a data acquisition system is the frequency span orFmax (which are used interchangeably), which is the maximum

frequency that can be represented. Typically this number is in hertz and normally several orders of magnitude larger than running speed; however this can vary based of the type of machinery that is being analyzed. For instance a 3 blade axial fan with rolling element bearings and a 6 pole motor would require a much lower Fmax than

a high speed gearbox. Typically not any value can be selected for a Fmax. Usually

there are options of discrete values such as 500, 1000, 2000, 4000, or 5,000 hz. The reason for these specific values is discussed in section 4.1.2. Table 4.1 provides a list of commonly used frequency spans (Fmax).

With most DAQ’s the sampling rate is indirectly selected through the choosing of a frequency span. That is the sampling rate is some multiple of Fmax, typically 2.56.

The reason for using these specific discrete values for Fmax is that when multiplied

by 2.56 the amount of data that is sampled is a power of two. This allows for a computationally efficient calculation of the Fourier transform. This topic is covered in detail in section 4.3.1. However the multiple of the sampling rate must be at least greater two because of the Nyquist Criterion. The Nyquist criterion shown in equation 4.1 states the minimum sampling rate to be able to accurately recreate an analog signal

For the purposes of the data acquisition system and as an example to showcase the importance of this ratio, the sampling rate has been defined in equation 4.2 where n is a rational number.

fs =nFmax (4.2)

The following examples will showcase the importance of the selection of this number and errors that occur with under sampling signals where the recreation of the signal is known as aliasing. To help prevent this low pass filters known as anti-aliasing filters can be used. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) -5 0 5 Amplitude

Figure 4.1: Simulated Analog Signal of a Sine Wave with an Amplitude of 5 and Frequency of 50Hz

A simulated analog sine wave is created with an Amplitude of 5 at a frequency of 50 hz. This analog wave will now be sampled at several different rates for n.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) -5 0 5 Amplitude Analog Signal Sampled Signal (n=1.5)

The signal is sampled at a rate of 75 hz (n=1.5) where each red marker indicates a discrete value recorded. This frequency is below the Nyquist Criterion and will result in an error when reconstructing the signal.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) -5 0 5 Amplitude Sampled Signal (n=1.5) Recreated Signal

Figure 4.3: Reconstructed Signal From Discrete Samples at n=1.5 Sam- pling Rate

In figure 4.3 the digital signal sampled in figure 4.2 is reconstructed using the curve fitting toolbox within MatLab. In all scenarios presented in this section a ”fourier8” fit was used within the Matlab curve fitting toolbox. The resulting reconstructed curve is significantly different than the original simulated analog signal and has a fundamental frequency much lower than the original.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) -5 0 5 Amplitude Analog Signal Sampled Signal (n=2)

In the scenario shown in figure 4.4 the simulated analog signal is sampled at a rate of 100 hz (n=2). The Nyquist Criterion must be greater than 2 so the Nyquist Criterion is not met in this scenario either

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) -5 0 5 Amplitude Sampled Signal (n=2) Recreated Signal

Figure 4.5: Reconstructed Signal From Discrete Samples at N=2 Sampling Rate

With a sampling rate at twice the frequency of the input signal that reconstructed signal shown in figure 4.5 is approximately zero and the original signal was not accu- rately reconstructed 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) -5 0 5 Amplitude Analog Signal Sampled Signal (n=2.56)

Figure 4.6: Signal Sampled above Nyquist Threshold Rate N=2.56

In figure 4.6 the original signal is sampled at 2.56 times the fundamental frequency of the simulated analog signal. This sampling rate is in agreement with the Nyquist Criterion described in equation 4.1.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) -5 0 5 Amplitude Sampled Signal (n=2.56) Recreated Signal

Figure 4.7: Reconstructed Signal From Discrete Samples at N=2.56 Sam- pling Rate

The reconstructed signal from the sampled data matches the simulated analog signal and shows that the sampling rate was sufficient to accurately analyze a signal of this frequency.

While in the development of a data acquisition system the 2.56 ratio is important, the end user of the software typically selects an Fmax only. However, it is important

to know the effects of Fmax selection and ensuring that theFmax is much higher than

any frequencies of interest for the machinery being analyzed. This emphasizes the importance of have sufficient understanding of the machine being analyzed as have an Fmax set too low could obscure energy in higher frequency bands that indicates

machinery faults or anomalies.