Sensors - July 1999 - Understanding Frequency Domain Measurements

There are times when analyzing signal components in the time domain
just doesn't provide the whole picture. On the other hand, the
frequency domain offers an easy way of accessing
the information you need.

When faced with the task of measuring dynamic signals, some engineers specify measurement hardware before carefully considering what the software requires. After all, it's the software that turns the raw data into a meaningful formthe measurement of interest. Ultimately, measurement techniques and algorithms dictate your hardware requirements because the algorithms are based on assumptions of how the hardware transforms signals from the physical world (see Photo 1). Measurements made by hardware that fails to meet software requirements can corrupt an entire measurement project, even when combined with valid measurements.

Many of us are familiar with time domain signals, such as those provided by oscilloscopes. We intuitively understand that these signals provide such information as rise time, periodicity, and amplitude. Unfortunately, sometimes when we try to analyze the components of such a signal, we find that time domain analysis doesn't provide the whole signal picture. For example, try to imagine measuring the following characteristics using only time-domain techniques:

It soon becomes apparent that there are really only a few choices for these tasks. If the signal is periodic, you could certainly determine its periodicity by measuring the time between zero crossings and so determine its fundamental frequency (f = 1/T). Having determined the fundamental frequency of the signal, you then have few tools left to determine the rest of the frequency components, and these may yield important information--such as distortion, SNR, system response, and the phase relationship between components.

So, while the time domain does contain all the relevant information needed to measure these signal characteristics, the analysis is often tedious and difficult. Small variations in the characteristics are not easily determined. They do, however, have a dramatic effect in the frequency domain. Therefore, measuring the frequency domain provides an easier way of accessing the information (see Figure 1).

Modern instruments make measurements in the frequency domain by carefully digitizing a signal and using FFT techniques. But FFT-based measurements place important selection criteria on the hardware in your acquisition system. An FFT measurement is only as good as the digitization of the signal. Information lost due to noise, resolution of the digitizer, or improper sampling techniques cannot be recovered.

You have to understand the requirements imposed by the analysis on the acquisition. To select the right hardware for FFT-based measurements, first focus on some essential considerations for single-channel measurements, where you're interested in signal content. Some of the more common applications for single-channel measurements are determining distortion and tracking unwanted frequency components. After you've established those requirements, you'll look at dual-channel measurements, where you're interested in the relationship between channels. Dual-channel measurements have all the requirements of single-channel measurements with the added need of channel matching (see Figure 2).

Figure 1. Dynamic signal characteristics are often difficult to obtain in the time domain, but they do have a dramatic effect in the frequency domain. This figure demonstrates a minimally distorted signal (1%) in the time domain (A) and frequency domain (B). The nondistorted signal and spectrum are represented by the black outline, and the distorted signal and spectrum are shown in green. (A) Time Waveform [V vs s](B) Power Spectrum [dB vs Hz]

Aliasing is the misinterpretation of high-frequency components as low-frequency components. Specifically, frequency components in the top half of the sampling rate appear to fall in the bottom half.

Aliasing is one of the least understood aspects of data acquisition (DA) systems and the one that's most responsible for errors in converting analog signals to the digital domain. Although many errors can be minimized or corrected after the data have been collected, aliasing corrupts the signal in a way that cannot be fixed. Once a signal contains significant aliased frequency components, it's impossible to distinguish between the signals you intend to measure and those that have aliased into your measurement band.

Because FFT-based measurements operate on data in the frequency domain, you must protect the measurement band from unwanted, aliased frequency content. Aliasing also ruins time-domain measurements.

So, how do frequencies alias? Components just above the Nyquist rate alias to just below the Nyquist rate. Components approaching the sampling rate appear to approach 0 (DC). As the frequency continues to increase, the alias version will bounce back and forth between DC and Nyquist. Figure 3 shows a frequency component aliasing back and forth as it increases, where the system's sampling rate is 1000 sps.

If your signals are band-limited to below the Nyquist rate, you won't have any aliasing problems. If you don't have this confidence, you must limit the frequency content of your sampled signals with anti-aliasing (low-pass) filters. These filters don't eliminate all aliases, but they can significantly reduce the aliases to a level below the desired measurement accuracy.

Perhaps the first characteristic specified in FFT-based measurement systems is the frequency spanthe operating range of alias-free frequency components that can be accurately measured. Because the operating range of frequencies must be alias-free, it's important that the anti-aliasing filter matches the desired frequency span. As long as the anti-aliasing filter protects the frequency span from significant aliased components, you can acquire your signals and make FFT-based measurements with confidence.

For example, assume that you want to measure the low-frequency noise (e.g., rumble and hum) of an automobile driving down a cobblestone road. Your measurement system makes 2500 sps--well above your low-frequency noise. Suppose that during your measurement normal braking causes the brake assembly to squeal and that the squeal noise is composed of a fundamental frequency at 750 Hz and harmonics at 1500 Hz and 2250 Hz. Because the squeal produces frequencies above the Nyquist frequency of 1250 Hz, improper anti-aliasing filters will allow your data to become corrupted (see Figure 4A). Using correct anti-aliasing filters, low-frequency measurements will be not be corrupted (see Figure 4B).

Shannon vs. Nyquist

When discussing aliasing effects and sampling theorems, it can become difficult to distinguish between Shannon's sampling theorem and the Nyquist criterion and to understand the difference between the Nyquist rate and the Nyquist frequency.

In his paper on early communications theory, "Certain Topics in Telegraph Transmission Theory," Nyquist uses the Fourier series to discuss the minimum transmission rate for telegraph waves and explain how that rate is related to the bandwidth of the transmitted information. Nyquist stated, "The minimum bandwidth required for unambiguous interpretation is substantially equal, numerically, to the speed of signaling . . . ."

In some respects, Nyquist clearly states the basic idea that the minimum signal rate must contain the bandwidth of the message, or sampled signal. For sampled signals, the bandwidth really ranges from fs/2 to fs/2, where fs is the sampling rate. So the minimum sampling rate, fs (known as the Nyquist rate), does indeed contain the signal bandwidth, whose highest frequency is limited to the Nyquist frequency, fs/2. Having the Nyquist rate contain the sampled signal is known as meeting the Nyquist criterion.

How does Shannon's sampling theorem come into the picture? In his paper "Communications in the Presence of Noise," Shannon gives Nyquist credit for "pointing out the fundamental importance of the time interval 1/(2W) seconds in connection with telegraphy." Shannon did us all a favor by putting the basic sampling theory in a form we can recognize: "If a function f(t) contains no frequencies higher than W cps (Hz), it is completely determined by giving its ordinates at a series of points spaced ¹/ ₂W seconds apart."

This has come to be known as Shannon's sampling theorem, or simply the Sampling theorem, often misnamed the Nyquist theorem. It can be simply restated: an analog signal containing components up to some maximum frequency f Hz can be completely represented by regularly spaced samples, provided the sampling rate is at least 2f sps.

A major difference between DA systems and FFT-based instrumentation is the way in which you set the frequency span and resolution of a measurement. With DA systems, you usually set the sampling rate directly. Whereas with FFT-based instrumentation, you simply specify the desired alias-free bandwidth.

In DA systems, the sampling rate divided by the size of the FFT determines the frequency resolution of the measurement. In instrumentation lingo, the frequency span divided by the number of frequency lines (i.e., the alias-free portion of the FFT) determines the resolution. For example, a typical sampling rate of 25.6 Ksps and an FFT size of 1024 leads to a resolution of 25 Hz. If you want better resolution, you'll have to either increase the FFT size or decrease the sampling rate. Not only is the resolution determined by the sampling rate, but the location of the frequencies is, too. In this example, the measurement frequencies are located at multiples of 25 Hz (e.g., 0, 25, and 50).

If you want to measure a 1 kHz component, Shannon's theorem would demand that you sample at >2 * the component (see the sidebar). How much greater is dependent on the specifications of the anti-aliasing filter of your DA system; a common ratio is 2.56 * the component to be measured. At this ratio, your highest component frequency will be sampled at a rate of 2.56 samples every cycle. Lower frequency components will contain more samples per cycle, as set by the ratio of the sampling rate to the frequency. The closer the alias-free ratio approaches the theoretical limit of 2, the more efficient the DA system becomes because you'll be able to collect less data to analyze your frequency components.

Some may be concerned about representing components with so few samples because the time-domain waveform doesn't seem to accurately represent the underlying sinusoid. However, that digitized time signal contains all the information necessary for complete frequency-domain analysis.

Suppose you digitize a 2000 Hz frequency component using a sampling rate of 2.56 * 2000 = 5120 sps. Figure 5A (page 46) shows the original continuous signal and the signal digitized at 2.56 samples per cycle. Although the time-domain signal looks bad, it does so only because the samples are connected with straight lines. The digital spectrum shown in Figure 5B accurately reproduces the original spectrum in the alias-free range.

Another important hardware characteristic is the variability of the sampling rate. Most systems are designed with an anti-aliasing filter and a nonvariable sampling rate, and they use digital resampling techniques to offer halved multiples (e.g., ¹/ ₂, ¹/ ₄, and ¹/ ₈) of the original bandwidth. In these types of systems, you're restricted to a discrete set of possible resolutions and FFT bin locations; in many cases, this is sufficient for the frequency analysis needs.

Some advanced systems allow you to choose any desired sampling rate and therefore offer full flexibility in setting your resolution and FFT bin locations. What makes this variability possible is an anti-aliasing filter that tracks the sampling rate, providing consistent alias-free protection relative to the sampling rate. Another important advantage in variable sampling rate systems is that you can adapt your measurements to a variety of standard rates (i.e., 44.1 Ksps and 51.2 Ksps) and update your measurements to future standards.

The easiest way to corrupt an analysis is to introduce unwanted noise into the system. Although there are many post-processing techniques to remove noise, the best way to remove noise is to prevent it from entering the measurement system. A proper FFT analysis requires a low-noise system. Proper cabling, grounding, and shielding of the signal path into the measurement system is a must, but even these techniques have to be complemented by the proper use of differential or single-ended connections.

Noise can be fairly insidious because it's difficult to recognize without proper frequency domain analysis. What may seem to be a clean signal in the time domain may actually have enough noise to bury many frequency components of interest. Being able to choose between a ground-referenced measurement and a differential measurement is important to achieve the best signal quality in the system.

For example, if you place an accelerometer on an electric motor, you'll have to work hard to get a clean signal. The motor noise superimposed on both signal lines could effectively ruin the measurement unless you use differential connections to eliminate the common-mode noise. Some measurements require ground-referenced inputs for better signal quality. Having the ability to choose between differential and ground-referenced connections is critical for achieving the best signal quality under varying measurement environments.

The demands of your analysis will determine your application's dynamic range and establish how you will map the input range to cover the full signal; the same demands will also decide whether you can detect when measurements are clipped. Although some analysis may require no more than a 60 dB dynamic range, others may call for more than 100 dB.

For instance, in predictive maintenance of rotating machinery, you may be interested in determining the health of the bearings. In this case, there's a large frequency component tracking the rotational speed of the main shaft and many smaller components that can give evidence of deterioration of the bearings. The ratio of the smaller frequencies to the large main rotational frequency determines the minimum dynamic range, which can be as high as 90 dB. The small components increase with time as the bearings wear. Thus, the better the dynamic range of your measurement system, the earlier you can detect bearing degradation (see Figure 6).

Theoretically, a dynamic range of 90 dB would require only 15 bits of resolution, as derived by the rule of 6 dB of dynamic range per bit. In reality, though, the application would require a minimum of 16 bits because of the effects of system noise and sampling nonlinearities. You have be careful when determining the resolution your system must have to represent its dynamic range. Resolution is only one of the factors that determines system performance. You must evaluate other hardware specifications, such as dynamic range, to meet your analysis requirements.

Other important features are user-selectable gain settings and overload detection. User-selectable gain settings allow for a wide variety of input voltage ranges. Matching the full range of your signal to the proper input voltage range will take full advantage of the system's dynamic range.

When averaging spectral data, one erroneous sample set can ruin the entire averaged measurement. Typically, a sample set is corrupted when the input signal exceeds the full-scale range. This condition is known as an overload, or clipping. The system must be able to detect this condition so that the spectral averaging process can properly reject corrupted data sets. Small amounts of clipping are difficult to detect in the time domain but are easy to see in the frequency domain. A clipped signal, which shows large amounts of high-order harmonics, is a common type of distortion. Overload-detection circuitry is the easiest way to detect an overload. Without overload detection, spectral averages can be corrupted. Then there is no way to recover and usually no way to tell that your data are invalid.

Calibration can take many forms, but the underlying principle is always the same: to ensure that the acquired data accurately reflect the real signal. Calibration helps you correct the effects of environmental factors (e.g., drift over temperature and time) that can otherwise alter the signal being analyzed.

All your analysis depends on the system's ability to accurately represent the real-world signals. Without calibration, you can never be sure of the results of your analysis.

FFT measurements based on the relationship between two channels must meet all the single-channel requirements as well as provide accurate channel-matching in both amplitude and phase characteristics. In measuring the frequency response of a system, you measure the input and output amplitude and phase at every frequency. The ratio of the input and output amplitudes is the gain of the system. The difference between the input and output phase is the phase of the system. To accurately represent the system gain, be sure that the measurement channels' gain stages are well matched. Any mismatch appears as a gain error in the final measurement.

Phase mismatch has a similar effect on measurements. The system specifications must determine the gain and phase matching of the acquisition hardware. To perform phase matching, the hardware must be able to simultaneously sample both channels. A typical phase error specification in a high-performance system is 1 degree or better. This specification is given for the maximum sampling rate. The phase error decreases as the sampling rate decreases.

When measuring phase in general, you may be concerned with sample clock jitter and stability. Significant jitter in the sample clock can cause the analysis to incorrectly identify the frequency of a component because of the frequency modulation from an unstable clock. Spectral smearing or spreading is often evidence of a jittering clock.

To perform quality measurements, whether in the time domain or the frequency domain, select your hardware carefully, using the guidelines we just discussed. Many of us are used to selecting measurement hardware first and thinking about the software second, but it's actually your measurement techniques and algorithms that dictate your hardware requirements. By taking this approach, you'll be guaranteed more accurate measurements and higher quality signal analysis.

Michael Cerna is a DSA Group Manager and Mendy Ouzillou is a Hardware Engineer at National Instruments Corp., 11500 N. Mopac Expwy., Austin, TX 78759; 512-794-0100.