Sampling rate, frequency aliasing, and number of bits per sample as parameters must be considered when evaluating the performance of digital polysomnographs.
Digital polysomnography refers to the collecting of polysomnography data, digitizing the data, and then displaying the data on a computer monitor or printer. Polysomnography data is analog, obtained from analog transducers and analog amplifiers. The preamplifiers are analog, not digital, and most of the performance specifications of the newer preamplifiers are not superior or even equal to those of the older polygraph preamplifiers. The feature that distinguishes digital polysomnography systems from analog systems is that the data is digitized, processed, and then displayed. This digitization adds the sampling rate, frequency aliasing, and number of bits per sample as parameters that must be considered in evaluating the performance of digital polygraphs. The data sampling rate is one of the most important of the additional performance specifications. The sampling rate to be used is partly a question of how much of the higher frequency information is needed from the EEG/EMG/ECG channels, but whatever the rate used, frequency aliasing must be minimized, or the digital data will be a distorted version of the analog data. Frequency aliasing refers to the data distortion created by the data not being sampled fast enough. It is reduced, but not eliminated, by sufficiently filtering the data before it is digitized. Frequency aliasing can be seen in the wagon wheels in old western movies that appear to be moving slowly backwards or forwards. This type of frequency aliasing does not occur in modern cameras with a faster frame rate. Frequency aliasing usually cannot be recognized by viewing the data, nor can it be corrected for once the aliasing has occurred. Unrecognizable frequency aliasing can also occur in displaying the data on a computer monitor, again providing a distorted version of the analog signal. In fact, if adequate precautions are not taken, the displayed data will not be an adequate representation of the original signal. Frequency aliasing also is a reason why it is not possible to display digitized data on today’s computer monitors in 30-second epochs per screen with the same quality that is possible with analog polysomnography using the recommended1 30-second pages (paper speed of 10 mm/sec). This display limitation is a result of the limited number of pixels available on the computer monitor, not the physical size of the monitor.
This discussion is restricted to systems using computer monitors (analog monitors are rarely used as they are much more expensive). Also, the discussion is limited to the higher frequency signals including the EEG and EMG. The information also applies to lower frequency signals such as airflow and respiratory effort, but the digital filtering and sampling rate requirements to accurately reproduce these lower frequency signals are less stringent.
Standards have yet to be established for the quality of digitized polysomnography data. The American Heart Association (AHA) has had established standards for digital electrocardiography for more than 25 years.2 For EEG recording using analog polysomnography, a minimum paper speed of 10 mm/sec is recommended as the slowest, which will permit clear resolution of alpha and sleep spindle frequency; the low-frequency time constant should be .3 seconds or greater. A high-frequency filter setting in the range of 30-35 cps is recommended.3 Filter settings depend somewhat on what information is to be extracted from the EEG. The AHA recommends a bandwidth of .1 to 100 Hz for ECG interpretation.2
Sampling rate
The sampling rate is normally determined using an interpretation of Shannon’s Sampling Theorem.4 Shannon’s theorem states that a signal can theoretically be digitized and then restored to its analog value if the signal is sampled at twice the highest frequency contained in the signal. This does not mean the bandwidth of the signal, but instead the highest frequency contained in the signal. This paraphrasing is correct for signals such as the EEG (and audio). If the signal is sampled at a slower rate, frequency aliasing can occur. Frequency aliasing distorts the image so that the signal is no longer an accurate reproduction of the original signal, and the original signal can no longer be recovered. To prevent frequency aliasing, the signal must be adequately filtered before it is digitized, not after the signal is digitized. Shannon’s theorem applies only to band limited signals, but it is routinely applied to transient signals, such as the EEG, which are not band limited. Just because an amplifier has a bandwidth of 30 Hz does not mean that higher frequency signals do not pass through the amplifier—witness the ubiquitous 60-Hz activity at the output of many EEG amplifiers with a 30-Hz bandwidth. The actual upper frequency limit of EEG signals is not known. There will always be some frequency aliasing of EEG data, no matter how high the digital sampling rate. An approximation often used is that the sampling rate be 10 times the -3 dB bandwidth. A 250-Hz sampling rate is about the minimum rate to use for accurate representation of the higher frequency (alpha, spindles, and muscle artifact) EEG components. The faster the sampling rate, the less signal distortion introduced by the antialiasing filters. A sampling frequency of 200 Hz is adequate, particularly if some data interpretation is performed between samples. Of course, the bandwidth of the signal to be digitized is determined by the information that is to be extracted from the signal. If the system does not use the higher frequency information, a smaller bandwidth and lower sampling rate can be used, but the same precautions must be taken to prevent frequency aliasing. In order to evaluate system performance, one must know the characteristics of the analog filters used before the digital conversion takes place. It is interesting to note that the AHA, more than 25 years ago, specified a minimum sampling rate of 500 Hz for digitizing ECG data.5
The sampling rate is an important factor in the acquisition of digital polysomnography data; it is even more important in the display of the data on a digital computer monitor. The conventional method of displaying computerized polysomnography data is to display 30 seconds of data on the computer screen, with the time axis being the abscissa. A high-quality computer display uses a resolution of 1,600 x 1,200. There are 1,600 horizontal points used to display 30 seconds of data. This is equivalent to a sampling rate of 53.33 samples per second. Precautions must be taken to prevent frequency aliasing, or the displayed waveforms will not necessarily be an accurate representation of the original data. The effective sampling rate will be much lower if a lower display resolution is used. Digitized EEG data cannot be accurately displayed on a computer monitor displaying 30 seconds of data in one horizontal trace. If 5 seconds of data are displayed on the full computer screen, then the display resolution of 1,600 horizontal pixels corresponds to a display of 320 samples per second. With adequate sampling, high-quality data can be accurately displayed on a computer monitor in 5-second epochs. Ten-second epoch displays will have an equivalent sampling rate of 160 samples per second. However, frequency aliasing (waveform distortion) will occur if EEG data is displayed in 30-second epochs on a digital monitor. Either the data must be further lowpass filtered or some other means must be taken to prevent antialiasing. The human’s ability to interpret between samples of displayed data does allow for a lower sampling rate for displayed data, but frequency aliasing must first be prevented. The methods used to prevent frequency aliasing of the data display should be clearly specified since it is normally not possible to recognize frequency aliasing after it has occurred.
Figure 1. The effect of the sampling rate on the appearance of a sleep
The effects of frequency aliasing can be shown in a variety of ways. For the purposes of illustrating frequency aliasing, a section EEG containing a sleep spindle was sampled at 500 Hz and then reproduced at reduced sampling rates. The reproduction consisted of reconverting the digitized data back to analog data with a digital to analog converter and then reproducing the data on an analog recorder. Figure 1 shows a section of data containing a sleep spindle reproduced at four sampling rates (250 Hz, 125 Hz, 62.5 Hz, and 31.25 Hz). Some of the higher frequency features of the waveform are distorted as the sampling rate of the data is reduced. An important point is that the nature of the analog data cannot be deduced from a reconstruction of the digitized data. The quality of the data must be known a priori, or the sampling rate and system frequency response must be known in order to have a good impression of the original data. Any reconstruction of the data, such as to a computer monitor, must include means to prevent antialiasing caused by undersampling, or else the displayed data is not necessarily an accurate reproduction of the original signal. Digital printers, with a display rate of 600 dots per inch, can more accurately display 30 seconds of polysomnography data on a 12-inch page, than can a computer monitor.
Bits per sample
The number of bits per sample used in the digitizing process is not necessarily an issue since the data can be accurately reconstructed using high sampling frequency one-bit sampling (delta-sigma modulation). Delta sigma modulated data is then converted to an eight-bit sample at a much lower equivalent sampling frequency. The efficient use of an analog to digital converter requires that the least significant bit of the conversion process should not be smaller than the noise level of the signal or of the converter. The computer monitor again presents a limitation when the number of bits used to represent the signal is considered. For a monitor with a display resolution of 1,600 x 1,200, 1,200 pixels or lines are available for the vertical display. If 20 channels are displayed simultaneously, 60 vertical pixels are available per channel. Sixty pixels are less than what are required to display six bits of data; less than six bits can be displayed per channel. If more channels are displayed, the amplitude display resolution deteriorates even further. The amplitude display resolution should be clearly defined, and ideally, means should be provided to increase the display resolution by “blowing up” a section of the waveform.
Frequency Filtering
Filtering for analog polysomnography includes a low-frequency filter to block out DC and any other very low-frequency artifact that might saturate the preamplifier, and a high-frequency filter to limit the amount of high-frequency activity. Digital polysomnography also requires an antialiasing filter, which is normally included in the high-frequency filter. It is necessary that the high-frequency filter be chosen to minimize frequency aliasing. This filtering must be done before the data is digitized and hence is analog filtering. It is preferable to do the remainder of the filtering, including 50 or 60 Hz notch filtering, after the data is digitized. Compared to analog filters, it is as easy, or easier, to tailor the high and low frequency filter characteristics of digital filters available for use in digital polysomnography systems. In fact, it is possible to design digital filters with sharper attenuation characteristics and less phase distortion than analog filters, but most digital systems simply simulate the characteristics of the filters used in analog polysomnography. Whatever the filters used, as with analog systems, the frequency response, including the filter phase characteristics, should be specified if the data are to be completely described.
Low-frequency filter: A system’s low-frequency response is normally specified by the system time constant TC, and at the high-frequency end by the high-frequency filter cutoff frequency. Early sleep recording criteria1 called for a time constant of at least .3 seconds. If the time constant is .3 seconds, significant attenuation of delta components in the EEG occurs; the time constant should be about .6 seconds in order to have less than 1-dB delta attenuation at .5 Hz. There have been no changes in the time constant specification for digital polysomnography. It is not practical to measure the system’s low-frequency performance by applying a sinusoidal signal generator as the frequencies needed are too slow. The normal procedure is to determine the system time constant by examining the system response to a pulse or square wave input, similar to what is done with analog polygraphs. The response to a square wave input will decay exponentially toward zero. It will decay to 10% of the input amplitude in a time T=2.3TC, or .69 seconds for a time constant of .3 seconds (see reference 6). It takes almost 1 second for the response to the pulse input to decay to the 10% value, so the half period of the square wave should be about 1 second, or the maximum frequency of the square wave used for measuring the time constant should be about .5 Hz if the time constant is to be easily measured. It is much more difficult to accurately measure the time constant if the calibration signal frequency is greater than .5 Hz. Digital acquisition systems can easily be designed to meet the .3 seconds or greater time constant specification. The AHA recommends the amplitude response be flat to within ±6% for ECG signals over the 1.0-30 Hz frequency range.5
High frequency cutoff: This frequency determines the upper frequency above which the system attenuates input frequencies. The attenuation is typically 30% at the cutoff frequency, and the filter attenuation increases by a factor of four for each doubling of the input frequency. Signal components at frequencies above the cutoff frequency do pass through the system, but they are attenuated by the filter. The high cutoff frequency must be less than the cutoff frequency of the input antialiasing filter, which must be appreciably less than one half the sampling frequency if significant frequency aliasing is to be avoided. Excessively sharp analog filters will create phase distortion, which should be included in the filter specification. It is best for any additional high-frequency filtering to be done using linear phase digital filters. The high cutoff frequency can also be determined by examining the system’s response to a square wave input, but it is much easier to measure the upper frequency characteristics using a sine wave generator. The AHA recommends a high cutoff frequency of 100 Hz for ECG signals.5
Jack R. Smith, PhD, is professor emeritus, Department of Electrical Engineering, University of Florida, Gainesville.
References
1. Rechtschaffen A, Kales A. A Manual of Standardized Terminology, Techniques and Scoring System for Sleep Stages of Human Subjects. NIH Publication No. 204. Washington, DC: US Government Printing Office; 1968.
2. Carskadon M, Rechtschaffen A. Monitoring and staging human sleep. In: Kryger MH, Roth T, Dement WC, eds. Principles and Practice of Sleep Medicine. 3rd ed. Philadelphia: WB Saunders; 2000:1197-1215.
3. Bailey JJ, Berson AS, Garson A, et al. Recommendation for standardization and specifications in automated electrocardiography: bandwidth and digital signal processing. Circulation. 1990;81:730-739.
4. Shannon CE. A mathematical theory of communications. Bell Syst Tech J. 1948;27:379-423, 623-656.
5. Pipberger HV, Arzbaecher RC, Berson AS, et al. Recommendations for standardization of leads, and of specifications for instruments in electrocardiography and vectorcardiography. Report of the Committee on Electrocardiography, American Heart Association. 1975;52:11-31.
6. Smith JR. Modern Communication Circuits. 2nd ed. New York: McGraw-Hill; 1998.
