INTRODUCTION

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IN MANY PHYSIOLOGICAL investigations, the electromyogram (EMG) is recorded as a complex signal, but only the total electrical activity, ideally reflecting global activity in a pool of motor neurons of the muscle, is of interest. In these cases, it is common to process the raw EMG signal with a device called a moving averager. This device produces an output signal that tracks a scaled version of the envelope of the raw EMG signal. This process is called amplitude demodulation. This processing technique is desirable, because when the slow-output waveform is digitized it requires much less storage space than the original raw signal. More importantly, the output waveform is readily interpretable as a continuous indication of the total electrical activity of the muscle. A moving averager is commonly constructed around a filter called a modified Paynter filter. We studied the properties of this filter and found that it has poor high-frequency attenuation. This permits higher frequency signal components to pass through to the output-demodulated waveform as interfering noise. Therefore, we designed and constructed a new filter to replace the modified Paynter filter and compared the performance of these filters.

The apparatus for amplitude demodulation consists of two major parts: 1) a precision full-wave rectifier, which mathematically generates the absolute value of the raw signal, and 2) a low-pass filter, which smoothes the jagged edges of the rectified signal. This creates a new signal representing the envelope of the original raw signal. The choice of low-pass filter for smoothing entails a trade-off between smoothing efficiency and sensitivity for genuine change in amplitude. The structure and implementation of the filter are not important, but the transfer function that describes the dynamic behavior is vital. Such a filter can be implemented with analog or digital circuitry.

A filter that has become a standard for EMG signal processing is often referred to as the Paynter filter, but it is better described as a modified, third-order, linear phase shift filter (13), here referred to as the "modified" Paynter filter. Use of this filter for EMG processing was proposed by Gottlieb and Agarwal (10) in 1970 and by Kreifeldt (12) in 1971. This filter has become a standard for EMG signal processing.

The design of the modified Paynter filter is recorded in an applications manual that was published as a promotional aid for an electronic amplifier manufacturer (13). The modified Paynter filter is derived from a simpler filter called a third-order delay line filter, which we will call the "original" Paynter filter. The original Paynter filter is a low-pass filter with a phase response that is approximately linear throughout the pass band, which means that the time delay is approximately constant for all pass-band frequencies

(1)

where T(s) is the output-over-input voltage transfer function of an original Paynter filter, R is resistance, C is capacitance, and s is a complex frequency variable.

The original Paynter filter of Eq. 1 was altered by placing a transmission zero pair at frequencies of plus and minus 1/RC radians per second, which results in the transfer function in Eq. 2, which in turn describes the modified Paynter filter

(2)

Schematics of single-amplifier implementations of the original and modified Paynter filters are shown in Fig. 1. Transfer functions in Eqs. 1 and 2 have a negative sign, because they are implemented around a single inverting amplifier. However, when placed in a circuit, another inverting amplifier that cancels the negative sign typically follows them. Therefore, a characterization of these transfer functions can omit the negative sign. Figure 2 shows a complete characterization of both Paynter transfer functions as well as our new, modified Bessel filter, which is the subject of this report.

View larger version (20K): [in this window] [in a new window]   Fig. 1.   Schematic of original (A) and modified Paynter filter (B). Modified Paynter filter has an additional high-pass feedforward leg going into inverting terminal of operational amplifier. Arrow in B indicates where a cut will return modified Paynter filter to original Paynter filter. R, resistance; C, capacitance; Vin, input voltage; Vout, output voltage.

View larger version (16K): [in this window] [in a new window]   View larger version (9K): [in this window] [in a new window]   Fig. 2.   Frequency response of modified Paynter filter (short-dashed lines), original Paynter filter (long-dashed lines), and modified Bessel filter (solid lines). Paynter filters have RC time constants normalized to 1 s; modified Bessel filter has a normalized delay of 2 seconds. Magnitude response is shown on linear (A) and logarithmic scales (B). Modified Paynter filter has a magnitude response curve with a notch at a frequency of 1 rad/s. C: delay. Modified Paynter filter has an abrupt 180° phase reversal at notch frequency. Stop band of modified Bessel filter begins at 1.6 rad/s, minimum stop-band attenuation is 74.5 dB, and notch frequencies are 1.64, 2.07, and 3.76 rad/s.

In Fig. 2 we consider the responses of the modified and original Paynter filters. The magnitude responses are shown with a linear (Fig. 2A) and a logarithmic scale (Fig. 2B). The phase-delay function is shown in Fig. 2C. We calculated the delay function from the phase response as phase divided by frequency. The modified Paynter filter has a clear discontinuity in the magnitude and delay responses at the notch frequency of 1/RC radians per second. The transmission zero pair has a large impact on the magnitude response. In the pass band, which is below the 1/RC notch frequency, the modified filter has a faster roll-off, but in the stop band there is an unfortunate rebound in transmission that peaks at ~1.5 times the notch frequency. The magnitudes of the filters cross at ~1.4 times the notch frequency, beyond which the original Paynter filter has higher attenuation than the modified Paynter filter. The logarithmically scaled magnitude plot in Fig. 2B shows that the rate of attenuation of the original Paynter filter is 60 dB/decade, but the addition of the transmission zero pair reduces the rate of attenuation of the modified filter to just 20 dB/decade.

The addition of the notch does not affect the phase or delay of the filter below the notch frequency. Therefore, the modified and original Paynter filters have identical phase and delay responses in the pass band. Figure 2C shows that the delays of both filters are approximately constant, but there is some ripple in the delay. This equiripple delay response of the Paynter filters contrasts with the delay response of the Bessel or Thompson filter, which is maximally flat (16, 18).

The modified Paynter filter approximates an idealized moving-average window filter, which has an output that represents the average of a signal over some time period (T). In pursuit of a better and more selective filter than the first-order low-pass filter, the perfect moving-average filter was considered to be an ideal solution (9, 10)

(3)

The operation of such a filter is described by Eq. 3, where x and y are input and output functions, respectively.

By constructing a modified Paynter filter with a notch frequency of 1/T Hz, a perfect moving-average filter with a window of T seconds is approximated (13). The modified Paynter filter has been widely used, because it approximates a moving-average filter (10, 12), which is why amplitude demodulators with modified Paynter filters are often called moving averagers. Real moving averagers typically have a selection of filter settings that correspond to a selection of different notch frequencies.

	FILTER REQUIREMENTS

Although the filters in some amplitude demodulators do approximate a perfect moving-average filter, is the perfect moving-average filter a worthy ideal to model? What features are desirable in a demodulation filter? Obviously, a low-pass filter should pass low frequencies and reject high frequencies, but to what degree should the rejected signals be attenuated? After rectification, a signal becomes harmonically rich, because at every zero crossing in the original signal there is a high-speed change of direction in the rectified signal. This means that the high-frequency components that need to be removed are similar in amplitude to the low-frequency components of the signal. Choosing the required amount of attenuation of these large-amplitude, high-frequency components requires a judgment, but if we assume that the filtered signal will be digitally sampled, we can generate a practical number. To reduce an unwanted component to <1 least significant bit in amplitude on a 12-bit analog-to-digital conversion system, we require attenuation of 2¹² or 4,096 times, which is 72.2 dB. The perfect moving-average filter and its approximation, the modified Paynter filter, have an ultimate roll-off that is first order, which is 20 dB/decade. This means that 72.2 dB of attenuation are achieved at about four decades above the edge frequency of the filter. The region beyond the 1/T notch should be considered the stop band of these filters. However, the side lobe of the modified Paynter filter rises to 0.1, which is an attenuation of only 20 dB, and the perfect moving-average filter rises to 0.2, which is an attenuation of only ~14 dB. Ironically, the approximation of the moving-average filter actually has preferable performance in this regard. Clearly, the rectangular or perfect moving-average filter is poor in terms of stop-band attenuation.

Step response is the next important criterion, because the output of the filter is the value we are using to quantify the EMG amplitude. Significant overshoot in the step response is not acceptable, because this will add artifactual peaks to the output signal that may be misinterpreted as being physiological. Overshoot should be considered a type of noise (12). It is preferable to have a sluggish rise and even lose some signal rather than generate new signal features. The step responses of the original and modified Paynter filters are shown in Fig. 3. The original filter has ~2% overshoot, and the modified filter has essentially none; these are acceptable values.

View larger version (15K): [in this window] [in a new window]   Fig. 3.   Time domain responses of modified Paynter filter (short-dashed lines), original Paynter filter (long-dashed lines), and modified Bessel filter (solid lines). Paynter filters have RC time constants normalized to 1 s; modified Bessel filter has a normalized delay of 2 seconds. A: step response; B: impulse responses. Original Paynter filter has a rise that is initially slower but increases and has ~2% overshoot. Modified Paynter filter has a rise that is initially faster and has little overshoot. Impulse response of modified Paynter filter has a peak at time 0.

Delay is the next important criterion. EMG amplitude demodulators are used to monitor onset and cessation of muscle activation. The low-pass filter will cause delay of signal, but we can compensate for this if 1) the delay time is similar for all frequencies and 2) we know the value of the time delay. Delay is phase lag divided by frequency, so linear phase implies constant time delay. The original and modified Paynter filters are approximately linear phase filters with an approximately constant delay (Fig. 2C). A convenient feature of these filters is that the delay is  times the RC time constant value. A filter with a normalized 1/RC frequency of 1/1 rad/s or 1/(2) H2 will have a delay of  or 3.14 s. This is convenient, because a Paynter filter with a time constant of 100 ms will have an approximately fixed delay of 50 ms for which we can readily compensate.

Thus the original and modified Paynter filters have excellent phase properties that make them ideal filters for EMG demodulation. Unfortunately, these filters have poor stop-band attenuation. The original Paynter filter has a third-order roll-off, but the addition of the transmission zero pair in the modified Paynter filter reduces the roll-off to first order. The benefit of the conveniently placed transmission zero pair comes at the cost of a rebound in stop-band attenuation. The transmission zero pair of the modified Paynter filter drastically reduces attenuation, because the overall degree of the filter is low: only third degree. However, if a filter with higher-order attenuation is used initially, then high and rapid attenuation can be achieved. This was the approach we used in this project. We used a seventh-order Bessel transfer function with three transmission zero pairs, which we call a modified Bessel transfer function. This generates a filter with pass-band characteristics comparable to the modified Paynter filters but with far greater stop-band attenuation. We constructed and tested an analog version of this filter and compared its performance with that of the modified Paynter filter.

	METHODS

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Filter design, construction, and characterization. The filter design process was carried out in three steps: 1) to find a transfer function that approximated the desired response, 2) to generate a prototype filter design from the transfer function, and 3) to design a practical filter that matches the performance of the prototype filter.

View larger version (18K): [in this window] [in a new window]   Fig. 4.   A: schematic of frequency-dependent negative resistance (FDNR) implementation of modified Bessel filter with low-frequency compensation resistors Ra and Rc added. B: active network implementation of 1 FDNR circuit element. Small-compensation capacitors are connected by dotted lines. D2, D4, and D6, FDNR elements; Drr and DRx, resistors, where x is 2, 4, or 6; Cs and Cl, capacitors; Ra and Rc, low-frequency compensation resistors.

Demodulator comparison and EMG recording. The new modified Bessel demodulators were compared with demodulators with modified and original Paynter filters. We obtained two commercial moving-average units (model MA-821RSP, CWE Instruments, Aardmore, PA). Each unit had four channels with rectifiers and modified Paynter filters. In one of the units, we converted each of the modified Paynter filters back to original Paynter filters by removing the 3C capacitor, which is next to the solid arrow in Fig. 1B. This disconnects the high-pass feedforward leg of the filter, which removes the 1/RC notch. On each unit we set the filter time constants to 20, 50, 100, and 200 ms. With 4 original Paynter, 4 modified Paynter, and 4 modified Bessel filter demodulators, we had a total of 12 amplitude demodulators for comparison.

	RESULTS

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Filter performance. Figure 5 shows the simulated and measured performances of the modified Bessel filter. The simulated magnitude response curve of the transformed network (lines in Fig. 5, A and B) closely matches the original transfer function magnitude response curve (Fig. 2, A and B). The simulated delay response of the transformed network (line in Fig. 5C) is marginally less smooth than the delay response of the original theoretical transfer function (Fig. 2C). The actual performance of the real transformed filter (triangles in Fig. 5) closely matches that predicted by computer simulation.

View larger version (14K): [in this window] [in a new window]   View larger version (10K): [in this window] [in a new window]   Fig. 5.   Frequency response of 100-ms modified Bessel filter. Lines, simulated response of transformed network; , measured values from real filter. Magnitude response is shown in linear (A) and logarithmic units (B). C: delay.

Demodulator performance. Figure 6 is a strip chart with a raw EMG of a parasternal intercostal muscle and several demodulated forms of that signal. Demodulated waveforms derived from all three types of filters with 50- and 100-ms time constants are shown. These signals are shown as recorded originally and after time correction for their different time delays. A vertical line is drawn through the cardiac impulse that appears in the EMG. The cardiac impulse appears in the demodulated signals and is delayed in all the signals that were not time corrected but is properly synchronized in the signals that were time corrected. The corrected signals were shifted by the amount predicted theoretically, with the modified Bessel filter having twice the delay of both Paynter filters. For example, the 100-ms modified Bessel filter was shifted back 100 ms and the 100-ms Paynter filters were shifted back 50 ms.

View larger version (38K): [in this window] [in a new window]   Fig. 6.   Strip chart showing raw and several demodulated versions of an inspiratory EMG obtained from parasternal intercostal muscle of an awake dog breathing room air. Demodulated waveforms are in groups of 3 with 50- or 100-ms time constants. In each group, top trace, modified Bessel filter; middle trace, modified Paynter filter; bottom trace, original Paynter filter. Two groups of traces above raw EMG are shown as recorded, including inherent time shift of each filter. Two groups of traces below raw EMG have been time corrected.

View larger version (11K): [in this window] [in a new window]   Fig. 7.   Averaged power spectra of 100-ms demodulated EMG. Short-dashed line, modified Paynter filter; long-dashed line, original Paynter filter; solid line, modified Bessel filter. Demodulated signals were sampled at 1 kHz, and power spectra were computed over 256 points using a Blackman window.

View larger version (9K): [in this window] [in a new window]   View larger version (10K): [in this window] [in a new window]   View larger version (8K): [in this window] [in a new window]   Fig. 8.   Plots of EMG amplitude compared with airflow using 100-ms filters: modified Paynter filter (A), original Paynter filter (B), and modified Bessel filter (C).

View larger version (27K): [in this window] [in a new window]   View larger version (37K): [in this window] [in a new window]   Fig. 9.   Strips charts showing raw and demodulated EMG of crural diaphragm during room air breathing (A) and during CO2 rebreathing with an end-tidal CO2 of 6% (B). Traces from top to bottom: electrocardiogram, demodulated EMG (100-ms modified Bessel filter), demodulated EMG (100-ms modified Paynter filter), raw EMG, and airflow with inspiration shown by upward deflection.

	DISCUSSION

Top Abstract Introduction Methods Results Discussion References

In summary, we characterized the modified Paynter filter and showed a constant time delay that is one-half of the rated time constant. We also demonstrated that the modified Paynter filter has poor high-frequency attenuation, degrading its performance for physiological application. Therefore, we designed and constructed a set of modified Bessel filters that can be used as replacements for the Paynter filter. These filters have similar pass bands, superior stop-band attenuation, and a time delay that, although double that of the Paynter filters, is nearly constant and can be easily corrected. Although these filters are more complex than the Paynter filter, the increased consistency for quantitative EMG justifies the modest additional expense and effort.

The original Paynter filter has better stop-band attenuation than the modified Paynter filter, but it has a wider pass band. We demonstrated a simple modification (Fig. 1B) to return the modified Paynter filter to the original Paynter filter to reduce high-frequency noise but widen the pass band.

Filter description using time delay. We have chosen to classify our new filter using its time delay. Usually, low-pass filters are identified by critical frequencies in their magnitude response. For example, the Butterworth filter is identified by the critical frequency at which the transmitted power is one-half that of the input signal. This frequency is called the cutoff frequency. However, this may not be the most useful way to classify the modified Paynter filter or the modified Bessel filter, because these filters have pass bands that taper. Conventional filters like the Butterworth, Chebychev, and elliptic have pass bands that approximate, to some degree, a rectangle, but the modified Paynter filter and the modified Bessel filter have pass bands with a triangular shape. They have a disproportionate amount of signal transmission above the one-half power frequency, because the one-half power frequency occurs so early in the pass band. Therefore, the one-half power frequency is a poor way to classify these filters.

Analog implementation. It is noteworthy that we have chosen to design and construct analog filters when digital signal-processing techniques are becoming common. However, the choice between analog and digital technology is only a choice of implementation. Comparable filter types can be built using either type of technology. For example, it is possible to design an infinite impulse response filter with characteristics that match the analog version we have described. Whatever the implementation, the particular characteristics of the filter are important. One filter that is easily implemented digitally is the moving-average window filter. This is done by computing successive averages over a fixed number of data samples (3). However, the moving-average or rectangular window filter has poor stop-band attenuation, which makes it a poor low-pass filter for EMG demodulation, because higher-frequency components will leak through. Careful analysis should be applied to digital and analog filter design.

Time delay correction. A linear phase filter has a constant time delay. The Paynter and Bessel filters have approximately constant time delay; the plot of time delay vs. frequency of the Paynter filter has small ripples, whereas that of the Bessel filter is maximally flat. Filters with stop-band zeros that are based on these linear phase filters inherit these desirable phase properties. This important property keeps the signal from being distorted when different frequency components are shifted relative to each other. However, the introduction of a time delay may have to be addressed in some applications. The modified Paynter filter has a time delay that is one-half of the time constant, and the modified Bessel filters that we designed have a time delay that is equal to the time constant. A 100-ms Paynter filter has a 50-ms delay, whereas a 100-ms modified Bessel filter has a 100-ms delay. If the time delay is of concern, it can be compensated for by shifting the demodulated EMG signal backwards in time relative to other recorded signals before analysis, as we did.

Single solution. We were able to construct filters with responses similar in magnitude to our new modified Bessel filter by cascading an elliptical filter and a low-Q notch filter, although this is not detailed in RESULTS. The transient response of this filter cascade was acceptable, as was the stop-band attenuation. However, we do not believe that use of cascading filters is an optimal solution, because the characteristics of two separate filters have to be understood in detail. In addition, a filter cascade does not make optimal use of filter hardware. For example, our cascade consisted of a seven-pole FDNR elliptical filter followed by the notch filter. In contrast, the entire modified Bessel filter was accommodated in a single seven-pole FDNR implementation.