Patent Publication Number: US-7587442-B2

Title: Method of determining the derivative of an input signal

Description:
TECHNICAL FIELD 
     The present invention relates to determining the derivative (i.e., the rate of change with respect to time) of a noise-containing input signal without introducing phase delay due to filtering. 
     BACKGROUND OF THE INVENTION 
     It is commonly necessary in control applications to determine the derivative of a measured input signal. In the control of a motor vehicle transmission, for example, a controller produces a digital speed signal based on the output of a shaft speed sensor, and then determines the acceleration of the shaft by computing the derivative of the speed signal. In such an application, conventional derivative calculations are problematic due to the presence of spurious noise in the measured signal, as the derivative of the noise is typically much larger than the derivative of the signal. The usual approach is to low-pass filter either the input signal or its derivative to remove or severely attenuate the noise-related component. However, the filtering introduces phase-delay, which is particularly undesirable if the derivative is to be used for control purposes. Accordingly, what is needed is a way of effectively determining the derivative of a noise-containing input signal without introducing any significant phase delay. 
     SUMMARY OF THE INVENTION 
     The present invention is directed to an improved method of determining the derivative of a noise-containing input signal where an aliased derivative is used to periodically reset a filtered version of a normally determined derivative. The aliased derivative is calculated using a slower update or sampling rate than the normally determined derivative, and the filtered version of the normally determined derivative is reset to an alias-based reset value at each update of the aliased derivative. Preferably, the reset value is determined according to a weighted sum of the aliased derivative and the filter output. The periodically reset filter output closely follows an idealized derivative of the input signal, substantially eliminating the phase delay introduced by conventional filtering. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of a method for determining the derivative of a digital input signal according to this invention; 
         FIG. 2  graphically depicts a noise-containing sinusoidal waveform, an idealized derivative of the waveform, an actual derivative of the waveform and a filtered version of the actual derivative, all as a function of time; and 
         FIG. 3  graphically depicts the idealized derivative of  FIG. 2  along with an output waveform generated by the block diagram of  FIG. 1  when the input is the noise-containing sinusoidal waveform of  FIG. 2 . 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     While the method of the present invention may be applied to any type of digital or analog input signal, it is disclosed herein in the context of an application involving a speed transducer and a rotary shaft such as a transmission input or output shaft. In a typical implementation, the speed transducer is positioned in proximity to the periphery of a toothed wheel fixed to the shaft and generates an electrical pulse with the passage of each tooth of the wheel. The pulses are fed to a circuit that produces a digital signal with logic level transitions corresponding to leading and/or trailing edges of the pulses, and the value of a free-running counter is sampled at specified transitions (i.e., zero-to-one, or one-to-zero) of the digital signal. Successive counter samples are differenced to form a succession of time intervals that are inversely proportional to the rotary speed of the shaft. In a dynamic system, the succession of time intervals defines a digital input signal that may be numerically differentiated to determine the acceleration of the shaft. In an analog implementation, the magnitude of the input signal is periodically sampled and converted to a corresponding digital value by an analog-to-digital converter to form a succession of digital values. 
     Referring to  FIG. 1 , the method of this invention is illustrated as a block diagram where a noise-containing input signal (IN) is applied to line  10 , and an output signal (OUT) representing the derivative of IN is produced on line  12 . The functionality of the various blocks may be implemented in various ways, the most common of which is with a microprocessor programmed to carry out the underlying mathematical operations. 
     The input IN comprises a succession of digital values representative of input signal magnitudes sampled at a rate sufficiently great with respect to the input signal frequency to prevent aliasing. For example, if the input signal has a maximum frequency f, the sampling rate is at least (2*f). The block  14  produces a normal derivative (ND) of IN on line  16 , and the block  18  produces an aliased derivative (AD) of IN on line  20 . The normal derivative ND is based on a conventional derivative calculation, exemplified by:
 
 ND ( k )=[IN( k )−IN( k− 1)]/Δ t   (1)
 
where k is the sample number, IN(k) and IN(k−1) are the current and previous samples of input IN, and Δt is the time interval between successive samples. The aliased derivative AD is computed in much the same way, except that the time interval between successive samples is an integer multiple (R) of Δt. Using the same notation as in equation (1), the aliased derivative AD is given by:
 
 AD ( R*k )=[IN( R*k )−IN( R ( k− 1))]/( R*Δt )  (2)
 
For example, if R=5, the aliased derivative AD will be calculated based on every fifth sample of the input IN, and will updated one-fifth as fast as the normal derivative ND. At each such update of the aliased derivative AD, the block  18  produces a trigger signal (TRIGGER) on line  22 .
 
     The block  24  designates a conventional first-order or second-order low-pass filter that can be reset to a supplied reset value on command. The normal derivative ND is applied to the input (INP) of block  24 , and a filtered version of ND is produced as the output OUT on line  12 . The TRIGGER signal produced by block  18  on line  22  is applied to the reset terminal (RST) of filter block  24 , and a reset value (RESET_VALUE) on line  26  is applied to the reset value (RV) terminal of filter block  24 . The RESET_VALUE signal on line  26  is a weighted sum of the filter output OUT on line  12  and the aliased derivative AD on line  20 . The block  28  applies a calibrated fractional gain G (which may have a value of 0.5, for example) to OUT, and the block  30  applies the complement of gain G (that is, 1-G) to AD, and the results are summed by block  32  to form the RESET_VALUE signal on line  26 . Thus, the output signal OUT is a filtered version of ND that is periodically reset to a RESET_VALUE based on a weighted sum of OUT and AD. 
     As indicated by the block diagram of  FIG. 1 , the method of the present invention combines the information gleaned from both the normal derivative ND and the aliased derivative AD. The normal derivative ND contains the correct phase information, while the aliased derivative AD is statistically insensitive to random noise superimposed on the input IN. Combining the aliased derivative AD and the filtered normal derivative to form the filter reset value (RESET_VALUE) produces a filter output (OUT) that retains the phase attribute of the normal derivative ND and the noise insensitivity of the aliased derivative AD. 
     The advantages of the present invention over a conventional filtering approach are illustrated by the graphs of  FIGS. 2-3  for an example in which the input signal IN is a noise-containing sinusoidal waveform, designated by the trace  36  in  FIG. 2 . The input signal IN is smoothly varying, and the noise is represented by the relatively high frequency, low magnitude, undulation. The trace  38  represents an idealized derivative of trace  36 ; that is, a derivative based on the value of the smoothly varying component of IN, without regard to the noise. However, calculating the actual derivative of trace  36  produces a very noisy signal such as designated by the reference numeral  40  in  FIG. 2 . If the noisy actual derivative is low-pass filtered to remove or severely attenuate the content attributable to noise, the result is a signal such as illustrated by the trace  42  in  FIG. 2 . While the noise content of trace  42  is sufficiently attenuated, it is also significantly phase-delayed with respect to trace  38  (the idealized derivative), as can be easily seen in  FIG. 2 . A similar result occurs when the input signal IN is low-pass filtered prior to computing the derivative. In contrast, the trace  44  of  FIG. 3  represents the output signal (OUT) produced by block  24  of  FIG. 1  when the input (IN) is the noise-containing waveform represented by trace  36  of  FIG. 2 . The trace  44  has a residual undulation similar to the filtered actual derivative trace  42  of  FIG. 2 , but the periodic resetting of OUT to the RESET_VALUE (which occurs approximately twice per second in the example of  FIGS. 2-3 ) causes the trace  44  to substantially conform in phase with the idealized derivative (i.e., trace  38 ). 
     In summary, the present invention provides a way of effectively determining the derivative of a noise-containing input signal without introducing any significant phase delay. Essentially, an aliased derivative is used to periodically reset a filtered version of a normally determined derivative, with the filter output being reset to a weighted sum of the aliased derivative and the filter output at each update of the aliased derivative. The periodically reset filter output closely follows an idealized derivative of the input signal, substantially eliminating the phase delay introduced by conventional filtering. 
     While the method of the present invention has been described with respect to the illustrated embodiment, it is recognized that numerous modifications and variations in addition to those mentioned herein will occur to those skilled in the art. For example, the gain G may be different than mentioned herein, and so on. Accordingly, it is intended that the invention not be limited to the disclosed embodiment, but that it have the full scope permitted by the language of the following claims.