Jing Bai

In the classic article written by Savitzky and Golay (1), which has been cited more than 3800 times according to ISI Web of Science, a kind of digital filter for smoothing and differentiation was developed. In their approach, each successive subset of 2

+ 1 points is fitted by a polynomial of degree 

) differentiation (zeroth differentiation = smoothing) of the original data at the midpoint is obtained by performing the differentiation on the fitted polynomial rather than on the original data. Finally, the running least-squares polynomial fitting can be performed simply and automatically by convolving the entire input data with a digital filter of length 2

The history and development of the Savitzky–Golay (SG) smoothing and differentiation filter have been reviewed in brief (2). The original tabular coefficients of the SG filter have been corrected by Steinier et al. (3) and Madden (4) subsequently. The typo of a formula by Madden (4) has been spotted recently (5). As pointed out by Mark and Workman (5), one limitation is that the existing SG filter is subject to using an odd number of data subset. With a matrix approach, the SG filter was extended recently for even number (2

) data (i.e., the SG filter could have an even length [6]). The feasibility of the even-length SG filter has been validated with computer simulations (6). In the least-square strain estimator (LSQSE) of ultrasound elastography, it was concluded that a data subset of number 8 could obtain the best performance (7). This method is equivalent to the SG differentiation filter with a linear fitting (8), and the data subset of even number corresponds to an even length of the filter. As discussed in (6), for an even order differentiation, the SG filter of even length has a better characteristic of attenuation at the high frequency of ω= π than that of odd length. However, the matrix approach (6) is not very convenient to calculate the filter coefficients. In this article, a more general approach based upon Gram polynomials is developed to calculate the convolution coefficients of the even-length SG filter for all polynomial degrees, all differentiation orders, and all data points.  

) of data subset and assume the index of the data samples ranges from –

. The data samples can be regarded equivalently as being symmetrical about the origin, by shifting the origin one half. Therefore, the input signal are represented by (6) 

  contains the convolution coefficients of the SG filter for different order differentiation at the origin (that is, the imaginary midpoint or the center of symmetry), given by (6) 

 is the fitting polynomial degree, S is the 2

The convolution coefficients for the s-th differentiation evaluated at the imaginary midpoint are given by (6) 

-th column vector of G, the subscript 0 represents the evaluated midpoint. With the matrix approach mentioned earlier, some closed-form solutions as well as the numerical table can be obtained (6). However, only the filter coefficients for the imaginary midpoint are convenient to obtain.  

It is expected that it is difficult to obtain the matrix inversion (S

  in Equation 2 with an explicit formula if 

 is large. Therefore, one would prefer numerical computation to symbolic computation. If 

 S tends to be ill conditioned (9). Hence, a more successful and general approach with the orthogonal polynomials (Gram polynomials or Legendre polynomials) has been proposed (9–11). In this article, such an approach also is extended for an even-length SG filter. All the equations are similar to those in (10), but the effects of the even number data and the aforementioned translation of the origin have been taken into account.  

The smoothing and the differentiation are expressed as 

) differentiation value evaluated at position 

				 are the corresponding coefficients for smoothing and differentiation, respectively. Note that 

 = 0 for the imaginary midpoint of the data subset, while 

– 1/2 for the first, second, and last points, repetitively.  

The convolution coefficients in Equations 6 and 7 are calculated as  

Articles

In various fields such as signal processing, imaging processing, analytical chemistry, and spectroscopic analysis, smoothing and differentiation is important and necessary. With a matrix approach, the Savitzky–Golay smoothing and differentiation filter was extended recently to even length. In this article, a more general approach is proposed for convenient computation.

Jing Bai

Articles

Savitzky–Golay Smoothing and Differentiation Filter of Even Length: A Gram Polynomial Approach