A class of finite-impulse-response (FIR) digital filters has been developed to perform certain frequency-limiting, decimation, and differentiation (with respect to time) functions on a time series of data samples. The method is implemented by use of design equations that contain parameters that can be adjusted to obtain the desired functionality while limiting such undesired effects as aliasing and gain ripple. The original application is processing of a time series of raw range data from the proposed Gravity Recovery and Climate Experiment (GRACE), in which microwave phase tracking between two small spacecraft orbiting the Earth would yield the time-tagged raw range data, which would be processed to extract information on the structure of the gravitational field of the Earth. The method is general enough to be applicable in other situations that involve similar signal-processing requirements.
Consider a time series wherein Ri' denotes the raw datum at the jth sampling period. One seeks an FIR filter that can be convolved with the raw data in the time domain over a time window of an odd number, Nf, of sampling periods to obtain low-pass filtering plus decimation by a factor of Nf. The low-pass filtered, decimated time series is to be given by
where the Fn terms are the FIR filter coefficients and Nh = (Nf –1)/2. One also seeks low-pass-filtering and decimating FIR filters and to obtain the first and second derivatives of the data with respect to time (range rate and range acceleration in the original application). The corresponding equations are
Each FIR filter is required to differentiate to the desired order and to exhibit a nearly rectangular low-pass frequency response. To prevent aliasing of out-of-band noise into the desired low-pass band, the low-pass cutoff frequency should be set at or near the applicable Nyquist value, which is half the output data sampling frequency. The well-known window-function approach is used to formulate the FIR filter. The time-domain window function consists of a rectangular time-domain window self-convolved Nc times. The frequency-domain response of such a time-domain window is approximately given by a simple closed-form expression of the form [sin x/x]Nc+1. This class of filters is classified as CRN filters designating N convolutions of a rectangle.
In designing the filter, one must choose values for the nominal cutoff frequency (bandwidth), for Nc, and for the filter length Tf = Nf/fs (where fs is the raw-data sampling frequency). The filter is first constructed in the frequency domain by convolving the desired rectangular low-pass frequency response with the known discrete Fourier transform, [sin x/x]Nc+1, of the selected Nc-self-convolution time-domain window function. The result of this convolution is then discrete-Fourier-transformed to the time domain to obtain the FIR coefficients. The advantage of this class of FIR filter is that the frequency-domain response can be approximately assessed "in advance" on the basis of the simple [sin x/x]Nc+1 function. Further differentiation can be easily applied by multiplying by 2π f in the frequency domain.
The upper part of the Figure 1 depicts the FIR coefficients and frequency response of a range filter designed according to this method for a test case, using Nc = 6. As one would expect, the FIR amplitude vs. time resembles a sin(x)/x function, except that it tapers toward zero in the outer time regions. This taper is caused by the window function. The lower part of Figure 1 shows the predicted frequency response of the range filter. Figure 2 depicts the FIR coefficients of a range-rate (first-derivative) filter for the same test case.
This work was done by J. B. Thomas of Caltech for NASA's Jet Propulsion Laboratory. For further information, access the Technical Support Package (TSP) free on-line at www.nasatech.com/tsp under the Information Sciences category.