filter.ascii

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ENTROPIC RESEARCH LABORATORY







ESPS APPLICATIONS NOTE: Filtering Sampled Data





Brian Sublett and David Burton

Revised by: Derek Lin





Entropic Research Laboratory, Inc.

600 Pennsylvania Ave., S.E. Suite 202

Washington, DC 20003

[phone]







1 .  INTRODUCTION





	The Entropic Signal Processing System (ESPS) contains a variety of 
programs used in the digital filtering of sampled data. This document 
introduces the reader to programs that deal with filtering and 
illustrates their use through some common examples.



	Digital filters are characterized by their coefficients.   In ESPS, 
these coefficients are usually stored in FEAFILT (5-ESPS) files. Programs 
that create FEAFILT (5-ESPS) files are called filter design programs.   
These are discussed in Section 2.   Programs that use FEAFILT (5-ESPS) 
files to perform the digital filtering operation are called filtering 
programs, and Section 3 describes these programs. Finally, Section 4 
discusses programs used to calculate and plot the frequency response or 
transfer function of the coefficients in a FEAFILT (5-ESPS) file.





1 .1 .  User Level Programs



	ESPS includes the following user level programs that support the 
filtering of sampled data. Each of these programs is described in this 
document.



notch_filt (1-ESPS) - Design a second order notch filter.

wmse_filt (1-ESPS)- Design a FIR filter with minimum weighted MS error.

win_filt (1-ESPS) - Design a FIR filter by the Kaiser windowing method.

pkmc_filt(1-ESPS) - Design a FIR filter by Parks-McClellan method.

cb_filt (1-ESPS) - Design a FIR filter by constraint-based method.

iir_filt (1-ESPS)- Design a Recursive (IIR) filter by various methods.

xpz (1-ESPS)   - Interactive Pole/Zero filter design.

zero_pole (1-ESPS) - Convert ASCII zero and pole locations to an ESPS 
FEAFILT (5-ESPS) file.

impulse_resp (1-ESPS) - Compute unit impulse responses of filters in 
FEAFILT (5-ESPS) file.

atofilt (1-ESPS)- Convert ASCII files to an ESPS FEAFILT (5-ESPS) file.

filter2 (1-ESPS)- Perform digital filtering on a sampled data file.

fft_filter (1-ESPS)- Perform FIR digital filtering using frequency 
domain convolution.

sfconvert (1-ESPS)- Converts sampling frequency in a sampled data file.

filtspec (1-ESPS)- Generate spectral record file for specified records 
in a FEAFILT (5-ESPS) file.





1 .2 .  Library Functions



	The following ESPS library functions support the use of filters. 
Library functions are not discussed in this document. The user should 
refer to the appropriate Section 3 manual page for further help with 
filtering library functions.



init_feafilt_hd (3-ESPS)- Initialize a FEA file header for subtype 
FEAFILE.

allo_feafilt_rec (3-ESPS)- Allocate memory for a FEAFILT (5-ESPS) file 
record.

block_filter2 (3-ESPS)- Filter data from a file or memory

get_feafilt_rec (3-ESPS)- Get the next data record from an ESPS FEAFILT 
(5-ESPS) file.

interp_filt (3-ESPS)- Perform interpolation filtering on a data array.

print_feafilt_rec (3-ESPS)- Print an ESPS FEAFILT (5-ESPS) record.

put_feafilt_rec (3-ESPS)- Put an ESPS FEAFILT (5-ESPS) record into a 
file.

pz_to_co (3-ESPS)   - Convert pole or zero locations to filter 
coefficients.





2 .  FILTER DESIGN



	There are two kinds of filters often discussed in the literature 
and text books:   finite impulse response (FIR) and infinite impulse 
response (IIR). FIR filters have only numerator coefficients, and are 
sometimes called moving-average or all-zero filters. IIR filters have 
denominator (and possibly numerator) coefficients, and are also called 
recursive filters.   ESPS programs provide design support for both types 
of filters, and this section describes those programs that calculate 
filter coefficients and store them in FEAFILT (5-ESPS) files.



	These programs use ESPS parameter files as means of user interface. 
Typically such parameter files contain parameters such as sampling rate, 
desired band edges, values, and weighting factors, etc... In IIR filter 
design, four types of filter designs are available: lowpass, highpass, 
bandpass, and bandstop. Their bandedges are controlled by only 4 
parameters. Eparam and xeparam(1-ESPS) may be used to invoke the IIR 
design program to prompt user for parameter values. 



	In FIR filter design, however, N-band filter design is possible, 
and the number of parameters for bandedges, desired band values vary with 
N.   Although eparam and xeparam(1-ESPS) can be invoked to prompt for 
parameter values, the default parameter file templates under the 
directory $ESPS_BASE/lib/params are pre-written and fixed to parameters 
applicable for either 2 or 3 band FIR filters.   Therefore, customize 
these template parameter files to suit your needs. The cover shell script 
xfir_filt in the directory $ESPS_BASE/bin is a good example for 
interactive filter design tool based on ESPS parameter file processing 
and XIG interface. (See ESPS Application Notes: X Interface Generation 
in ESPS and xwaves+)





2 .1 .  notch_filt



	The program notch_filt (1-ESPS) designs a second order IIR filter 
that removes or "notches out" a narrow region in the frequency domain 
and passes all other frequencies. This is done by placing a zero on the 
unit circle in the Z-plane at a point corresponding to this notch 
frequency, and then placing a pole very close to it and just inside the 
unit circle.   The complex conjugates of this pole and zero are also 
included to guarantee a filter with real coefficients. The frequency 
response will therefore be zero when evaluated at the zero frequencies, 
but nearly everywhere else on the unit circle, the effects of the poles 
and zeroes will cancel.



	To use notch_filt, the user must specify the sampling frequency, 
the notch frequency, the notch bandwidth, and, of course, the output file 
name. The notch bandwidth is defined as the distance (in the frequency 
domain) between the -6 dB. points on either side of the notch. The three 
critical frequencies may be specified on the command line or in a 
parameter file. If all three are specified on the command line, no 
parameter file is necessary.



2 .1 .1 .  notch_filt Example.



	To illustrate the use of notch_filt, a filter will be designed to 
remove 120 Hz noise from 8 kHz. sampled data. The notch bandwidth will 
be 20 Hz. The command for this is:



           notch_filt -s8000 -n120 -b20 notch120.filt



	The response for the resulting filter (in the file notch120.filt) 
is shown in Figure 2-1. The commands used to obtain the plot in Figure 
2-1 are covered in Section 4.









2 .2 .  wmse_filt





          The program wmse_filt (1-ESPS) designs an FIR filter of specified 
length using the weighted mean-square-error criterion.   The user 
specifies a desired frequency response and a weighting function and the 
program finds the set of filter coefficients whose response minimizes 
the mean square error with respect to the desired response, when weighted 
by the weighting function.   In other words, for desired response D(f) 
and weighting function W(f), the program finds the coefficients of the 
filter H(f) which minimizes:

 If uniform weighting is used (i.e. W(f) = 1), the solution to the above 
equation will simply be the Inverse Discrete Fourier Transform of D(f), 
truncated to the desired number of coefficients.



	The user may specify the desired response and weighting function in 
one of two modes: band and points.



	In "band mode" the program assumes that the desired response and 
the weighting function are constant throughout a set of adjacent 
frequency bands from zero to half the sampling frequency. Bands are 
specified by parameters in a parameter file.



	In "points mode" the program allows the desired response and 
weighting function to be arbitrarily shaped and to be specified at a 
number of uniformly spaced points from zero to half the sampling 
frequency.   The number of points must be one plus a power of two. The 
point values of the desired response and the weighting function are 
specified in two ASCII files. The first number in each of these files is 
an integer telling how many point values are in the file, and the point 
values follow.



	Among ESPS FIR design programs, this is the fastest and the most 
numerically stable program. However, due to the Gibbs phenomenon, 
overshoot in the passband near the transition bandedges exists.





2 .2 .1 .  Band Mode Example



	To illustrate the "band mode" of operation, a simple bandpass filter 
will be designed. The following command reads design parameters from a 
parameter file.



           wmse_filt -P param1.wmse band_pass1.filt



The parameter file is shown below. In this example, it creates a bandpass 
filter with passband edges at 1000 and 2000 Hz.



Content of the parameter file param1.wmse:



	float samp_freq = 8000: "Sampling frequency:";

	int filt_length = 201: "filter length - odd:";

	int nbands = 3: "Number of bands:";

	float band_edge1 = 0;

	float band_edge2 = 1000;

	float band_edge3 = 2000;

	float band_edge4 = 4000;

	float band1_des = 0: "desired value for band1:";

	float band2_des = 1 "desired value for band2:";

	float band3_des = 0 "desired value for band3:";

	#uniform weighting

	float band1_wt = 1: "weighting value for band1:";

	float band2_wt = 1: "weighting value for band2:";

	float band3_wt = 1: "weighting value for band3:";





The response for this filter is shown in Figure 2-2.



	Incidentally, it may be easier to use eparam or xeparam(1-ESPS) with 
wmse_filt(1-ESPS) to prompt for design parameters. By default, the 
parameter file $ESPS_BASE/lib/params/Pwmse_filt contains design 
parameters for a 3-band filter. You may use this as a template and 
construct a parameter file for designing a N-band filter. See next 
example for constructing a 5-band filter.



	You can obtain better stop band rejection by allowing the filter 
response to rolloff more slowly than in the above example. This can be 
accomplished with wmse_filt by specifying transition bands on either side 
of the passband, where the error is weighted with a factor of zero. This 
allows the filter design algorithm to concentrate the zeroes in the 
regions of interest since the error in the transition region can grow 
arbitrarily large without affecting the total weighted error. Note that 
the desired response function is the same in this example as in the one 
above, even though it is specified by using more bands.



           wmse_filt -P param2.wmse band_pass1.filt



The parameter file is shown below. In this example, it creates a bandpass 
filter with passband edges at 1000 and 2000 Hz, a 100 Hz transition region 
is provided on either side of the passband with zero error weighting.



Content of the parameter file param2.wmse:



	float samp_freq = 8000: "Sampling frequency:";

	int filt_length = 201: "filter length - odd:";

	int nbands = 5: "Number of bands:";

	float band_edge1 = 0;

	float band_edge2 = 900;

	float band_edge3 = 1000;

	float band_edge4 = 2000;

	float band_edge5 = 2100;

	float band_edge6 = 4000;

	float band1_des = 0: "desired value for band1:";

	float band2_des = 0 "desired value for band2:";

	float band3_des = 1 "desired value for band3:";

	float band4_des = 0 "desired value for band4:";

	float band5_des = 0 "desired value for band5:";

	float band1_wt = 1: "weighting value for band1:";

	float band2_wt = 0: "weighting value for band2:";

	float band3_wt = 1: "weighting value for band3:";

	float band4_wt = 0: "weighting value for band4";

	float band5_wt = 1: "weighting value for band5:";



          The response for this filter is shown in Figure 2-3.   Note 
that the passband ripple has been reduced and the stop band attenuation 
has been increased by about 30 dB.









2 .2 .2 .  Points Mode Example.



	In this section, a more complicated filter will be designed using 
the  "points  mode" method. Suppose a filter with the amplitude response 
described as follows: over 0 to 8200 Hz, the amplitude response is 
proportional to frequency with a slope of +(1/820); over 8201 to 9200 
Hz, with the slope of -(1/100); over 9201 to 10000Hz, the amplitude 
response is zero. And suppose the error weight over 0 to 8200 Hz is 1; 
over 8201 to 9200 Hz, 0; over 9201 to 10000 Hz, 1. This is a 
differentiator and could be used as an FM demodulation filter. 



	Before the filter can be designed, ASCII files must be created 
containing the desired response and the weighting function. For best 
results, the number of points in these functions should be large. These 
ASCII files can be created easily by using a simple awk (1) command ( or 
Masscomp utilities gas (1G) and af (1G).   ).   



       cat  >  fm_des   (This puts the number of points at the beginning 
of the file.)

       1025

       ^d

       awk '{x=0;print x;for(i=1;i<820;i++) print x+=0.012195121}' - >> 
fm_des

       <ret>

       ^d

       awk '{x=10;print x;for(i=1;i<100;i++) print x+=-0.1}' - >> fm_des

       <ret>

       ^d

       awk '{for(i=0;i<105;i++) print x=0}' - >>fm_des

       <ret>

       ^d





	The weighting function is created below. The transition band 
receives low weighting to improve performance in the pass band and stop 
band.



       cat  >  fm_wt

       1025

       ^d

       awk '{for(i=0;i<820;i++) print x=1}' - >> fm_wt

       <ret>

       ^d

       awk '{for(i=0;i<820;i++) print x=0.1}' - >> fm_wt

       <ret>

       ^d

       awk '{for(i=0;i<820;i++) print x=1}' - >> fm_wt

       <ret>

       ^d





The filter is designed with the following command:



           wmse_filt -c "FM demodulator." -n 401 -d fm_des -w fm_wt 
fm_demod.filt



The comment string following the -c option will be copied into the 
comment portion of the output file header.   The response for this filter 
is shown on a linear scale in Figure 2-6.







2 .3 .  win_filt



	The program win_filt(1-ESPS) allows you to design a FIR filter by 
the Kasier windowing method. This method windows the ideal frequency 
response with the Kaiser window which controls the sharpness of the 
design filter. Windowing technique in FIR design often requires long 
filter length to meet the requirement of a sharp filter.



	The ideal frequency response of the filter is described by the 
parameters in a parameter files. To illustrate its usage, suppose a 
highpass filter with the cutoff frequency at 1500 Hz, sampled at 8000 
Hz. Since the parameter template in $ESPS_BASE/lib/params is in a form 
of 2-band filter, we'll use eparam(1-ESPS) to invoke win_filt.



	eparam win_filt high.filt



	Sampling frequency: [8000] :

	Number of bands: [2] :

	ransition bandwidth: [100] :

	rejection in dB from one band to another: [20] :

	band_edge1 [0] :

	band_edge2 [1000] :1500

	band_edge3 [4000] :

	desired value for band1: [1] :0

	desired value for band2: [0] :1



We required the transition bandwidth of 100 Hz and stopband response to 
be at least 20 dB down from passband. The filter response is shown below 
in Figure 2.5







2 .4 .  pkmc_filt



	The design method is based on the Parks-McClellan algorithm. The 
resulting filter minimizes the maximum weighted approximation error. In 
other words, for desired response D(f) and error weighting function W(f), 
the program finds the coefficients of the filter H(f) which minimizes:

One of the four standard FIR types is produced.   For a filter with 
multiple stopbands and passbands, the Type 1 (symmetric and odd length) 
or Type  2  (symmetric and even length) filter is designed. For a 
differentiator or a Hilbert Transformer, the Type 3 (anti-symmetric and 
odd length) or the Type 4 (anti-symmetric and even length) filter is 
designed.



	The Parks-McClellan algorithm is known to run to numerically 
difficulty, especially if the filter length is over 100. However, since 
the algorithm can distribute error equally among passband and stopband 
by equirippling to avoid excessive overshoot near transition bandedges, 
filter designed by this method is preferred for certain application.



	Suppose a differentiator in Section 2.2.2 - over the 0 to 8200 Hz 
region, the amplitude response is proportional to frequency with a slope 
of +(1/820); over 8201 to 9200 Hz, with the slope of -(1/100); over 9201 
to 10000Hz, the amplitude response is zero. And suppose the error weight 
over 0 to 8200 Hz is 1; over 8201 to 9200 Hz, 0; over 9201 to 10000 Hz, 
1. A parameter file, param.diff based on the template, $ESPS_BASE/lib/
params/Ppkmc_filt, follows: