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.he 'FILTERING APPLICATIONS NOTE''page %'
.fo '1.4'\ERL'7/28/89'
.nf
.ce 
\s+4\fBENTROPIC RESEARCH LABORATORY, INC.\fR\s-4
.sp .75i
.ce 100
\s+1\fBESPS APPLICATIONS NOTE: Filtering Sampled Data\fP\s-1
.sp .3i
\fIBrian Sublett and David Burton\fP
.sp .3i
Entropic Research Laboratory, Inc.
600 Pennsylvania Ave SE Suite 202
Washington DC 20003
[phone]
.sp .5i
.fi
.ft LR
.RM
.EQ
delim $$
.EN
.sh 1 "INTRODUCTION"
.sp 1
.pp
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.
.(f
\(co Copyright 1987, 1988 Entropic Speech, Inc., 1991 Entropic Research Laboraroty, Inc.; All rights reserved.  
.)f
.pp
Digital filters are characterized by their coefficients.  In ESPS,
these coefficients are usually stored in FILT (5\-\s-1ESPS\s+1) files.  Programs
that create FILT (5\-\s-1ESPS\s+1) files are called filter design programs.  These are 
discussed in Section 2.  Programs
that use FILT (5\-\s-1ESPS\s+1) 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 FILT (5\-\s-1ESPS\s+1) file.
.sp 1
.sh 2 "User Level Programs"
.pp
ESPS includes the 
following user level 
programs that support the filtering of sampled data.  
Each of these programs is described in this document.
.nf
.ta 1.5i

\fInotch_filt\fP (1\-ESPS)	- Design a second order notch filter.
\fIwmse_filt\fP (1\-ESPS)	- Design an FIR filter using the weighted mean square error criterion.
\fIzero_pole\fP (1\-ESPS)	- Convert ASCII zero and pole locations to an ESPS FILT (5\-\s-1ESPS\s+1) file.
\fIimpulse_resp\fP (1\-ESPS)	  - Compute unit impulse responses of filters in FILT (5\-\s-1ESPS\s+1) file.
\fIatofilt\fP (1\-ESPS)		- Convert ASCII files to an ESPS FILT (5\-\s-1ESPS\s+1) file.
\fIiir_filt\fP (1\-ESPS)	- Design a Recursive (IIR) filter.
\fIfilter\fP (1\-ESPS)		- Perform digital filtering on a sampled data file.
\fIfft_filter\fP (1\-ESPS)	- Perform FIR digital filtering using frequency domain convolution.
\fIfiltspec\fP (1\-ESPS)	- Generate spectral record file for specified records in a FILT (5\-\s-1ESPS\s+1) file.

.fi
.sh 2 "Library Functions"
.pp
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.
.nf
.ta 2i

\fIallo_filt_rec\fP (3\-ESPS)	- Allocate memory for a FILT (5\-\s-1ESPS\s+1) file record.
\fIblock_filter\fP (3\-ESPS)	- Filter a data array.
\fIget_filt_rec\fP (3\-ESPS)	- Get the next data record from an ESPS FILT (5\-\s-1ESPS\s+1) file.
\fIinterp_filt\fP (3\-ESPS)	- Perform interpolation filtering on a data array.
\fIprint_filt_rec\fP (3\-ESPS)	- Print an ESPS FILT (5\-\s-1ESPS\s+1) record.
\fIput_filt_rec\fP (3\-ESPS)	- Put an ESPS FILT (5\-\s-1ESPS\s+1) record into a file.
\fIpz_to_co\fP (3-ESPS)		- Convert pole or zero locations to filter coefficients.

.fi
.sh 1 "FILTER DESIGN"
.pp
There are two kinds of filters often 
dicsussed 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 \fImoving-average\fR or \fIall-zero\fR filters.
IIR filters have denominator (and possibly numerator) coefficients,
and are also called \fIrecursive\fR 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 FILT (5\-\s-1ESPS\s+1) files. 
.sp 1
.sh 2 "notch_filt"
.pp
The program 
.i "notch_filt (5\-\s-1ESPS\s+1)" 
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.
.pp
To use 
.i 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.
.sh 3 "notch_filt Example."
.pp
To illustrate the use of 
.i 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:
.nf
.sp 1
.ul 2
    notch_filt -s8000 -n120 -b20 notch120.filt
.bp
.fi
.pp
The response for the resulting filter (in the file 
.i notch120.filt
) is
shown in Figure 2-1.  The commands used to obtain the plot in Figure 2-1
are covered in section 4.
.GP fig2-1.gps 3 6 0 0
.GE
.sp 1
.ce
.b "Figure 2-1: Filter designed by notch_filt."
.sp 1
.sh 2 "wmse_filt"
.sp 1 
.pp
The program 
.i "wmse_filt (5\-\s-1ESPS\s+1)" 
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
.i D(f)
and weighting function
.i W(f),
the program finds the coefficients of the filter
.i H(f)
which minimizes:
.nf
.ce
.if t \{
.EQ
int from W to -W ^{{left |^ D(f) - H(f) right | } sup 2 ~W(f)}~ df
.EN
\}
.if n \{
integral (from -W to W) of |D(f)-H(f)|(squared) * W(f) df
\}
.fi
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
.i D(f),
truncated to the desired number of coefficients.
.pp
The user may specify 
the desired response and weighting function in one of two
modes: \fIprompt\fR and \fIpoints\fR.  
.pp
In "prompt 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.
The user is prompted on
standard output for the sampling frequency, the number of 
frequency bands,
the edges of each band, the desired
magnitude response of each band, and the weighting value of each band. 
The user enters these values on standard input.
The first band always starts at zero and the last band always ends at
half the sampling frequency.
Since the bands are assumed contiguous, the user will only enter 
.i "nbands+1"
edges for
.i nbands
bands.
.pp
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.
Data in both files may be separated by any combination
of spaces or newlines.  The desired response data
and the weighting function data may
be in the same file, in which case the desired response comes
first and the same file name must be given twice on the command line.
.pp
In either mode,
points on the desired response may be negative in order to cause the
resulting filter gain to be negative in that particular region.
No part of the weighting function may be negative, however.
.pp
Besides being easier, the prompt mode is more accurate than the 
points mode if the desired response
and the weighting function are in the right form  
(\fIi.e.\fR, piecewise constant functions).
This is because the design 
algorithm must calculate the convolution of two infinite
time domain functions.  The prompt mode uses an analytical
solution for this convolution, but the points mode only estimates it
using truncated versions of the two functions.  When using the points mode,
the accuracy of the solution can be increased by increasing the number of
points in the desired response and the weighting function.
.sp 1
.sh 3 "Prompt Mode Example."
.pp
To illustrate the "prompt mode" of operation, a simple band pass filter will
be designed.  The initial command line is shown below with the subsequent
computer prompts following in bold face and the user's responses in regular
type.  This example uses uniform weighting.
.sp 1
.ul
    wmse_filt  band_pass1.filt
.sp 1
.pp
.ft B
Please enter the number of coefficients: 
.ft
201
.pp
.ft B
Please enter the number of bands: 
.ft
3
.pp
.ft B
Please enter the sampling rate: 
.ft
8000
.pp
.ft B
Please enter the bandedges.
.ft
0 1000 2000 4000
.pp
.ft B
Please enter the band gains: 
.ft
0 1 0
.pp
.ft B
Please enter the band weights: 
.ft
1 1 1
.pp
.nf
.ft B
wmse_filt: Please enter comments.
.pp
.ft B
wmse_filt: Must be less than 2047 characters and terminated with ^D.
.ft
.pp
This is a bandpass filter with passband from 1 Khz.
.pp
to 2 Khz. The error was weighted uniformly.
.fi
.sp 1
.pp
The response for this filter is shown in Figure 2-2.
.pp
The user can obtain better stop band rejection by allowing the filter
response to roll off more slowly than in the above example.
This can be accomplished with 
.i 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.
.sp 1
.ul
    wmse_filt band_pass2.filt
.sp 1
.pp
.ft B
Please enter the number of coefficients: 
.ft
201
.pp
.ft B
Please enter the number of bands: 
.ft
5
.pp
.ft B
Please enter the sampling rate: 
.ft
8000
.pp
.ft B
Please enter the bandedges.
.ft
0 900 1000 2000 2100 4000
.pp
.ft B
Please enter the band gains: 
.ft
0 0 1 0 0
.pp
.ft B
Please enter the band weights: 
.ft
1 0 1 0 1
.pp
.nf
.ft B
wmse_filt: Please enter comments.
.pp
.ft B
wmse_filt: Must be less than 2047 characters and terminated with ^D.
.ft
.pp
This is a bandpass filter with passband from 1 Khz.
.pp
to 2 Khz. A 100 Hz. transition region was provided 
.pp
on either side of the passband with zero error weighting.
.fi
.sp 1
.pp
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.
.bp
.GP fig2-2.gps 3.5 6 0 0
.GE
.sp 1
.ce
.b "Figure 2-2: Filter designed by wmse_filt."
.GP fig2-3.gps 3.5 6 0 0
.GE
.sp 1
.ce
.b "Figure 2-3: Filter designed by wmse_filt."
.bp
.sh 3 "Points Mode Example."
.pp
In this section, a more complicated filter will be designed using the
"points mode" method.  
Suppose a filter with the characteristics shown in Figure 2-4
is desired.  This filter, with the amplitude response proportional to
frequency over the range of interest, could be used as an FM
demodulation filter.  
The weighting function shown in Figure 2-5 will again allow for a
transition band.
.pp
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 the Masscomp
utilities 
.i "gas (1G)" 
and 
.i "af (1G)."  
The 
.i gas 
command creates an additive
sequence of numbers and the 
.i af 
command performs arithmetic functions
on these sequences of numbers.  For this example, only 
.i gas 
is necessary.
The desired response function shown in Figure 2-4 contains 1025 points and
is created as follows:
.sp 1
.nf
cat  >  fm_des   (This puts the number of points at the beginning of the file.)
1025
^d
gas  -s0  -n820  -c1  -i0.012195121  >>  fm_des
gas  -s10  -n100  -c1  -i-0.1  >>  fm_des
gas  -s0  -n105  -c1  -i0  >>  fm_des
.fi
.sp 1
.pp
The weighting function shown in Figure 2-5 is created
below.
The transition band receives low weighting to improve performance
in the pass band and stop band.
.sp 1
.nf
cat  >  fm_wt
1025
^d
gas  -s1  -n820  -c1  -i0  >>  fm_wt
gas  -s0.1  -n100  -c1  -i0  >>  fm_wt
gas  -s1  -n105  -c1  -i0  >>  fm_wt
.fi
.sp 1
.pp
The filter is designed with the following command:
.sp 1
.nf
.ul
    wmse_filt -c "FM demodulator." -n 401 -d fm_des -w fm_wt fm_demod.filt
.fi
.sp
The comment
string following the \fI-c\fR 
option will be copied into the comment portion of the
output file header.  If the \fI-c\fR 
option had not been specified, the program
would have prompted for comments.
This allows the user to conveniently enter