cfirpm
Complex and nonlinear-phase equiripple FIR filter design
Syntax
b = cfirpm(n,f,@fresp)
b = cfirpm(n,f,@fresp,w)
b = cfirpm(n,f,a)
b = cfirpm(n,f,a,w)
b = cfirpm(...,'sym')
b = cfirpm(...,'skip_stage2')
b = cfirpm(...,'debug')
b = cfirpm(...,{lgrid})
[b,delta] = cfirpm(...)
[b,delta,opt] = cfirpm(...)
Description
cfirpm allows arbitrary frequency-domain
constraints to be specified for the design of a possibly complex FIR filter. The Chebyshev (or minimax)
filter error is optimized, producing equiripple FIR filter designs.
b = cfirpm(n,f,@ returns
a length fresp)n+1 FIR filter with the best approximation
to the desired frequency response as returned by function fresp,
which is called by its function handle (@fresp). f is
a vector of frequency band edge pairs, specified in the range -1 and 1, where 1 corresponds to the normalized Nyquist frequency.
The frequencies must be in increasing order, and f must
have even length. The frequency bands span f(k) to f(k+1) for k odd;
the intervals f(k+1) to f(k+2) for k odd
are “transition bands” or “don't care”
regions during optimization.
Predefined fresp frequency response functions
are included for a number of common filter designs, as described below.
(See Create Function Handle for more information on how
to create a custom fresp function.) For all of
the predefined frequency response functions, the symmetry option 'sym' defaults
to 'even' if no negative frequencies are contained
in f and d = 0; otherwise 'sym' defaults
to 'none'. (See the 'sym' option
below for details.) For all of the predefined frequency response functions, d specifies
a group-delay offset such that the filter response has a group delay
of n/2+d in units of the sample interval. Negative
values create less delay; positive values create more delay. By default d = 0:
@lowpass,@highpass,@allpass,@bandpass,@bandstopThese functions share a common syntax, exemplified below by
@lowpass.b = cfirpm(n,f,@lowpass,...)andb = cfirpm(n,f,{@lowpass,d},...)design a linear-phase (n/2+ddelay) filter.Note
For
@bandpassfilters, the first element in the frequency vector must be less than or equal to zero and the last element must be greater than or equal to zero.@multibanddesigns a linear-phase frequency response filter with arbitrary band amplitudes.b = cfirpm(n,f,{@multiband,a},...)andb = cfirpm(n,f,{@multiband,a,d},...)specify vectoracontaining the desired amplitudes at the band edges inf. The desired amplitude at frequencies between pairs of pointsf(k)andf(k+1)forkodd is the line segment connecting the points(f(k),a(k))and(f(k+1),a(k+1)).@differentiatordesigns a linear-phase differentiator. For these designs, zero-frequency must be in a transition band, and band weighting is set to be inversely proportional to frequency.b = cfirpm(n,f,{@differentiator,fs},...)andb = cfirpm(n,f,{@differentiator,fs,d},...)specify the sample ratefsused to determine the slope of the differentiator response. If omitted,fsdefaults to 1.@hilbfiltdesigns a linear-phase Hilbert transform filter response. For Hilbert designs, zero-frequency must be in a transition band.b = cfirpm(n,f,@hilbfilt,...)andb = cfirpm(N,F,{@hilbfilt,d},...)design a linear-phase (n/2+ddelay) Hilbert transform filter.@invsincdesigns a linear-phase inverse-sinc filter response.b = cfirpm(n,f,{@invsinc,a},...)andb = cfirpm(n,f,{@invsinc,a,d},...)specify gainafor the sinc function, computed as sinc(a*g), where g contains the optimization grid frequencies normalized to the range [–1,1]. By default,a= 1. The group-delay offset isd, such that the filter response will have a group delay of N/2 +din units of the sample interval, where N is the filter order. Negative values create less delay and positive values create more delay. By default,d= 0.
b = cfirpm(n,f,@ uses
the real, nonnegative weights in vector fresp,w)w to weight
the fit in each frequency band. The length of w is
half the length of f, so there is exactly one weight
per band.
b = cfirpm(n,f,a) is a synonym
for b = cfirpm(n,f,{@multiband,a}).
b = cfirpm(n,f,a,w) applies
an optional set of positive weights, one per band, for use during
optimization. If w is not specified, the weights
are set to unity.
b = cfirpm(..., imposes
a symmetry constraint on the impulse response of the design, where 'sym')'sym' may
be one of the following:
'none'indicates no symmetry constraint. This is the default if any negative band edge frequencies are passed, or iffrespdoes not supply a default.'even'indicates a real and even impulse response. This is the default for highpass, lowpass, allpass, bandpass, bandstop, inverse-sinc, and multiband designs.'odd'indicates a real and odd impulse response. This is the default for Hilbert and differentiator designs.'real'indicates conjugate symmetry for the frequency response
If any 'sym' option other than 'none' is
specified, the band edges should be specified only over positive frequencies;
the negative frequency region is filled in from symmetry. If a 'sym' option
is not specified, the fresp function is
queried for a default setting. Any user-supplied fresp function
should return a valid 'sym' option when
it is passed 'defaults' as the filter order N.
b = cfirpm(...,'skip_stage2') disables
the second-stage optimization algorithm, which executes only when cfirpm determines
that an optimal solution has not been reached by the standard firpm error-exchange.
Disabling this algorithm may increase the speed of computation, but
may incur a reduction in accuracy. By default, the second-stage optimization
is enabled.
b = cfirpm(..., enables
the display of intermediate results during the filter design, where 'debug')'debug' may
be one of 'trace', 'plots', 'both',
or 'off'. By default it is set to 'off'.
b = cfirpm(...,{lgrid}) uses
the integer lgrid to control the density of the
frequency grid, which has roughly 2^nextpow2(lgrid*n) frequency
points. The default value for lgrid is 25.
Note that the {lgrid} argument must be a 1-by-1
cell array.
Any combination of the 'sym', 'skip_stage2', 'debug',
and {lgrid} options may be specified.
[b,delta] = cfirpm(...) returns
the maximum ripple height delta.
[b,delta,opt] = cfirpm(...) returns
a structure opt of optional results computed by cfirpm and
contains the following fields.
Field | Description |
|---|---|
| Frequency grid vector used for the filter design optimization |
| Desired frequency response for each point in |
| Weighting for each point in |
| Actual frequency response for each point in |
| Error at each point in |
| Vector of indices into |
| Vector of extremal frequencies |
User-definable functions may be used, instead of the predefined
frequency response functions for @fresp.
The function is called from within cfirpm using
the following syntax
[dh,dw] = fresp(n,f,gf,w,p1,p2,...)
where:
nis the filter order.fis the vector of frequency band edges that appear monotonically between -1 and 1, where 1 corresponds to the Nyquist frequency.gfis a vector of grid points that have been linearly interpolated over each specified frequency band bycfirpm.gfdetermines the frequency grid at which the response function must be evaluated. This is the same data returned bycfirpmin thefgridfield of theoptstructure.wis a vector of real, positive weights, one per band, used during optimization.wis optional in the call tocfirpm; if not specified, it is set to unity weighting before being passed tofresp.dhanddware the desired complex frequency response and band weight vectors, respectively, evaluated at each frequency in gridgf.p1,p2,..., are optional parameters that may be passed tofresp.
Additionally, a preliminary call is made to fresp to
determine the default symmetry property 'sym'.
This call is made using the syntax:
sym = fresp('defaults',{n,f,[],w,p1,p2,...})
The arguments may be used in determining an appropriate symmetry default as necessary. You can
use the local function lowpass as a template for generating new
frequency response functions. To find the lowpass function, type
edit cfirpm at the command line and search for
lowpass in the cfirpm code. You can copy the
function, modify it, rename it, and save it in your path.
Examples
Equiripple Lowpass Filter
Design a 31-tap linear-phase lowpass filter. Display its magnitude and phase responses.
b = cfirpm(30,[-1 -0.5 -0.4 0.7 0.8 1],@lowpass); fvtool(b,1,'OverlayedAnalysis','phase')
FIR Approximation to Allpass Response
Design a nonlinear-phase allpass FIR filter of order 22 with frequency response given approximately by , where .
n = 22; % Filter order f = [-1 1]; % Frequency band edges w = [1 1]; % Weights for optimization gf = linspace(-1,1,256); % Grid of frequency points d = exp(-1i*pi*gf*n/2 + 1i*pi*pi*sign(gf).*gf.*gf*(4/pi)); % Desired frequency response
Use cfirpm to compute the FIR filter. Plot the actual and approximate magnitude responses in dB and the phase responses in degrees.
b = cfirpm(n,f,'allpass',w,'real'); % Approximation freqz(b,1,256,'whole') subplot(2,1,1) % Overlay response hold on plot(pi*(gf+1),20*log10(abs(fftshift(d))),'r--') subplot(2,1,2) hold on plot(pi*(gf+1),unwrap(angle(fftshift(d)))*180/pi,'r--') legend('Approximation','Desired','Location','SouthWest')
Algorithms
An extended version of the Remez exchange method is implemented
for the complex case. This exchange method obtains the optimal filter
when the equiripple nature of the filter is restricted to have n+2 extremals.
When it does not converge, the algorithm switches to an ascent-descent
algorithm that takes over to finish the convergence to the optimal
solution. See the references for further details.
References
[1] Demjanjov, V. F., and V. N. Malozemov. Introduction to Minimax. New York: John Wiley & Sons, 1974.
[2] Karam, L.J. Design of Complex Digital FIR Filters in the Chebyshev Sense. Ph.D. Thesis, Georgia Institute of Technology, March 1995.
[3] Karam, L.J., and J. H. McClellan. "Complex Chebyshev Approximation for FIR Filter Design." IEEE® Transactions on Circuits and Systems II, March 1995, pp. 207–216.