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cfirpm() - Signal Processing

```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,@fresp) returns
a length 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.
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
the string 'lowpass'.b = cfirpm(n,f,@lowpass,...) andb = cfirpm(n,f,{@lowpass,d},...) design
a linear-phase (n/2+d delay) filter.Note:

For @bandpass filters, 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 [email protected] designs a linear-phase
frequency response filter with arbitrary band amplitudes.b = cfirpm(n,f,{@multiband,a},...) andb = cfirpm(n,f,{@multiband,a,d},...) specify
vector a containing the desired amplitudes at the
band edges in f. The desired amplitude at frequencies
between pairs of points f(k) and f(k+1) for k odd
is the line segment connecting the points (f(k),a(k)) and (f(k+1),a(k+1))[email protected] designs 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 rate fs used to determine the slope
of the differentiator response. If omitted, fs defaults
to [email protected] designs 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+d delay) Hilbert
transform [email protected] designs a linear-phase
inverse-sinc filter response.b = cfirpm(n,f,{@invsinc,a},...) andb = cfirpm(n,f,{@invsinc,a,d},...) specify
gain a for 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 is d,
such that the filter response will have a group delay of N/2 + d in 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,@fresp,w) uses
the real, nonnegative weights in vector 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(...,'sym') imposes
a symmetry constraint on the impulse response of the design, where '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 if fresp does 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 responseIf 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' string when
it is passed the string '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(...,'debug') enables
the display of intermediate results during the filter design, where '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.FieldDescription
opt.fgridFrequency grid vector used for the filter design optimization
opt.desDesired frequency response for each point in opt.fgrid
opt.wtWeighting for each point in opt.fgrid
opt.HActual frequency response for each point in opt.fgrid
opt.errorError at each point in opt.fgrid
opt.iextrVector of indices into opt.fgrid for
extremal frequencies
opt.fextrVector 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:n is the filter order.f is the vector of frequency band
edges that appear monotonically between -1 and 1,
where 1 corresponds to the Nyquist frequency.gf is a vector of grid points that
have been linearly interpolated over each specified frequency band
by cfirpm. gf determines the
frequency grid at which the response function must be evaluated. This
is the same data returned by cfirpm in the fgrid field
of the opt structure.w is a vector of real, positive
weights, one per band, used during optimization. w is
optional in the call to cfirpm; if not specified,
it is set to unity weighting before being passed to fresp.dh and dw are
the desired complex frequency response and band weight vectors, respectively,
evaluated at each frequency in grid gf.p1, p2, ...,
are optional parameters that may be passed to fresp.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. The function private/lowpass.m may
be useful as a template for generating new frequency response functions.```

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(...)`

Example

`[dh,dw] = fresp(n,f,gf,w,p1,p2,...)`