ParksMcClellan optimal FIR filter design
b = firpm(n,f,a)
b = firpm(n,f,a,w)
b = firpm(n,f,a,'ftype'
)
b = firpm(n,f,a,w,'ftype'
)
b = firpm(...,{lgrid})
[b,err] = firpm(...)[b,err,res] = firpm(...)
b =
firpm(n,f,@
fresp
,w)
b =
firpm(n,f,@
fresp
,w,'ftype'
)
firpm
designs a linearphase FIR filter using the ParksMcClellan algorithm [1]. The ParksMcClellan
algorithm uses the Remez exchange algorithm and Chebyshev approximation theory
to design filters with an optimal fit between the desired and actual
frequency responses. The filters are optimal in the sense that the
maximum error between the desired frequency response and the actual
frequency response is minimized. Filters designed this way exhibit
an equiripple behavior in their frequency
responses and are sometimes called equiripple filters. firpm
exhibits
discontinuities at the head and tail of its impulse response due to
this equiripple nature.
b = firpm(n,f,a)
returns
row vector b
containing the n+1
coefficients
of the order n
FIR filter whose frequencyamplitude
characteristics match those given by vectors f
and a
.
The output filter coefficients (taps) in b
obey
the symmetry relation:
$$b(k)=b(n+2k),\text{}k=1,\mathrm{...},n+1$$
Vectors f
and a
specify
the frequencymagnitude characteristics of the filter:
f
is a vector of pairs of normalized
frequency points, specified in the range between 0 and 1, where 1
corresponds to the Nyquist frequency. The frequencies must be in increasing
order.
a
is a vector containing the desired
amplitudes at the points specified in f.
The desired amplitude at frequencies between pairs of points (f(k), f(k+1)) for k odd is the line segment connecting the points (f(k), a(k)) and (f(k+1), a(k+1)).
The desired amplitude at frequencies between pairs of points (f(k), f(k+1)) for k even is unspecified. The areas between such points are transition or "don't care" regions.
f
and a
must
be the same length. The length must be an even number.
The relationship between the f
and a
vectors
in defining a desired frequency response is shown in the illustration
below.
firpm
always uses an even filter order for
configurations with even symmetry and a nonzero passband at the Nyquist
frequency. This is because for impulse responses exhibiting even symmetry
and odd orders, the frequency response at the Nyquist frequency is
necessarily 0. If you specify an oddvalued n
, firpm
increments
it by 1.
b = firpm(n,f,a,w)
uses
the weights in vector w
to weight the fit in each
frequency band. The length of w
is half the length
of f
and a
, so there is exactly
one weight per band.
Note

b = firpm(n,f,a,
and'ftype'
)
b = firpm(n,f,a,w,
specify
a filter type, where 'ftype'
)'ftype'
is
'hilbert'
, for linearphase filters with odd symmetry (type III and type
IV)
The output coefficients in b
obey the relation b(k) = –b(n + 2 – k), k = 1, ..., n + 1. This class of filters includes the
Hilbert transformer, which has a desired amplitude of 1 across the
entire band.
For example,
h = firpm(30,[0.1 0.9],[1 1],'hilbert');
designs an approximate FIR Hilbert transformer of length 31.
'differentiator'
, for type III and type IV filters, using a special weighting technique
For nonzero amplitude bands, it weights the error by a factor of 1/f so that the error at low frequencies is much smaller than at high frequencies. For FIR differentiators, which have an amplitude characteristic proportional to frequency, these filters minimize the maximum relative error (the maximum of the ratio of the error to the desired amplitude).
b = firpm(...,{lgrid})
uses
the integer lgrid
to control the density of the
frequency grid, which has roughly (lgrid*n)/(2*bw)
frequency
points, where bw
is the fraction of the total frequency
band interval [0,1] covered by f
. Increasing lgrid
often
results in filters that more exactly match an equiripple filter, but
that take longer to compute. The default value of 16
is
the minimum value that should be specified for lgrid
.
Note that the {lgrid}
argument must be a 1by1
cell array.
[b,err] = firpm(...)
returns
the maximum ripple height in err.
returns
a structure [b,err,res] = firpm(...)
res
with the following fields.
 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 
You can also use firpm
to write a function
that defines the desired frequency response. The predefined frequency
response function handle for firpm
is @firpmfrf
,
which designs a linearphase FIR filter.
b
returns
row vector =
firpm(n,f,@
fresp
,w)b
containing the n+1
coefficients
of the order n
FIR filter whose frequencyamplitude
characteristics best approximate the response returned by function
handle @
fresp
. The function
is called from within firpm
with the following
syntax.
[dh,dw] = fresp(n,f,gf,w)
The arguments are similar to those for firpm
:
n
is the filter order.
f
is the vector of normalized frequency
band edges that appear monotonically between 0 and 1, where 1 is the Nyquist frequency.
gf
is a vector of grid points that
have been linearly interpolated over each specified frequency band
by firpm
. gf
determines the
frequency grid at which the response function must be evaluated, and
contains 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 firpm
; 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
.
b
designs
antisymmetric (odd) filters, where =
firpm(n,f,@
fresp
,w,'ftype'
)'ftype'
is
either 'd'
for a differentiator or 'h'
for
a Hilbert transformer. If you do not specify an ftype
,
a 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 n
, f
, w
,
etc., may be used as necessary in determining an appropriate value
for sym
, which firpm
expects
to be either 'even'
or 'odd'
.
If fresp
does not support this calling
syntax, firpm
defaults to even symmetry.
If your filter design fails to converge, it is possible that the filter design is correct. Verify the design by checking the frequency response.
If your filter design fails to converge and the resulting filter design is not correct, attempt one or more of the following:
Increase the filter order
Relax the filter design by reducing the attenuation in the stopbands and/or broadening the transition regions
firpm
designs type I, II, III, and IV linearphase
filters. Type I and type II are the defaults for n
even
and n
odd, respectively, while type III (n
even)
and type IV (n
odd)
are obtained with the 'hilbert'
and 'differentiator'
flags.
The different types of filters
have different symmetries and certain constraints on their frequency
responses (see [5] for more details).
Linear Phase Filter Type  Filter Order  Symmetry of Coefficients  Response H(f), f = 0  Response H(f), f = 1 (Nyquist) 

Type I  Even  even: $$b(k)=b(n+2k),\text{\hspace{1em}}k=1,\mathrm{...},n+1$$  No restriction  No restriction 
Type II  Odd  even: $$b(k)=b(n+2k),\text{\hspace{1em}}k=1,\mathrm{...},n+1$$  No restriction  H(1)=0

Type III  Even  odd: $$b(k)=b(n+2k),\text{\hspace{1em}}k=1,\mathrm{...},n+1$$  H(0) = 0  H(1) = 0 
Type IV  Odd  odd: $$b(k)=b(n+2k),\text{\hspace{1em}}k=1,\mathrm{...},n+1$$  H(0) = 0  No restriction 
[1] Digital Signal Processing Committee of the IEEE Acoustics, Speech, and Signal Processing Society, eds. Programs for Digital Signal Processing. New York: IEEE Press, 1979, algorithm 5.1.
[2] Digital Signal Processing Committee of the IEEE Acoustics, Speech, and Signal Processing Society, eds. Selected Papers in Digital Signal Processing. Vol. II. New York: IEEE Press, 1976.
[3] Parks, Thomas W., and C. Sidney Burrus. Digital Filter Design. New York: John Wiley & Sons, 1987, p. 83.
[4] Rabiner, Lawrence R., James H. McClellan, and Thomas W. Parks. "FIR Digital Filter Design Techniques Using Weighted Chebyshev Approximation." Proceedings of the IEEE^{®}. Vol. 63, Number 4, 1975, pp. 595–610.
[5] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. DiscreteTime Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1999, p. 486.
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