Complex and nonlinear-phase equiripple FIR filter design

`b = cfirpm(n,f,@`

* fresp*)

b = cfirpm(n,f,@

`fresp`

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(...)

`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`

`@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'`

`'none'`

. (See the `'sym'`

`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`

,`@bandstop`

These functions share a common syntax, exemplified below by the string

`'lowpass'`

.`b = cfirpm(n,f,@lowpass,...)`

and`b = 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 zero.`@multiband`

designs a linear-phase frequency response filter with arbitrary band amplitudes.`b = cfirpm(n,f,{@multiband,a},...)`

and`b = 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))`

.`@differentiator`

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},...)`

and`b = 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 1.`@hilbfilt`

designs a linear-phase Hilbert transform filter response. For Hilbert designs, zero-frequency must be in a transition band.`b = cfirpm(n,f,@hilbfilt,...)`

and`b = cfirpm(N,F,{@hilbfilt,d},...)`

design a linear-phase (`n/2+d`

delay) Hilbert transform filter.`@invsinc`

designs a linear-phase inverse-sinc filter response.`b = cfirpm(n,f,{@invsinc,a},...)`

and`b = 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,@`

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'`

`'none'`

indicates no symmetry constraint. This is the default if any negative band edge frequencies are passed, or ifdoes not supply a default.`fresp`

`'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'`

`fresp`

`fresp`

`'sym'`

`'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'`

`'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'`

`{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:

`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'`

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.

[1] Karam, L.J., and J.H. McClellan. "Complex
Chebyshev Approximation for FIR Filter Design." *IEEE ^{®} Trans.
on Circuits and Systems II,*March 1995. Pgs. 207-216.

[2] Karam, L.J. *Design of Complex
Digital FIR Filters in the Chebyshev Sense, *Ph.D. Thesis,
Georgia Institute of Technology, March 1995.

[3] Demjanjov, V.F., and V.N. Malozemov.* Introduction
to Minimax, *New York: John Wiley & Sons, 1974.

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