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# 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,@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. 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, @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,@fresp,w) uses the real, non-negative 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, invsinc, 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' 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.

Field

Description

opt.fgrid

Frequency grid vector used for the filter design optimization

opt.des

Desired frequency response for each point in opt.fgrid

opt.wt

Weighting for each point in opt.fgrid

opt.H

Actual frequency response for each point in opt.fgrid

opt.error

Error at each point in opt.fgrid

opt.iextr

Vector of indices into opt.fgrid for extremal frequencies

opt.fextr

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'. 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.

## Examples

### Example 1

Design a 31-tap, linear-phase, lowpass filter:

```b = cfirpm(30,[-1 -0.5 -0.4 0.7 0.8 1],@lowpass);
fvtool(b,1)
```

Click the Magnitude and Phase Response button.

### Example 2

Design a nonlinear-phase allpass FIR filter:

```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
```

Vector d now contains the complex frequency response that we desire for the FIR filter computed by cfirpm.

Now compute the FIR filter that best approximates this response:

```b = cfirpm(n,f,'allpass',w,'real');  % Approximation
freqz(b,1,256,'whole');
subplot(2,1,1); hold on              % Overlay response
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')
```

expand all

### 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] 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.