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Constrained equiripple FIR filter

`B = firceqrip(n,Fo,DEV)`

B = firceqrip(...,'slope',r)

B = firceqrip('minorder',[Fp Fst],DEV)

B = firceqrip(...,'passedge')

B = firceqrip(...,'stopedge')

B = firceqrip(...,'high')

B = firceqrip(...,'min')

B = firceqrip(...,'invsinc',C)

B = firceqrip(...,'invdiric',C)

`B = firceqrip(n,Fo,DEV)`

designs
an order `n`

filter (filter length equal `n`

+
1) lowpass FIR filter with linear phase.

`firceqrip`

produces the same equiripple
lowpass filters that `firpm`

produces
using the Parks-McClellan algorithm. The difference is how you specify
the filter characteristics for the function.

The input argument `Fo`

specifies the frequency
at the upper edge of the passband in normalized frequency (0<`Fo`

<1).
The two-element vector `dev`

specifies the peak or
maximum error allowed in the passband and stopbands. Enter ```
[d1
d2]
```

for `dev`

where `d1`

sets
the passband error and `d2`

sets the stopband error.

`B = firceqrip(...,'slope',r)`

uses
the input keyword '`slope`

' and input argument `r`

to
design a filter with a nonequiripple stopband. `r`

is
specified as a positive constant and determines the slope of the stopband
attenuation in dB/normalized frequency. Greater values of `r`

result
in increased stopband attenuation in dB/normalized frequency.

`B = firceqrip('minorder',[Fp Fst],DEV)`

designs filter with the
minimum number of coefficients required to meet the deviations in
`DEV`

= [*d1*
*d2*] while having a transition width no greater than
`Fst`

– `Fp`

, the difference between the
stopband and passband edge frequencies. You can specify `'mineven'`

or `'minodd'`

instead of `'minorder'`

to design
minimum even order (odd length) or minimum odd order (even length) filters,
respectively. The `'minorder'`

option does not apply when you specify
the `'min'`

(minimum-phase), `'invsinc'`

, or the
`'invdiric'`

options.

`B = firceqrip(...,'passedge')`

designs
a filter where `Fo`

specifies the frequency at which
the passband starts to rolloff.

`B = firceqrip(...,'stopedge')`

designs
a filter where `Fo`

specifies the frequency at which
the stopband begins.

`B = firceqrip(...,'high')`

designs
a high pass FIR filter instead of a lowpass filter.

`B = firceqrip(...,'min')`

designs
a minimum-phase filter.

`B = firceqrip(...,'invsinc',C)`

designs
a lowpass filter whose magnitude response has the shape of an inverse
sinc function. This may be used to compensate for sinc-like responses
in the frequency domain such as the effect of the zero-order hold
in a D/A converter. The amount of compensation in the passband is
controlled by `C`

, which is specified as a scalar
or two-element vector. The elements of `C`

are specified
as follows:

If

`C`

is supplied as a real-valued scalar or the first element of a two-element vector,`firceqrip`

constructs a filter with a magnitude response of 1/sinc(`C`

*`pi`

*`F`

) where`F`

is the normalized frequency.If

`C`

is supplied as a two-element vector, the inverse-sinc shaped magnitude response is raised to the positive power`C(2)`

. If we set`P=C(2)`

,`firceqrip`

constructs a filter with a magnitude response 1/sinc(`C`

*`pi`

*`F`

)^{P}.

If this FIR filter is used with a cascaded integrator-comb (CIC)
filter, setting `C(2)`

equal to the number of stages
compensates for the multiplicative effect of the successive sinc-like
responses of the CIC filters.

**Note**

Since the value of the inverse sinc function becomes unbounded at
`C=1/F`

, the value of `C`

should be greater
the reciprocal of the passband edge frequency. This can be expressed as
`Fo<1/C`

. For users familiar with CIC decimators,
`C`

is equal to 1/2 the product of the differential delay and
decimation factor.

`B = firceqrip(...,'invdiric',C)`

designs a lowpass filter with a passband
that has the shape of an inverse Dirichlet sinc function. The frequency response of the
inverse Dirichlet sinc function is given by $$\{rC{(\frac{\mathrm{sin}(f/2r)}{\mathrm{sin}(Cf/2)}\}}^{p}$$where *C*, *r*, and
*p* are scalars. The input `C`

can be a scalar or
vector containing 2 or 3 elements. If `C`

is a scalar,
`p`

and `r`

equal 1. If `C`

is a
two-element vector, the first element is `C`

and the second element is
`p`

, `[C p]`

. If `C`

is a
three-element vector, the third element is `r`

, ```
[C p
r]
```

.

To introduce a few of the variations on FIR filters that you
design with `firceqrip`

, these five examples cover
both the default syntax `b = firceqrip(n,wo,del)`

and
some of the optional input arguments. For each example, the input
arguments `n`

, `wo`

, and `del`

remain
the same.

`firceqrip`

Design a 30th order FIR filter using `firceqrip`

.

b = firceqrip(30,0.4,[0.05 0.03]); fvtool(b)

Design a minimum order FIR filter using `firceqrip`

. The passband edge and stopband edge frequencies are 0.35$\pi $ and 0.45$\pi $ rad/sample. The allowed deviations are 0.02 and 1e-4.

`b = firceqrip('minorder',[0.35 0.45],[0.02 1e-4]); fvtool(b)`

Design a 30th order FIR filter with the `stopedge`

keyword to define the response at the edge of the filter stopband.

`b = firceqrip(30,0.4,[0.05 0.03],'stopedge'); fvtool(b)`

Design a 30th order FIR filter with the `slope`

keyword and r = 20.

b = firceqrip(30,0.4,[0.05 0.03],'slope',20,'stopedge'); fvtool(b)

Design a 30th order FIR filter defining the stopband and specifying that the resulting filter is minimum phase with the `min`

keyword.

b = firceqrip(30,0.4,[0.05 0.03],'stopedge','min'); fvtool(b)

Comparing this filter to the filter in Figure 1. The cutoff frequency `wo = 0.4`

now applies to the edge of the stopband rather than the point at which the frequency response magnitude is 0.5.

Viewing the zero-pole plot shown here reveals this is a minimum phase FIR filter - the zeros lie on or inside the unit circle, z = 1

`fvtool(b,'polezero')`

Design a 30th order FIR filter with the `invsinc`

keyword to shape the filter passband with an inverse sinc function.

`b = firceqrip(30,0.4,[0.05 0.03],'invsinc',[2 1.5]); fvtool(b)`

The inverse sinc function being applied is defined as 1/sinc(2*w)^1.5.

Design two order 30 constrained equiripple FIR filters with inverse-Dirichlet-sinc-shaped passbands. The cutoff frequency in both designs is pi/4 radians/sample. Set `C=1`

in one design `C=2`

in the second design. The maximum passband and stopband ripple is 0.05. Set `p=1`

in one design and `p=2`

in the second design.

Design the filters.

b1 = firceqrip(30,0.25,[0.05 0.05],'invdiric',[1 1]); b2 = firceqrip(30,0.25,[0.05 0.05],'invdiric',[2 2]);

Obtain the filter frequency responses using `freqz`

. Plot the magnitude responses.

[h1,~] = freqz(b1,1); [h2,w] = freqz(b2,1); plot(w,abs(h1)); hold on; plot(w,abs(h2),'r'); axis([0 pi 0 1.5]); xlabel('Radians/sample'); ylabel('Magnitude'); legend('C=1 p=1','C=2 p=2');

Inspect the stopband ripple in the design with `C=1`

and `p=1`

. The constrained design sets the maximum ripple to be 0.05. Zoom in on the stopband from the cutoff frequency of pi/4 radians/sample to 3pi/4 radians/sample.

figure; plot(w,abs(h1)); set(gca,'xlim',[pi/4 3*pi/4]); grid on;

`diric`

| `fdesign.decimator`

| `firhalfband`

| `firnyquist`

| `firgr`

| `ifir`

| `iirgrpdelay`

| `iirlpnorm`

| `iirlpnormc`

| `fircls`

| `firls`

| `firpm`

| `sinc`