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# kaiserord

Kaiser window FIR filter design estimation parameters

## Syntax

```[n,Wn,beta,ftype] = kaiserord(f,a,dev) [n,Wn,beta,ftype] = kaiserord(f,a,dev,fs) c = kaiserord(f,a,dev,fs,'cell') ```

## Description

`kaiserord` returns a filter order `n` and `beta` parameter to specify a Kaiser window for use with the `fir1` function. Given a set of specifications in the frequency domain, `kaiserord` estimates the minimum FIR filter order that will approximately meet the specifications. `kaiserord` converts the given filter specifications into passband and stopband ripples and converts cutoff frequencies into the form needed for windowed FIR filter design.

`[n,Wn,beta,ftype] = kaiserord(f,a,dev)` finds the approximate order `n`, normalized frequency band edges `Wn`, and weights that meet input specifications `f`, `a`, and `dev`. `f` is a vector of band edges and `a` is a vector specifying the desired amplitude on the bands defined by `f`. The length of `f` is twice the length of `a`, minus 2. Together, `f` and `a` define a desired piecewise constant response function. `dev` is a vector the same size as `a` that specifies the maximum allowable error or deviation between the frequency response of the output filter and its desired amplitude, for each band. The entries in `dev` specify the passband ripple and the stopband attenuation. You specify each entry in `dev` as a positive number, representing absolute filter gain (not in decibels).

### Note

If, in the vector `dev`, you specify unequal deviations across bands, the minimum specified deviation is used, since the Kaiser window method is constrained to produce filters with minimum deviation in all of the bands.

`fir1` can use the resulting order `n`, frequency vector `Wn`, multiband magnitude type `ftype`, and the Kaiser window parameter `beta`. The `ftype` output argument is intended for use with `fir1`; it is equal to `'high'` for a highpass filter and `'stop'` for a bandstop filter. For multiband filters, it can be equal to `'DC-0'` when the first band is a stopband (starting at f = 0) or `'DC-1'` when the first band is a passband.

To design an FIR filter `b` that approximately meets the specifications given by `kaiser` parameters `f`, `a`, and `dev`, use the following command.

```b = fir1(n,Wn,kaiser(n+1,beta),ftype,'noscale') ```

`[n,Wn,beta,ftype] = kaiserord(f,a,dev,fs)` uses a sampling frequency `fs` in Hz. If you don't specify the argument `fs`, or if you specify it as the empty vector `[]`, it defaults to 2 Hz, and the Nyquist frequency is 1 Hz. You can use this syntax to specify band edges scaled to a particular application's sampling frequency. The frequency band edges in `f` must be from 0 to `fs`/2.

`c = kaiserord(f,a,dev,fs,'cell')` is a cell-array whose elements are the parameters to `fir1`.

### Note

In some cases, `kaiserord` underestimates or overestimates the order `n`. If the filter does not meet the specifications, try a higher order such as `n+1`, `n+2`, and so on, or a try lower order.

Results are inaccurate if the cutoff frequencies are near 0 or the Nyquist frequency, or if `dev` is large (greater than 10%).

## Examples

collapse all

Design a lowpass filter with passband defined from 0 to 1 kHz and stopband defined from 1500 Hz to 4 kHz. Specify a passband ripple of 5% and a stopband attenuation of 40 dB.

```fsamp = 8000; fcuts = [1000 1500]; mags = [1 0]; devs = [0.05 0.01]; [n,Wn,beta,ftype] = kaiserord(fcuts,mags,devs,fsamp); hh = fir1(n,Wn,ftype,kaiser(n+1,beta),'noscale'); freqz(hh)```

Design an odd-length bandpass filter. Note that odd length means even order, so the input to `fir1` must be an even integer.

```fsamp = 8000; fcuts = [1000 1300 2210 2410]; mags = [0 1 0]; devs = [0.01 0.05 0.01]; [n,Wn,beta,ftype] = kaiserord(fcuts,mags,devs,fsamp); n = n + rem(n,2); hh = fir1(n,Wn,ftype,kaiser(n+1,beta),'noscale'); [H,f] = freqz(hh,1,1024,fsamp); plot(f,abs(H)) grid```

Design a lowpass filter with a passband cutoff of 1500 Hz, a stopband cutoff of 2000 Hz, a passband ripple of 0.01, a stopband ripple of 0.1, and a sample rate of 8000 Hz. Design an equivalent filter using the `'cell'` option.

```[n,Wn,beta,ftype] = kaiserord([1500 2000],[1 0],... [0.01 0.1],8000); b = fir1(n,Wn,ftype,kaiser(n+1,beta),'noscale'); c = kaiserord([1500 2000],[1 0],[0.01 0.1],8000,'cell'); bcell = fir1(c{:}); hfvt = fvtool(b,1,bcell,1); legend(hfvt,'b','bcell')```

## Tips

Be careful to distinguish between the meanings of filter length and filter order. The filter length is the number of impulse response samples in the FIR filter. Generally, the impulse response is indexed from n = 0 to n = L–1 where L is the filter length. The filter order is the highest power in a z-transform representation of the filter. For an FIR transfer function, this representation is a polynomial in z, where the highest power is zL–1 and the lowest power is z0. The filter order is one less than the length (L–1) and is also equal to the number of zeros of the z polynomial.

## Algorithms

`kaiserord` uses empirically derived formulas for estimating the orders of lowpass filters, as well as differentiators and Hilbert transformers. Estimates for multiband filters (such as bandpass filters) are derived from the lowpass design formulas.

The design formulas that underlie the Kaiser window and its application to FIR filter design are

`$\beta =\left\{\begin{array}{ll}0.1102\left(\alpha -8.7\right),\hfill & \alpha >50\hfill \\ 0.5842{\left(\alpha -21\right)}^{0.4}+0.07886\left(\alpha -21\right),\hfill & 21\le \alpha \le 50\hfill \\ 0,\hfill & \alpha <21\hfill \end{array}$`

where α = –20log10δ is the stopband attenuation expressed in decibels (recall that δp = δs is required).

The design formula is

`$n=\frac{\alpha -7.95}{2.285\left(\Delta \omega \right)}$`

where n is the filter order and Δω is the width of the smallest transition region.

## References

[1] Kaiser, James F. “Nonrecursive Digital Filter Design Using the I0-sinh Window Function.” Proceedings of the 1974 IEEE International Symposium on Circuits and Systems. 1974, pp. 20–23.

[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, pp. 123–126.

[3] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1999.