Least-squares linear-phase FIR filter design
b
=
firls(n,f,a)
b = firls(n,f,a,w)
b = firls(n,f,a,'ftype'
)
b = firls(n,f,a,w,'ftype'
)
firls
designs a linear-phase FIR filter
that minimizes the weighted, integrated squared error between an ideal
piecewise linear function and the magnitude response of the filter
over a set of desired frequency bands.
b
returns
row vector =
firls(n,f,a)b
containing the n+1
coefficients
of the order n
FIR filter whose frequency-amplitude
characteristics approximately match those given by vectors f
and a
.
The output filter coefficients, or "taps," in b
obey
the symmetry relation.
$$b(k)=b(n+2-k),\text{}k=1,\mathrm{...},n+1$$
These are type I (n
odd) and type II (n
even)
linear-phase filters. Vectors f
and a
specify
the frequency-amplitude characteristics of the filter:
f
is a vector of pairs of frequency
points, specified in the range between 0 and 1, where
1 corresponds to the Nyquist frequency. The frequencies must be in
increasing order. Duplicate frequency points are allowed and, in fact,
can be used to design a filter exactly the same as those returned
by the fir1
and fir2
functions with a rectangular (rectwin
) window.
a
is a vector containing the desired
amplitude at the points specified in f
.
The desired amplitude function 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 function at frequencies between pairs of points (f(k), f(k+1)) for k even is unspecified. These are transition or "don't care" regions.
f
and a
are
the same length. This length must be an even number.
firls
always uses an even filter order for
configurations with a passband at the Nyquist frequency. This is because
for odd orders, the frequency response at the Nyquist frequency is
necessarily 0. If you specify an odd-valued n
, firls
increments
it by 1.
The figure below illustrates the relationship between the f
and a
vectors
in defining a desired amplitude response.
b = firls(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.
b = firls(n,f,a,
and'ftype'
)
b = firls(n,f,a,w,
specify
a filter type, where 'ftype'
)'ftype'
is:
'hilbert'
for linear-phase filters
with odd symmetry (type III and type IV). The output coefficients
in b
obey the relation
$$b(k)=-b(n+2-k),\text{\hspace{1em}}k=1,\dots ,n+1.$$
This class of filters includes the Hilbert transformer, which has a desired amplitude of 1 across the entire band.
'differentiator'
for
type III and type IV filters, using a special weighting technique.
For nonzero amplitude bands, the integrated squared error has a weight
of (1/f)^{2} 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, the filters minimize the relative integrated squared
error (the integral of the square of the ratio of the error to the
desired amplitude).
One of the following diagnostic messages is displayed when an incorrect argument is used:
F must be even length. F and A must be equal lengths. Requires symmetry to be 'hilbert' or 'differentiator'. Requires one weight per band. Frequencies in F must be nondecreasing. Frequencies in F must be in range [0,1].
A more serious warning message is
Warning: Matrix is close to singular or badly scaled.
This tends to happen when the product of the filter length and transition width grows large. In this case, the filter coefficients b might not represent the desired filter. You can check the filter by looking at its frequency response.
Reference [1] describes the
theoretical approach behind firls
. The function
solves a system of linear equations involving an inner product matrix
of size roughly n/2
using the MATLAB^{®} \
operator.
This function designs type I, II, III, and IV linear-phase filters.
Type I and II are the defaults for n even and odd respectively, while
the 'hilbert'
and 'differentiator'
flags
produce type III (n even) and IV (n odd) filters. The various filter
types have different symmetries and constraints on their frequency
responses (see [2] for details).
Linear Phase Filter Type | Filter Order | Symmetry of Coefficients | Response H(f), f = 0 | Response H(f), f = 1 (Nyquist) |
---|---|---|---|---|
Type I | Even | $$b(k)=b(n+2-k),\text{\hspace{1em}}k=1,\mathrm{...},n+1$$ | No restriction | No restriction |
Type II | Even | $$b(k)=b(n+2-k),\text{\hspace{1em}}k=1,\mathrm{...},n+1$$ | No restriction | H(1) = 0 |
Type III | Odd | $$b(k)=-b(n+2-k),\text{\hspace{1em}}k=1,\mathrm{...},n+1$$ | H(0) = 0 | H(1) = 0 |
Type IV | Odd | $$b(k)=-b(n+2-k),\text{\hspace{1em}}k=1,\mathrm{...},n+1$$ | H(0) = 0 | No restriction |
[1] Parks, Thomas W., and C. Sidney Burrus. Digital Filter Design. New York: John Wiley & Sons, 1987, pp. 54–83.
[2] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1999.
fir1
| fir2
| firpm
| rcosdesign