Analog and Digital Filters 583
R.6.77 The MATLAB function [numL, demL] = lp2lp (numN, denN, wc) is used to transform
a normalized LPF prototype given by the transfer function H(s) = numN(s)/denN(s)
into a denormalized LPF with a numerator numL, denominator denL, and cutoff fre-
quency wc. This function fi le returns the coeffi cients of the polynomials numL and
denL as a row vector arranged in descending powers of s.
R.6.78 Similarly, the MATLAB function [numH, denH] = lp2hp (numN, denN, wc) is used to
transform a low-pass normalized fi lter into a denormalized HPF with the pass
band given by wc.
R.6.79 The function fi le [numbp, denbp] = lp2bp (numN, denN, wo, BW) transforms an LP nor-
malized fi lter into a BPF or BSF (lp2bs), where wo is its center frequency with BW.
R.6.80 Some practical observations of the resulting fi lter complexity and their associated
costs are summarized as follows:
a. The complexity of a fi lter is generally given by its order n. The higher the order,
the greater the complexity and cost in term of elements.
b. Generally, for a particular set of fi lter specs, the elliptic fi lter implementation is
less complex than the equivalent Chebyshev (type 1 or 2).
c. The Chebyshev (type 1 or 2) cost is considerably less than its Butterworth equiv-
alent counterpart.
d. An estimate of the fi lter complexity is given by the following example: an elliptic
fi lter of order 6 requires an equivalent Chebyshev fi lter (type 1 or 2) of order 10
and a Butterworth equivalent implementation of order 29.R.6.81 Recall that digital fi lters as well as analog fi lters have as their primary objective
to fi lter, meaning to pass or reject, certain band of frequencies. Analog fi lters are
expressed as functions of w (radians per second), whereas digital fi lters are func-
tions of W = w * T called the scaled or digital frequency, where T is the sampling
period (sometimes indicated by Ts).
R.6.82 Recall that by using the digital frequency W (some textbooks use Ω), the frequency
response of the digital fi lter is always periodic with period 2π.
R.6.83 Digital fi lters can be classifi ed based on their impulse response length as follows:
a. FIR
b. IIR
R.6.84 An FIR fi lter returns an impulse response with an amplitude that decays to zero in
a fi nite number of steps (samples) or duration.
R.6.85 An IIR fi lter returns an impulse response that has, in theory, an infi nite duration or
continues forever because of its recursive nature (feedback).
This defi nition is not accurate because the impulse response of any system
including IIR fi lters in real life eventually decay to zero in a fi nite time, which may
be long.
R.6.86 IIR fi lters are used to closely approximate the gain and phase response of analog
fi lters and the specs are usually given in terms of analog variables such as BW,
minimum pass-band ripple (Rp ), and maximum stop-band ripple (Rs). These types
of fi lters can substitute analog fi lters.
R.6.87 The approach followed in the design of IIR fi lters are as follows:
a. Design an analog prototype with the desired frequency response (transfer func-
tion) given by its transfer function H(s).