In Digital Signal Processing, filters play the most important role which

is to remove unwanted or interfering signal from a system. They can either

be in a form of analog or digital. Four classes of analog filters exist to

process an input signal: low pass, high pass, band pass, and band stop. On

the other hand, digital filters are classified as Finite Impulse Response and

Infinite Impulse Response.

IIR filters are called recursive filters because their impulse response

are composed of decaying exponential while FIR filters are carried out by

convolution. Also, IIR filters have much better frequency response than FIR

filters of the same order. Hence, their phase characteristic is not linear which

can cause a problem to the systems which need phase linearity (Milivojević,

2009).

IIR filters are generally used in applications where some phase

distortion is tolerable. Of the class of IIR filters, elliptic filters are the most

efficient to implement in the sense that for a given set of specifications, an

elliptic filter has a lower order or fewer coefficients than any other IIR filter

type. When compared with FIR filters, elliptic filters are considerably more

efficient (Proakis and Manolakis, 2006).

Margaris (2014) said that elliptic filters, also known as Cauer filters, are

noted for having an equiripple passband magnitude response similar to

Chebyshev Type I filters, and an equiripple stopband magnitude response

similar to Chebyshev Type II filters while, at the same time offer a very

narrow transition band. However, they have the most nonlinear phase

response over their passband. Also, the passband as well as the stopband

ripple amplitude can be adjusted independently and the filter reduces to

Chebyshev I or Chebyshev II filter if one of them tends to zero. Elliptic filters

give the smallest filter order with respect to the other filter types for the

same values of the filter design parameters.

Unlike the Chebyshev filter, an elliptic filter has an extremely sharp

cutoff frequency. This is why an elliptic filter is ideal for filter design cases

where frequencies just entering the stopband of the filter requires great

attenuation and where there are very close signals and exact cut off

frequencies. Also, it suits best for a lowpass filter where errors need to be

minimized on both sides of the cutoff frequency because of the distribution

of the rippling effect across both the passband and stopbands in it (Brewer

and Leach, 2003).

An elliptic filter provides the largest ratio of the passband gain to

stopband gain for a given transition band and requires the smallest transition

band for a given ratio of passband to stopband gain (Green,2014).

One of its defining roles in our society, is its application for biomedical

technology, specifically for ECG signal processing. The ECG signal, which

preliminary represents the condition of the heart, operates at a very low

frequency. In an environment wherein noise is omnipresent, obtaining an

accurate and clean signal would be very difficult and pose multiple

difficulties for the physician. Filters, specifically elliptic filters, work

exceptionally well with filtering out these unnecessary noise without

tampering or destroying the signal at interest (Chavan et al, 2005).

According to Brewer and Leach (2003), “There are several common

uses for an elliptic filter in engineering design. One is separating audio

signals by type. A subwoofer typically handles low frequency signals, and

thus it is important that it not try to process high frequency signals. In this

case, an elliptic filter going into the input of the subwoofer would filter out

these high frequency signals and leave the device with the sound it should

output. Another common use would be in television signal processing. There

is often high frequency noise in a video signal which must be filtered out.

The sharp roll-off of the elliptic filter would be necessary to ensure that the

television receives all of the signal that it should but nothing that would go

over its maximum refresh rate, which could cause hardware problems.”

One of the four classes of analog filters is the band stop filters. Band

stop filters or band reject filters reject one band of frequency and allow all

other frequency band to pass. They consist of two passbands and one

stopband. Band stop filters are classified as wide band band-reject filter and

narrow band band-reject filter.

Wide band band-reject filters consist of a summing amplifier and low

pass and high pass filter sections. In contrast, narrow band band-reject

filters, also called as notch filters, have a very narrow stopband. A notch

filter has three cases, the standard notch, low pass notch, and the high pass

notch. It can be determined with the relationship of the pole frequency to the

zero frequency.

Band stop filters are used in the biomedical instrumentation and

blanking of control tones for telephone lines. It is also used in the rejection of

a single frequency, such as 50 Hz power line frequency hum. Also, they are

important components in communication systems and in the design of

duplexers by incorporating them with couples (Brewer and Leach, 2003).

An elliptic band-stop filter is a band-stop filter displaying the

characteristics of an elliptic filter that is allowing or having ripples in its pass-

band and stop-band.

One note-worthy application of this type of filter is its participation as a

filter option for the investigation on the control system for seismic isolation

of Advanced LIGO. The LIGO or Laser Interferometer Gravitational-Wave

Observatory, is a large scale physics experiment aiming to directly detect

gravitational waves. Filters were employed to reduce the effects of

mechanical resonances of the plant at frequencies which are above the

normal upper unity gain frequency. Operation of LIGO started in 2002 and

ended in 2010 with nothing to show for their efforts. But as of 2015 the

former LIGO detectors are currently being replaced by improved versions of

the devices, one of which has twice the sensitivity of the original (East and

Lantz, 2005).

There are various ways of designing a filter. One of the most commonly

used method is using the reference analog prototype filter. The

specifications and requirements of the desirable filter is used for prototyping

the analog filter design. This is followed by scaling the frequency range of

analog prototype filter into desirable frequency range. And the last step is

mapping the analog filter into its digital counterpart filter.

The analysis of these filters could be done in the time domain or

frequency domain. And a filter system analyzed in frequency domain uses

the transfer function H(s). This transfer function is determined by getting the

Laplace transform of h(t) or the impulse response. The transfer

function ,H(s), is characterized by ratio of polynomials wherein the

numerator has degree of 'm' and the denominator has the degree of 'n'. 'G'

represents the overall gain of the filter system.

It can also be represented by its first order factors or second order

factors. The z's are called zeroes, the roots of the polynomial in numerator,

and p's are called poles, the roots of the polynomial in denominator.

Elliptic filter is the most difficult filter to design because of its

complexity in mathematical functions. Fortunately, many great minds have

worked on these mathematical functions and their work is the basis of the

researchers throughout the whole analysis.

Elliptic filter’s magnitude frequency response is

where Rn is the Chebyshev rational function of order n, based on the elliptic

integral and the Jacobian elliptic functions. Another key concept in designing

elliptic filter is the complete elliptic integral (CEI). This is used in determining

the normalized analog transfer function. It is followed by scaling the

normalized transfer function into its desired frequency range. The impulse

response invariant design method (or impulse invariant transformation) is

used to map the analog filter into digital filter. This is based on creating a

digital filter with an impulse response from the frequency response. Inverse

laplace transform is used to convert the transfer function in frequency

domain into impulse response h(t). The system’s discrete-time impulse

response h(nT) is determined by sampling the impulse response h(t). The

last step is taking the z-transform of sampled response to come up with the

digital filter transfer function H(z).