Introduction: A network which freely passes desired band of frequencies while almost suppress other band of frequencies is called filter. This type of network was first examined by G.A Campbell and O.J Zobel of bell telephone laboratories. Filters are networks that process signals in a frequency-dependent manner. The basic concept of a filter can be explained by examining the frequency dependent nature of the impedance of capacitors and inductors. In filters, attenuation changes suddenly as the frequency is varied. Thus, filters have ability to discriminate between signals which differ in frequency.
Filters are widely employed in signal processing and communication systems in applications such as channel equalization, noise reduction, radar, audio processing, video processing, biomedical signal processing, and analysis of economic and financial data. For example in a radio receiver band-pass filters, or tuners, are used to extract the signals from a radio channel. In an audio graphic equalizer the input signal is filtered into a number of sub-band signals and the gain for each sub-band can be varied manually with a set of controls to change the perceived audio sensation. In a Dolby system pre-filtering and post filtering are used to minimize the effect of noise. In hi-fi audio a compensating filter may be included in the preamplifier to compensate for the non-ideal frequency-response characteristics of the speakers. Filters are also used to create perceptual audio-visual effects for music, films and in broadcast studios.
The primary functions of filters are one of the followings: To confine a signal into a prescribed frequency band as in low-pass, high-pass, and band-pass filters. To decompose a signal into two or more sub-bands as in filter-banks, graphic equalizers, sub-band coders, frequency multiplexers. To modify the frequency spectrum of a signal as in telephone channel equalization and audio graphic equalizers. To model the input-output relationship of a system such as telecommunication channels, human vocal tract, and music synthesizers. Investigates various type of filter:

Radio frequency filter:

An RF (radio frequency) filter is a device that is utilized to allow or stop selected signals or frequencies, or used to eliminate (filter out) any unwanted signals. In other words, an RF filter is designed to allow for the attenuation or transmission of a range of frequencies that would be applied. For instance, an RF filter helps to cut out RF interference that could occur if a hairdryer, lamp, or other "noisy" device is activated.
Generally, there are four types of RF filters:

A high pass filter has a cut-off frequency, thereby allowing minimal or no loss in transmission for high frequencies, but considerably attenuating any low frequencies. A low pass filter is the opposite that of a high pass filter - that is, allowing for the transmission frequencies below the cut-off frequency, but attenuating any frequencies above the cut-off frequency. A band pass filter will allow for the transmission of a selected range or band of frequencies with no attenuation, but will attenuate frequencies below or above(lower or higher than) the desired or allowed band. Examples of band pass filters include cavity filters, surface acoustic wave (SAW) filters and crystal filters. A band reject filter will attenuate a frequency range or band while allowing all other frequencies to pass un attenuated. Examples of band reject filters include notch filters and band stop filters.

Various types of RF filters can be found in air traffic control and communications systems, medical alert systems, telemetry applications, two-way pagers, and satellite communications. Other uses include garage door openers, fire control radars, keyless locks, and radar and missile guidance systems.
Microwave filters:

Microwave Filters are two-port networks used to control the frequency response in a system by permitting good transmission of wanted signal frequencies while rejecting unwanted frequencies. Generally there are four types of filters. They are, Low-pass High-pass Band-pass Band-stop.
Microwave filter design has been a persistent and productive field for investigation from the very beginning of microwave engineering. Nowadays, high performance filters are needed in many microwave systems. Because of the importance of microwave filters, a great deal of material on the theory and design of filters is widely available in the literature. The purpose of this chapter is to introduce the basic theory of microwave filters, to describe how to design practical microwave filters, and to investigate ways of implementing high performance filters for modern communication systems.
Passive filters:

In passive filters, passive elements such as resistors, capacitors and inductors are used. The disadvantage of passive filter is the inductors are heavy and bulky and hence passive filters become costly. However they do not require additional power supplies for their operation. The range of frequencies over which attenuation by filter is called pass band. The range of frequencies over which attenuation is infinite is called stop band or attenuation band of the filter. Butterworth Filters Chebyshev Filters Bessel Filters Elliptical Filters
Butterworth filter:

The Butterworth filter is a medium-Q filter that is used in designs that require the amplitude response of the filter to be as flat as possible. The Butterworth response is the flattest passband response available and contains no ripple. Since the Butterworth response is only a medium-Q filter, its initial attenuation steepness is not as good as some filters but it is better than others. This characteristic often causes the Butterworth response to be called a “middle-of-the-road” design.
The attenuation of Butterworth filter is given by,
A_(dB=10 10g[ 1+( ω/ωc)2n])
Where:
ω = the frequency at which the attenuation is desired ωC = the cutoff frequency ω(3dB) of the filter n = the number of elements in the filter In most cases for Butterworth filters, ωc is defined as the frequency of the 3-dB pass band edge point. Figure:1 Butterworth low pass filter

Chebyshev filters:

The Chebyshev filter is a high-Q filter that is used when: A steeper initial descent into the stop band is required The pass band response is no longer required to be float.
With this type of requirement, ripple can be allowed in the pass band. As more ripples is introduced, the initial slope at the beginning of the stop band is increased and produces a more rectangular attenuation curve when compared to the rounded Butterworth response. Both curves are for n = 3 filters. The Chebyshev response shown has 3 dB of pass band ripple and produces a 10-dB improvement in stop band attenuation over the Butterworth filter. The poles would lie on an ellipse of the unit circle. That means that like Butterworth filters, Chebyshev filters contain only poles. However, while the poles of the Butterworth filter lie on a circle in the s-plane, those of the Chebyshev filter lie on an ellipse. The Chebyshev phase response exhibits more linearity than the Elliptic one and less linearity than the Butterworth one. Figure:2 chebyshev low pass filter

Elliptical filters:

Another name for the Elliptic Filter is ’Cauer’ filter. Compared with Butterworth and Chebyshev filters, Elliptic filters have the most rapid transition (narrow transition band). However, this does not come without a price. Elliptic filters have a ripple in both the pass band and stop band. This is the result of a pole-zero configuration which consists of both poles and zeros. An Elliptic filter is notorious for introducing large phase distortions, especially near the edge of the pass-band where the sharp amplitude characteristic implies a strongly non-linear phase characteristic. Figure: 3 elliptical low pass filters
Bessel filters:

Butterworth filters have fairly good amplitude and transient behavior. The Chebyshev filters improve on the amplitude response at the expense of transient behavior. The Bessel filter is optimized to obtain better transient response due to a linear phase (i.e. constant delay) in the pass band. This means that there will be relatively poorer frequency response (less amplitude discrimination).The poles of the Bessel filter can be determined by locating all of the poles on a circle and separating their imaginary parts by: 2/n where n is the number of poles. Note that the top and bottom poles are distanced by where the circle crosses the jω axis by: 1/nor half the distance between the other poles. Design procedure of the filter:

The various type of performances of passive filters has been investigated above and the Bandpass filter has been designed below by using chebyshev filter design with pass band ripple of 1.0 dB with theses specifications:

Parameter Units
Centre frequency 200 MHz
Bandwidth 20 MHz
Return loss @ Centre frequency < - 10 dB
Attenuation < -48 @ 150 MHz and 250 MHz
VSWR ≤ 2

Table:1 parameters and units
Given,
Centre of frequency fc = 200 MHz,
Attenuation < -48 @ 150 MHz and 250 MHz
Bandwidth = 20 MHz
BPF = (250-150)/4 = 5

Figure 4: Attenuation characteristics for a chebyshev filter with 1.0 dB ripple.

Where BPF = 5 and the attenuation is – 48
By using this table, it has been found, n = 3
Take RS = 50 ohm and
RL = 50 ohm
RS/ RL = 50 / 50
= 1

Figure: 5 chebyshev low pass element values for 1.0 dB
Look at the table, Where n = 3 and Rs/RL = 1
So the values are,
C1 = 2.216
L2 = 1.088
C3 = 2.216

Low pass filter prototype design:

Figure: 6 prototype of chebyshev low pass filter

Transformation low pass to Bandpass filter:

Figure: 7 transformation of low pass filter to Bandpass filter.

The values had been calculated for capacitance and inductance by using parallel resonant branch and series resonant branch formulas:
Parallel resonant branch:
C = cn/(2πRL Bw 3dB)
L = (RL Bw 3 dB)/(2πLnfc^2 )
Series resonant branch:
L = RlLn/(2π Bw 3dB)
C = (Bw 3dB)/(2π Cn fc^2 Rl)
Where parallel resonant branch are,
C1 = 2.216
L1 = 2.216
C3 = 2.216
L3 = 2.216

Parallel resonant branch:

C1 = cn/(2πRL Bw 3dB)
C1 = 2.216/(2π*50*20MHz )
C1 = 352. 687 Pf
L1 = (RL Bw 3 dB)/(2πLnfc^2 )
L1 = (50* 20MHz)/(2π*2.216〖*(200MHz)〗^2 )
L1 = 1.795nH
C3 = cn/(2πRL Bw 3dB)
C3 = 2.216/(2π*50*20MHz )
C3 = 352.687pf
L3 = (RL Bw 3 dB)/(2πLnfc^2 )
L3 = (50* 20MHz)/(2π*2.216〖*(200MHz)〗^2 )
L3 = 1.795nH
Series resonant branch:
L = RlLn/(2π Bw 3dB)
L2 = (50*1.088)/(2π*20MHz)
L2 = 432.901nH

C2 = (Bw 3dB)/(2π Cn fc^2 Rl)
C2 = 20MHz/(2π*1.088* (200MHz)^2 50)
C2 = 1.462pf
Complete chebyshev Bandpass filter design is shown below

Figure: 8 the complete circuit design

Simulation and results:

Figure: 9 filter design by using rfsim 99 software

Figure 10: complete chebyshev Bandpass filter design.

Figure: 11 the Bandpass filter design for 1dB chebyshev filter
Centre frequency:
According to the given parameters the Centre of frequency is 200 MHz Figure: 12 Centre frequency of Bandpass filter
The start band is at 150MHz and stop band at 250MHz.
Where f= 200.1MHz at -0.0072dB
Therefore, it achieves the give requirements.
Bandwidth:

Figure: 13 bandwidth diagram
The attenuation of -3dB at f1 = 190.481MHz
The attenuation of -3dB at f2 = 210.521 MHz
Bandwidth = f2-f1
= (210.521MHz) – (190.481MHz)
= 20.04MHz
So, it achieves the given requirement bandwidth of 20MHz
Return loss @ Centre frequency:

Figure: 14 return loss
Return loss should be < -10
Here, it gets -42.69dB.
Calculation for VSWR:

Return loss = -42.69 dB
20 log = - 42.69 dB
〖 log〗^(-1)= -42.69dB/20
〖 log〗^(-1) = - 2.1345
= 0.00733
VSWR = 1 + 0.00733/ 1- 0.00733
VSWR = 1.01
The given VSWR is ≤2
Therefore the VSWR is also achieves the given requirements.

Q- factor: (f2-f1)/Bw
= ((210.521MHz)-(190.481))/20MHz
= 20.04MHz/20MHz
= 1.0012
Alternate filter designs:

Insertion loss method:

The insertion loss method allows a high degree of control over the pass band and stop band amplitude and phase characteristics, with a systematic way to synthesize a desired response. The necessary design tradeoffs can be evaluated to best meet the application requirements. If, for example, a minimum insertion loss is most important, a binomial response could be used; a Chebyshev response would satisfy a requirement for the sharpest cutoff. If it is possible to sacrifice the attenuation rate, a better phase response can be obtained by using a linear phase filter design. And in all cases, the insertion loss method allows filter performance to be improved in a straightforward manner, at the expense of a higher order filter. For the filter prototypes to be discussed below, the order of the filter is equal to the number of reactive elements.

Given, Centre of frequency fc = 200 MHz,
Attenuation < -48 @ 150 MHz and 250 MHz
Bandwidth = 20 MHz
BPF = (250-150)/4 = 5

Figure 15: Attenuation characteristics for a chebyshev filter with 0.5 dB ripple.
Where BPF = 5 and the attenuation is – 48
By using this table, it has been found, n = 3
Take RS = 50 ohm and
RL = 50 ohm
RS/ RL = 50 / 50 = 1

Figure: 16 elements values for chebyshev 0.5dB low pass filter

From the table 15 the elements values for low pass filter prototype circuit are given us, gl = 1.5963 = L1, g2 = 1.0967 = C2, g3 = 1.5963 = L3, g4=1.000 =RL

The ratio of 0.5dB ripple is
Center frequency is 200MHz and the bandwidth is 20MHz
So the ratio is ∆=0.1

A series inductor is transformed to a series LC circuit with impedance and frequency scaled values,

A shunt capacitor is transformed to a shunt LC circuit with impedance and frequency scaled values,

Where, ∆ =0.1
R_(0=) 50 ohm ω_(0=2π200MHz) L1 = (1.5963*50)/(2π*200MHz*0.1) = 635.14nH

C1 = 0.1/(2π*200MHz*1.5963*50) = 0.9777pF

L2 = (0.1*50)/(2π*200MHz*1.0967) = 3.62nH

C2 = 1.0967/(2π*200MHz*0.1*50) = 174.5pF

L3 = (1.5963*50)/(2π*200MHz*0.1) = 635.14nH

C3 = 0.1/(2π*200MHz*1.5963*50) = 0.9777pF

Figure: 17 Band pass circuit design

Figure: 18 the Bandpass filter design for 0.5dB chebyshev filter by insertion loss method

Center of frequency:
According to the given parameters the center of frequency is 200MHz.

Figure: 19 Centre frequency of Bandpass filter
The start band is at 150MHz and stop band at 250MHz.
Where f= 200MHz at -0.0000042dB
Therefore center of frequency is = 0

Bandwidth:

Figure: 20 bandwidth diagram
The attenuation of -2.93dB at f1 = 188.677MHz
The attenuation of -2.88dB at f2 = 211.924 MHz
Bandwidth = f2-f1
= (211.924MHz) – (188.677MHz)
= 23.24MHz

Return loss at center of frequency:

Figure: 21 return loss
Return loss should be < -10
Here, it gets -42.94dB

Calculation for VSWR:

Return loss = -42.94 dB
20 log = - 42.94 dB
〖 log〗^(-1)= -42.94dB/20
〖 log〗^(-1) = - 2.147
= 0.00712
VSWR = 1 + 0.00712/ 1- 0.00712
VSWR = 1.01
The given VSWR is ≤2
Therefore the VSWR is also achieves the given requirements.
Q-factor:

(f2-f1)/Bw
= ((211.924MHz)-(188.677))/20MHz
= 23.24MHz/20MHz
= 1.162

Evaluate performances of both designs:

Parameters Units Main design Alternative design
Center frequency 200MHz 200.1MHz at -0.0072dB 200MHz at -0.0000042dB

Bandwidth 20MHz 20.04MHZ 23.04MHz
Center frequency at return loss