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One-Port Network 

• The basic idea of a one-port network is shown below. The one-port is a ``black box'' with a single pair of input/output terminals,

referred to as a port. Network theory is normally described interms of circuit theory elements, in which case a voltage isapplied at the terminals and a current flows as shown.

• A one-port network characterized by its driving point impedance

 Z(s) or admittance G(s). For any applied voltage V(s), theobserved current  I(s)= V(s)G(s).

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A one-port network characterized by its driving pointimpedance  Z(s) or admittance G(s). For any appliedvoltage V(s), the observed current  I(s)= V(s)G(s).

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• Two-port network

• A two-port network (a kind of four-terminal network orquadripole) is an electrical circuit or device with two pairs ofterminals connected together internally by an electrical network.

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A linear circuit • is an electronic circuit in which, for a sinusoidal input 

voltage of frequency f , any output of the circuit (the current through any component, or the voltage between any twopoints) is also sinusoidal with frequency f . Note that theoutput need not be in phase with the input.

•  An equivalent definition of a linear circuit is that it obeysthe superposition principle. This means that the output of

the circuit F(x) when a linear combination of signals ax1(t)+ bx2(t) is applied to it is equal to the linear combination ofthe outputs due to the signals x1(t) and x2(t) appliedseparately:

• Informally, a linear circuit is one in which the values of theelectronic components, the resistance, capacitance,i

nductance, gain, etc. don't change with the level of voltageor current in the circuit.

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The superposition theorem states:

• The current or voltage associated with a branchin a linear network equals the sum of the currentor voltage components set up in that branch dueto each of the independent sources acting one atthe time on the circuit

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Define nonlinear circuit?

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Define nonlinear circuit?

If the system is non-linear, the current or voltage responsewill contain harmonics of the excitation frequency. Aharmonic is a frequency equal to an integer multiplied by

the fundamental frequency. For example, the “secondharmonic” is a frequency equal to two times thefundamental frequency.Some researchers have made use of this phenomenon.Linear systems should not generate harmonics, so thepresence or absence of significant harmonic responseallows one to determine the systems linearity. Otherresearchers have intentionally used larger amplitudeexcitation potentials. They use the harmonic response toestimate the curvature in the cell's current voltage curve.

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Transfer functions • are commonly used in the analysis of systems such as

single-input single-output filters, typically within the fieldsof signal processing, communication theory, and controltheory. The term is often used exclusively to refer to linear,time-invariant systems (LTI), as covered in this article.

Most real systems have non-linear input/outputcharacteristics, but many systems, when operated withinnominal parameters (not "over-driven") have behavior thatis close enough to linear that LTI system theory is anacceptable representation of the input/output behavior.

• In its simplest form for continuous-time input signal x (t )and output y (t ), the transfer function is the linear mappingof the Laplace transform of the input, X (s), to the outputY (s):

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or

where H (s) is the transfer function of the LTI system.

In discrete-time systems, the function is similarly written as

(see Z transform) and is often referred to as the pulse-transferfunction.

We can express transfer function in the frequency domain also:

TV(jω)=Vout(jω)/ Vin(jω) 

or

TI(jω)=Iout(jω)/ Iin(jω) 

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Frequency spectrum

The frequency spectrum of a time-domain signal is a

representation of that signal in the frequency domain. Thefrequency spectrum can be generated via a Fouriertransform of the signal, and the resulting values are usuallypresented as amplitude and phase, both plotted versusfrequency 

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Spectrum analysis is the technical process of decomposing acomplex signal into simpler parts. As described above,many physical processes are best described as a sum ofmany individual frequency components. Any process thatquantifies the various amounts (e.g. amplitudes, powers,intensities, or phases), versus frequency can be called

spectrum analysis.

Spectrum analysis can be performed on the entire signal.Alternatively, a signal can be broken into short segments(sometimes called frames), and spectrum analysis may beapplied to these individual segments. Periodic functions 

(such as sin(t )) are particularly well-suited for this sub-division. General mathematical techniques for analyzingnon-periodic functions fall into the category of Fourieranalysis.

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Spectrum of the saw-toof type signal

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Electronic filters are electronic circuits which perform signal processing functions,specifically to remove unwanted frequencycomponents from the signal, to enhancewanted ones, or both. Electronic filters can

be:• passive or active • analog or digital • high-pass, low-pass, bandpass, band-

reject (band reject; notch), or all-pass.• discrete-time (sampled) or continuous-

time 

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Other filter technologies• Quartz filters and piezoelectrics• In the late 1930s, engineers realized that small mechanical systems made of rigid materials such

as quartz would acoustically resonate at radio frequencies, i.e. from audible frequencies (sound) up

to several hundred megahertz. Some early resonators were made of steel, but quartz quicklybecame favored. The biggest advantage of quartz is that it is piezoelectric. This means that quartzresonators can directly convert their own mechanical motion into electrical signals. Quartz also hasa very low coefficient of thermal expansion which means that quartz resonators can produce stablefrequencies over a wide temperature range. Quartz crystal filters have much higher quality factorsthan LCR filters. When higher stabilities are required, the crystals and their driving circuits may bemounted in a "crystal oven" to control the temperature. For very narrow band filters, sometimesseveral crystals are operated in series.

• Engineers realized that a large number of crystals could be collapsed into a single component, bymounting comb-shaped evaporations of metal on a quartz crystal. In this scheme, a "tapped delayline" reinforces the desired frequencies as the sound waves flow across the surface of the quartzcrystal. The tapped delay line has become a general scheme of making high-Q filters in manydifferent ways. 

• SAW filters• SAW (surf ace acoustic wave) filters are electromechanical devices commonly used in radio

frequency applications. Electrical signals are converted to a mechanical wave in a deviceconstructed of a piezoelectric crystal or ceramic; this wave is delayed as it propagates across thedevice, before being converted back to an electrical signal by further electrodes. The delayedoutputs are recombined to produce a direct analog implementation of a finite impulse response filter. This hybrid filtering technique is also found in an analog sampled filter. SAW filters arelimited to frequencies up to 3 GHz. 

• BAW filters• BAW (Bulk Acoustic Wave) filters are electromechanical devices. BAW filters can implement ladder

or lattice filters. BAW filters typically operate at frequencies from around 2 to around 16 GHz, andmay be smaller or thinner than equivalent SAW filters. Two main variants of BAW filters are makingtheir way into devices, Thin film bulk acoustic resonator or FBAR and Solid Mounted Bulk AcousticResonators.

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Passive filters

Passive implementations of linear filters are based oncombinations of resistors (R), inductors (L) and capacitors (C). These types are collectively known as passive filters,because they do not depend upon an external power supplyand/or they do not contain active components such astransistors.

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Single element types

A low-pass electronic filter realised by an RC circuit The simplest passive filters, RC and RL filters, include onlyone reactive element, except hybrid LC filter which is characterized by inductance and capacitanceintegrated in one element.

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L filter

An L filter consists of two reactive elements, one in series

and one in parallel.

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T and π filters

Low-pass π filter 

High-pass T filter

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• Three-element filters can have a 'T' or 'π' topology and ineither geometries, a low-pass, high-pass, band-pass, orband-stop characteristic is possible. The components can bechosen symmetric or not, depending on the requiredfrequency characteristics. The high-pass T filter in theillustration, has a very low impedance at high frequencies,and a very high impedance at low frequencies. That meansthat it can be inserted in a transmission line, resulting in

the high frequencies being passed and low frequenciesbeing reflected. Likewise, for the illustrated low-pass π filter, the circuit can be connected to a transmission line,transmitting low frequencies and reflecting highfrequencies. Using m-derived filter sections with correcttermination impedances, the input impedance can beincreased.

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Digital filter

• A digital filter system usually consists of an analog-to-digitalconverter to sample the input signal, followed by a microprocessorand some peripheral components such as memory to store dataand filter coefficients etc. Finally a digital-to-analog converter tocomplete the output stage.

• A general finite impulse response filter with n stages, each with anindependent delay, d i, and amplification gain, ai.

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FILTER SPECIFICATIONS

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FILTER SPECIFICATIONS

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FILTER SPECIFICATIONS

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FILTER SPECIFICATIONS

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Classification by topology

• Electronic filters can be classified by the technology used toimplement them. Filters using passive filter and active filter technology can be further classified by the particularelectronic filter topology used to implement them.

• Any given filter transfer function may be implemented inany electronic filter topology.

• Some common circuit topologies are:

• Cauer topology - Passive

• Sallen Key topology - Active

• Multiple Feedback topology - Active• State Variable Topology - Active

• Biquadratic topology biquad filter - Active

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Ladder topologies

Ladder topology, often called Cauer topology after Wilhelm

Cauer (inventor of the Elliptical filter)

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Unbalanced

L Half section T Section Π SectionLadder network 

Ladder network

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Modified ladder topologies

Fig. m-derived topology

Image filter design commonly uses modifications of the basic ladder topology.

These topologies, invented by Otto Zobel have the same passbands as theladder on which they are based but their transfer functions are modified toimprove some parameter such as impedance matching, stopband rejectionor passband-to-stopband transition steepness.

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Bridged-T topologies

• Typical bridged-T Zobel network equaliser used to correct highend roll-off  

• The bridged-T topology is also used in sections intended toproduce a signal delay but in this case no resistive componentsare used in the design.

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Lattice topology

Lattice topology X-section phase 

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ACTIVE FILTERS

Sallen–Key topology 

The Sallen–Key topology is an electronic filter topology usedto implement second-order active filters that is particularlyvalued for its simplicity.

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The generic unity-gain Sallen–Key filter topology implemented with aunity-gain operational amplifier is shown in Figure 1. The

following analysis is based on the assumption that the operationalamplifier is ideal.

Figure 1: The generic Sallen–Key filter topology.

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and v - inputs must match (i.e., v + = v -). However, the inverting input v - isconnected directly to the output v out, and so

• By Kirchhoff's current law (KCL) applied at the v x node,

• By combining Equations (1) and (2),• Applying Equation (1) and KCL at the OA's non-inverting input v + gives

• which means that

• Combining Equations (2) and (3) gives

• Rearranging Equation (4) gives the transfer function 

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• Combining Equations (2) and (3) gives

• Rearranging Equation (4) gives the transfer function 

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ELECTRONIC FILTERSSallen-Key Lowpass filter example

Sallen-Key Highpass filter example

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Multiple feedback topology

Multiple feedback topology circuit

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ELECTRONIC FILTERSThe transfer function of the multiple feedback topology circuit, like all second-order linear filters, is:

.

In an MF filter,

is the Q factor .

is the DC voltage gain 

is the corner frequency

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Biquad filter

A biquad filter is a type of linear filter that implements atransfer function that is the ratio of two quadratic functions.

The name biquad  is short for biquadratic .Biquad filters are typically active and implemented with a

single-amplifier biquad (SAB) 

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Tow-Thomas Biquad Example

For example, the basic configuration in Figure 1 can be used as either a low-pass or bandpass 

filter depending on where the output signal is taken from.

Figure 1: The common Tow-Thomas biquad filter topology.

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Network synthesis filters

Butterworth filter

Butterworth filters are described as maximally flat, meaning thatthe response in the frequency domain is the smoothest possible

curve of any class of filter of the equivalent orderThe Butterworth

class of filter was first described in a 1930 paper by the Britishengineer Stephen Butterworth after whom it is named. The filterresponse is described by Butterworth polynomials 

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Chebyshev filterA Chebyshev filter has a faster cut-off transition than a

Butterworth, but at the expense of there being ripples in thefrequency response of the passband. There is a compromise to

be had between the maximum allowed attenuation in the

passband and the steepness of the cut-off response. This isalso sometimes called a type I Chebyshev, the type 2 being a

filter with no ripple in the passband but ripples in the stopband.The filter is named after Pafnuty Chebyshev whose Chebyshevpolynomials are used in the derivation of the transfer function

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Cauer (Eliptical) filter

Cauer filters have equal maximum ripple in thepassband and the stopband. The Cauer filterhas a faster transition from the passband to

the stopband than any other class of network

synthesis filter. The term Cauer filter can beused interchangeably with elliptical filter, butthe general case of elliptical filters can have

unequal ripples in the passband and stopband.The filter is named after Wilhelm Cauer and

the transfer function is based on elliptic

rational functions.

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Bessel filter

The Bessel filter has a maximally flat time-delay (groupdelay) over its passband. This gives the filter a linear

phase response and results in it passing waveforms withminimal distortion. The Bessel filter has minimal

distortion in the time domain due to the phase responsewith frequency as opposed to the Butterworth filter which

has minimal distortion in the frequency domain due tothe attenuation response with frequency. The Besselfilter is named after Friedrich Bessel and the transfer

function is based on Bessel polynomials.

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As is clear from the image, elliptic filters are sharper than all the others, but they showripples on the whole bandwidth. 

In the particular implementation – analog or digital, passive or active – makesno difference; their output would be the same.