IN THE UNITED STATES PATENT AND TRADEMARK OFFICE
`
`In re Patent of: Webster et al.
`
`U.S. Patent No.:
`Issue Date:
`
`6,754,195
`June 22, 2004
`
`Attorney Docket No.: 27410—002lIP1
`
`Appl. Serial No.: 10/143,134
`Filing Date:
`May 10, 2002
`Title:
`WIRELESS COMMUNICATION SYSTEM CONFIGURED TO
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`COMMUNICATE USING A MIXED WAVEFORM CONFIGURATION
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`DECLARATION OF ELIZABETH SHEEHAN
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`published in 1999 by CRC Press (hereinafter the “Jamal reference”), Print ISBN: 978-0-8493-
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`8347—2, eBook ISBN: 978~0—4l5-87617-9.
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`Dated:
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`l-7-’7~5'-10”’
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`Elizabeth Sheehan
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`Page 3 of 3
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`
`
`Exhibit A
`
`
`
`Copy of Exhibit 1014 (“Jamal”)
`
`

`
`
`
`.Afioodmmflacm:o.>>>>>>=_3Ev.U._._mmm:n_omococaEm_._>aoo
`
`
`
`
`
`
`
`
`
`
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`...w.B__u_...._m.3._mEm_.smegma
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`MARVELL 1014
`
`

`
`Filters
`
`82.1
`82.2
`
`82.3
`
`82.4
`
`82.5
`
`82.6
`827
`
`Introduction
`Filter Classification
`
`The Filter Approximation Problem
`Butterworth Filters ' Chebyshev Filters of Chebyshev I
`Filters Elliptic or Cauer Filters ' Bessel Filters
`
`Design Examples for Passive and Active Filters
`Passive R, L, C Filter Design ° Active Filter Design
`Discrete—Time Filters
`
`Digital Filter Design Process
`FIR Filter Design
`Windowed FIR Filters ° Optimum FIR Filters - Design of
`Narrowband FIR Filters
`
`82.8
`
`IIR Filter Design
`Design of Arbitrary IIR Filters - Cascade-Form IIR Filter
`Structures
`82.9 Wave Digital Filters
`. 82.10 Antialiasing and Smoothing Filters
`82.11 Switched Capacitor Filters
`82.12 Adaptive Filters
`
`Rahman Jamal
`National Instruments Germany
`Robert Steer
`Frequency Devices
`
`82.1
`
`Introduction
`
`In its broadest sense, a filter can be defined as a signal processing system Whose output signal, usually
`called the response, differs from the input signal, called the excitation, such that the output signal has
`some prescribed properties. In more practical terms an electric filter is a device designed to suppress,
`pass, or separate a group of signals from a mixture of signals according to the specifications in a particular
`application. The application areas of filtering are manifold, for example to band—lirnit signals before
`sampling to reduce aliasing, to eliminate unwanted noise in communication systems, to resolve signals
`into their frequency components,
`to convert discrete—time signals into continuous—time signals,
`to
`demodulate signals, etc. Filters are generally classified into three broad classes: continuous—time, sampled-
`data, and d1'screte—time filters depending on the type of signal being processed by the filter. Therefore,
`the concept of signals are fundamental in the design of filters.
`A signal is a function of one or more independent variables such as time, space, temperature, etc. that
`carries information. The independent variables of a signal can either be continuous or discrete. Assuming
`that the signal is a function of time, in the first case the signal is called continuous—time and in the
`second, discrete—time. A continuous—time signal is defined at every instant of time over a given interval,
`whereas a discrete—time signal is defined only at a discrete—time instances. Similarly, the values of a signal
`can also be classified in either continuous or discrete.
`
`© 1999 by CRC Press LLC
`
`

`
`ln real-world signals, often referred to as analog signals, both amplitude and time are continuous,
`These types of signals cannot be processed by digital machines unless they have been converted into
`discrete—time signals. By contrast, a digital signal is characterized by discrete signal values, that are defined
`only at discrete points in time. Digital signal values are represented by a finite number of digits, which
`are usually binary coded. The relationship between a continuous-time signal and the corresponding
`discrete~time signal can be expressed in the following form:
`
`X(kT)=x(tLkT,
`
`k=0,1,2,...,
`
`(82.1)
`
`where Tis called the sampling period.
`Filters can be classified on the basis of the input, output, and internal operating signals. A continuous
`data filter is used to process continuous-time or analog signals, whereas a digital filter processes digital
`signals. Continuous data filters are further divided into passive or active filters, depending on the type
`of elements used in their implementation. Perhaps the earliest type of filters known in the engineering
`community are LC filters, which can be designed by using discrete components like inductors and
`capacitors, or crystal and mechanical filters that can be implemented using LC equivalent circuits. Since
`no external power is required to operate these filters, they are often referred to as passive filters. In contrast,
`active filters are based on active devices, primarily RC elements, and amplifiers. In a sampled data filter,
`on the other hand, the signal is sampled and processed at discrete instants of time. Depending on the
`type of signal processed by such a filter, one may distinguish between an analog sampled data filter and
`a digital filter, In an analog sampled data filter the sampled signal can principally take any value, whereas
`in a digital filter the sampled signal is a digital signal, the definition of which was given earlier. Examples
`of analog sampled data filters are switched capacitor (SC) filters and charge—transfer device (CTD) filters
`made of capacitors, switches, and operational amplifiers.
`
`82.2 Filter Classification
`
`Filters are commonly classified according to the filter function they perform. The basic functions are:
`low—pass, high-pass, bandpass, and bandstop. If a filter passes frequencies from zero to its cutoff frequency
`QC and stops all frequencies higher than the cutoff frequencies, then this filter type is called an ideal low-
`pass filter, In contrast, an ideal higlbpass filter stops all frequencies below its cutoff frequency and passes
`all frequencies above it. Frequencies extending from Q, to Q2 are passed by an ideal bandpass filter,
`while all other frequencies are stopped. An ideal bandstop filter stops frequencies from £2, to Q2 and
`passes all other frequencies. Figure 82.1 depicts the magnitude functions of the four basic idea} f.‘1lt'r.~.r types.
`So far we have discussed ideal filter characteristics having rectangular magnitude responses. These
`characteristics, however, are physically not realizable. As a consequence, the ideal response can only be
`approximated by some nonideal realizable system. Several classical approximation schemes have been
`developed, each of which satisfies a different criterion of optimization. This should be taken into account
`when comparing the performance of these filter characteristics.
`
`82.3 The Filter Approximation Problem
`
`Generally the input and output variables of a linear, tirne~invariant, causal filter can be characterized
`either in the time—domain through the convolution integral given by
`
`,(.)= J.0[ha(t—1:)X(t)dt
`
`(82.2)
`
`or, equivalently, in the frequency—domain through the transfer function
`
`© 1999 by CRC Press LLC
`
`

`
`IH..on>|I
`
`1
`
`IH.<jr2>|I
`
`1
`
`go.
`
`o
`
`o, _.
`(2
`
`42.
`
`0
`
`0
`
`.1
`Q
`
`|H.o'rz>|T
`
`—L—:L.
`
`1
`0 M
`
`1H.o'm1T
`
`%|
`~01 -0.
`
`1
`
`0
`
`£1.
`
`3'
`
`7;
`
`FIGURE 82.1 The magnitude function of an ideal filter is l in the passband and U in the stopband as shown for
`(a) low—pass, (b) high—pass. (c) bandpass, and (cl) stopband filters.
`
`<2
`
`[=0
`
`(32.3)
`
`where Ha(s) is the Laplace transform of the impulse response ha(t) and X(5), Y(s) are the Laplace transforms
`of the input signal X(t) and the output or the filtered signal y(t). X(s) and Y(s) are polynomials in s =
`(5 +jQ and the overall transfer function Ha(s) is a real rational function of s with real coefficients. The
`zeroes of the polynomial X(s) given by 5 = smi are called the poles of Ha(s) and are commonly referred to
`as the natural frequencies of the filter. The zeros of Y(s) given by s = so, which are equivalent to the zeroes
`of Ha(s) are called the transmission zeros of the filter. Clearly, at these frequencies the filter output is zero
`for any finite input. Stability restricts the poles of Ha(s) to lie in the left half of the s—plane excluding the
`jQ—axis, that is Re{sM.} < 0. For a stable transfer function Ha(s) reduces to HBUQ) on the jQ—axis, which
`is the continuous—time Fourier transform of the impulse response ha(t) and can be expressed in the
`following form:
`
`Ha(jo)=lHa(jr2)ld’°l°l
`
`(82.4)
`
`where ‘HE is called the magnitude function and 0(9) = arg Ha(jQ) is the phase function. The gain
`magnitude of the filter expressed in decibels (dB) is defined by
`
`0c(Q) = 20 1oglHa ( = 10 ioglira (jg)
`
`(82.5)
`
`2
`
`Note that a filter specification is often given in terms of its attenuation, which is the negative of the gain
`function also given in decibels. While the specifications for a desired filter behavior are commonly given
`in terms of the loss response 0c(§2), the solution of the filter approximation problem is always carried
`out with the help of the characteristic function C (]'Q) giving
`
`oc(Q)=10 1og[1+lC(jo) 2]
`
`(82.6)
`
`© 1999 by CRC Press LLC
`
`

`
`IH.<ia)1‘
`
`1/z//
`
`1/1+9}
`
`IPassband ripple
`
`
`
`Transifidn
`band 1
`
`......_________...~_._._._:\
`Passband 1/AL
`
`P
`
`o
`
`not)
`
`Q
`
`FIGURE 82.2 The squarec magnitude function of an analog filter can have ripple in the passband and in the
`stopband.
`
`is not a rational function, but C(jQ) can be a polynomial or a rational function and
`Note that oc(§2)
`approximation with polynomial or rational functions is relatively convenient. It can also be shown that
`frequency—dependent properties of lC(jQ) I are in many ways identical to those of oL(£2). The approximation
`problem consists of determining a desired response
`such that the typical specifications depicted
`in Figure 82.2 are met. This so—called tolerance scheme is characterized by the following parameters:
`
`PS C
`
`°°{OlO{O
`
`Passband cutoff frequency (rad/s)
`Stopband cutoff frequency (rad/S)
`—3 dB cutoff frequency (rad/s)
`Permissible error in passband given by 8 = (10”1“ ~ 1)“, where r is the maximum acceptable
`attenuation in dB; note that 10 log 1/ (1 + 82)“ = —r
`1/A Permissible maximum magnitude in the stopband, ie, A = 10°‘/2°, where OL is the minimum
`acceptable attenuation in dB; note that 20 log (NA) = —0L.
`
`The passbanti of a low—pass filter is the region in the interval [0,§2P] where the desired characteristics of
`a given signal are preserved. In contrast, the stopband of a low—pass filter (the region [.Qs,o<>]) rejects
`signal components. The transition band is the region between (Qx — Qp), which would be 0 for an ideal
`filter. Usually, the amplitudes of the permissible ripples for the magnitude response are given in decibels.
`The following sections review four different classical approximations: Butterworth, Chebyshev Type I,
`elliptic, and Bessel.
`
`Butterworth Filters
`
`The frequency response of an Nth—order Butterworth low—pass filter is defined by the squared magnitude
`function
`
` Ha (jg)
`
`2=—.1~.
`1+(s2/52¢)”
`
`(82.7)
`
`It is evident from the Equation 82.7 that the Butterworth approximation has only poles, i.e., no finite
`zeros and yields a maximally flat response around zero and infinity. Therefore, this approximation is also
`
`© l999 by CRC Press LLC
`
`

`
`called maximally flat magnitude (MFM). In addition, it exhibits a smooth response at all frequencies
`and a monotonic decrease from the specified cutoff frequencies.
`Equation 82.7 can be extended to the complex s—domain, resulting in
`
`_
`
`1
`
`H (s)H (—s)_;—P+(s/Joan)”
`
`The poles of this function are given by the roots of the denominator
`
`s -9 e’"l‘/“(Ml/2”],
`k— c
`
`k=O, 1,
`
`2N—1
`
`(82.8)
`
`(82.9)
`
`Note that for any N, these poles lie on the unit circle of radius QC in the s—plane. To guarantee stability,
`the poles that lie in the left half—plane are identified with
`As an example, we will determine the
`transfer function corresponding to a third—order Butterworth filter, i.e., N = 3.
`
`Ha(s)Ha(—s)§;+—(_—S¥= 1_S6
`
`(82.10)
`
`The roots of denominator of Equation 82.10 are given by
`
`s :9 e”‘l1/“(“"‘l/6],
`
`k=0, 1, 2, 3, 4,5
`
`(82.11)
`
`Therefore, We obtain
`
`(82.12)
`
`The corresponding transfer function is obtained by identifying the left half—plane poles with Ha Note
`that for the sake of simplicity we have chosen QC = l.
`
`1
`1
`H”(S)=(s+1)(s+1/2qfi/2)(s+1/2+jJ§/2)21+2s+2s2+s3
`
`(82.13)
`
`Table 82.}. gives the Butterworth denominator polynomials up N: 5,
`Taijie
`gives the Butterworth poles in real and imaginary components and in frequency and Q.
`
`© 1999 by CRC Press LLC
`
`

`
`TABLE 82.1 Butterworth Denominator Polynomials
`
`Order(N)
`
`Butterworth Denominator Polynomials of H(5)
`s + 1
`5“ + J5 s + 1
`53 + 253 + 25 + 1
`54 + 2.613153 + 3.414252 + 2.6131s+1
`:5 + 3.236154 + 5.236153 + 5.236152 + 3.23615 +1
`
`U'i>&C/-JlV>—-
`
`TABLE 82.2 Butterworth and Bessel Poles
`
`Butterworth Poles
`
`Bessel Poles (-3 dB)
`
`Re
`a
`
`1m (ij)
`b
`
`N
`
`9
`
`Q
`
`Re
`a
`
`lm (ij)
`b
`
`Q
`
`Q
`
`1
`
`3
`
`4
`
`5
`
`1.000
`0.000
`-1.000
`——
`1.000
`0.000
`-1.000
`0.577
`1.272
`0.636
`-1.102
`0.707
`1.000
`0.707
`-0.707
`—
`1.323
`0.000
`-1.323
`—-
`1.000
`0.000
`-1.000
`0.691
`1.448
`0.999
`-1.047
`1.000
`1.000
`0.866
`-0.500
`0,522
`1.430
`0.410
`-1.370
`0.541
`1.000
`0.383
`-0.924
`0.805
`1.603
`1.257
`-0.995
`1.307
`1.000
`0.924
`-0.383
`-
`1.502
`0.000
`-1.502
`—
`1.000
`0.000
`-1.000
`0.564
`1.556
`0.718
`-1.381
`0.618
`1.000
`0.588
`-0.809
`0.916
`1.755
`1.471
`-0.958
`1.618
`1.000
`0.951
`-0.309
`0.510
`1.604
`0.321
`-1.571
`0.518
`1.000
`0.259
`-0.966
`0.611
`1.689
`0.971
`-1.382
`0.707
`1.000
`0.707
`-0.707
`1.023
`1.905
`1.662
`-0.931
`1.932
`1.000
`0.966
`-0.259
`—
`1.684
`0.000
`-1.684
`-
`1.000
`0.000
`-1.000
`0.532
`1.716
`0.589
`-1.612
`0.555
`1.000
`0.434
`-0.901
`0.661
`1.822
`1.192
`-1.379
`0.802
`1.000
`0.782
`-0,623
`1.126
`2.049
`1.836
`-0.910
`2.247
`1.000
`0.975
`-0.223
`0.506
`1.778
`0.273
`-1.757
`0.510
`1.000
`0.195
`-0.981
`0.560
`1.832
`0.823
`-1.637
`0.601
`1.000
`0.556
`-0.831
`0.711
`1.953
`1.388
`-1.374
`0.900
`1.000
`0.831
`-0.556
`
`
`
`
`
`
`
`2.563 -0.893 1.998 2.1891.0000.981-0.195 1.226
`
`6
`
`7
`
`8
`
`In the next example. the order N of a low—pass Butterworth filter is to be determined whose cutoff
`frequency (-3 dB) is QC = 2 kHz and stopband attenuation is greater than 40 dB at £25 = 6 kHz. Thus
`the desired filter specification is
`
`201ogHa(jo)ls—40,
`
`
`£2295
`
`.Ha(j§2)1S0.01,
`
`92:25
`
`-.1———=(o.01)2
`1+(s25/sac)”
`
`or equivalently,
`
`It follows from Equation 82.7
`
`© 1999 by CRC Press LLC
`
`(82.14)
`
`(82.15)
`
`(82.16)
`
`

`
`Solving the above equation for N gives N = 4.19. Since N must be an integer, a fifth—order filter is required
`for this specification.
`
`Chebyshev Filters or Chebyshev I Filters
`
`The frequency response of an Nth—order Chebyshev low—pass filter is specified by the squared~magnitude
`frequency response function
`
`1
`H :2 51.1
`a(J
`T1+z~:2TNZ(§2/Qp)
`
`T
`
`(82.17)
`
`where TN (X) is the Nth—order Chebyshev polynomial and 8 is a real constant less than 1 which determines
`the ripple of the filter. Specifically, for nonnegative integers N, the Nth—order Chebyshev polynomial is
`given by
`
`~ cos(Ncos‘1 X),
`T
`N(X)T cosh(Ncosh'1 X),
`
`M31
`1x121
`
`High—order Chebyshev polynomials can be derived from the recursion relation
`
`T,,+,(x) = ZXTN (X) ~ TN_,(x)
`
`(82.18)
`
`(82.19)
`
`where T0(X) = l and T1(x) = X.
`The Chebyshev approximation gives an equsiripple characteristic in the passband and is maximally
`flat near infinity in the stopband. Each of the Chebyshev polynomials has real zeros that lie within the
`interval (~l,l) and the function values for X e [—l,l] do not exceed +1 and «l.
`The pole locations for Chebyshev filter can be determined by generating the appropriate Chebyshev
`polynomials, inserting them into Equation 82.17, factoring, and then selecting only the left half plane
`roots. Alternatively, the pole locations Pk of an Nth—order Chebyshev filter can be computed from the
`relation, for 1<= l ——> N
`
`Pk = ——sin (Bk sinh B+j cos Gk coshB
`
`(82.20)
`
`where ®k = (ZI(— l)1'c/ZN and B: sinh"1(1/8).
`Note.’ PM,” and Pk are complex conjugates and when N is odd there is one real pole at
`
`PNH = -2 sinh [3
`
`For the Chebyshev polynomials, Qp is the last frequency where the amplitude response passes through
`the value of ripple at the edge of the passband. For odd N polynomials, where the ripple of the Chebyshev
`polynomial is negative going, it is the [—l/ (1 + £2)](“) frequency and for even N, where the ripple is
`positive going, is the 0 dB frequency.
`The Chebyshev filter is completely specified by the three parameters 8, Q9, and N. In a practical design
`application, 8 is given by the permissible passband ripple and Qp is specified by the desired passband
`cutoff frequency, The order of the filter, i.e., N, is then chosen such that the stopband specifications are
`satisfied.
`
`© 1999 by CRC Press LLC
`
`

`
`Elliptic or Cauer Filters
`
`The frequency response of an Nth~order elliptic low—pass filter can be expressed by
`
`lHa(Je)l2=
`
`1
`
`Fgfldyf)
`
`(82.21)
`
`is called the Jacobian elliptic function. The elliptic approximation yields an equiripple passband
`where FN
`and an equiripple stopband. Compared with the sarne—0rder Butterworth or Chebyshev filters, the elliptic
`design provides the sharpest transition between the passband and the stopband. The theory of elliptic filters,
`initially developed by Cauer, is involved, therefore for an extensive treatment refer to Reference 1.
`Elliptic filters are completely specified by the parameters 8, 0L, Qp, £25, and N
`
`= passband ripple
`where 8
`= stop band floor
`a
`Qp = the frequency at the edge of the passband (for a designated passband ripple)
`$25 = the frequency at the edge of the stopband (for a designated stopband floor)
`N = the order of the polynomial
`
`In a practical design exercise, the desired passband ripple, stopband floor, and 95 are selected and N
`is determined and rounded up to the nearest integer value. The appropriate Jacobian elliptic function
`must be selected and
`must be calculated and factored to extract only the left plane poles. For
`some synthesis techniques, the roots must expanded into polynomial form.
`This process is a formidable task. While some filter manufacturers have written their own computer
`programs to carry out these calculations, they are not readily available, However, the majority of appli-
`cations can be accommodated by use of published tables of the pole/zero configurations of low—pass
`elliptic transfer functions. An extensive set of such tables for a common selection of passband ripples,
`stopband floors, and shape factors is available in Reference 2.
`
`Bessel Filters
`
`The primary objectives of the preceding three approximations were to achieve specific loss characteristics.
`The phase characteristics of these filters, however, are nonlinear. The Bessel filter is optimized to reduce
`nonlinear phase distortion, i.e., a maximally flat delay. The transfer function of a Bessel filter is given by
`
`_ B
`_
`B
`Ha(s)—£TN%§—~§:3kSk,
`
`k=0
`
`_
`(2N—I<)!
`Bk————§H(N_k)!
`
`_
`k—0,1,...,N
`
`(82.22)
`
`is the Nth—order Bessel polynomial. The overall squared—magnitude frequency response
`where BN (s)
`function is given by
`
`lHa(js2)l2=1—
`
`2(N—1)o4
`+
`Q2
`BN4 (2N—1)2(2N—3)
`
`+--.
`
`To illustrate Equation 82.22 the Bessel transfer function for N = 4 is given below:
`
`105
`(S)
`1o5+105s+45sZ+1os3+s4
`H =mw———_~
`
`(82.23)
`
`(32.24)
`
`© 1999 by CRC Press LLC
`
`

`
`Table 82.2 lists the factored pole frequencies as real and imaginary parts and as frequency and Q for
`Bessel transfer functions that have been normalized to QC = —3 dB.
`
`82.4 Design Examples for Passive and Active Filters
`
`Passive R, L, C Filter Design
`
`The simplest and most commonly used passive filter is the simple, first~order (N = 1) R—C filter shown
`in l3H'.g11re
`Its transfer function is that of a first—order Butterworth low~pass filter. The transfer
`function and -3 dB QC are
`
`H(s)=
`
`1
`RCs+l
`
`where
`
`S2C=—L
`RC
`
`(82.25)
`
`While this is the simplest possible filter implementation, both source and load impedance change the dc
`gain and/or corner frequency and its rolloff rate is only first order, or -6 dB/octave.
`To realize higher—order transfer functions, passive filters use R, L, C elements usually configured in a
`ladder network. The design process is generally carried out in terms of a doubly terminated two—port
`network with source and load resistors R, and R2 as shown in l3*‘i.gure. 824. Its symbolic representation is
`given below.
`The source and load resistors are normalized in regard to a reference resistance RB = R1, i.e.,
`
`,
`
`r2=—RA=52—
`RB
`‘[31
`
`RB
`
`2
`
`(82.26)
`
`The values of L and C are also normalized in respect to a reference frequency to simplify calculations.
`Their values can be easily scaled to any desired set of actual elements.
`
`9 L
`Iv = I:
`B
`
`,
`
`CV =S2BCvRB
`
`(82.27)
`
`R
`
`Vi
`
`C
`
`Vo
`
`,
`FIGURE 82.3 A passive first—order RC filter can serve as an
`antialiasing filter or to minimize high—frequency noise.
`
`
`
`
`'7 "‘
`
`Filter
`
`—'
`
`"1
`
`FIGURE 82.4 A passive filter can have the symbolic
`representation of a doubly terminated filter.
`
`© 1999 by CRC Press LLC
`
`10
`
`

`
`.
`Q
`r1 <-- :2
`0
`o
`1
`
`WW I
`T
`3
`
`2
`
`O l Nvx
`I‘, <——
`O T
`1
`
`2
`
`,
`::
`o—
`3
`
`Jvvw I O
`—> F;
`I o
`N
`
`N-1
`
`0 AAA
`I‘, +—~
`o
`
`l
`
`.
`::
`+
`2
`
`3
`
`.
`::
`e
`N-1
`
`e./vvw O
`—> 1'2
`o
`
`N
`
`rvvx
`O
`r, <———
`o
`
`l
`
`,
`::
`——o
`2
`
`rvvw
`
`3
`
`.
`::
`o
`N—l
`
`Jvv\__O
`—> T2
`o
`
`N
`
`/vvx
`
`. O
`::—> 1
`.
`o
`N-1 N
`
`FIGURE 82.5 Even and odd N passive all—po1e filter networks can be realized by several circuit configurations
`(N odd, above; N even, below).
`
`Low—pass filters, whose magnitude—squared functions have no finite zero, i.e., whose characteristic func-
`tions C(jQ) are polynomials, can be realized by lossless ladder networks consisting of inductors as the
`series elements and capacitors as the shunt elements. These types of approximations, also referred to as
`a11—poIe approximations, include the previously discussed Butterworth, Chebyshev Type I, and Bessel
`filters. Figure
`shows four possible ladder structures for even and odd N, where N is the filter order.
`In the case of doubly terminated Butterworth filters, the normalized values are precisely given by
`
`(2v—1)n
`aV=2sin —~—— , V=l,...,N
`2N
`
`(82.28)
`
`where av is the normalized L or C element Value. As an example we will derive two possible circuits for
`a doubly terminated Butterworth low—pass of order 3 with RB = 100 S2 and a cutoff frequency Q = QB =
`10 kHz. The element values from Equation 82.28 are
`
`1
`I1=2s1n
`
`(2—1)n
`
`RB
`=1=>Ll=E=159mH
`
`c2=2sin
`
`(4—1)7t
`6
`
`=2=>C2=
`
`2
`QCRB
`
`=3.183 nF
`
`(82.29)
`
`_
`
`I
`13=2s1n
`
`(5 —l)n:
`6
`
`RB
`==1=>L3=——=l.59mH
`QC
`
`A possible realization is shown in Figure 82.8.
`
`1009
`
`L,
`
`nl/in
`
`v,()
`
`c, T
`
`v.,>
`
`1009
`
`FIGURE 82.6 A third—order passive all—po1e filter can be realized by a doubly terminated third—order circuit.
`
`© 1999 by CRC Press LLC
`
`11
`
`

`
`Table 82.3. Element Values for low—pass filter circuits
`
`N = 2, Element Number
`
`N = 3, Element Number
`
`Filter Type
`Butterworth
`
`r2
`«xv
`1
`
`1
`1.4142
`1.4142
`
`2
`0.7071
`1.4142
`
`1
`1.5000
`1.0000
`
`2
`1.3333
`2.0000
`
`3
`0.5000
`1.0000
`
`0.5158
`1.0864
`1.0895
`0.4215
`0.7159
`co
`Chebyshev type 1
`1.0316
`1.1474
`1.0316
`——
`-
`1
`0.1—dB ripple
`0.7981
`1.3001
`1.3465
`0.7014
`0.9403
`oo
`Chebyshev type I
`1.5963
`1.0967
`1.5963
`——
`—
`1
`0.5 dB ripple
`0.1667
`0.4800
`0.8333
`0.3333
`10000
`on
`Bessel
`0.1922
`0.5528
`1.2550
`0,4227
`1.5774
`1
`._:_:_:_—
`
`".l."a.ble 823 gives normalized element values for the various all—pole filter approximations discussed in
`the previous section up to order 3 and is based on the following normalization:
`
`r, = 1;
`1.
`2. All the cutoff frequencies (end of the ripple band for the Chebyshev approximation) are QC = 1 rad/s;
`3.
`1'2 is either 1 or <><>, so that both singly and doubly terminated filters are included.
`
`The element values in Table 82.3 are numbered from the source end in the same manner as in Figure 82.4.
`In addition, empty spaces indicate unrealizable networks. In the case of the Chebyshev filter, the amount
`of ripple can be specified as desired, so that in the table only a selective sample can be given. Extensive
`tables of prototype element values for many types of filters can be found in Reference 4.
`The example given above, of a Butterworth filter of order 3, can also be verified using Table 82.3. The
`steps necessary to convert the normalized element values in the table into actual filter values are the same
`as previously illustrated.
`In contrast to all—pole approximations, the characteristic function of an elliptic filter function is a
`rational function. The resulting filter will again be a ladder network but the series elements may be
`parallel combinations of capacitance and inductance and the shunt elements may be series combinations
`of capacitance and inductance.
`Figure 82.5 illustrates the general circuits for even and odd N, respectively. As in the case of all—pole
`approximations, tabulations of element values for normalized low—pass filters based on elliptic approxi~
`mations are also possible. Since these tables are quite involved the reader is referred to Reference 4.
`
`Active Filter Design
`
`Active filters are widely used and commercially available with cutoff frequencies from millihertz to
`megahertz. The characteristics that make them the implementation of choice for several applications are
`small size for low frequency filters because they do not use inductors; precision realization of theoretical
`transfer functions by use of precision resistors and capacitors; high input impedance that is easy to drive
`and for many circuit configurations the source impedance does not effect the transfer function; low
`output impedance that can drive loads without effecting the transfer function and can drive the transient,
`switched capacitive, loads of the input stages of A/D converters and low (N4-THD) performance for pre-
`A/D antialiasing applications (as low as -100 dBc).
`Active filters use R, C, A (operational amplifier) circuits to implement polynomial transfer functions.
`They are most often configured by cascading an appropriate number of first— and second—order sections.
`The simplest first—order (N = 1) active filter is the first—order passive filter of Figure 82.3 with the
`addition of a unity gain follower amplifier. Its cutoff frequency
`is the same as that qiven in
`Equation 82.25. Its advantage over its passive counterpart is that its operational amplifier can drive
`whatever load that it can tolerate without interfering with the transfer function of the filter.
`
`© 1999 by CRC Press LLC
`
`12
`
`

`
`
`
`(C)
`
`Second—order active filters can be realized by common filter circuits: (A) Sallen and Key low—pass,
`FIGURE 82.7
`(B) multiple feedback bandpass, (C) state variable.
`
`The vast majority of higher—order filters have poles that are not located on the negative real axis in
`the s~plane and therefore are in complex conjugate pairs that combine to create second—order pole pairs
`of the form:
`
`I‘I(S)=S2+~03£S+(1): <:>s2+2as+a2+b2
`
`(82.30)
`
`II
`
`a ijb
`SIDm + O‘
`ll l ll
`
`—I
`
`l\'D
`
`where p1,p2
`mg
`
`Q
`
`The most commonly used two—pole active filter circuits are the Sallen and Key low—pass resonator, the
`multiple feedback bandpass, and the state Variable implementation as shown in Figure 82.723, E), and c. In
`the analyses that follow, the more commonly used circuits are used in their simplest form. A more
`comprehensive treatment of these and numerous other circuits can be found in Reference 20.
`The Sallen and Key circuit of Figure 82.7a is used primarily for its simplicity. Its component count is
`the minimum possible for a two—pole active filter. It cannot generate stopband zeros and therefore is
`limited in its use to monotonic roll-off transfer functions such as Butterworth and Bessel filters. Other
`
`limitations are that the phase shift of the amplifier reduces the Q of the section and the capacitor ratio
`becomes large for high—Q circuits. The amplifier is used in a follower configuration and therefore is
`subjected to a large common mode input signal swing which is not the best condition for low distortion
`performance. It is recommended to use this circuit for a section Q < 10 and to use an amplifier whose
`gain bandwidth product is greater than 100 f;,.
`The transfer function and design equations for the Sallen and Key circuit of Figure 82.73 are
`
`1
`
`H(s)=
`
`cc
`
`1
`2
`IRRZ 1
`2 T. #
`S+I%CzS+1%R2C1C2
`
`=
`
`of
`
`to P
`‘2+”p”‘”'°°5
`
`(82.31)
`
`© 1999 by CRC Press LLC
`
`13
`
`

`
`
`
`FIGURE 82.8 A three—pole Butterworth active can be configured with a buffered first~order RC in cascade with a
`two—pole Sallen and Key resonator.
`
`from which obtains
`
`2:
`
`1
`R1R2C1C2
`
`,
`
`Q=o3pR1C2 =
`
`RICE
`R261
`
`24
`li 1—4QC2
`C1
`
`4T1'fpQC2
`
`which has valid solutions for
`
`In the special case where
`
`3 2 4Q2
`C2
`
`(82.32)
`
`(82.33)
`
`(82.34)
`
`(82.35)
`
`=
`
`= R,
`
`then
`
`R2
`R1
`C=1/27tRfp, C1=2QC, and C2=C/2Q
`
`The design sequence for Sallen and Key low—pass of Figure 82.7a is as follows:
`
`For a required I; and Q, select C1, C2 to satisfy Equation 82.34. Compute R1, R2 from Equation 82.33
`(or Equation 82.35 if R1 is chosen to equal R2) and scale the Values of C1 and C2 and R1 and R2 to
`desired impedance levels.
`
`As an example, a three~pole low—pass active filter is shown in igure 82.8. It is realized with a buffered
`single—pole RC low—pass filter section in cascade with a two-pole Sallen and Key section.
`To construct a three—po1e Butterworth filter, the pole locations are found in Table 82.2 and the element
`values in the sections are calculated from Equation 82.25 for the single real pole and in accordance with
`the Sallen and Key design sequence listed above for the complex pole pair.
`From Table 82.2, the normalized pole locations are
`
`rp1=1.0oo,
`
`rp2=1.000,
`
`and Qp2=1.000
`
`For a cutoff frequency of 10 kHz and if it is desired to have an impedance level of 10 k9, then the
`capacitor values are computed as follows:
`
`© 1999 by CRC Press LLC
`
`14
`
`

`
`For R1: 10 k9:
`
`from Equation 82.25, C1:
`
`1
`2nR1rp,
`
`=
`
`1
`2n(10,000)(10,000)
`
`= 10
`2007»;
`
`=0.00159 uF
`
`-0
`
`For R2=R3= R= 10 k9:
`
`from Equation 82.35, C:
`
`1
`1
`2