Filters and Equalizers

Transcription

1 Filters and Equalizers By Raymond L. Barrett, Jr., PhD, PE CEO, American Research and Develoment, LLC

2 . Filters and Equalizers Introduction This course will define the notation for roots of olynomial exressions describing Linear- Time Invariant (LTI systems in the frequency domain, and relate the oerator notation to the time-domain resonse using comlex exonential notation. A single ole circuit will be introduced and resonses analyzed in the frequency domain and time domain. An ideal delay will be introduced for comarison and to set a reference for ste resonse behaviors. Polynomial root locations will be described in the comlex s-lane and comlex conjugate airs lotted and described using (ω, ζ notation as well as (τ, Q notation. Phasor notation will be introduced for evaluation of steady-state sinusoidal excitation of transfer functions. Second-order, comlex conjugate ole airs are introduced and the asymtotic behaviors develoed and contrasted to the single-ole behaviors in magnitude, hase, and grou delay attributes. Straight-line aroximations are roduced and the errors of aroximation discussed. Classical Butterworth, Chebyshev, and Bessel filters are introduced and the construction formulae develoed. The Cauer filter is also illustrated, but mathematical develoment not included. Frequency domain and time domain resonses are develoed using a th design form as reresentative of even-order forms and a 5 th order design as reresentative of oddorder forms. Only the 5 th order Bessel filter examle will be synthesized from the equations. A 5 th order equalizer will be develoed for the 5 th order Chebyshev and Cauer filter and shown to rovide equivalent results. An additional ole air will be added to the 5 th order equalizer and the justification and imrovements noted. Transformations will be discussed to convert low-ass rototye designs to high-ass and band-ass filters. The course is designed for a racticing engineer seeking a caability for designing and secifying filters and equalizers for frequency domain and time domain alications.. Notation Following in the ath of intellectual giants, we adot the Lalace exonential and Fourier comlex notations for describing the signals and the systems we emloy. Using the notations, we can accurately reresent signals in the time domain, as well as transform to the frequency domain for sectral analysis. We define a signal s imulse resonse as: x st ( t A e V [.] x Coyright Raymond L. Barrett, Jr. Page

3 Allowing the coefficient s to be a comlex number ermits the describing equation [.] to be reresentative of growing or declining exonential waveforms emloying the real comonent, as well as eriodic sinusoidal waveforms emloying the imaginary comonent, both in the time and frequency domains. We emloy the imulse resonse as a time-domain descritor, but evaluate instead the integral of the imulse resonse or ste resonse because of the difficulty in forming the infinite-magnitude/unit-area form of the imulse excitation. The unit ste is more intuitive and easier to formulate for time-domain simulation. We describe Linear, Time-Invariant (LTI systems using a rational olynomial descritive equation of the form given in equation [.] below. V V out in ( s ( s Coyright Raymond L. Barrett, Jr. Page 3 ( s ( N H ( s [.] D s We adot the classical aroach to system descrition with the roots of N(s defined as zeroes because the transfer function is zero at those root locations, and the roots of D(s defined as oles because the limit of the transfer function is infinite at those root locations and ossibly because the magnitude resembles an infinite-height tent-ole under the fabric of the s domain magnitude surface around that location. The degree of N(s is less than or equal to the degree of D(s for a realizable system, and each can be factored into first or second order factors. ( s σ [.] A first-order factor in s transforms to the time domain with the exonential imulse resonse exressed in equation [.] above. The second order factor in s exressed in equation [.3] below has three ossible cases, deending on the solution of its quadratic equation. s bs c [.3] ( b b c s ± [.] In equation [.], the square root contains a term known as the discriminant. There are three cases for the value of that discriminant, including ositive, zero, or negative values. If the discriminant is ositive, the second-order term can be factored into two distinct first-order factors with unequal real roots. If the discriminant is zero, the second-order term can be

4 factored into two equal first-order factors with two identical reeated roots. If, however, the discriminant is negative, the solution requires a square root of a negative number with an imaginary solution; hence two comlementary conjugate root locations. If c > b, we have: b c b s ± j [.5] To cover all cases, we define every root to be of the form given in equation [.6] below, with either the real σroot or ωroot, ossibly zero. s root σ j [.6] root root Root locations contain information about the transfer function behavior in both time and frequency domains. To characterize the resonse of a system, we emloy the ste resonse of the system in the time domain and the magnitude-hase resonse of the Bode lot in the frequency domain. To illustrate, we utilize first a single ole resistor-caacitor (RC system shown in figure. below, and contrast its behavior with an ideal delay.. The Single-Pole RC Circuit Schematic For the circuit shown in figure. above, we analyze the resonse as a voltage-divider using the ratio of imedances as follows: 6 Z R R Ω [.7] Ω Cs s Z C 6 [.8] V V ca in ( s H ( s Z C Z Z C R Cs R Cs [.9] RCs s Coyright Raymond L. Barrett, Jr. Page

5 The initial value theorem allows us to calculate: Vca ( Lim s s s [.] Because the result is indeterminate in the ratio, we aly LHoital s rule, as follows: ( s s ( Lim s s ( s ( s V Lim Lim s s [.] ca s s To evaluate the ste resonse, we use the integral of the imulse resonse and emloy the final value theorem as follows. First, the time domain integral of the imulse resonse is found as follows: t t λ t τ λ V ( [ ] t x λ dλ Ax e dλ e dλ e [.] The s domain integral is found as follows: H s s ( s ( s [.] The final value theorem allows us to calculate: V ( t Lim s Lim ca s s s( s ( s [.3] We redict that the final value from both the time-domain and frequency domain will agree and be a value of unity. We show the time-domain ste resonse behavior in the simulation of figure. below.. The Single-Pole RC Circuit Ste Resonse Coyright Raymond L. Barrett, Jr. Page 5

6 For the steady-state AC resonse, we evaluate the transfer function with the s jω substitution as follows: Vca ( j H ( j [.] V j We evaluate the magnitude and hase exressions as follows: in H ( [.5] ( ( Φ H ( tan [.6] We show grou-delay as the derivative of the hase exression as follows: Φ H ( tan ( [.7]. The Single-Pole RC Circuit Magnitude, Phase, and Grou Delay Resonses For contrast, we emloy an ideal delay and show its time domain resonse as follows:.3 The Ideal One Second Delay Time-Domain Ste Resonse Coyright Raymond L. Barrett, Jr. Page 6

7 . The Ideal One Second Delay Magnitude, Phase, and Grou Delay Resonses From figure. above, we see that the ideal delay has a uniform magnitude, regardless of the frequency but the single ole shown in figure. has an asymtotic magnitude of unity only u to its ole frequency associated with the one-second time-constant. The ideal delay has a constant hase exression derivative or grou-delay but the single ole has an asymtotic grou delay of unity only u to its ole frequency. The ideal delay is reresented as the multilier e -st for any ositive time index. 3. Root Locations The location of a ole (as well as a zero, in the comlex s lane reveals information about its behavior in both time and frequency domains as we have demonstrated in the section above using a single-ole examle. 3. The Comlex S Plane Real oles are located only on the Real axis, but we are mostly interested in oles with a real art that lie in the Left Half-Plane (LHP, because they describe the decaying exonentials found in hysical systems. Similarly, comlex conjugate ole airs are most often found in the left half lane for ractical systems, but some lossless ideal systems may Coyright Raymond L. Barrett, Jr. Page 7

8 also include the imaginary axis with a real comonent of zero. We shall see later that the LHP restriction is not as stringent for the location of zeroes as it is for oles. We emloy two of several conventions for describing the comlex-conjugate airs; following algebraic maniulation to the first canonic form: τ τ s s [3.] Q τ τ ± τ Q Q s root [3.] τ s root s root τ τ ± Q Q Q [3.] τ Qτ [ ± Q ] [ ± j ] [3.3] s root Q [3.] Q τ We have a set of equivalence relations ζ /Q, and τ /ω for the alternate canonic form of equation [3.5] as follows: s ς s [3.5] With the alternate form, we have: s root ς ± j ς [3.6] ς s root ς ± j ς [3.7] Coyright Raymond L. Barrett, Jr. Page 8

9 We shall see that these two equivalent canonical forms may be used interchangeably, but one or the other may have some advantage for imlementation issues. We show a comlex conjugate ole-air in the s-lane with each canonic reresentation as follows: 3. Comlex-Conjugate Pole-Pair in the Comlex S Plane (Both Forms Shown Note the indicated angle θ in figure 3. above is arameterized by the tan - relationshis as follows: θ tan [3.8] Q ς θ tan [3.9] ς We will find the θ angle useful in referring to root-attern loci in filter designs. We will utilize a hasor aroach to system behavior for frequency domain evaluation. It leads to a few intuitive tools for the classical system behaviors. In figure 3. below, we illustrate two hasors or hase-vectors denoted as and to reresent the system resonse from the two roots shown caused by the excitation from the steady-state excitation locus on the imaginary axis designated as the jω oint. Coyright Raymond L. Barrett, Jr. Page 9

10 3. Phasor Resonse for a Comlex-Conjugate Pair We reresent an evaluation of the artial roduct shown in equation [3.8] below. X ( s ( s ( σ j ( s ( σ j [3.8] The two root locations are: s σ j [3.9] The two hasors are: s σ j [3.] ( σ j [3.] We obtain the magnitudes as follows: ( σ j [3.] ( σ [3.3] ( σ [3.] Coyright Raymond L. Barrett, Jr. Page

11 We obtain the hase contributions as follows: From this reliminary work, we see that: Φ ( tan [3.5] σ Φ ( tan [3.6] σ X ( s s j [3.7] Φ Φ ( Φ ( [3.8] X ( If we convert the magnitudes to a logarithmic deendency, such as db, equation [3.7] also becomes a summation. If the hasors describe ole roots, the magnitudes are inverted and result in negative or subtractive db, whereas zeroes result in ositive or additive db. Similarly, oles subtract their hase contributions and zeroes add their hase contributions.. Second-Order Pole Behaviors The locus of the comlex-conjugate ole airs with constant characteristic frequency and differing daming or Q resents insights into the lacement of loci mas for filters and will be develoed in the following sections.. Comlex-Conjugate Pole Pair with Q.5,., and. Parameters (Concet We have shown in figure. above a concetual relationshi for the ole locations of a comlex-conjugate ole air with three different Q values. The angles are for relative Coyright Raymond L. Barrett, Jr. Page

12 direction only, but the loci are all on a circle of constant radius. We evaluate the location angles to find the resective values for θ as follows: θ tan tan (, tan, tan Q.5,.,. Q 3 5 [.] Q.5,.,. θ [.] o o o 9,3,. 5 Q.5,.,. In figure. above, we did not show the recise angles for the Q values indicated because the Q.5 value is actually a reeated double-ole exactly on the real axis and difficult to resolve as a ole-air. In figure. below, we show the magnitude, hase, and delay characteristics for the same ole-air collection.. Pole Pair Magnitude, Phase, and Grou Delay with Q.5,., and. The transfer function descrition of the behaviors show in figure. above is given with equation [3.] as the denominator of N(s/D(s and its roots as the solution of the D(s olynomial and N(s in the numerator as shown in equation [.] below. ( s ( N D s τ τ s Q s Coyright Raymond L. Barrett, Jr. Page [.] For the examle, τ, which imlies ω, and because ω πf, the characteristic frequency f in Hertz is /π.59 Hertz, as can be identified on the anels of figure. above. We exlore the asymtotic behaviors of figure. while referring to equation [3.9] to develo those asymtotes. We evaluate equation [.3] below at frequencies below, at, and above the characteristic frequency in the following using the notation of Ω ωτ as a normalizing arameter.

13 D ( Ω s j Ω ( Ω Q [.3] Φ / Ω D ( Ω tan [.] Q Ω GD ( Ω Ω Φ/ ( Ω Q D Ω Ω Ω Ω Q Ω GD τ ( ( Ω Ω Q Ω ( Ω GD Q Ω τ ( ( Ω Ω Q ( Ω Ω Q [.5] [.6] [.7] We evaluate the asymtotes and characteristic using the limits of Ω,, and aroaching infinity as follows:, Q, [.8] D Ω ( Ω,, We observe from equation [.] above that in the region with values of Ω >>, we can reduce equation [.] to the aroximation in equation [.9] as follows: D ( Ω Ω Ω>> Ω Ω Q Ω [.9] It is this exonential relationshi that rovides the asymtotic sloe of two decades magnitude decrease for every decade of frequency increase and leads us to resent figure. Coyright Raymond L. Barrett, Jr. Page 3

14 using the logarithmic frequency and magnitude scales. The ratio is the constant sloe of negative two decades magnitude er frequency decade. In contrast, the single ole behavior shown in figure. roduced a single decade er decade sloe. In our normalized notation, we reroduce the single ole magnitude relationshi with equation [.] as follows: H ( Ω [.] ( Ω The asymtote for Ω >> roduces the result: H ( Ω Ω>> [.] Ω Ω We see that equation [.] algebraically confirms the single-ole asymtotic behavior reviously shown in figure.. We will use the characteristic of a one-decade magnitude decrease er decade frequency increase er ole to hel define frequency selectivity of a collection of oles in a filter. For the Φ(Ω hase resonse, we observe in figure., a relatively symmetrical sloe with a nearly arithmetic deendence of hase er decade of frequency variation above and below the characteristic frequency, always with 9 o for Ω, indeendent of the Q value as indicated in equation [.] below. Φ Ω o o / D (, 9, 8 Ω,, [.] The grou delay is determined by the hase sloe in equation [.7], and is evaluated over the range of frequencies in equation [.3] below. τ GD( Ω, Qτ, [.3] Ω,, Q We see that the hase sloe is roortional to Q at Ω, and the least sloe aears for the real oles with the Q.5 value. It is instructive to note that the single-ole behavior exhibits the same shae hase deendency as shown in figure. above, but with the asymtotes given by equation [.] below. Φ [.] o o Pole ( Ω, 5, 9 Ω,, We observe in both figure. and figure. with Q.5, that the arithmetic hase sloe can be aroximated by a straight-line segment over a two-decade san. We restate equation [.] over those restricted limits in equation [.] below. Coyright Raymond L. Barrett, Jr. Page

15 Φ [.5] o o o Pole( Ω 5.7, 5, 8. 3 Ω,, We are able to use the results of equation [.5] to infer that a straight-line aroximation with the asymtotic value of o for frequencies lower than one decade below a ole s characteristic frequency, -9 o for frequencies greater than one decade above that ole s characteristic frequency, and -9 o /decade sloe within the two-decade range is a close aroximation of each real-ole s hase characteristic. Further, using the imlications we inferred from equation [.], we extend the aroximation to increase the sloe to twice that of a single ole for the comlex-conjugate ole air, but with the sloe increased by a multile of Q times greater sloe magnitude. We have reroduced the single-ole hase anel from figure. and the ole-air hase anel from figure. into figure. below and suerimosed the aroximate straight-line segments in red for comarison.. Single-Pole and Pole Pair Phase with Q.5,., and. Aroximations We have already discussed the asymtotic behaviors of the ole magnitude functions and resent similar straight-line aroximations for the magnitudes in figure.3 below. The left anel is the magnitude lot from figure. and the right anel is the magnitude lot from figure. and reroduced with the asymtotes from rior discussion..3 Single-Pole and Pole Pair Magnitudes with Q.5,., and. Aroximations Coyright Raymond L. Barrett, Jr. Page 5

16 The aroximations suorted by the results resented in figure. and figure., are sufficiently accurate to ermit construction of selectivity and hase aroximations for comlicated filter structures. In the time domain, we resent the results for the comlex ole-air and reroduce the results from the rior figure.3 and figure. shown together in figure. below.. Delay for Ideal, Single Pole, and Pole Pair with Q.5,., and. As shown in figure. above, the case with two real oles reresented with Q.5 resents twice the delay of the single ole reroduced in the center anel above. The resonant eaking and grou delay associated with Q values of. and. resents a resonant overshoot and ringing of the ste resonse that become imortant in dealing with digital signals in bandlimited filter structures. 5. Some Classical Filter Forms The classical aroach identifies the locus of ole ositions according to some desired behavior required for the filter in the time or frequency domain. In this section, we will briefly introduce the Butterworth filter, Chebyshev filter, Inverse Chebyshev filter leading to the Cauer filter, and Bessel filter and show some behaviors in time and frequency domains. We begin with all-ole filters and introduce zeroes. We introduce the maximally flat delay filter and introduce the need for equalizers to obtain good erformance in both time and frequency domains. The N th order Butterworth filter [S. Butterworth, On the theory of Filter Amlifiers, Wireless Engineer, vol. 7, , Oct., 93] has a monotonically decreasing magnitude function as the frequency increases. The characteristic frequency and order, or number of oles arameterizes the lowass Butterworth filter. A straight-line aroximation as discussed in the rior section is usually emloyed to determine the required number of oles to reject some frequency comonent at a higher frequency than the characteristic. The ratio of the rejected frequency to the characteristic is calculated and the base- logarithm Coyright Raymond L. Barrett, Jr. Page 6

17 obtained to determine the number of decades that searate the two frequencies. The required attenuation at the rejected frequency used to determine the requisite attenuation er decades of searation, and thus the minimum number of oles required. For examle, we will calculate the order of a Butterworth filter required to rovide attenuation by a factor of at a frequency that is seven times the characteristic frequency. The frequency we must reject is.8 decades above the characteristic frequency (log[7].8. The rejection factor of reresents three decades (log[] 3. We need a filter with 3 decades magnitude decrease for.8 decades frequency increase, so the minimum ratio is 3/ oles. We do not have the otion of fractional oles, so we require a th order Butterworth filter. The Butterworth function is defined as: BN ( Ω N Ω [5.] The Butterworth function is used to define the squared-magnitude of a Butterworth filter. Ω [5.] Ω ( j ( Ω N H B N B N The definition used in equation [5.] is obtained indirectly from geometrical constructions in the comlex s-lane. We are using the entire s-lane, and exress the magnitude function with right-half-lane (RHP, and mirrored left-half-lane (LHP factors for construction as follows: H N s H N s s j H j s B N B H N ( jω [5.] s s H N H N N s [5.3] The ole locations are defined as the denominator roots: s N [5.] s N e j( k π [5.5] Coyright Raymond L. Barrett, Jr. Page 7

18 The key to Butterworth s derivation is the exression of the higher-order roots of (- as an exonential. The exonent of N describes the total collection of oles in both the RHP and LHP. We construct the mirror image distribution and use only the stable LHP oles for the filter. The location of oles must consist of a single real ole for N an odd number, or comlex conjugate ole-airs for N an even number, due to the nature of oles themselves. The hysical realization imoses symmetry on the s-lane locations. The unit magnitude of the exonential in equation [5.5] imoses a locus of a unit circle. We can construct the s- lane ma from the information above. We show ole location examles for th order and 5 th order Butterworth filters in figure 5. below. 5. Butterworth Pole Locations for th and 5 th Order Filters From the geometry of the oles on a unit circle, the normalized ole locations are calculated for the LHP locations only. The Q for each comlex conjugate ole air can be recalculated and the radius of the circle is the characteristic ω frequency. Starting from the ole with the highest Q factor in each diagram, we see that the incremental angle is always bisected by the imaginary axis. That angle can be deduced to always be π/n radians. We increment by π/n radians until we reach or exceed π/ radians and cease calculations without using the last result. From figure 3., we see that a ratio can be used to identify the Q factor for a comlex conjugate ole air: Qτ tan( θ [5.6] Q Qτ Q ( ( Coyright Raymond L. Barrett, Jr. Page 8

19 ( Q [5.7] tan( θ Q [5.8] tan( θ We calculate the Q values for the two sets of comlex conjugate ole-airs in table 5. below using equation [5.8] above. Table 5. - Butterworth Pole Q factors th Order 5th Order Angle Radians Q Angle Radians Q We use the same imlementation for the Butterworth filters that we emloyed for the comlex conjugate ole-air characterization but using two instances. We arameterize two sets, one set for the th order filter, and the other set for the 5 th order filter. We introduce a single ole into the set for the 5 th order filter. We then roduce the exected results below and obtain the same characteristic frequency as rior examles. 5. Magnitude, Phase, and Grou Delay for th Order Butterworth Filter Coyright Raymond L. Barrett, Jr. Page 9

20 5. Magnitude, Phase, and Grou Delay for 5 th Order Butterworth Filter In contrast, the 5 th order filter achieves a five-decade magnitude decrease er decade of frequency increase where the th order filter achieves a four-decade magnitude decrease er decade of frequency increase. That four-decade decrease is sufficient to meet the examle requirement of a filter with 3 decades magnitude decrease for.8 decades frequency increase. Both the th order filter and 5 th order filter aear to rovide the maximally-flat resonse magnitude of a Butterworth filter. 5.3 Ste Resonse for th and 5 th Order Butterworth Filters As shown in figure 5.3 above, both the th order filter and 5 th order filter exhibit the exected resonant eaking and grou delay associated with the Q emloyed. The resonant overshoot and ringing of the ste resonse must be considered in dealing with digital signals in bandlimited filter structures. The Butterworth filter has a monotonically decreasing magnitude function as the frequency increases. It is maximally flat in the sense that all derivatives tend to zero as the frequency tends to zero. In contrast, the Chebyshev filter constructs the assband from zero to the characteristic frequency with an equirile magnitude by emloying Chebyshev olynomials. The rile, as well as the characteristic frequency and order are arameters of the lowass Chebyshev filter. We shall develo a 5 th order Chebyshev filter and contrast its behavior with the Butterworth filter above. The Chebyshev olynomial is defined as: ( ( Ω cos N cos ( Ω T N [5.9] The olynomial is difficult to evaluated directly, but is attacked using a recursive relationshi develoed from trigonometric identities. First, for N, and N, we see: ( Ω cos( T [5.] Coyright Raymond L. Barrett, Jr. Page

21 ( Ω ( Ω cos ( Ω Then, by use of the recursive trigonometric identity: cos T cos [5.] (( n x cos( nx cos( x cos( ( n x [5.] The higher order olynomials can be defined recursively by: ( Ω ΩT ( Ω T ( Ω T [5.3] N N N The olynomials are emloyed much like the Butterworth case, over the entire s-lane using the magnitude-squared function. H T N ( jω [5.] ε T The ε arameter rovides the rile scaling value, and deends on the eak-to-eak rile allowed. For a rile value AMax exressed in db (the common measure, we have: ( A N ( Ω Max ε [5.5] We will develo a 5 th order filter with db rile to contrast with the Butterworth examle reviously exlored. First, we find the ε arameter using equation [5.5] as follows: ( ε [5.6] db We require the 5 th order Chebyshev olynomial, so we invoke the recursive definition to extend from the lower orders we have: ( Ω ΩT ( Ω T ( Ω Ω T [5.7] 3 ( Ω ΩT ( Ω ( Ω Ω( Ω Ω Ω 3Ω T [5.8] 3 T 3 ( Ω ΩT ( Ω T ( Ω Ω( Ω 3Ω ( Ω T [5.9] 3 ( Ω ( 8Ω 6Ω ( Ω ( 8Ω 8Ω T [5.] From equation [5.] and equation [5.] we have: Coyright Raymond L. Barrett, Jr. Page

22 T ( jω [5.] ε H N ( 8Ω 8Ω.589( 8Ω 8Ω Armed with equation [5.], we can find the oles for this 5 th order Chebyshev filter, but we do not have a great insight from a numerical solution. As an alternative, we re-visit the geometry of the s-lane ole locations from a different, but still algebraic direction. We roceed by making a substitution from equation [5.9] that defines the Chebyshev olynomial into equation [5.] that defines the filter magnitude with the result in equation [5.] that follows. H T N ( jω [5.] ε T Ω ε cos N cos Ω N [ ( ] ( ( We know that the Lalace oerator s becomes the Fourier oerator jω in the steady-state AC analysis, but we are using the normalized Ω ω/ω substitution. We can make the definition: s j Ω [5.3] We solve for the relationshi: Ω s [5.] j We define a comlex number ξ as an intermediate variable, but retain the relationshi to the roots that define s σ jω as a location in the s- lane: ξ α jβ cos s cos ( Ω [5.5] j Inverting the defined relationshi in equation [5.5], we can roduce: j s cos ( α jβ [5.6] j ( σ j cos( α jβ [5.7] σ j j cos ( α jβ [5.8] Coyright Raymond L. Barrett, Jr. Page

23 Using equation [5.8], we can solve for the ole locations in the s-lane using an identity to rovide the needed relationshi: cos cos cos ( α jβ cos ( α jβ cos ( α j β ( α j β cos ( x jx jx e e [5.9] ( α jβ j( α jβ j β jα β jα e e e e e e [5.3] ( α jβ ( cosh β sinh β ( cosα j sinα ( cosh β sinh β ( cosα j sinα [5.3] ( cosh β sinh β j sinα( cosh β sinh β cosα cosα ( cosh β sinh β j sinα( cosh β sinh β ( cosh β sinh β cosα( cosh β sinh β cosα j sinα ( cosh β sinh β j sinα( cosh β sinh β [5.3] [5.33] cosα cosh β cosα cosh β j sinα sinh β j sinα sinh β [5.3] cos ( α jβ cosα cosh β j sinα sinh β [5.35] We can substitute the relationshi derived in equation [5.35] into the rior equation [5.8] to obtain equation [5.36] below. σ j j cos( α jβ j [ cosα cosh β j sinα sinh β ] [5.36] j We equate the comonents as follows: sinα sinh β cosα cosh β σ j [5.37] Coyright Raymond L. Barrett, Jr. Page 3

24 sinα sinh β σ [5.38] Coyright Raymond L. Barrett, Jr. Page cosα cosh β [5.39] From equation [5.38] and equation [5.39], we can locate the oles in the s-lane, given the corresonding α and β values. ( j Ω [5.] ε [ ] [ cos ( N cos ( Ω ] ε cos ( Nξ H T N From equation [5.], we identify the oles of the magnitude-squared as: We factor equation [5.] as follows: [ cos ( ] ε N ξ [5.] ( ε cos( Nξ ( jε cos( Nξ ± jε cos( Nξ We solve equation [5.] as follows: j [5.] We again emloy equation [5.35] as follows: cos ( N ξ ± cos N( α jβ [5.3] jε cos ( Nξ cos Nα cosh Nβ j sin Nα sinh Nβ ± [5.] jε We equate real and imaginary arts as follows: cos N α cosh Nβ [5.5] sin N α sinh Nβ ± [5.6] ε We solve equation [5.5] by noting that cos(x for x /-π/ and oints with increments of π added. We simlify as follows:

25 cos N α π [5.7] Nα ± kπ π N α ± k [5.8] ( ( k π α ± [5.9] N Because we have imosed the cos(nα condition, we know that sin(nα and we are able to solve equation [5.6] as: sinh N β ± [5.5] ε N β ± sinh [5.5] ε β ± sinh N ε [5.5] We return to equation [5.38], and equation [5.39] with the values of a identified in equation [5.9] above and b identified in equation [5.5] above and roduce the LHP ole location equations as follows: ( k sin π sinh sinh sinα sinh β N ε σ N [5.5] ( k cos π cosh sinh cosα cosh β N ε N [5.53] We are enabled to synthesize a Chebyshev filter as needed. We use equation [5.5] and equation [5.53] above because we have a closed form algebraic exression for the ole locations for all values of the ossible arameters. We can also maniulate the two equations into a slightly different form, as follows: Coyright Raymond L. Barrett, Jr. Page 5

26 Coyright Raymond L. Barrett, Jr. Page 6 β σ α sinh sin [5.5] β α cosh cos [5.55] Using the following identity, we have: cosh sinh cos sin β β σ ω α α [5.56] sinh cosh sinh sinh ε ε σ ω N N [5.57] Equation [5.57] exresses the locus of an ellise in the s-lane with a semi-minor axis given by equation [5.58] below, and a semi-major axis given by equation [5.59] below: ε ω sinh sinh N a [5.58] ε ω sinh cosh N b [5.59] Algebraic maniulation allows equation [5.58] and equation [5.59] above to be redefined as follows N N a ε ε ε ε ω [5.6] N N b ε ε ε ε ω [5.6]

27 The definitions above ermit the construction in the s-lane of the Chebyshev filter from two Butterworth circles with radii equal to the semi-axes given and shown in figure 5. below. 5. Pole Locations for 5 th Order Chebyshev Filter We note that the oles on the ellise are entirely within the ω circle. We find the unnormalized ole osition as ωeff as follows: σ ω [5.6] eff From the characteristics of a comlex-conjugate ole-air, we find the Q factor from the ratio of σ/ω as follows: σ tan( θ [5.63] Q ( ( Q σ [5.6] Q σ Coyright Raymond L. Barrett, Jr. Page 7 [5.65] Table 5. below summarizes the ole locations for the 5 th order Chebyshev filter with db rile. Table th Order Chebyshev Poles σ ω E-7

28 ω effective Q Magnitude, Phase, and Grou Delay for 5 th Order Chebyshev Filter In figure 5.5 above, we show the magnitude, hase, and grou delay resonses for the th order Chebyshev filter, and in figure 5.6 below, we show its ste resonse. 5.6 Ste Resonse for 5 th Order Chebyshev Filter To enable comarison of the 5 th order Chebyshev magnitude resonse with the 5 th order Butterworth, we resent the results in the same anel as shown in figure 5.7 below. We see that the Chebyshev filter with db of rile below its characteristic frequency is nearly a decade better in attenuation above a common characteristic frequency in comarison to the Butterworth filter. The Chebyshev design is used where the rile can be tolerated and imroved selectivity near the characteristic frequency is needed. 5.7 Magnitude Comarison for 5 th Order Butterworth and Chebyshev Filter Coyright Raymond L. Barrett, Jr. Page 8

29 To enable comarison of the 5 th order Chebyshev ste resonse with the 5 th order Butterworth, we resent the results in the same anel as shown in figure 5.8 below. 5.8 Ste Resonse for 5 th Order Butterworth and Chebyshev Filter We see that greater grou delay of the Chebyshev does not severely alter the overshoot, but the ringing of the higher Q factor oles makes its ringing much longer duration. The inverse Chebyshev filter can be constructed following the same derivation ath for the Chebyshev filter itself, but using the magnitude-squared definition in equation [5.66] below: H T N ( jω ε TN ε T ( Ω ( N Ω ε [ cos ( cos ( ] N k π Ω sec N [5.68] Coyright Raymond L. Barrett, Jr. Page 9 Ω [5.66] The Inverse Chebyshev filter is maximally flat in the ass-band, much like the Butterworth, but equal-rile in its sto-band rejection. The notion of the inverse relationshi extends to the ole and zero locations. The oles of equation [5.56] are derived as we have already done for the Chebyshev itself, but with the argument inverse. The oles of equation [5.56] are obtained by inverting the oles of the original Chebyshev filter in olar notation and each location. The oles are exressed as a magnitude and angle the inverse retains the angle but the magnitude is inverted. With the angle retained, the Q factor is invariant, but the ωeff transforms to outside the enclosing semi-major axis circle. The zeroes occur at frequencies determined by the numerator equation with the roots at: ( ± ( k Ω π N cos [5.67] (

30 Table th Order Inverse Chebyshev Roots ω effective Q ω z e6 We lot the results for the Inverse-Chebyshev design based on the rior 5 th order Chebyshev ole locations in figure 5.9 below. 5.9 Magnitude Resonse for 5 th Order Inverse Chebyshev Filter The inverse Chebyshev filter is discussed here because it is the first design to introduce zeroes, but it is not used widely. The characteristic frequency defines the edge of the stoband, rather than the ass-band. Also, the rile factor for an inverse Chebyshev design must be re-calculated from the desired sto-band characteristic and used in the redecessor rototye Chebyshev rior to inversion. The Inverse Chebyshev has a rejection in the stoband bounded by the value given in equation [5.69] below. Atten [5.69] ε for attenuation of 6dB., we solve for ε as follows: ε [5.7] Atten We see that ε is a small number and the rototye redecessor Chebyshev filter is much different from the examle we have exlored. Also, for N an odd order, the attenuation tends to a single ole roll-off, and to a constant for N an even order. There are much better filters Coyright Raymond L. Barrett, Jr. Page 3

31 for selectivity than the Inverse Chebyshev, but they, too, are based on ellitic loci in the s- lane. The best selectivity of a ole-zero design can be obtained using a Cauer filter. Unfortunately, the derivation of the ole locations is even more involved than that for a Chebyshev design and will not be included here. Instead, we show that we can use a synthesized Cauer filter and show that we can aroach much of its behavior as similar to the Chebyshev design. 5. Magnitude, Phase, and Grou Delay for 5 th Order Cauer Filter The Cauer design uses Ellitic functions to determine the ole ositions in the ass-band, as well as in the sto-band. More arameters are required to determine the two deendent rile behaviors. The 5 th order design ermits two zero airs in the sto band, as well as the zero at infinity that roduces the residual single-ole asymtote at the high end of the sto-band. The discontinuities in the hase resonse and grou delay are artifacts of the lotting routine that is unable to interret the raid hase sloes through infinite-q zeroes. Phase resonse, as well as grou delay in the sto-band should be interreted as continuous. 5. Ste Resonse for 5 th Order Cauer Filter The ste resonse for the 5 th order Cauer filter design resembles that for the 5 th order Chebyshev filter to which it is related. In figure 5. below, we show the magnitude resonse for each of the 5 th order filters we have discussed (excet the Inverse Chebyshev, alongside a detail anel. Coyright Raymond L. Barrett, Jr. Page 3

32 5. Magnitude Resonses for 5 th Order Butterworth, Chebyshev, and Cauer Filters In figure 5. above, we see that the Butterworth filter has a monotonically decreasing magnitude function and tends to a 5-decade magnitude decrease er decade of frequency increase as exected in a 5 th order design. In the detail, we see that the Butterworth filter is truly maximally flat rior to its characteristic frequency. The Chebyshev filter, also has a similar asymtotic rate, but near the characteristic frequency, it dros to a greater attenuation for every frequency above the same characteristic frequency as the Butterworth does. In the detail, it is also clear that the Chebyshev filter fluctuates between unity and a.9 magnitude, a value that translates to a db relative attenuation. We see in the detail in figure 5. above that the Cauer filter has a magnitude fluctuation over the same san and with nearly identical resonse, in comarison to the Chebyshev filter. In the first decade of frequency increase above the characteristic frequency, though, the Cauer filter has a magnitude that decreases at a faster rate than the Chebyshev filter, reaching the equal rile level caused by the Cauer filter zero attern before the Chebyshev filter. The Chebyshev filter continues with the constant 5-decade magnitude decrease er decade of frequency increase as exected in a 5 th order design, but the Cauer filter rovides its equal rile level, ultimately returning to a single-ole -decade magnitude decrease er decade of frequency increase. Desite the differences in erformance at frequencies above the characteristic frequency, the 5 th order Chebyshev filter and 5 th order Cauer filter rovide an excellent match in the assband and also the time domain resonses as shown in the ste resonse shown in figure 5.3 below. 5.3 Ste Resonse for 5 th Order Chebyshev, and Cauer Filters Coyright Raymond L. Barrett, Jr. Page 3

33 The excellent domain ste resonses shown in figure 5.3 above are related to the close match in grou delay resonses shown in figure 5. below. 5. Grou Delay for 5 th Order Chebyshev, and Cauer Filters We will use the closely matched grou delay resonses to use the Chebyshev filter behavior as a temlate for grou delay equalization of the Cauer filter. Each of the filter designs we have discussed thus far have increased the discrimination between the ass-band and the sto-band and imroved selectivity. As each imrovement was made, the grou delay, overshoot and time-domain resonse got worse. If we aroach the frequency searation issue between the ass-band and sto-band with the intent of keeing the grou delay as constant as ossible, we see there is another family of filter designs. We first introduced the ideal delay, reresented as the multilier e -st for any ositive time index and showed the result in figure. for the one-second delay. We reeat that illustration here as figure 5.5 below for convenience The Ideal One Second Delay Magnitude, Phase, and Grou Delay Resonses If we start from the definition of the delay function, we can exress a transfer function as: H T ( s cosh( sinh( st e st st [5.7] Coyright Raymond L. Barrett, Jr. Page 33

34 In equation [5.7] above, we have the entire transfer function, not the magnitude-squared function that we have emloyed in rior derivations. Thus, we can find the oles of equation [5.7] above by setting the denominator to zero and solving the resulting D(s equation as follows. cosh( st sinh( st [5.7] We form a series exansion of the hyerbolic functions and erform a continued fraction exansion of the series at the degree we require. ( 6 8 ( st ( st ( st ( st ( st cosh st [5.73]!! 6! 8!! ( st ( st ( st ( st ( st sinh( st st [5.7] 3! 5! 7! 9!! We illustrate by forming an aroximation by exanding the denominator using a continued fraction exansion to the 5 th order. To emloy the method, we form the ratio with the greater degree olynomial in the denominator and erform long division. We begin the rocess with equation [5.75] below. cosh sinh ( st ( st st ( st ( st ( st 6! st 3! ( st ( st ( st! 6! 6 st 5! 7! 6 ( st ( st ( st ( st ( st ( st (! 3!! 5! 6! 7! 6 8 st ( st ( ( ( ( st st st st st 3! 5! 7! 9!! 6 [5.76] cosh sinh ( st ( st st ( 3!! ( ( 5!! ( 3 ( 7! 6! ( 5 ( 9! 8! ( 7 st st st st 3!! 5!! 7! 6! 9! 8! 6 8 ( st ( st ( st ( st ( st 3! 5! 7! 9!! Coyright Raymond L. Barrett, Jr. Page 3

35 Filters and Equalizers 6 8 st 3 5! 7! 9!! 3! 6 8 st ( st ( st ( st ( st ( st 3! 5! 7! 9!! ( st ( st ( st ( st ( st [5.77] After the series of transformations above, we are left with a remainder with the greater degree olynomial in the numerator, so we invert the fraction as follows, and iterate the same stes as above, adding terms from the original series as needed. cosh sinh ( st 6 8 ( st st ( st ( st ( st ( st ( st st 3 5! ( st ( st ( st ( st ( st 3! 6 7! 5! 7! 8 9! 9!!! 3! st 3 st st 3 3! 3 5! 5! 6 7! 6 7! 7! 8 9!! 6 ( st 3 ( st 3 ( st 3 ( st 5! 8 3 9!! 3! 6 8 ( st 3 ( st 3 ( st 3 ( st 3 ( st 8 9! st 3 st st 3 8 5! 5! 7! 8 7! 8 9! 9! 8! 6 8 ( st ( st ( st ( st ( st 6 8 ( st ( st ( st ( st ( st 3! 36 3! 3! st 3 st st 5 5! 7 7! 8 7! 9! 9!! 3! ( st ( st ( st ( st 6 8 ( st ( st ( st ( st ( st 36 3! 36 3! 9 [5.78] We continue the fraction exansion as: Coyright Raymond L. Barrett, Jr. Page 35

36 ( ( st Filters and Equalizers cosh st [5.79] sinh 3 st st 5 st 7 st 9 st st We choose a 5th order solution, so we truncate and solve: cosh sinh cosh sinh cosh sinh ( st ( st ( st ( st ( st ( st [5.8] st 3 st 3 st 5 st 5 st 7 st st 63 ( st st 9 9sT 9sT [5.8] st 3 st 3 st 5 9sT st 5 st ( st 63 ( st 35 9( st 3 ( st 63sT [5.8] st 3 st 3 st 5 9sT st 5 st ( st 63 cosh sinh cosh sinh ( st ( st 3 ( st st 3 ( st 63( st st ( st 35 ( st st 3 ( ( st 35 ( st 63( st st ( ( st 35 ( st 35 9( st 3 ( st 63sT [5.83] [5.8] Coyright Raymond L. Barrett, Jr. Page 36

37 cosh sinh cosh sinh ( st ( st ( st ( st st 3 ( st 35( st ( st 5( st 95 5( st ( st ( st 5( st 95( st [5.85] [5.86] Therefore, we sum the cosh(st and sinh(st olynomials to obtain the oles: 5 3 ( 5( st 5( st ( st 95( st 95 st [5.87] ( ( st ( st ( st ( st st [5.88] H T ( s [5.89] 5 3 ( st 5( st 5( st ( st 95( st 95 The transfer function in equation [5.89] above defines a 5 th order Bessel filter. We find the roots numerically and resent the results in table 5.3 below. Table 5.3-5th Order Bessel Roots σ ω ω effective Q Magnitude, Phase (Log and Lin, and Grou Delay for 5 th Order Bessel Filter We see in figure 5.6 above that the Bessel filter has a monotonic decrease in magnitude as the frequency increases and asymtotic 5 th order decrease associated with a five-ole behavior. For frequencies below its characteristic frequency at ~.7Hz for the -second Coyright Raymond L. Barrett, Jr. Page 37

38 delay, the magnitude is not as flat as the 5 th order Butterworth, and certainly not as selective as the Chebyshev or Cauer filters. The Bessel filter, however, does exhibit the flat grou delay characteristic that the design objective set. The flat grou delay is shown in the far right anel and the relationshi to the linear hase is seen in the anel adjacent to the grou delay. The result of the flat delay characteristic is to obtain the ste resonse shown in figure 5.7 below. 5.7 Ste Resonse for 5 th Order Bessel Filter with Ideal for Reference We see in figure 5.7 above that the resonse curve asses the 5% magnitude very near the -second delay as romised by the design objective. The resonse is obtained with nearly no over-shoot as exected from all frequency comonents exeriencing the same delay. The Bessel filter may offer sufficient selectivity for some alications, but for those alications requiring both good frequency domain selectivity and time domain resonse, the solution may require adding a delay equalizer to a filter with the requisite selectivity. 6. Delay Equalizers Delay equalizers are constructed from the class of All-Pass filters. All ass filters are constructed with a frequency-deendent hase behavior, but a unity-magnitude transfer function. To achieve the behavior, the all ass filter emloys oles in the LHP and zeroes in the RHP at mirror image locations as shown in figure 6. below. In figure 6. below we show the ole-zero lot in the s-lane with the oles taken from the Butterworth th and 5 th order filter examle. We have added RHP zeros in mirror image locations and shown the hasors for the highest frequency Pole-Zero (PZ air for discussion in each anel. Coyright Raymond L. Barrett, Jr. Page 38

39 6. All Pass Filter th Order and 5 th Order Pole Locations with Phasors for PZ Pairs Because the oles are the roots of the denominator D(s olynomial and the zeroes are the roots of the numerator N(s olynomial, we can construct hasor airs for each PZ air in the all ass filter. Each hasor must have the same magnitude or length from the mirror image location to the jω excitation locus on the imaginary axis. Therefore, the magnitude ratio for mirror image PZ airs is unity for all airs. We see also that the hasor angles are comlement to each other. We equate θ -θz for the hasor airs, but note that the contribution from the numerator of a transfer function adds angles and from the denominator of subtracts angles. The result is a transfer function with unity magnitude and twice the hase contribution from the LHP oles alone. 6. All Pass Filter th Order Magnitude, Phase, and Grou Delay We see in figure 6. above that the magnitude is indeed unity across the entire frequency san for the th order all ass filter. We see also that the center anel of figure 6. above indicates twice the hase angles as the th order Butterworth filter with the same ole locations as indicated in figure 5. in the revious section. Likewise, the consequent grou delay is twice that of the th order Butterworth filter also as shown in the similar rightmost anels of the figures for each imlementation. Coyright Raymond L. Barrett, Jr. Page 39

40 6. All Pass Filter 5 th Order Magnitude, Phase, and Grou Delay We see in figure 6. above that the magnitude is also unity across the entire frequency san for the 5 th order all ass filter, too. We see also that the center anel of figure 6. above indicates twice the hase angles as the 5 th order Butterworth filter with the same ole locations as indicated in figure 5. in the revious section. Likewise, the consequent grou delay is twice that of the 5 th order Butterworth filter also as shown in the similar rightmost anels of the figures for each imlementation. 6.3 All Pass Filter th and 5 th Order Ste Resonse The ste resonses of the th and 5 th order all ass filters are shown for comleteness in figure 6.3 above. For these realizations there is no aarent use for this behavior. We use a goal-seeking algorithm to roduce an equal rile aroximation to a flat delay characteristic for the 5 th order Chebyshev and Cauer filter designs we have exlored in the revious section. The goal seeking algorithm roduced the set of 5 th order equalizer oles shown in table 6. below, along with an additional ole air for a 7 th order equalizer. The goal for the 5 th order equalizer was set to roduce an equal rile delay u to and including the delay for the 5 th order Chebyshev and Cauer filters. The goal for the additional ole air was to include its delay at a frequency higher than the combined rior effects. Coyright Raymond L. Barrett, Jr. Page

41 Table th & 7th Order Equalizer Pole # 3 ω effective Q We show the results for the 5 th order sections\ grou delay contributions in figure 6. below. 6. All Pass Filter Section Grou Delay Plot for Table 6. Pole, Pole, and Pole 3 We combine a cascade of the sections found in figure 6. above to roduce the 5 th order equalizer shown in figure 6.5 below. 6.5 All Pass 5 th Order Equalizer Magnitude, Phase, and Grou Delay The 5 th order equalizer grou delay we found does not have an intuitive delay deendency in figure 6.5 above, until we add the delay to the grou delay for the 5 th order Chebyshev or Cauer filter to roduce the magnitude and grou delay results in figure 6.6 below. Coyright Raymond L. Barrett, Jr. Page

42 6.6 All Pass 5 th Order Equalizer Alied to Chebyshev, and Cauer Filters The 5 th order equalizer grou delay we found comlements the filter delay to make the sum nearly constant, as shown in figure 6.6 above, and in detail in figure 6.7 below. 6.7 All Pass 5 th Order Equalized Detail for Chebyshev, and Cauer Filters 6.8 Grou Delay for 5 th Order Chebyshev, and Cauer Filters without the Equalizer We see in figure 6.8 above that both the Chebyshev and Cauer filters exhibit a grou delay variation that sans from about seconds to about 3 seconds, or a 9 second difference. The 5 th order equalizer has increased the total grou delay by adding delay at low frequencies in a attern that makes the sum more nearly constant. The variation is reduced to near a second delay difference in the ass band, or a reduction to about times smaller variation. Higher order equalizers can achieve flatter results with less variation than the 5 th order examle above. With the equalizer, and without it, we show the ste resonse behaviors in figure 6.9 below. 6.9 Ste Resonses for 5 th Order Chebyshev, and Cauer Filters Coyright Raymond L. Barrett, Jr. Page

43 We see in figure 6.9 above that both the Chebyshev and Cauer filters exhibit less overshoot and reduced ringing after the transition than the results without equalization. The Chebyshev result is on the left and the Cauer is on the right, but the difference is nearly indistinguishable. The 5 th order equalization utilizes the grou delay of the highest frequency ole of the filter, adding delay at lower frequencies to flatten the comosite delay. Because the highest frequency ole of the original filter has its highest Q factor also, it contributes the eak delay to the summation. Unfortunately, the filter bandwidth extends to higher frequencies than that eak grou delay and the lower grou delay at higher frequencies allows high frequency comonents to arrive with less delay than lower frequency comonents. The lack of grou delay above the highest frequency ole is the cause of the under-shoot and ringing before the equalized transition occurs. We have added another le air all ass section, making the equalizer a 7 th order solution. That ole is listed as ole number in table 6. and occurs at a higher frequency than the eak grou delay of the original 5 th order filters. In figure 6. below we show the grou delay contribution for this new ole air. 6. All Pass Section Grou Delay for 7 th Order Equalizer Augmentation 6. Grou Delay of All Pass 7 th Order Equalizer Alied to Cauer Filter We see in figure 6. above that the additional grou delay adds another eak to the resonse at a higher frequency and above the highest rior eak delay frequency. The effect of the additional delay bandwidth is a slight imrovement in both the undershoot and ringing as shown in figure 6. below. Coyright Raymond L. Barrett, Jr. Page 3