0300Smith46 - The fundamentals of linear-phase filters for digital communications.pdf

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46 www.rfdesign.com March 2000

N umerous sources of technical infor-mation on linear-phase filters canbe found. However, many of thesesources label systems as distortionlessor dispersionless without any formaldefinition of either. Moreover, the condi-

tion where a linear-phase filter has aphase bias (an inverter) also is rar elyaddressed. This article presents a defini-tion for a distortionless system and dis-cusses the n eed for normal iz ing th ephase-response curve of an inverting lin-ear-phase or inverting equalization filter.

If one were to plot the phase-delay(also known as propagation delay) vs.f requency fo r t he i nve r t i ng l i nea r -phase filter, an erroneous result willfol l ow . One wou ld see a dominan thyperbolic curve that could be inter-preted as increasing in nonlinearity asthe frequency gets smaller, asymptoti-

cally approaching infinite phase-delayat DC. This result shows up even whenan ideal inverter i s considered (seebelow), indicating dispersive behavior.

This result is wrong. In addition, thenotion of phase at DC ma kes no sense.

If a computer simulation was per-form ed ove r a f requency ran ge fa renough above DC (to a point where thehyperbolic shape could no longer bevisually detected), one could easi lybelieve the results and modify their fil-

ter design incorrectly. The simulationdata could still be greatly in error dueto the hyperbolic behavior. Phase-delayerrors of 50 ns at 10 MHz and 5 ns at100 MHz can be en count ered.

To design an ideal distortionless fil-ter, a designer needs to implement notonly a filter with linear-phase but alsoa f i l t e r w i t h a p h a s e - s h i f t o f z e r od e g r e e s a t D C . T h i s ca n b e e a s i l yshown by injecting a saw-tooth wavethr ough an ideal inverter (phase shift = for all frequencies). (See Figure 2.)The resulting waveform has only beeninverted, but one cannot superimpose a

time-delayed output waveform with theinput waveform and get a match. Theoutput waveform is not an exact replicaof the input and, thu s, in a strict sense,

is distorted. The inversion, however, isnon-dispersive.Other tha n the consta ntphase-shift, the signal has not under-gone an y time-sprea ding distort ion.

The linear-phase and zero DC phase-shift conditions (distortionless) are syn-onymous with the more conventionaldefinition of the linear-phase (constant-

group-delay) filter. The phase-delay andgroup-delay are both consta nt and equa lto each other for all f [1][2][3][4]. Thesituation of linear-phase with a magni-tude inversion at DC, however, does notviolate this constraint. It only mandatesnormalization of phase at zero frequency(DC) to zero-ra dians.

The ph ase -de l ay , or p ropaga t iondelay, is the time it takes a sinusoid (ofa given frequency) to traverse a certainamount of phase shift. However, in thesi tuation where signals are a sum ofsinusoids, the pha se-delay concept mu stbe applied carefully. A constant phase

shift is not a constant phase-delay ofthe aggregate s ignal . Each spect ra lcomponent of the signal ma y be delayedby d i fferent amounts . The exampleabove is an ideal inversion of a SAW-tooth signal. This is a condition whereeach spectral component is delayed bythe sam e amount of time, but has a dif-ferent am ount of phase sh ift.

Phase offset at DCThe concept of phase offset (or just

phase) at DC is hard to understand,but it does represent the sign of magni-tude. That is, an ideal inverter can be

viewed as having a phase shift of radian s a t frequencies, including DC.Remember, tha t in phasor nota t ion ,exp. (-j ) = 1. This accounts for themagnitude inversion

The fundamentals of linear-phase

filters for digital communicationsN orm alizat ion of an inverting linear-phase filter at DC to zero-degrees ph ase sh ift foranalysis of the time-delay of individ ual spectral com ponents.

By Marc Smith

sig nal processing

Figure 1. Phase () and phase delay (tp) plot of an ideal inverter where tp = /(2f). Figure 2. SAW-tooth signals


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48 March 2000

Most circuit simulators used todayrepresent signals via plots of absolutemagnitude and phase plots. Thus, aninverter circuit simulation at DC typi-

cally shows pha se plots with

radiansphase sh i f t . I t i s th is s imula ted DCphase shift (or bias) that needs to benormalized to zero radians to properlyanalyze the spectral components of dig-ital signals at frequency.

Before moving on, lets look at a pha-sor diagram. (See Figure 3.) Note thatwhen the frequency is equal to zero (= 0), the phasor can only be at the zeroor radian position. This is in agree-ment with what we knowinvert ingand non-inverting filters exist at DC aswell as at frequ ency.

Phase distortionPhase d i s to r t i on (A .K .A . De lay

Distortion) results in t ime-dispersion orspreading as it is sometimes referredto. Phase-delay and group-delay aretwo important quanti t ies consideredwhen a na lyzing the effects of phase dis-tortion. These concepts are typicallyshown via an ampli tude modulationexample where a high-frequency carri-er is m odulated (mult iplied) by a lower-frequency sinusoid (envelope). The AMsignal can a lso be derived from th esummation of two steady-state sinu-soidal signals with near but differentfrequencies (small .] The modulated

signal is then subjected to a channelcharacterized by non-linear phase. Theenvelope of the resu lting composite sig-na l will be delayed by an a mount calledthe group-delay. The carrier signal willbe delayed by a different amount calledthe phase-delay.

The definitions of phase-delay andgroup-delay ar e a s follows:

(1)

(2)

AM propagationthough the non-linear filter

The pr opagat ion of an AM signalthrough a non-linear filter is a classicexample that exemplifies the distinctionbetween the effects on the envelopewave and on the carrier wave. It can beseen tha t th e delay of the car rier wave isdifferent than the delay of the envelope.(See Figure 4.) As long as the carrier

wave and its sidebands a re su bjected toa constant-group-delay (i.e. operatingfrequency region of approximate linearphase), the envelope wave will not dis-tort. Note, however, that the relativephasing (positioning) of the carrier waveto the envelope wave has changed.

Even th ough ph ase- and group-delay

have their roots founded in the realm ofsteady stat e AM and FM systems, theyare much needed tools for the design ofequalizers a nd linear-phase filters u sedin digital communication systems. Inequalizers, resynchronization of bi-phase digital signals requires knowl-edge of the dispersive channel charac-teristics for all spectral components ofthe signal. Group- and phase-delay foreach spectral component of interest canbe used to design a filter with the prop-er phase response. Proper constraintson the filters pha se- and gr oup-delaycan yield a near dispersionless linear-

phase system.

Linear phase filtersThe phase response of a linear-phase

f il t e r (or sy s t em) can be desc r ibedmathematically using the well-knowny=mx+b equation form as follows:

(3)

If the phase-delay and the group-delay are equal to each other over a fre-quency range of interest, signals withspectral components within this fre-

quency range will pass without disper-s ion ( t ime-sp read ing d i s to r t i on ) .Substituting the linear-phase equation(Equation 3) into the phase- and group-delay equations (Equations 1 and 2)yields:

(4)

(5)

tf

fg =

{ }=

1

2 2

1

( )

tf

f fp = = +

( )

2 2 2

1 0

( )f f= +1 0

tf

fg =

1

2

( ( ))

tf

pf

=( )

2

Figure 3. Phasors at DC ( = 0).


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50 March 2000

Note that the phase-delay equation(Equation 4) shows the error-inducinghyperbolic function. For a dispersionless fil-ter (or channel), the phase- and group-delay

must be equal to each other. For this condi-tion to hold, the previous phase- and group-delay equations dictate that the frequency(f) be infinite or, more reasonably, the phasebias (0) be zero.

Figure 5 shows the mathemat ica lconsequences of dealing with an un-normalized phase plot of an invertingchannel . The phase characterist ic islinear in frequency, but t he pha se- andgroup-delay are not equal to each other.In fact, the group-delay is flat indicat-ing a channel tha t cont r ibutes zerophase distortion. However, if one wereto calculate the phase-delay of several

sinus oids, a digita l pulses fundam enta lwave and several of its harmonics, amisleading result occurs: the sinusoidsof different frequency have differentphase-delays. When the phase bias isnot set to zero (normalized), the phase-delay expression becomes err oneous.

As discussed previously, the phase-bias, 0, can only be zero or radiansfor real-valued signals. An ideal phaseinversion of radians can be interpret-ed as a constant multiplication of value -1 for all frequency including DC. Thisinversion does not cause dispersion.

One source [5] dealt with the phase

inversion (phase bias) as a consta nt pha-sor (exp{j0}). This phasor was lumpedin with the amplitude term and referredto as a constant multiplier in a distor-tion-free system. Even though t he defin-ition of distortion-free is different thanthe one t he au tho r p re sen t s i n t h i spaper (only in the manner in which theinversion is handled), the major point isif a system is distortionless, it is by defi-nition dispersionless. An ideal invertinglinear-phase filter is non-dispersive. Thephase-delay for this case needs to be

normalized to zero degrees at DC.Before moving on , le t s redefine

pha se-delay to eliminat e phas e offset:

(6)

where:

Now things make sense! Phase- andgroup-delay are equa l to each other,and a simple phase inversion does notimply dispersion. All we have to do nowis define the conditions for a dispersion-less and a distortionless system. They

are as follows:

Definition of a non-dispersivesystem

A non-dispersive systems outputproduces a time-delayed, inverted ornon-inverted replica of th e input .

S t a t ed ma th ema t i ca l ly , a sy s t emhaving an impulse response, h(t), isnon-dispersive if and only if:

(7)

(0 = Radians).

Definition of a distortion-less system

All distortionless systems are alson o n - d i s p e r s i v e . T h e y a r e f u r t h e rrequired to have zero phase at DC (i.e.0 = 0). We define: Distortionless=Non-Dispersive, Non-Inverting. t p(phase-delay) = t g(group-delay)= con-stant . Pha se char acteristic is linear with fre-

where A: Re and 0,0 { } { }

h t s t A e s tj( ) * ( ) ( )= 0

0 1 00= = +, ( )an d f f

tf

fp =

=

( ) 0 1

2 2

Figure 4 . AM signal through a non-linear filter.

-Slope

=2t

p

-Slop

e=2tg

fc-fm fc fc+fmf

t

t p t g

t

0

Frequency

Output

Input

P

hase


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quency.Now that we have all our definitions, lets derive a very

useful expression that shows the relationship between thephase- and group-delay. Substitut ing Equations 3 and 6 intoEqua tion 2 yields:

(8)

To properly evaluate this expression using the unnormal-ized phase-delay expression for all frequencies including DC,LHopitals ru le mus t be a pplied as we ar e faced with a 0term as f0. If we evaluate this expression using the normal-ized phase-delay expression, we are faced with a 0constantterm as f0. The latter is easier to evaluate and m akes moresense. As th e frequen cy goes to zero, so does the gr oup-delay.

Expanding the previous expression via the product ruleyields:

(9)

(10)

(for f 0 and t p = F(f).)

This result is helpful in evaluating phase-distortion in lin-ear-phase filters, equalizers, and tra nsmission channels. Forideal linea r-pha se filters (non-dispersive), the ter m:

,must equal zero over the frequency range of interest. Whenth is condition is met , tp = t g = const an t, regar dless of whet heror not the linear-phase filter or system is inverting or non-inverting.

Constant group-delayfilter systemsThe phrase constant group delay is used synonymously

with the linear-phase definition. As long as a system channelhas a linear phase response, a constant-group-delay will pre-va i l y i e ld ing nea r d i s to r t i on l e ss s i gna l t ransmiss ion .

Unfortunately, this definition cannot be referenced verbatim.There ar e man y situations where an overall non-linear phase

response exists with frequency regions chara cterized by anear -linear relationship. The pr evious AM signal exampleconsidered this exact situation. The carrier a nd its sidebandswere const ra ined to a consta nt -group-delay region, resultingin a non-distorted en velope waveform . No dispersion of eitherth e carr ier, or the en velope waveform , occur red. However, therelative phase relationship between the carr ier and t he enve-lope is not preserved. In a sense, there is a spreading effectbetween t he carrier and the envelope because t heir time-rela-tionship to each other ha s been pushed apar t.

In digital communication systems, data is sent in the formof pulses which have harmonically related spectral compo-nents. Under non-linear phase conditions, dispersion readilyoccurs. Neglecting amplitude distortion (attenuation) for themoment, all harmonics of a digital pulse signal must propa-

gate with the same velocity in a system channel in order toarrive at its destination without dispersion. In other words,all harmonic components must have the same phase-delay.Strictly speaking, constant group delay does not imply con-stant-phase-delay.

Interpretation of the tg-tp equationReferring to the t g-tp equation (Equation 10), it is reassur-

ing to see that constant phase-delay implies constant group-delay but not vice-versa. A region of frequency can have afixed group delay while also having a varying phase-delay.This agrees with the previous AM signal example where theenvelope-delay (t g) and the carrier-delay (t p) had different val-ues. The ph ase dia gram (Figure 6) shows regions of opera tionwhere t g t p.

When the group-delay and the ph ase-delay are n ot equa l toeach other , two conditions a rise. One condition is ter med nor-mal dispersion and is characterized by a region of operationwhere the group-delay exceeds the phase-delay. The othercondition, ter med anomalous dispersion , is cha ra cterized by aregion of operation where the phase-delay exceeds the group-delay. In a dispersive medium, the group-delay can be zero,positive or negat ive in value.

It is interesting to note that Q3 in Figure 6 a lso representsa point that meets the cri teria for zero phase distort ion:group-delay is equal to phase-delay. Unfortunately, it is onlya point an d thus a limited frequen cy region (i.e. na rr owbandwidth) would be useful for data transmission given some

t

f

p

=

+t f

t

ftg

pp

tf

f t ft

ft

f

fg p

pp=

{ }=

+

tf

ff

f tf

f tg p p=

{ }=

+{ }=

{ }

1

2

1

22 0

( )

m

Figure 5. Un-normalized phase plot example of an inverting channel (note thefactor of 2 has been omitted for illustration purposes).

No Distortion

tp= tg

Anomalous Dispersion

tp> tg

Normal Dispersion

tg> tp

tp=tg

Q2

Q1

Q3

Frequency(0 Hz,0 Deg.)

Phase

Figure 6. Normal and anomalous dispersion.


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acceptable level of dispersion.Q1 and Q2 mark the inflection points

on the phase-frequency plot in Figure 6.At these points, the group-delay is zero

and the corresponding group-velocity isinfinite. While the group-velocity canexceed the speed of light, the actual sig-na l or energy velocity is always less thanthe speed of light [2]. When the group-delay is zero, the corr esponding criterion(derived form Equation 10) takes on thesame form as th at of the tp = t g condition:the r ate of change at a given point m ustequal the slope of a line from that pointth rough th e origin [2].

(11)

(Condition for t g = 0 (infinite group-velocity.)

Real world effectsDigital signals in the real world also

fall victim to other types of distortion.Amplitude distortion, amplitude jitter,phase j i t t e r , a s we l l a s d i spe rs ion(phase distortion) all contribute to per-forma nce degrada tion of tra nsmittedsignals. Amplitude and ph ase jitter ar etypically dominated by external noisesources. Amplitude distortion resultsfrom the nonlinear attenuating charac-t e r i s t i c o f a t ransmiss ion channe l .

Generally, harmonics with higher fre-quency componen t s a re a t t enua t edmore t han th ose of lower frequen cy.

Fourier ana lysis of d ig i ta l pu lsesreveals the well known fact that onlyodd-harmonics are present. Moreover,a s t h e amp l i t ude o f each ha r mon icreduces as th e har monic order increases,only the first few harmonics matter.Thus, an alysis can be greatly reduced bylimiting th e investigation to just th e fun-dament al and its significant ha rmonics.

If a transmission line is long enough,nearly all the harmonic content will begone, leaving only the analysis of fun-

dament al sinewaves. In digital systemscontaining bi-phase coded signals overlong t ran smission channels, a receiverwill see pseudo-random patterns of twoalterna ting near -sinewaves (one h alfthe pulse width of the other) modulatedby a transient. The higher-frequencypulse will be smaller in magnitude. Atransient is excited every time a pulse(near-sinewave) changes in pulse width(new frequency).

Turning our attention back to phasedistortion, lets look a t wh at pha se-fre-

quency characterist ics contribute tod i spe rs ion . A gene r a l equa t ion forpha se can be writt en as follows:

(12)

The first term (phase bias, 0) doesnot contribute to dispersion and as dis-cussed earlier, should be normalized tozero. Applying Equations 5 and 6, onecan see that the second term in equa-tion 12 (Linear phase term, 1f) pro-v ides a consta n t - t ime-delay term ingroup-delay and phase-delay expres-sions. The l inear phase term, as onemight expect, does not lend itself to dis-pers ive behavior . The th i rd te r m inEquation 12, however, does promotedispersion. This second-order (2f2)term supplies a linear characteristic insignal delay time. That is, different fre-quency components are subject to dif-ferent propagation delays (i.e. varyingphase-delay). Thus, second and higher-order phase t erms furnish th e conditionfor phase distortion. Although this isnot revelational, hopefully, a mentalimage of phase-frequency characteris-tics and its relationship to delay timesof spectral components, is reinforced (orfor some, introduced).

Points to remember Ph ase distortion (dispersion) causes

inter-symbol interference (ISI) and multi-FM channel systems to bleed into eachother (co-channel inter ference).

N or m a l iz e a p h a s e p l ot a t D Cbefore using phase values at frequencyfor phase-delay (propagation-delay)calculations.

The group-delay is the pha se delayof the groups or envelope.

Group-delay variation is typicallyu s e d a s a m e a s u r e i n e s t im a t i n gp h a s e n o n - l i n e a r i t y a n d e n s u i n gwaveform distortion. The strict mag-n i tu de o f t h e de l ay i s gene ra l l y of minimal consequence.

Group-delay, tg, of a digital signal(or an y signal composed of mult iple fre-quency components) becomes a func-tion of frequency in a dispersive situa-tion. In dispersive channels, the enve-lope (group) of a complex input signalundergoes a spreading effect.

Be ca r e fu l in p l ac ing t oo mu chimportan ce on t he concept of phase-and group-delay. These entities simplyrepresent relative phase arrangementsat various frequencies for steady-statesignals. For digital communication sys-

tems, the propagation delays must bedetermined for t ransien t s ignals asopposed to stea dy-sta te signals.

In data t ran smission systems, dis-

persion is only part of the distortionpictu re. It is typically accompa nied byamplitude distortion (attenuation) and

jit t er effects. Any of these effect s canbe a l imit ing factor in transmissionperformance.

A dispersive cha nn el is not th e endof th e road. Equalization filters (equa l-izers) exist that counteract the effectsof the phase-distorting medium by lin-earizing the phase response over thebandwidth of interest.

References:1. William s, Art hur & Taylor, Fr ed.

Electronic Fil ter Design Ha ndbook, 3rd-

Edition, New York, NY: McGraw-HillInc . , 1995 - Sec . 2 .2 Trans i en tResponse, Pg. 2.21-2.24.

2. Matick, Richard. TransmissionL in es for Digita l an d Com m unica ti onNetworks, New York, NY: McGraw-HillInc . , 1969 - Chapter 3 , Veloci ty of Propagation, Sec. 3.1-3.7, Pg. 57-81.

3. Ramo, Whinnery & Van Duzer.Fields and Waves in Communication

E lect ron ics , 2nd-Ed, New York, NY:John Wiley & Sons, Inc., 1984 - Sec.5.12 Group and Energy Velocities, Pg.254-256.

4 . R o d e n , M a r t i n . D ig i t a l

C o m m u n i c a t i o n S y s t e m D e s i g n ,En glewood Cliffs, NJ : Pr ent ice-HallInc., 1988 - Sec. 2.3 Distortion, Pg. 66-70.

5. Collin, Robert. Foundations forM icr ow ave E n gi n eer in g, New York,NY: McGraw-Hill Inc., 1966 - Sec. 3.11Wave Velocities, Pg. 132 -134.

( ) ...f f f= + + +0 1 22

=

t

f

t

f

p p

About the authorMarc Smith gra duat ed in 1986 with

a BS degree in Electr ical En gineeringand Computer Science from the Uni-versity of California, Berkeley. He hasworked for 10 years in the area of

Inertial Measurement Systems and 3years in t he ar ea of Digital Commun i-c a t i on S y s t e m s . H e i s c u r r e n t l yemployed as a Senior ASIC Develop-ment Engineer a t Syst r on DonnerInertial Division (a BEI Sensors andS y s t e m s C o m p a n y ). H e c a n b ereached at m sm [email protected] . Theauthor would like to extend a specialthanks to Matt Taylor (Tut Systems)and Marc Loyer (Level One Commu-nications) for their insights and con-structive criticisms.

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0300Smith46 - The fundamentals of linear-phase filters for digital communications.pdf