signal processing
The fundamentals of linear-phase
filters for digital communications
Normalization of an inverting linear-phase filter at DC to zero-degrees phase shift for
analysis of the time-delay of individual spectral components.
By Marc Smith
N
umerous sources of technical information on linear-phase filters can
be found. However, many of these
sources label systems as “distortionless
or dispersionless” without any formal
definition of either. Moreover, the condition where a linear-phase filter has a
phase bias (an inverter) also is rarely
addressed. This article presents a definition for a distortionless system and discusses the need for normalizing the
phase-response curve of an inverting linear-phase or inverting equalization filter.
If one were to plot the phase-delay
(also known as propagation delay) vs.
frequency for the inverting linearphase filter, an erroneous result will
follow. One would see a dominant
hyperbolic curve that could be interpreted as increasing in nonlinearity as
the frequency gets smaller, asymptotically approaching infinite phase-delay
at DC. This result shows up even when
an ideal inverter is considered (see
below), indicating dispersive behavior.
This result is wrong. In addition, the
notion of phase at DC makes no sense.
If a computer simulation was performed over a frequency range far
enough above DC (to a point where the
hyperbolic shape could no longer be
visually detected), one could easily
believe the results and modify their filter design incorrectly. The simulation
data could still be greatly in error due
to the hyperbolic behavior. Phase-delay
errors of 50 ns at 10 MHz and 5 ns at
100 MHz can be encountered.
To design an ideal distortionless filter, a designer needs to implement not
only a filter with linear-phase but also
a filter with a phase-shift of zero
degrees at DC. This can be easily
shown by injecting a saw-tooth wave
through an ideal inverter (phase shift =
–π for all frequencies). (See Figure 2.)
The resulting waveform has only been
inverted, but one cannot superimpose a
time-delayed output waveform with the
input waveform and get a match. The
output waveform is not an exact replica
of the input and, thus, in a strict sense,
is distorted. The inversion, however, is
non-dispersive.Other than the constant
phase-shift, the signal has not undergone any “time-spreading” distortion.
The linear-phase and zero DC phaseshift conditions (distortionless) are synonymous with the more conventional
definition of the linear-phase (constantgroup-delay) filter. The phase-delay and
group-delay are both constant and equal
to each other for all “f’” [1][2][3][4]. The
situation of linear-phase with a magnitude inversion at DC, however, does not
violate this constraint. It only mandates
normalization of phase at zero frequency
(DC) to zero-radians.
The phase-delay, or propagation
delay, is the time it takes a sinusoid (of
a given frequency) to traverse a certain
amount of phase shift. However, in the
situation where signals are a sum of
sinusoids, the phase-delay concept must
be applied carefully. A constant phase
shift is not a constant phase-delay of
the aggregate signal. Each spectral
component of the signal may be delayed
by different amounts. The example
above is an ideal inversion of a SAWtooth signal. This is a condition where
each spectral component is delayed by
the same amount of time, but has a different amount of phase shift.
Phase offset at DC
The concept of phase offset (or just
phase) at DC is hard to understand,
but it does represent the sign of magnitude. That is, an ideal inverter can be
viewed as having a phase shift of ±π
radians at frequencies, including DC.
Remember, that in phasor notation,
exp. (-j•π) = –1. This accounts for the
magnitude inversion
Φ) and phase delay (tp) plot of an ideal inverter where tp = Φ/(2π
πf).
Figure 1. Phase (Φ
46
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Figure 2. SAW-tooth signals
March 2000
Most circuit simulators used today
represent signals via plots of absolute
magnitude and phase plots. Thus, an
inverter circuit simulation at DC typically shows phase plots with ±π radians
phase shift. It is this simulated DC
phase shift (or bias) that needs to be
normalized to zero radians to properly
analyze the spectral components of digital signals at frequency.
Before moving on, let’s look at a phasor diagram. (See Figure 3.) Note that
when the frequency is equal to zero (ω
= 0), the phasor can only be at the zero
or ±π radian position. This is in agreement with what we know—inverting
and non-inverting filters exist at DC as
well as at frequency.
Phase distortion
Phase distortion (A.K.A. Delay
Distortion) results in time-dispersion or
“spreading” as it is sometimes referred
to. Phase-delay and group-delay are
two important quantities considered
when analyzing the effects of phase distortion. These concepts are typically
shown via an amplitude modulation
example where a high-frequency carrier is modulated (multiplied) by a lowerfrequency sinusoid (envelope). The AM
signal can also be derived from the
summation of two steady-state sinusoidal signals with near but different
frequencies (small δω.] The modulated
signal is then subjected to a channel
characterized by non-linear phase. The
envelope of the resulting composite signal will be delayed by an amount called
the group-delay. The carrier signal will
be delayed by a different amount called
the phase-delay.
The definitions of phase-delay and
group-delay are as follows:
Φ (f )
tp =
2πf
 1   ∂ (Φ (f )) 
tg =   
 2π   ∂f 
(1)
(2)
AM propagation
though the non-linear filter
The propagation of an AM signal
through a non-linear filter is a classic
example that exemplifies the distinction
between the effects on the envelope
wave and on the carrier wave. It can be
seen that the delay of the carrier wave is
different than the delay of the envelope.
(See Figure 4.) As long as the carrier
48
ω = 0).
Figure 3. Phasors at DC (ω
wave and it’s sidebands are subjected to
a constant-group-delay (i.e. operating
frequency region of approximate linear
phase), the envelope wave will not distort. Note, however, that the relative
phasing (positioning) of the carrier wave
to the envelope wave has changed.
Even though phase- and group-delay
have their roots founded in the realm of
steady state AM and FM systems, they
are much needed tools for the design of
equalizers and linear-phase filters used
in digital communication systems. In
equalizers, resynchronization of biphase digital signals requires knowledge of the dispersive channel characteristics for all spectral components of
the signal. Group- and phase-delay for
each spectral component of interest can
be used to design a filter with the proper phase response. Proper constraints
on the filter’s phase- and group-delay
can yield a near dispersionless linearphase system.
Linear phase filters
The phase response of a linear-phase
filter (or system) can be described
mathematically using the well-known
y=mx+b equation form as follows:
Φ (f ) = Φ1 • f + Φ 0
(3)
If the phase-delay and the groupdelay are equal to each other over a frequency range of interest, signals with
spectral components within this frequency range will pass without dispersion (time-spreading distortion).
Substituting the linear-phase equation
(Equation 3) into the phase- and groupdelay equations (Equations 1 and 2)
yields:
Φ (f ) Φ1 Φ 0
tp =
=
+
2πf
2π 2πf
(4)
tg =
Φ1
1 ∂
• {Φ (f )} =
2π ∂f
2π
(5)
March 2000
Frequency
fc-fm fc fc+fm
0
f
Input
t
tp
tg
tg
π•
2•
e=
lop
-S
Phase
-S
lop
e=
2•
π•
tp
Output
t
−Φ
Figure 4 . AM signal through a non-linear filter.
Note that the phase-delay equation
(Equation 4) shows the error-inducing
hyperbolic function. For a dispersionless filter (or channel), the phase- and group-delay
must be equal to each other. For this condition to hold, the previous phase- and groupdelay equations dictate that the frequency
(f) be infinite or, more reasonably, the phase
bias (Φ0) be zero.
Figure 5 shows the mathematical
consequences of dealing with an unnormalized phase plot of an inverting
channel. The phase characteristic is
linear in frequency, but the phase- and
group-delay are not equal to each other.
In fact, the group-delay is flat indicating a channel that contributes zero
phase distortion. However, if one were
to calculate the phase-delay of several
sinusoids, a digital pulse’s fundamental
wave and several of its harmonics, a
misleading result occurs: the sinusoids
of different frequency have different
phase-delays. When the phase bias is
not set to zero (normalized), the phasedelay expression becomes erroneous.
As discussed previously, the phasebias, Φ0, can only be zero or ±π radians
for real-valued signals. An ideal phase
inversion of ±π radians can be interpreted as a constant multiplication of value 1 for all frequency including DC. This
inversion does not cause dispersion.
One source [5] dealt with the phase
inversion (phase bias) as a constant phasor (exp{j•Φ0}). This phasor was lumped
in with the amplitude term and referred
to as a constant multiplier in a distortion-free system. Even though the definition of distortion-free is different than
the one the author presents in this
paper (only in the manner in which the
inversion is handled), the major point is
if a system is distortionless, it is by definition dispersionless. An ideal inverting
linear-phase filter is non-dispersive. The
phase-delay for this case needs to be
50
normalized to zero degrees at DC.
Before moving on, let’s redefine
phase-delay to eliminate phase offset:
tp =
Φ (f ) − Φ 0 Φ1
=
2πf
2π
(6)
where:
Φ 0 = 0, ± π and Φ (f ) = Φ1 • f + Φ 0
Now things make sense! Phase- and
group-delay are equal to each other,
and a simple phase inversion does not
imply dispersion. All we have to do now
is define the conditions for a dispersionless and a distortionless system. They
are as follows:
Definition of a non-dispersive
system
A non-dispersive system’s output
produces a time-delayed, inverted or
non-inverted replica of the input.
Stated mathematically, a system
having an impulse response, h(t), is
non-dispersive if and only if:
h(t) * s(t) = A • e− j•Φ 0s(t − τ)
(7)
where : A ∈ {Re} and Φ 0 ∈ {0,π}
(Φ0 = Radians).
Definition of a distortionless system
All distortionless systems are also
non-dispersive. They are further
required to have zero phase at DC (i.e.
Φ0 = 0). We define:
•Distortionless=Non-Dispersive, NonInverting.
•tp(phase-delay) = tg(group-delay)= constant.
•Phase characteristic is linear with fre-
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Phase
No Distortion
tp= tg
Anomalous Dispersion
tp > tg
Normal Dispersion
tg > tp
Q1
m
Q2
Q3
tp= tg
(0 Hz,0 Deg.)
Frequency
Figure 5. Un-normalized phase plot example of an inverting channel (note the
π has been omitted for illustration purposes).
factor of 2π
Figure 6. Normal and anomalous dispersion.
quency.
Now that we have all our definitions, lets derive a very
useful expression that shows the relationship between the
phase- and group-delay. Substituting Equations 3 and 6 into
Equation 2 yields:
Unfortunately, this definition cannot be referenced verbatim.
There are many situations where an overall non-linear phase
response exists with frequency regions characterized by a
near-linear relationship. The previous AM signal example
considered this exact situation. The carrier and its sidebands
were constrained to a “constant-group-delay” region, resulting
in a non-distorted envelope waveform. No dispersion of either
the carrier, or the envelope waveform, occurred. However, the
relative phase relationship between the carrier and the envelope is not preserved. In a sense, there is a spreading effect
between the carrier and the envelope because their time-relationship to each other has been pushed apart.
In digital communication systems, data is sent in the form
of pulses which have harmonically related spectral components. Under non-linear phase conditions, dispersion readily
occurs. Neglecting amplitude distortion (attenuation) for the
moment, all harmonics of a digital pulse signal must propagate with the same velocity in a system channel in order to
arrive 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 constant-phase-delay.
tg =
1 ∂
1 ∂
∂
• {Φ (f )} =
• {2πf • tp + Φ 0} = {f • tp}
2π ∂f
2π ∂f
∂f
(8)
To properly evaluate this expression using the unnormalized phase-delay expression for all frequencies including DC,
L’Hopital’s rule must be applied as we are faced with a “0•∞”
term as f→0. If we evaluate this expression using the normalized phase-delay expression, we are faced with a “0•constant”
term as f→0. The latter is easier to evaluate and makes more
sense. As the frequency goes to zero, so does the group-delay.
Expanding the previous expression via the product rule
yields:
tg =
∂
∂tp
∂f
+ tp •
{f • tp} = f •
∂f
∂f
∂f
∂tp
∴ tg = f •
+ tp
∂f
(9)
Interpretation of the tg-tp equation
(10)
(for f ≥ 0 and tp = F(f).)
This result is helpful in evaluating phase-distortion in linear-phase filters, equalizers, and transmission channels. For
ideal linear-phase filters (non-dispersive), the term:
∂tp
∂f ,
must equal zero over the frequency range of interest. When
this condition is met, tp = tg = constant, regardless of whether
or not the linear-phase filter or system is inverting or noninverting.
Constant group-delayfilter systems
The phrase “constant group delay” is used synonymously
with the linear-phase definition. As long as a system channel
has a linear phase response, a constant-group-delay will prevail yielding near distortionless signal transmission.
54
Referring to the tg-tp equation (Equation 10), it is reassuring to see that constant phase-delay implies constant groupdelay but not vice-versa. A region of frequency can have a
fixed group delay while also having a varying phase-delay.
This agrees with the previous AM signal example where the
envelope-delay (tg) and the carrier-delay (tp) had different values. The phase diagram (Figure 6) shows regions of operation
where tg ≠ tp.
When the group-delay and the phase-delay are not equal to
each other, two conditions arise. One condition is termed normal dispersion and is characterized by a region of operation
where the group-delay exceeds the phase-delay. The other
condition, termed anomalous dispersion, is characterized by a
region of operation where the phase-delay exceeds the groupdelay. In a dispersive medium, the group-delay can be zero,
positive or negative in value.
It is interesting to note that Q3 in Figure 6 also represents
a point that meets the criteria for zero phase distortion:
group-delay is equal to phase-delay. Unfortunately, it is only
a point and thus a limited frequency region (i.e. narrow
bandwidth) would be useful for data transmission given some
www.rfdesign.com
March 2000
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 is
infinite. While the group-velocity can
exceed the speed of light, the actual signal or energy velocity is always less than
the speed of light [2]. When the groupdelay is zero, the corresponding criterion
(derived form Equation 10) takes on the
same form as that of the tp = tg condition:
the rate of change at a given point must
equal the slope of a line from that point
through the origin [2].
∂tp
tp
=−
∂f
f
(11)
(Condition for tg = 0 (infinite groupvelocity.)
Real world effects
Digital signals in the real world also
fall victim to other types of distortion.
Amplitude distortion, amplitude jitter,
phase jitter, as well as dispersion
(phase distortion) all contribute to performance degradation of transmitted
signals. Amplitude and phase jitter are
typically dominated by external noise
sources. Amplitude distortion results
from the nonlinear attenuating characteristic of a transmission channel.
Generally, harmonics with higher frequency components are attenuated
more than those of lower frequency.
Fourier analysis of digital pulses
reveals the well known fact that only
odd-harmonics are present. Moreover,
as the amplitude of each harmonic
reduces as the harmonic order increases,
only the first few harmonics matter.
Thus, analysis can be greatly reduced by
limiting the investigation to just the fundamental and its significant harmonics.
If a transmission line is long enough,
nearly all the harmonic content will be
gone, leaving only the analysis of fundamental sinewaves. In digital systems
containing bi-phase coded signals over
long transmission channels, a receiver
will see pseudo-random patterns of two
alternating near-sinewaves (one half
the pulse width of the other) modulated
by a transient. The higher-frequency
pulse will be smaller in magnitude. A
transient is excited every time a pulse
(near-sinewave) changes in pulse width
(new frequency).
Turning our attention back to phase
distortion, let’s look at what phase-fre-
56
quency characteristics contribute to
dispersion. A general equation for
phase can be written as follows:
Φ (f ) = Φ 0 + Φ1f + Φ 2f 2 + ...
(12)
The first term (phase bias, Φ0) does
not contribute to dispersion and as discussed earlier, should be normalized to
zero. Applying Equations 5 and 6, one
can see that the second term in equation 12 (Linear phase term, Φ1•f) provides a constant-time-delay term in
group-delay and phase-delay expressions. The linear phase term, as one
might expect, does not lend itself to dispersive behavior. The third term in
Equation 12, however, does promote
dispersion. This second-order (Φ2•f2)
term supplies a linear characteristic in
signal delay time. That is, different frequency components are subject to different propagation delays (i.e. varying
phase-delay). Thus, second and higherorder phase terms furnish the condition
for phase distortion. Although this is
not revelational, hopefully, a mental
image of phase-frequency characteristics and its relationship to delay times
of spectral components, is reinforced (or
for some, introduced).
Points to remember
•Phase distortion (dispersion) causes
inter-symbol interference (ISI) and multiFM channel systems to “bleed” into each
other (co-channel interference).
•Normalize a phase plot at DC
before using phase values at frequency
for phase-delay (propagation-delay)
calculations.
•The group-delay is the phase delay
of the groups or envelope.
•Group-delay variation is typically
used as a measure in estimating
phase non-linearity and ensuing
waveform distortion. The strict magnitude of the delay is generally of
minimal consequence.
•Group-delay, tg, of a digital signal
(or any signal composed of multiple frequency components) becomes a function of frequency in a dispersive situation. In dispersive channels, the envelope (group) of a complex input signal
undergoes a “spreading” effect.
•Be careful in placing too much
importance on the concept of phaseand group-delay. These entities simply
represent relative phase arrangements
at various frequencies for steady-state
signals. For digital communication sys-
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tems, the propagation delays must be
determined for transient signals as
opposed to steady-state signals.
•In data transmission systems, dispersion is only part of the distortion
picture. It is typically accompanied by
amplitude distortion (attenuation) and
jitter effects. Any of these effects can
be a limiting factor in transmission
performance.
•A dispersive channel is not the “end
of the road.” Equalization filters (equalizers) exist that counteract the effects
of the phase-distorting medium by “linearizing” the phase response over the
bandwidth of interest.
References:
1. Williams, Arthur & Taylor, Fred.
Electronic Filter Design Handbook, 3rdEdition, New York, NY: McGraw-Hill
Inc., 1995 - Sec. 2.2 Transient
Response, Pg. 2.21-2.24.
2. Matick, Richard. Transmission
Lines for Digital and Communication
Networks, New York, NY: McGraw-Hill
Inc., 1969 - Chapter 3, Velocity of
Propagation, Sec. 3.1-3.7, Pg. 57-81.
3. Ramo, Whinnery & Van Duzer.
Fields and Waves in Communication
Electronics, 2nd-Ed, New York, NY:
John Wiley & Sons, Inc., 1984 - Sec.
5.12 Group and Energy Velocities, Pg.
254-256.
4.
Roden,
Martin.
Digital
Communication System Design,
Englewood Cliffs, NJ: Prentice-Hall
Inc., 1988 - Sec. 2.3 Distortion, Pg. 6670.
5. Collin, Robert. Foundations for
Microwave Engineering, New York,
NY: McGraw-Hill Inc., 1966 - Sec. 3.11
Wave Velocities, Pg. 132-134.
About the author
Marc Smith graduated in 1986 with
a BS degree in Electrical Engineering
and Computer Science from the University of California, Berkeley. He has
worked for 10 years in the area of
Inertial Measurement Systems and 3
years in the area of Digital Communication Systems. He is currently
employed as a Senior ASIC Development Engineer at Systron Donner
Inertial Division (a BEI Sensors and
Systems Company). He can be
reached at [email protected]. The
author would like to extend a special
thanks to Matt Taylor (Tut Systems)
and Marc Loyer (Level One Communications) for their insights and constructive criticisms.
March 2000