Upload
cecihaan
View
227
Download
0
Embed Size (px)
Citation preview
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 1/16
I 2011
Phase Delay, Group
Delay, Non-minimum
PhaseTrabajo de Investigación para DSP
Paola Salinas y Faviola Romero
U N I V E R S I D A D P R I V A D A B O L I V I A N A
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 2/16
Procesamiento Digital de Señales
Phase Delay
BASICS
The frequency-dependent complex transfer function H(f) of an arbitrary device under test (DUT)
can be expressed as follows:
where A(f) denominates the magnitude and(f) the phase response of the DUT. Vector network
analyzers are able to measure both magnitude and phase response.
Group delay measurements are based on phase measurements.
The measurement procedure corresponds to the definition of group delay as the negative
derivative of the phase (in degrees) with respect to frequency f.
For practical reasons, Vector Network Analyzers ZVR measure a difference quotient rather than a
differential quotient of (2), which yields a good approximation to the wanted group delay
provided that the variation of phase is not too nonlinear in the observed frequency range f,
which is called the aperture.
Fig. 1 shows an illustration of the terms
= 2- 1 and f = f2- f1 for linearly decreasing phase response, e.g. of a delay line.
The aperture f should be chosen in accordance with
the desired measurement accuracy and
the variation of the group delay of the DUT versus frequency.
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 3/16
Measurement accuracy can be increased by making the aperture broader. On the other hand, the
resolution of group delay measurements with respect to frequency suffers from a wide aperture
and fine details of group delay variations versus frequency can no longer be observed. (This is a
similar effect as with the well-known SMOOTHING function, which takes the average of
measurement values at adjacent frequency points. If the smoothing aperture is too broad, it will
cause a flat measurement trace without any details left.)
The appropriate choice for the aperture f always depends on the group delay characteristics of
the DUT
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 4/16
PHASE DELAY MEASUREMENTS
For a non-dispersive DUT, e.g. a coaxial cable, group delay is not a function of frequency at all but
constant. Consequently, phase (f) is a linear function of frequency:
where is the delay time of the cable which is directly related to its mechanical length Lmech via
the permittivity of the dielectric material within the cable and the velocity of light c.
The velocity of light is: c 2.9979
108
m/s 30 cm/ns 1 ft/ns and can be easily remembered as"one light foot", which is approximately the distance which light travels within 1 ns.
The product Lmech is called the electrical length Lel of the cable, as it denotes the effective
length of the cable as "seen" by the electrical signal to travel. In practice, the electrical length is
always longer than the mechanical length, because of the permittivity being >1 for all practical
dielectrics. (For a pure vacuum the permittivity is equal to one, which results in an electrical
length identical with the mechanical length. Within a finite frequency range even <1 is possible
for a plasma, which causes a shorter electrical length.)
As an example, a cable with a length of 10.34 m filled with a typical dielectric, e.g. teflon ( = 2.1),
causes a delay time of 50 ns. The electrical length is approximately Lel 15 m. The velocity of
propagation for an electrical signal within the cable is c/ 0.69c in this example.
As for all non-dispersive DUTs and also for the cable of this example, there is no difference
between the delay time and the group delay gr as the phase is a linear function of frequency
(4).This can be shown analytically and by measurement.
With (f) = -360°·f· it follows from (2) that gr = .
For a measurement example, a coaxial cable as described can be used. A cable with 50 ns delay
time produces a phase shift of 360° for a frequency shift of 1 / 50 ns = 20 MHz. For an accurate
tracking of the phase shift versus frequency, the frequency span and the number of points for the
network analyzer must be chosen so that the phase shift between each two adjacent frequency
points does not exceed 180°. If this is not the case, it is impossible to distinguish between a phase
shift of o or o + n·360° (n is an arbitrary integer), because of phase ambiguity.
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 5/16
The described phenomenon is illustrated in Fig. 2, where calculation of the phase difference2 -1 yields an incorrect result for the phase shift of approximately -90°, because the phase jump
of360° is simply overlooked. The correct value for is approximately -450°, i.e. -90°-360°.
The wrong value is caused by too large step width between adjacent frequency points. (This
phenomenon is similar to sampling of an audio signal with too small sampling frequency, thus
causing undersampling errors.)
To take account of this problem, either the number of frequency points should be increased or
the frequency span reduced.
Note: Be aware of the phase-tracking phenomenon when performing phase or delaymeasurements. Make sure that the phase shift between each two adjacent frequency points is
always smaller than 180° to obtain proper phase tracking.
Please be especially careful when using nonlinear sweep. In the described example of a 50 ns cable
the frequency difference between two adjacent frequency points has to be smaller than 10 MHz.
For linear sweep and a START frequency of e.g.
1 MHz and a STOP frequency of e.g. 4 GHz this means that the number of points must be greater
than 400. For example, 500 points would be a good choice.
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 6/16
Fig. 3 shows measurement results for the phase of a coaxial cable with approximately 50 ns delay
time. The measured phase starts with -20° at 1 MHz and linearly decreases down to -72246° at 4
GHz.
To show the linear regression of the displayed phase response of distinctly more than ±180°, a
special feature of ZVR is utilized, called PHASE UNWRAP
[FORMAT] PHASE: PHASE UNWRAP
[SCALE] MAX VALUE = 0: MIN VALUE = -100 k
This function forces the phase to be displayed linearly without the usual sawtooth character which
could otherwise be seen.
As explained above, phase unwrap needs a phase shift of less than 180° between adjacent
frequency points to work properly.
The phase unwrap feature is illustrated in Fig. 4 for the example of a 2.5 ns coaxial cable.
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 7/16
The delay time of the DUT can simply be calculated from the measured phase shift: (6where 1 =
measured phase at the start frequency
f1 and 2 = measured phase at the stop frequency
f2. With actually measured values for the 50 ns cable, the result for in (6) is = 50.169 ns.
It is very convenient that Vector Network Analyzers ZVR offer this type of measurement as a
special formatting called PHASE DELAY.
[FORMAT] PHASE DELAY
For this purpose the phase is measured at the
START frequency, then automatically tracked and unwrapped during the sweep and finally
measured at the STOP frequency. The PHASE DELAY is eventually calculated from (6). A particular
advantage of phase delay measurements is the extraordinary accuracy of this technique, which
can simply be demonstrated for the example shown:
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 8/16
The phase measurement uncertainty stated in the data sheet is <0.4° (and is typically in the
range of a few hundreds of a degree). Neglecting the tiny frequency deviation of the ZVR, a
maximum uncertainty can be calculated from (6) for the measured delay time of
Converting the delay time into a length yield
which actually is a dramatically tiny inaccuracy for an electrical length measurement.
It is worthwhile keeping in mind that the determined accuracy is an absolute value and does not
depend on the length of the cable (if the phase can still be properly tracked). For a 100 ns cable,
for instance, the uncertainty is still approximately 0.28 ps. (To achieve proper phase tracking, the
number of points in that case should be doubled to 1000.)
The data conversion from PHASE DELAY to either ELECTRICAL LENGTH or MECHANICAL LENGTH
can be performed automatically by the ZVR itself if the appropriate softkey in the FORMAT menu
is activated by the user.
[FORMAT] ELECTRICAL LENGTH or
[FORMAT] MECHANICAL LENGTH
For the conversion to MECHANICAL LENGTH an arbitrary permittivity may be chosen from the SET
DIELECTRIC TABLE of the FORMAT menu, e.g. one of the predefined materials, as for instance
Teflon with a predefined permittivity = 2.1, or a newly edited user-defined dielectric.
The measured value is indicated on the display in the upper right hand corner beneath the marker
readout values, with a prefix PD, EL or ML for PHASE DELAY, ELECTRICAL LENGTH or MECHANICAL
LENGTH.
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 9/16
Group Delay
The Basics
Delay and absolut e delay
The finite time required for a signal to pass through a filteror any device for that matteriscalled delay
Absolute delay is the delay a signal experiences passing through the device at some reference
frequency
Delay versus f requency
If delay through a filter is plotted on a graph of frequency (x-axis) versus time delay (y-axis), the
plot often has a parabola- or bathtub-like shape
Network analyzer plot of Ch. T8 bandpass filter
The upper trace shows magnitude versus frequency: the filters bandpass characteristics. The x-
axis is frequency, the y-axis is amplitude.
The lower trace shows group delay versus frequency. The x-axis is frequency, the y-axis is time.
Note the bathtub-like shape of the curve.
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 10/16
Phase versus f requency
If propagation or transit time through a device is the same at all frequencies, phase is said to be
linear with respect to frequency
If phase changes uniformly with frequency, an output signal will be identical to the input signal
except that it will have a time shift because of the uniform delay through the device
If propagation or transit time through a device is different at different frequencies, the result is
delay shift or non-linear phase shift
If phase changes non-linearly with frequency, the output signal will be distorted
Delay and phase distort ion
Delay distortionalso known as phase distortionis usually expressed in units of time:
milliseconds (ms), microseconds (µs) or nanoseconds (ns) relative to a reference frequency
Phase distortion is related to phase delay
Phase distortion is measured using a parameter called envelope delay distortion, or group delay
distortion
The Formal Definit ion
Group delay is the derivative of radian phase with respect to radian frequency. It is equal to the
phase delay for an ideal non-dispersive delay device, but may differ greatly in actual devices where
there is a ripple in the phase versus frequency characteristic.
The Mat h
In its simplest mathematical representation:
where GD is group delay variation, is phase in radians, and is frequency in radians per second
For the purists, group delay , also is defined
And yet another definition is
The Translat ion
If phase versus frequency is non-linear, group delay exists.
In a system, network, device or component with no group delay, all frequencies are transmittedthrough the system, network, device or component in the same amount of timethat is, with
equal time delay.
If group delay exists, signals at some frequencies travel faster than signals at other frequencies.
The Analogy
Imagine a group of runners with identical athletic abilities on a smooth, flat track
[
N
d
d GD !
[
[N [X
x
x!
)()(
)()()( 8�(5( [
[[
[[ je H
d
d
d
d D
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 11/16
All of the athletes arrive at the finish line at exactly the same time and with equal time delay from
one end of the track to the other!
Now lets substitute a group of RF signals for the athletes. Here, the track is the equivalent of a
filters passband.
All of the frequencies arrive at the destination at exactly the same time and with equal time delay
through the filter pass band!
Back to athletes, but now there are some that have to run in the ditches next to the track.
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 12/16
Some athletes take a little longer than others to arrive at the finish line. Their time delay from one
end of the track to the other is unequal.
Substitute RF signals for the athletes again. The track is a filters passband, the ditches are the
filters band edges.
Group delay exists, because some frequenciesthe ones near the band edgestook longer than
others to travel through the filter!
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 13/16
Now take the dotted line connecting the frequencies and flip it on its side. The result is the classic
bathtub-shaped group delay curve.
Why Is Group Delay Import ant?
Group delay, a linear distortion, causes inter-symbol interference to digitally modulated signals.
This in turn degrades modulation error ratio (MER)the constellation symbol points get fuzzy.
This test equipment screen shot is from a cable networks upstream spectrum in which in-channel
group delay was negligibleabout 60 to 75 ns peak-to-peak.
Unequalized MER is 28.3 dB, well above the 17~20 dB MER failure threshold for 16-QAM
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 14/16
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 15/16
MATLAB® simulat ion f or 64-QAMwit h group delay
Wrapping Up
y Common sources of group delay in a cable network
y AC power coils/chokes (affects 5~10 MHz in the upstream)
y Node and amplifier diplex filters (affect frequencies near the diplex filter cutoff region in
the upstream and downstream)
y Band edges and roll off areas
y High-pass filters, data-only filters, step attenuators, taps or in-line equalizers with filters
y Group delay ripple caused by impedance mismatch-related micro-reflections and
amplitude ripple (poor frequency response)
The Fix?
y Use adaptive equalization available in DOCSIS 1.1 and 2.0 modems (not supported in
DOCSIS 1.0 modems)
y Avoid frequencies where diplex filter group delay is common
y Sweep the forward and reverse to ensure frequency response is flat (set equipment to
highest resolution available; use resistive test points or probe seizure screws to see
amplitude ripple)
y Identify and repair impedance mismatches that cause micro-reflections
y Use specialized test equipment to characterize and troubleshoot group delay (group delay
can exist even when frequency response is flat)
8/7/2019 phase y group delay
http://slidepdf.com/reader/full/phase-y-group-delay 16/16
Non- Minimum Phase
Systems that are causal and stable whose inverses are causal and unstable are known as non-
minimum-phase systems. A given non-minimum phase system will have a greater phase
contribution than the minimum-phase system with the equivalent magnitude response.
Maximum phase
A maximum-phase system is the opposite of a minimum phase system. A causal and stable LTI
system is a maximum-phase system if its inverse is causal and unstable. That is,
The zeros of the discrete-time system are outside the unit circle.
The zeros of the continuous-time system are in the right-hand side of the complex plane.
Such a system is called a maximum-phase system because it has the maximum group delay of the
set of systems that have the same magnitude response. In this set of equal-magnitude-response
systems, the maximum phase system will have maximum energy delay.
For example, the two continuous-time LTI systems described by the transfer functions
have equivalent magnitude responses; however, the second system has a much larger
contribution to the phase shift. Hence, in this set, the first system is the minimum-phase system
and the second system is the maximum-phase system.
Mixed phase
A mixed-phase system has some of its zeros inside the unit circle and has others outside the unitcircle. Thus, its group delay is neither minimum nor maximum but somewhere between the group
delay of the minimum and maximum phase equivalent system.
For example, the continuous-time LTI system described by transfer function
is stable and causal; however, it has zeros on both the left- and right-hand sides of the complex
plane. Hence, it is a mixed-phase system.