phase y group delay

8/7/2019 phase y group delay

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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



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.



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



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.



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.



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.



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:



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.



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.



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



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.



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!



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





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)



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.

Documents

phase y group delay