Introduction to Frequency

1
Instructor Richard Mellitz
Introduction to Frequency Domain Analysis (3
Classes) Many thanks to Steve Hall, Intel for the
use of his slides Reference Reading Posar Ch
4.5 http//cp.literature.agilent.com/litweb/pdf/59
52-1087.pdf
Slide content from Stephen Hall
2
Outline

• Motivation Why Use Frequency Domain Analysis
• 2-Port Network Analysis Theory
• Impedance and Admittance Matrix
• Scattering Matrix
• Transmission (ABCD) Matrix
• Masons Rule
• Cascading S-Matrices and Voltage Transfer
  Function
• Differential (4-port) Scattering Matrix

3
Motivation Why Frequency Domain Analysis?

• Time Domain signals on T-lines lines are hard to
  analyze
• Many properties, which can dominate performance,
  are frequency dependent, and difficult to
  directly observe in the time domain
• Skin effect, Dielectric losses, dispersion,
  resonance
• Frequency Domain Analysis allows discrete
  characterization of a linear network at each
  frequency
• Characterization at a single frequency is much
  easier
• Frequency Analysis is beneficial for Three
  reasons
• Ease and accuracy of measurement at high
  frequencies
• Simplified mathematics
• Allows separation of electrical phenomena (loss,
  resonance etc)

4
Key Concepts

• Here are the key concepts that you should retain
  from this class
• The input impedance the input reflection
  coefficient of a transmission line is dependent
  on
• Termination and characteristic impedance
• Delay
• Frequency
• S-Parameters are used to extract electrical
  parameters
• Transmission line parameters (R,L,C,G, TD and Zo)
  can be extracted from S parameters
• Vias, connectors, socket s-parameters can be used
  to create equivalent circuits
• The behavior of S-parameters can be used to gain
  intuition of signal integrity problems

5
Review Important Concepts

• The impedance looking into a terminated
  transmission line changes with frequency and line
  length
• The input reflection coefficient looking into a
  terminated transmission line also changes with
  frequency and line length
• If the input reflection of a transmission line is
  known, then the line length can be determined by
  observing the periodicity of the reflection
• The peak of the input reflection can be used to
  determine line and load impedance values

6
Two Port Network Theory

• Network theory is based on the property that a
  linear system can be completely characterized by
  parameters measured ONLY at the input output
  ports without regard to the content of the system
• Networks can have any number of ports, however,
  consideration of a 2-port network is sufficient
  to explain the theory
• A 2-port network has 1 input and 1 output port.
• The ports can be characterized with many
  parameters, each parameter has a specific
  advantage
• Each Parameter set is related to 4 variables
• 2 independent variables for excitation
• 2 dependent variables for response

7
Network characterized with Port Impedance

• Measuring the port impedance is network is the
  most simplistic and intuitive method of
  characterizing a network

I
I
I
I
1
2
1
2
port
2
-
port
2
-




Port 2
V
V
V
V
Port 1
1
2
Network
1
2
Network
-
-
-
-
Case 1 Inject current I1 into port 1 and measure
the open circuit voltage at port 2 and calculate
the resultant impedance from port 1 to port 2
Case 2 Inject current I1 into port 1 and measure
the voltage at port 1 and calculate the
resultant input impedance
8
Impedance Matrix

• A set of linear equations can be written to
  describe the network in terms of its port
  impedances
• Where
• If the impedance matrix is known, the response of
  the system can be predicted for any input

Or
Open Circuit Voltage measured at Port i
Current Injected at Port j
Zii ? the impedance looking into port i Zij ? the
impedance between port i and j
9
Impedance Matrix Example 2
Calculate the impedance matrix for the following
circuit
R2
R1
R3
Port 2
Port 1
10
Impedance Matrix Example 2
Step 1 Calculate the input impedance
R2
R1
-
R3
I1
V1
Step 2 Calculate the impedance across the network
R1
R2
-
R3
I1
V2
11
Impedance Matrix Example 2
Step 3 Calculate the Impedance matrix Assume
R1 R2 30 ohms R3150 ohms
12
Measuring the impedance matrix

• Question
• What obstacles are expected when measuring the
  impedance matrix of the following transmission
  line structure assuming that the micro-probes
  have the following parasitics?
• Lprobe0.1nH
• Cprobe0.3pF

Assume F5 GHz
13
Measuring the impedance matrix

• Answer
• Open circuit voltages are very hard to measure at
  high frequencies because they generally do not
  exist for small dimensions
• Open circuit ? capacitance impedance at high
  frequencies
• Probe and via impedance not insignificant

Without Probe Capacitance
0.1nH
T-line
Zo 50
Port 1
Port 2
Port 2
Z21 50 ohms
With Probe Capacitance _at_ 5 GHz
Zo 50
Port 2
Port 1
106 ohms
106 ohms
Z21 63 ohms
14
Advantages/Disadvantages of Impedance Matrix

• Advantages
• The impedance matrix is very intuitive
• Relates all ports to an impedance
• Easy to calculate
• Disadvantages
• Requires open circuit voltage measurements
• Difficult to measure
• Open circuit reflections cause measurement noise
• Open circuit capacitance not trivial at high
  frequencies

Note The Admittance Matrix is very similar,
however, it is characterized with short circuit
currents instead of open circuit voltages
15
Scattering Matrix (S-parameters)

• Measuring the power at each port across a well
  characterized impedance circumvents the problems
  measuring high frequency opens shorts
• The scattering matrix, or (S-parameters),
  characterizes the network by observing
  transmitted reflected power waves

a2
a1
2
-
port
2
-
port
Port 1
Port 2
R
R
Network
Network
b2
b1
ai represents the square root of the power wave
injected into port i
bj represents the power wave coming out of port j
16
Scattering Matrix

• A set of linear equations can be written to
  describe the network in terms of injected and
  transmitted power waves
• Where
• Sii the ratio of the reflected power to the
  injected power at port i
• Sij the ratio of the power measured at port j
  to the power injected at port i

17
Making sense of S-Parameters Return Loss

• When there is no reflection from the load, or the
  line length is zero, S11 Reflection coefficient

R50
RZo
Zo
Z-l
Z0
S11 is measure of the power returned to the
source, and is called the Return Loss
18
Making sense of S-Parameters Return Loss

• When there is a reflection from the load, S11
  will be composed of multiple reflections due to
  the standing waves

Zo
RL
Z0
Z-l

• If the network is driven with a 50 ohm source,
  then S11 is calculated using the input impedance
  instead of Zo

50 ohms
S11 of a transmission line will exhibit periodic
effects due to the standing waves
19
Example 3 Interpreting the return loss

• Based on the S11 plot shown below, calculate both
  the impedance and dielectric constant

R50
Zo
R50
L5 inches
0.45
0.4
0.35
0.3
S11, Magnitude
0.25
0.2
0.15
0.1
0.05
0
1.0
1.5
2.0
2.5
3..0
3.5
4.0
4.5
5.0
Frequency, GHz
20
Example Interpreting the return loss
0.45
1.76GHz
2.94GHz
0.4
Peak0.384
0.35
0.3
S11, Magnitude
0.25
0.2
0.15
0.1
0.05
0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Frequency, GHz

• Step 1 Calculate the time delay of the t-line
  using the peaks

• Step 2 Calculate Er using the velocity

21
Example Interpreting the return loss

• Step 3 Calculate the input impedance to the
  transmission line based on the peak S11 at
  1.76GHz
• Note The phase of the reflection should be
  either 1 or -1 at 1.76 GHz because it is aligned
  with the incident

• Step 4 Calculate the characteristic impedance
  based on the input impedance for x-5 inches

Er1.0 and Zo75 ohms
22
Making sense of S-Parameters Insertion Loss

• When power is injected into Port 1 with source
  impedance Z0 and measured at Port 2 with
  measurement load impedance Z0, the power ratio
  reduces to a voltage ratio

a20
a1
2
-
port
2
-
port
V1
V2
Zo
Zo
Network
Network
b2
b1
S21 is measure of the power transmitted from
port 1 to port 2, and is called the Insertion
Loss
23
Loss free networks

• For a loss free network, the total power exiting
  the N ports must equal the total incident power

• If there is no loss in the network, the total
  power leaving the network must be accounted for
  in the power reflected from the incident port and
  the power transmitted through network

• Since s-parameters are the square root of power
  ratios, the following is true for loss-free
  networks

• If the above relationship does not equal 1, then
  there is loss in the network, and the difference
  is proportional to the power dissipated by the
  network

24
Insertion loss example

• Question
• What percentage of the total power is dissipated
  by the transmission line?
• Estimate the magnitude of Zo (bound it)

25
Insertion loss example

• What percentage of the total power is dissipated
  by the transmission line ?
• What is the approximate Zo?
• How much amplitude degradation will this t-line
  contribute to a 8 GT/s signal?
• If the transmission line is placed in a 28 ohm
  system (such as Rambus), will the amplitude
  degradation estimated above remain constant?
• Estimate alpha for 8 GT/s signal

26
Insertion loss example

• Answer
• Since there are minimal reflections on this line,
  alpha can be estimated directly from the
  insertion loss
• S210.75 at 4 GHz (8 GT/s)

When the reflections are minimal, alpha can be
estimated

• If S11 lt 0.2 (-14 dB), then the above
  approximation is valid

• If the reflections are NOT small, alpha must be
  extracted with ABCD parameters (which are
  reviewed later)
• The loss parameter is 1/A for ABCD parameters
• ABCD will be discussed later.

27
Important concepts demonstrated

• The impedance can be determined by the magnitude
  of S11
• The electrical delay can be determined by the
  phase, or periodicity of S11
• The magnitude of the signal degradation can be
  determined by observing S21
• The total power dissipated by the network can be
  determined by adding the square of the insertion
  and return losses

28
A note about the term Loss

• True losses come from physical energy losses
• Ohmic (I.e., skin effect)
• Field dampening effects (Loss Tangent)
• Radiation (EMI)
• Insertion and Return losses include effects such
  as impedance discontinuities and resonance
  effects, which are not true losses
• Loss free networks can still exhibit significant
  insertion and return losses due to impedance
  discontinuities

29
Advantages/Disadvantages of S-parameters

• Advantages
• Ease of measurement
• Much easier to measure power at high frequencies
  than open/short current and voltage
• S-parameters can be used to extract the
  transmission line parameters
• n parameters and n Unknowns
• Disadvantages
• Most digital circuit operate using voltage
  thresholds. This suggest that analysis should
  ultimately be related to the time domain.
• Many silicon loads are non-linear which make the
  job of converting s-parameters back into time
  domain non-trivial.
• Conversion between time and frequency domain
  introduces errors

30
Cascading S parameter
3 cascaded s parameter blocks
a11
a21
b12
b22
a13
a13
b11
b21
a12
a22
b13
b13

• While it is possible to cascade s-parameters, it
  gets messy.
• Graphically we just flip every other matrix.
• Mathematically there is a better way ABCD
  parameters
• We will analyzed this later with signal flow
  graphs

31
ABCD Parameters

• The transmission matrix describes the network in
  terms of both voltage and current waves

I2
I1
2
-
port
2
-
port
V1
V2
Network
Network

• The coefficients can be defined using
  superposition

32
Transmission (ABCD) Matrix

• Since the ABCD matrix represents the ports in
  terms of currents and voltages, it is well suited
  for cascading elements

I3
I2
I1
V3
V1
V2

• The matrices can be cascaded by multiplication

This is the best way to cascade elements in the
frequency domain. It is accurate, intuitive and
simplistic.
33
Relating the ABCD Matrix to Common Circuits
Z
Assignment 6 Convert these to s-parameters
Port 1
Port 2
Y
Port 1
Port 2
Z1
Z2
Port 1
Port 2
Z3
Y3
Y1
Y2
Port 2
Port 1
Port 1
Port 2
34
Converting to and from the S-Matrix

• The S-parameters can be measured with a VNA, and
  converted back and forth into ABCD the Matrix
• Allows conversion into a more intuitive matrix
• Allows conversion to ABCD for cascading
• ABCD matrix can be directly related to several
  useful circuit topologies

35
ABCD Matrix Example 1

• Create a model of a via from the measured
  s-parameters

36
ABCD Matrix Example 1

• The model can be extracted as either a Pi or a T
  network

L2
L1
CVIA

• The inductance values will include the L of the
  trace and the via barrel (it is assumed that the
  test setup minimizes the trace length, and
  subsequently the trace capacitance is minimal
• The capacitance represents the via pads

37
ABCD Matrix Example 1

• Assume the following s-matrix measured at 5 GHz

38
ABCD Matrix Example 1

• Assume the following s-matrix measured at 5 GHz

• Convert to ABCD parameters

39
ABCD Matrix Example 1

• Assume the following s-matrix measured at 5 GHz

• Convert to ABCD parameters

• Relating the ABCD parameters to the T circuit
  topology, the capacitance and inductance is
  extracted from C A

Z1
Z2
Port 1
Port 2
Z3
40
ABCD Matrix Example 2

• Calculate the resulting s-parameter matrix if the
  two circuits shown below are cascaded

Port 1
Port 2
2
-
port
50
Network X
50
Network
Port 1
Port 2
2
-
port
50
Network Y
50
Network
2
-
port
2
-
port
50
Network Y
Network X
50
Network
Network
Port 2
Port 1
41
ABCD Matrix Example 2

• Step 1 Convert each measured S-Matrix to ABCD
  Parameters using the conversions presented
  earlier

• Step 2 Multiply the converted T-matrices

• Step 3 Convert the resulting Matrix back into
  S-parameters using thee conversions presented
  earlier

42
Advantages/Disadvantages of ABCD Matrix

• Advantages
• The ABCD matrix is very intuitive
• Describes all ports with voltages and currents
• Allows easy cascading of networks
• Easy conversion to and from S-parameters
• Easy to relate to common circuit topologies
• Disadvantages
• Difficult to directly measure
• Must convert from measured scattering matrix

43
Signal flow graphs Start with 2 port first

• The wave functions (a,b) used to define
  s-parameters for a two-port network are shown
  below. The incident waves is a1, a2 on port 1 and
  port 2 respectively. The reflected waves b1 and
  b2 are on port 1 and port 2. We will use as and
  bs in the s-parameter follow slides

44
Signal Flow Graphs of S Parameters
In a signal flow graph, each port is represented
by two nodes. Node an represents the wave coming
into the device from another device at port n,
and node bn represents the wave leaving the
device at port n. The complex scattering
coefficients are then represented as multipliers
(gains) on branches connecting the nodes within
the network and in adjacent networks.
Example Measurement equipment strives to be
match i.e. reflection coefficient is 0
See http//cp.literature.agilent.com/litweb/pdf/5
952-1087.pdf
45
Masons Rule Non-Touching Loop Rule

• T is the transfer function (often called gain)
• Tk is the transfer function of the kth forward
  path
• L(mk) is the product of non touching loop gains
  on path k taken mk at time.
• L(mk)(k) is the product of non touching loop
  gains on path k taken mk at a time but not
  touching path k.
• mk1 means all individual loops

46
Voltage Transfer function

• What is really of most relevance to time domain
  analysis is the voltage transfer function.
• It includes the effect of non-perfect loads.
• We will show how the voltage transfer functions
  for a 2 port network is given by the following
  equation.
• Notice it is not S21

47
Forward Wave Path
48
Reflected Wave Path
Vs
a1
b2
s21
s22
s11
GS
GL
s12
b1
a2
49
Combine b2 and a2
50
Convert Wave to Voltage - Multiply by sqrt(Z0)
51
Voltage transfer function using ABCD
Lets see if we can get this results another way
52
Cascade ABCD to determine system ABCD
53
Extract the voltage transfer function

• Same as with flow graph analysis

54
Cascading S-Parameter

• As promised we will now look at how to cascade
  s-parameters and solve with Masons rule
• The problem we will use is what was presented
  earlier
• The assertion is that the loss of cascade channel
  can be determine just by adding up the losses in
  dB.
• We will show how we can gain insight about this
  assertion from the equation and graphic form of a
  solution.

55
Creating the signal flow graph
B
A
C

• We map output a to input b and visa versa.
• Next we define all the loops
• Loop A and B do not touch each other

56
Use Masons rule
B
A
C
Masons Rule
A
B
C
A
B

• There is only one forward path a11 to b23.
• There are 2 non touching looks

57
Evaluate the nature of the transfer function
Assumption is that these are 0

• If response is relatively flat and reflection is
  relatively low
• Response through a channel is s211s212213

58
Jitter and dB Budgeting

• Change s21 into a phasor
• Insertion loss in db

i.e. For a budget just add up the dbs and jitter
59
Differential S-Parameters

• Differential S-Parameters are derived from a
  4-port measurement
• Traditional 4-port measurements are taken by
  driving each port, and recording the response at
  all other ports while terminated in 50 ohms
• Although, it is perfectly adequate to describe a
  differential pair with 4-port single ended
  s-parameters, it is more useful to convert to a
  multi-mode port

a
b
S
S
S
S
a
2
1
11
12
13
14
1
b
S
4
-
port
2
S
S
S
b
21
b
2
22
24
23
1
b
S
S
S
S
3
33
34
31
32
b
S
S
S
4
S
43
41
44
42
60
Differential S-Parameters

• It is useful to specify the differential
  S-parameters in terms of differential and common
  mode responses
• Differential stimulus, differential response
• Common mode stimulus, Common mode response
• Differential stimulus, common mode response (aka
  ACCM Noise)
• Common mode stimulus, differential response
• This can be done either by driving the network
  with differential and common mode stimulus, or by
  converting the traditional 4-port s-matrix

b
DS
DS
DCS
DCS
dm1
11
12
11
12
b
DS
dm2
DS
DCS
DCS
21
22
22
21
b
CS
CS
CDS
CDS
cm1
11
12
11
12
b
CS
CDS
CS
cm2
CDS
21
21
22
22
Matrix assumes differential and common mode
stimulus
61
Explanation of the Multi-Mode Port
Common mode conversion Matrix Differential
Stimulus, Common mode response. i.e., DCS21
differential signal (D)-(D-) inserted at port
1 and common mode signal (D)(D-) measured at
port 2
Differential Matrix Differential Stimulus,
differential response i.e., DS21 differential
signal (D)-(D-) inserted at port 1 and diff
signal measured at port 2
b
a
DS
DS
DCS
DCS
dm1
dm1
11
12
11
12
b
a
DS
dm2
dm2
DS
DCS
DCS
21
22
22
21

b
a
CS
CS
CDS
CDS
cm1
cm1
11
12
11
12
b
a
CS
CDS
CS
cm2
cm2
CDS
21
21
22
22
differential mode conversion Matrix Common mode
Stimulus, differential mode response. i.e.,
DCS21 common mode signal (D)(D-) inserted
at port 1 and differential mode signal
(D)-(D-) measured at port 2
Common mode Matrix Common mode stimulus, common
mode Response. i.e., CS21 Com. mode signal
(D)(D-) inserted at port 1 and Com. mode
signal measured at port 2
62
Differential S-Parameters

• Converting the S-parameters into the multi-mode
  requires just a little algebra

Example Calculation, Differential Return Loss
The stimulus is equal, but opposite, therefore
2
1
2
-
port
4
-
port
Network
Network
4
3
Assume a symmetrical network and substitute
Other conversions that are useful for a
differential bus are shown
Differential Insertion Loss
Differential to Common Mode Conversion (ACCM)
Similar techniques can be used for all multi-mode
Parameters
63
Next class we will develop more differential
concepts
64
backup review
65
Advantages/Disadvantages of Multi-Mode Matrix
over Traditional 4-port

• Advantages
• Describes 4-port network in terms of 4 two port
  matrices
• Differential
• Common mode
• Differential to common mode
• Common mode to differential
• Easier to relate to system specifications
• ACCM noise, differential impedance
• Disadvantages
• Must convert from measured 4-port scattering
  matrix

66
High Frequency Electromagnetic Waves

• In order to understand the frequency domain
  analysis, it is necessary to explore how high
  frequency sinusoid signals behave on transmission
  lines
• The equations that govern signals propagating on
  a transmission line can be derived from Amperes
  and Faradays laws assumimng a uniform plane wave
• The fields are constrained so that there is no
  variation in the X and Y axis and the propagation
  is in the Z direction

• This assumption holds true for transmission lines
  as long as the wavelength of the signal is much
  greater than the trace width

X
Direction of propagation
Z
For typical PCBs at 10 GHz with 5 mil traces
(W0.005)
Y
67
High Frequency Electromagnetic Waves

• For sinusoidal time varying uniform plane waves,
  Amperes and Faradays laws reduce to

Amperes Law A magnetic Field will be induced by
an electric current or a time varying electric
field
Faradays Law An electric field will be
generated by a time varying magnetic flux

• Note that the electric (Ex) field and the
  magnetic (By) are orthogonal

68
High Frequency Electromagnetic Waves

• If Amperes and Faradays laws are differentiated
  with respect to z and the equations are written
  in terms of the E field, the transmission line
  wave equation is derived

This differential equation is easily solvable for
Ex
69
High Frequency Electromagnetic Waves

• The equation describes the sinusoidal E field for
  a plane wave in free space

Note the positive exponent is because the wave
is traveling in the opposite direction
Portion of wave traveling In the z direction
Portion of wave traveling In the -z direction
permittivity in Farads/meter (8.85 pF/m for
free space) (determines the speed of light in
a material)
permeability in Henries/meter (1.256 uH/m for
free space and non-magnetic materials)
Since inductance is proportional to
capacitance is proportional to , then
is analogous to in a transmission line,
which is the propagation delay
70
High Frequency Voltage and Current Waves

• The same equation applies to voltage and current
  waves on a transmission line

Incident sinusoid
RL
Reflected sinusoid
z-l
z0
If a sinusoid is injected onto a transmission
line, the resulting voltage is a function of time
and distance from the load (z). It is the sum of
the incident and reflected values
Note is added to specifically
represent the time varying Sinusoid, which was
implied in the previous derivation
Voltage wave reflecting off the Load and
traveling towards the source
Voltage wave traveling towards the load
71
High Frequency Voltage and Current Waves

• The parameters in this equation completely
  describe the voltage on a typical transmission
  line

Complex propagation constant includes all the
transmission line parameters (R, L C and G)
(For the loss free case)
(lossy case)
Attenuation Constant (attenuation of the signal
due to transmission line losses)
(For good conductors)
Phase Constant (related to the propagation
delay across the transmission line)
(For good conductors and good dielectrics)
72
High Frequency Voltage and Current Waves

• The voltage wave equation can be put into more
  intuitive terms by applying the following
  identity

Subsequently

• The amplitude is degraded by
• The waveform is dependent on the driving function
  ( ) the delay of
  the line

73
Interaction transmission line and a load

• The reflection coefficient is now a function of
  the Zo discontinuities AND line length
• Influenced by constructive destructive
  combinations of the forward reverse waveforms

Zo
Zl
(Assume a line length of l (z-l))
Z-l
Z0
This is the reflection coefficient looking into a
t-line of length l
74
Interaction transmission line and a load

• If the reflection coefficient is a function of
  line length, then the input impedance must also
  be a function of length

Zin
RL
Z-l
Z0
Note is dependent on and
This is the input impedance looking into a t-line
of length l
75
Line load interactions

• In chapter 2, you learned how to calculate
  waveforms in a multi-reflective system using
  lattice diagrams
• Period of transmission line ringing
  proportional to the line delay
• Remember, the line delay is proportional to the
  phase constant
• In frequency domain analysis, the same principles
  apply, however, it is more useful to calculate
  the frequency when the reflection coefficient is
  either maximum or minimum
• This will become more evident as the class
  progresses

To demonstrate, lets assume a loss free
transmission line
76
Line load interactions
Remember, the input reflection takes the form
The frequency where the values of the real
imaginary reflections are zero can be calculated
based on the line length
Term 1
Term 2
Term 10 Term 2
Term 20 Term 1
Note that when the imaginary portion is zero, it
means the phase of the incident reflected
waveforms at the input are aligned. Also notice
that value of 8 and 4 in the terms.
77
Example 1 Periodic Reflections

• Calculate
• Line length
• RL
• (assume a very low loss line)

Er_eff1.0
RL
Zo75
Z-l
Z0
78
Example 1 Solution
Step 1 Determine the periodicity zero crossings
or peaks use the relationships on page 15 to
calculate the electrical length
79
Example 1 Solution (cont.)

• Note the relationship between the peaks and the
  electrical length
• This leads to a very useful equation for
  transmission lines

• Since TD and the effective Er is known, the line
  length can be calculated as in chapter 2

80
Example 1 Solution (cont.)

• The load impedance can be calculated by observing
  the peak values of the reflection
• When the imaginary term is zero, the real term
  will peak, and the maximum reflection will occur
• If the imaginary term is zero, the reflected wave
  is aligned with the incident wave and the phase
  term 1
• Important Concepts demonstrated
• The impedance can be determined by the magnitude
  of the reflection
• The line length can be determined by the phase,
  or periodicity of the reflection