Multipath Effects in Wireless Networks Calculus Topic: Taylor and Maclaurin Series

1
Multipath Effects in Wireless Networks Calculus
Topic Taylor and Maclaurin Series

• Damla Turgut, Ph.D.
• School of Electrical Engineering and Computer
  Science
• University of Central Florida
• Email turgut_at_eecs.ucf.edu
• Web http//www.cs.ucf.edu/turgut

2
Review Problem


Section 12.57 Find the sum of the following
series
3
Review Problem

• 1) Which one of the following is the correct
  Taylor expansion?

A.
B.
C.
D.
None of the above is correct
4
Review Problem

• Given the Taylor expansion
• 2) Which one of the following is the infinite
  sum?

A.
C.
D.
B.
None of the above is correct
5
Review Problem

• 3) Which one of the following f(x) will be used?

A.
B.
C.
D.
None of the above
6
Review Problem

• 4) For the expansion about a 0 is

A.
B.
C.
D.
None of the above
7
Review Problem

• 5) What is the infinite sum?

A.
C.
D.
B.
None of the above is correct
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Outline

• Background/Motivation
• Multipath Interference
• Multipath Example
• Plotting with MS Excel

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Background/Motivation

• Discussion of how Taylor series apply to
  multipath in wireless 802.11 networks
• Multipath propagation occurs when radio frequency
  (RF) signals go through different paths from
  source to destination
• Direct path -? no delay
• Indirect path (bounces off an obstruction such as
  metal, coated glass, or stone) -? delay
• This phenomenon is called multipath distortion

10
Background/Motivation

• The effects of multipath distortion can vary
  depending on the path taken
• Multipath interference opportunities
• in urban areas reflections from building
  structures and automobile
• at home the walls, appliances, and furniture

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Terminology Wave, Crest, and Amplitude

• A wave is a disturbance that propagates through
  space or often transferring energy
• A crest is the point on a wave with the greatest
  positive value or upward displacement in a cycle.
  A trough is the opposite of a crest.
• The amplitude, A is a nonnegative scalar measure
  of a wave's magnitude of oscillation

Courtesy of Dr. Winningham, Physics and Wikipedia
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Terminology Wavelength, Period

• The wavelength, l, is the distance from one crest
  to the next
• More generally, the wavelength is the minimum
  distance between any two identical points on
  adjacent waves
• The period, T, is the time interval required for
  two identical points of adjacent waves to pass by
  a point

Courtesy of Dr. Winningham, Physics and Wikipedia
13
Terminology Frequency and Period

• The frequency, , is the measurement of the
  number of occurrences of a repeated event per
  unit of time.
• The frequency and the period are related
• When the time interval is the second, the units
  of frequency are S-1 Hz (hertz)

Courtesy of Dr. Winningham, Physics and Wikipedia
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Terminology Frequency and Period
1
2
3
4
5
Courtesy of Wikipedia
Question According to the sine wave of
frequencies give above which of the following has
the correct order of wave having higher
frequencies to lower frequencies?

• 1, 2, 3, 4, 5 B. 1, 3, 5, 2, 4
• 5, 4, 3, 2, 1 D. 2, 4, 1, 3, 5

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Terminology Various Waves
Courtesy of Wikipedia
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Learning Objective 1 Multipath Interference

• To understand multipath interference, we have to
  look at what happens when two waves are added
  together
• From trigonometry, the cosine function is
  represented as

f frequency (Hz or 1/s or s-1) A is amplitude
(volts) is the phase angle (radians) t is
time
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Learning Objective 1 Multipath Interference

• The combined expression for the addition of two
  cosine functions is
• (radians) represent the difference in
  phase between the two cosine waves and there is a
  different amplitude associated with each wave

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Learning Objective 1 Question 1

• If the combined expression for the addition of
  two cosine functions is
• We get the following equation if two cosine
  waves are 180 degrees or radians out of
  phase,

A.
True
B.
False
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Learning Objective 1 Question 2

• According to the formula below, what is the
  resulting solution if the amplitudes are equal

A.
B.
Nothing happens
C.
It depends on the frequency
Solution becomes zero
D.
E.
None of the above
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Learning Objective 1 Multipath Interference
AB
A
B
Cancellation!
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Learning Objective 1 Multipath Interference

• Now, we need to put this all together
• In a multipath scenario, we are adding together
  cosine waves that are coming from different paths
• The time it takes for us to receive the waves
  from each path is different
• This results in phase shifts and possible
  cancellations

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Learning Objective 1 Multipath Interference

• Point of reflection ? angle of incidence angle
  of reflection
• d1 is the distance traveled directly
• d2 is the distance traveled by the reflected path
  directly
• d1 lt d2

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Learning Objective 1 Multipath Interference

• This difference of two paths creates interference
  and multipath effects
• We will use this figure to develop mathematical
  expressions for multipath and refer to it in the
  remaining of this lesson

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Learning Objective 2 Multipath Example

• We can compare the direct path distance to the
  receiving node versus the distance of the
  indirect path
• The delay of time of arrival can be written as a
  phase as follows

time delay
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Learning Objective 2 Multipath Example

• Phase of the received signal is a function of
  time delay and frequency of the signal
• The time delay can be computed given the distance
  and the fact that RF energy must travel at the
  speed of light, as follows

represents the distance in meters
is the speed of light which is equal to
meters/sec
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Learning Objective 2 Multipath Example
Given
and

• We are interested in the time delay difference
  between the direct and reflected energy as a
  function of the difference in distance
• Therefore, we replace with
• We obtain the following
• This formula shows that if we know the difference
  covered between the directed and reflected
  energy, we can compute the phase difference
  between the two signals

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Learning Objective 2 Multipath Example
Given

• The interference resulting from multipath
  reflection off of the ground can be written as

Question Is the above formula correct or not?
A.
B.
True
False
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Learning Objective 2 Multipath Example
What is the difference in distance traveled via
the direct path versus reflected path?
T
R

• Let us take another look at our original problem
• We want to find the path length difference
  between the two signals received by R
• This will give us the phase difference!

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Learning Objective 2 Multipath Example

• We can perform a trick. Let us reflect
  everything through the ground
• What is the benefit of doing this?

A. It makes the diagram easier to read B. The
signal really propagates through the ground C.
We create right triangles that are easier to deal
with
T
R
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Learning Objective 2 Multipath Example

• Reminder of Pythagoreans theorem
• Question What is ?
• A.
• B.
• C.

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Learning Objective 2 Multipath Example

• Using Pythagoreans theorem, we can now compute
  the distances

T
R
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Learning Objective 2 Multipath Example
Question What is
A.
B.
C.
D.
None of the above
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Learning Objective 2 Multipath Example

• Using the Taylors series, we can represent this
  function as an infinite series and by only using
  the first two terms of the infinite series, we
  can form an approximation to the function

34
Learning Objective 2 Multipath Example

• Let us try to write the equation for the received
  wave
• At a glance, can you predict what effect d has on
  r(t)?

35
Learning Objective 2 Multipath Example

• We need to simplify the expression
• Let us first rearrange the equation a little

36
Learning Objective 2 Multipath Example

• We can, then, write this as
• where,

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Learning Objective 2 Multipath Example

• We can, then, write this as
• Question What part of this equation would you
  choose to simplify with a Taylors series?
• The entire expression
• B.
• C.

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Learning Objective 2 Multipath Example

• It is really the part under the radical that is
  causing difficulty in analyzing our equation
• We will use what is referred to as a Maclaurin
  Series

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Learning Objective 2 Multipath Example

• Let us write out the first two terms of our
  series
• Question What is the first term of our series?
• A.
• B.
• C.

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Learning Objective 2 Multipath Example

• Question What is the second term of our series?
• A.
• B.
• C.

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Learning Objective 2 Multipath Example

• The result is
• Observe that we now have a simpler, 1st order
  polynomial
• We could use more terms of the Taylors series,
  but we would end up with a 2nd order, 3rd order,
  etc, polynomial

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Learning Objective 2 Multipath Example

• Now we can start using our simplification for the
  portion under the radical

43
Learning Objective 2 Multipath Example

• Now, we just need to do a little algebra

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Learning Objective 2 Multipath Example

• After all of that, we now have something that
  appears much simpler
• Question Why is there an approximation sign?
• A. To remind ourselves the equation is simpler
• B. Because we used a Taylor series
• C. Because the Taylor series was truncated

45
Learning Objective 2 Multipath Example

• Now we can work with a much simpler formula for
  the phase difference between the direct path and
  reflected path wave

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Learning Objective 2 Multipath Example

• This gives us the following for the received
  signal
• Question What happens as d becomes very large?
• A. The sine waves cancel
• B. The phase becomes nearly zero and the two sine
  waves add
• C. There is an oscillation as I go out in
  distance and sometimes they add and sometimes
  they cancel

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Learning Objective 2 Multipath Example

• To summarize
• We used a Taylor series to approximate a function
• The approximate function was far easier to make
  predictions about than the original function
• Even if you have a computer and can plot the
  results, the Taylor series allowed us to prove
  concepts
• The two signals no longer cancel each other when
  the distance between the transmitter and receiver
  is large
• Remember that we truncated the series, so the new
  formula was an approximation of the original!

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Review Problems

• 1) Which one of the following is the correct
  Taylor expansion?

A.
B.
C.
D.
None of the above is correct
49
Review Problems

• 2) Which one of the following topic was discussed
  as an application
• to Taylor series?

A.
Multipath interface
B.
Multipath interference
C.
None of the above is correct
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Review Problems
2.
3.
1.

• 3) Which one of the choices identify
• the following points (1-4) correctly?
• amplitude, crest, period, wavelength
• B. wavelength, period, crest, amplitude
• C. crest, wavelength, amplitude, period
• D. amplitude, period, crest, wavelength
• E. None of the above

4.
51
Review Problems

• 4) What is the relationship between frequency and
  period?

A.
B.
C.
D.
None of the above is correct
52
Review Problems

• 5) What is ?
• A.
• B.
• C.

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Plotting with MS Excel

• In this next part we are going to show
• 1. An example of using MS Excel to plot
• 2. Show a comparison of the plots for the
  received signal function before and after the
  Taylor series approximation

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Practical Demonstration

• First, we are going to plot

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Practical Demonstration

• First, lets establish our time increment
• In Cell F1, Type Frequency
• In Cell G1, Type Period
• In Cell F2, Type 2.4e9
• In Cell G2, Type 1/F2

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Practical Demonstration

• We have the period, and we want to plot two of
  those periods
• We need to choose how many points to plot, lets
  plot 100 points
• In Cell H1, Type Total Time
• In Cell H2, Type G22
• In Cell I1, Type NPoints
• In Cell I2, Type 100
• In Cell J1, Type Time Increment
• In Cell J2, Type H2/I2

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Practical Demonstration

• We now know the time increment, so we can start
  building our plot.
• In Cell A1, Type 0
• In Cell A2, Type 8.33e-12

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Practical Demonstration

• Next, we can use the auto fill feature of Excel
• Begin by clicking on cell A1
• While clicking, drag the cursor so that cells A1
  and A2 are highlighted
• Note the tiny square at the bottom right corner
  of the box surrounding the selected items
• When the mouse hovers over the square, it turns
  into a set of cross hairs

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Practical Demonstration

• Click and drag the square downward
• Youll notice a hash/border box surrounds a set
  of the cells and follows the cursor
• Continue dragging this box until up to cell A100
  has been selected and release the mouse

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Practical Demonstration

• Youll notice that cells will be filled in based
  upon the increment specified in the first two
  cells.
• We stopped just one cell short of what would have
  given us the total two periods of 8.33e-10.

61
Practical Demonstration

• Before we enter the formula, we need an
  additional cell.
• We are going to allow the distance, d, to be
  changed
• In Cell F4, Type Distance
• In Cell F5, Type 1
• While we are at it, we can compute the phase,
  which doesnt depend upon time!
• In Cell F6, Type Phase
• In Cell F7, type the formula in the seen in the
  formula bar

62
Practical Demonstration

• Now we are ready to enter the formula
• In Cell B1, type the formula in the formula bar
• Note that there is square on the bottom right
  corner of box highlighting cell B1
• Double-click this cell

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Practical Demonstration

• You should now have something that looks like
  this
• We are ready to plot

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Practical Demonstration

• Click on the column header for column A and drag
  to highlight column B as well. Both columns
  should be highlighted
• Select the graph button
• Select XY (Scatter)
• Select the plot type we have shown, here.
• Click Nextgt

65
Practical Demonstration

• At this point, you see step 2 of the chart wizard
• You can add titles and other features by clicking
  Next gt or just click Finish to see the graph
• Well click finish

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Practical Demonstration

• And the graph is complete.
• Try adjusting the distance over several values
  and note how the curve behaves
• What kind of conclusions can you make about the
  magnitude of this function as a distance varies?

67
Practical Demonstration

• Lets compare our approximation

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Practical Demonstration

• Well add a field for the phase approximation
• In Cell F9, Type Phase Approx
• In Cell F10, Type the formula in the formula bar

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Practical Demonstration

• In Cell C1, Type the formula in the formula bar
• Select cell C1 and double click on the square at
  the bottom right corner of the selection
  highlight

70
Practical Demonstration

• Right click on the chart and select Source
  Data from the menu options
• Were going to click on Add to bring up the
  option to add another series

71
Practical Demonstration

• After selecting Add, fill in the fields of the
  dialog as seen, here.
• Note the ranges can be selected by highlighting
  the cells with the cursor instead of entering the
  values
• When you are done, press Ok.

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Practical Demonstration

• Once again, try changing the distance and
  observing the results.
• Are there regions where our approximation is more
  accurate or less accurate
• What can say, in general, about the
  approximation?
• If it is not always accurate, why use it?