Empirically Based Statistical Ultra-Wideband (UWB) Channel Model

1
Project IEEE P802.15 Working Group for Wireless
Personal Area Networks (WPANs) Submission Title
Empirically Based Statistical Ultra-Wideband
Channel Model Date Submitted 24 June,
2002 Source Marcus Pendergrass, Time Domain
Corporation 7057 Old Madison Pike,
Huntsville, AL 35806 Voice256-428-6344 FAX
256-922-0387, E-Mail marcus.pendergrass_at_timedom
ain.com Re Ultra-wideband Channel Models IEEE
P802.15-02/208r0-SG3a, 17 April,
2002, Abstract An ultra-wideband (UWB) channel
measurement and modeling effort, targeted towards
the short-range, high data rate wireless personal
area network (WPAN) application space, is
described. Results of this project include a
measurement database of 429 UWB channel
soundings, including both line of sight and non
line of sight channels, a statistical description
of this database, and recommended models and
modeling parameters for several UWB WPAN
scenarios of interest. Purpose The information
provided in this document is for consideration in
the selection of a UWB channel model to be used
for evaluating the performance of a high rate UWB
PHY for WPANs. Notice This document has been
prepared to assist the IEEE P802.15. It is
offered as a basis for discussion and is not
binding on the contributing individual(s) or
organization(s). The material in this document is
subject to change in form and content after
further study. The contributor(s) reserve(s) the
right to add, amend or withdraw material
contained herein. Release The contributor
acknowledges and accepts that this contribution
becomes the property of IEEE and may be made
publicly available by P802.15.
2
Empirically Based Statistical Ultra-Wideband
(UWB) Channel Model

• Marcus Pendergrass and William C. Beeler
• 24 June 2002
• with thanks to Laurie Foss, Joy Kelly, James
  Mann, Alan Petroff, Alex Petroff, Mitchell
  Williams, and Scott Yano for assistance and
  support.

3
Executive Summary

• Important to characterize the Wireless Personal
  area network (WPAN) environment.
• 429 channel soundings taken in residential and
  office environments.
• Statistical multipath models for 3 environments
  described LOS 0-4 meters, NLOS 0-4 meters, NLOS
  4 - 10 meters.
• Channel response modeled as a sum of scaled and
  delayed versions template waveform.
• Good fit to measurement data. Distortion lt1dB.
• Recommendations offered

4
Outline

• Introduction
• Measurement Campaign
• Data Analysis
• Statistical Environmental Models
• Analytical Models
• Conclusions/Recommendations

5

• Introduction
• Measurement Campaign
• Data Analysis
• Statistical Environmental Models
• Analytical Models
• Conclusions/Recommendations

6
Introduction

• Channel Impulse Response (CIR) modeling of
  radio-frequency channels necessary for system
  design, trades.
• Multipath channel effects will be a key
  determinant of system performance, reliability.
• Large literature on channel modeling available,
  including work on the UWB channel in particular.
• Important to characterize the wireless personal
  area network (WPAN) environment in both line of
  sight (LOS) and non line of sight (NLOS) cases.
• Models should be tuned to WPAN applications and
  environments.

7
Approach

• Measurement Campaign
• Channel soundings taken in a variety of WPAN-type
  environments.
• Data Analysis
• Deconvolution of channel impulse response (CIR)
  from measurements.
• Assessment of channel distortion.
• Statistical analysis of UWB channel parameters as
  a function of environment type.
• Fit existing models to data
• IEEE 802.11 model.
• The D-K model.
• Assess goodness of fit
• Recommend models, parameters

8
Overview of Results

• 429 channels soundings taken from 11 different
  home and office environments.
• Data will be made available to SG3a.
• Environmental signal distortion estimated.
• Multipath channel parameters described
  statistically
• RMS delay
• Distribution of multipath arrival times.
• Average power decay profile.
• Ability of existing models to capture the
  phenomenology of the data assessed.
• Recommendations made for models and parameters.

9

• Introduction
• Measurement Campaign
• Data Analysis
• Statistical Environmental Models
• Analytical Models
• Conclusions/Recommendations

10
Purpose

• Support statistical analysis WPAN propagation
  environments by obtaining a well-documented set
  of diverse measurements of the UWB channel.
• Short range (0-4 meters), and medium range (4 -
  10 meters)
• LOS and NLOS channels
• office and residential environments

11
Measurement Plan

• NLOS and LOS measurements for WPAN multipath
  channel characterization.
• Metal stud and wooden stud environments.
• Metal studs typical of office environments
  wooden studs more typical of residential
  environments.
• 11 different office and home locations
• Detailed documentation for each channel sounding
• X,Y,Z coordinates of transmit/receive antenna
  locations.
• Channel categorized as LOS or NLOS

12
Test Setup Details

• Summary
• Approximately omni-directional transmit/receive
  antennas (roughly 3 dBi gain)
• PCS and ISM band pass rejection filter
• Effective noise figure 4.8 dB at receive antenna
  terminals
• Gain 19.8 dB
• Radiated power at approximately -10 dBm in the 3
  to 5 GHz spectrum (close to FCC UWB limit)

13
Test Setup Details

• Data recorded
• 100 ns channel record.
• 4096 data points per record.
• Effective sampling time is 24.14 ps (20 GHz
  Nyquist frequency).
• 350 averages per data point per channel record
  (for high SNR).
• Triggered sampling for accurate determination of
  effective LOS arrival time.
• Channel stimulus is UWB signal with 3 to 5 GHz 3
  dB bandwidth, approximately 1.7 ns pulse duration.

14
Channel Measurement Test Setup
15
Measurement Issues

• Received pulse distortion
• Need accurate received pulse templates for
  deconvolution analysis.
• Resolution assessment of waveform distortion
  due to the angle of arrival of the incoming
  signal.
• Determination of line of sight delay time in NLOS
  channels.
• Accurate determination of multipath intensity
  profiles for NLOS channels requires knowing where
  the line of sight path would have arrived, had it
  not been obstructed.
• Resolution careful design and characterization
  of test setup and parameters (group delays, NF,
  antenna pattern, etc.), along with periodic
  excitation of the environment. Utilize known
  delays of test equipment, known transmit/receive
  locations, and periodic triggering to estimate
  what the direct path arrival time would have been
  for a NLOS channel.

16
Measurement IssueReceived Pulse Distortion

• Accurate received waveform template needed for
  effective deconvolution of channel impulse
  response.
• Sources of waveform distortion
• environment (non-linear group delay,
  frequency-selective attenuation, etc.)
• interference (intermittent and steady state)
• antenna pattern
• Environmental distortion to be estimated in data
  analysis.
• Interference in minimized with appropriate
  filtering (PCS, ISM bands).
• Distortion due to non-ideal antenna pattern was
  assessed empirically.
• distortion as a function of elevation angle.

17
Typical Normalized Antenna Azimuth and Elevation
Patterns (omni-directional antennas)
18
Received Pulse Distortion Test Setup
19
Pulse Distortion Test Results
Normalized amplitudes

• For angles of elevation between -70 degrees and
  70 degrees, waveform distortion was found to be
  minimal.
• Significant distortion near 90 degrees
  elevation however, signal is severely attenuated
  in this region.
• Use of a single received pulse template was
  judged acceptable for deconvolution analysis.

20
Measurement IssueDetermination of LOS Delay

• In our test set-up, periodic excitation of the
  environment (non time-hopped) allowed for more
  accurate calculation of LOS delays.
• With periodic excitation the channel ring-down
  from previous pulse can add to the recorded
  response data if the record length is shorter
  than the ring-down time of the channel.
• Random excitation decorrelates the previous
  pulses ring-down from the recorded response
  through the DSO averaging process.
• Effect is most pronounced in channels with high
  RMS delay spread.

21
Periodic Channel Stimulus Example
22
Random Channel Stimulus Example
23
Minimal Effect on RMS Delay

• Ability to accurately determine LOS delay was
  judged important enough to utilize periodic (non
  time-hopped) pulse trains.

24
Channel Measurement Environments

• 11 different office and home environments
• Metal and wood stud constructions
• Distances less than or equal to 10 meters.
• 471 channel soundings taken in total.
• Complete documentation of measurement locations
  and environments.

25
Example Measurement Locations A Typical Office
Environment
26
Example Measurement Locations Conference Room
27
Example Measurement Locations Residential Living
Room
28
Measurement Database

• 471 channel soundings taken in total.
• Database consists of a subset of 429 of these
  channels
• All measurements vertically polarized.
• Includes received waveform scans and extracted
  channel impulse responses.
• Includes calculated channel parameters, including
  RMS delay and path loss.
• Also includes various measurement meta-data,
  including
• locations of transmitter and receiver
• channel categorized as LOS or NLOS.
• calculated line of sight delay time
• environment type (wood stud, metal stud)
• polarization
• number of intervening walls between transmitter
  and receiver.

29

• Introduction
• Measurement Campaign
• Data Analysis
• Statistical Environmental Models
• Analytical Models
• Conclusions/Recommendations

30
Analysis Goals

• Extract a description of the channel that is
  independent of the channel stimulus.
• Estimate distortion caused by the propagation
  environments.
• Produce a statistical description of channel
  parameters as a function of environment type.

31
Major Analysis Assumptions

• Channel modeled as a linear time-invariant (LTI)
  filter.
• assume that there are negligible changes to the
  channel on the time scale of a communications
  packet.
• Impulse response for the channel is assumed to be
  of the form
• channels effect on signal is modeled as a series
  of amplitude scalings ak and time delays tk.

(1)
32
CLEAN Algorithmused to deconvolve CIR from
channel record

• CLEAN is a variation of a serial correlation
  algorithm
• Uses a template received waveform to sift through
  an arbitrary received waveform
• Cross-correlation with template suppresses
  non-coherent signals and noise
• Result is aks and tks of CIR independent of
  measurement system

33
CLEAN AlgorithmCompared to Frequency Domain
De-Convolution
34
CLEAN Algorithmgeometric interpretation
Energy Capture Ratio
Relative Error
Least Squares Condition
(2)
35
CLEAN Algorithmestimation of signal distortion

• CLEAN returns the CIR in precisely the desired
  form (1).
• Convolution of CIR with pulse template p(t)
  produces the reconstructed channel record r(t)
• When the least squares condition (2) holds, the
  residual difference between the CLEAN
  reconstruction and original channel record is a
  measure of the distortion introduced by the
  channel (i.e. the amount of signal energy that is
  not of the form (1)).

36
CLEAN Residual Estimates of Signal Distortion

• Least squares condition met at 85 energy capture
  ratio, on average.
• Estimated signal distortion
• NLOS, 0 to 4 meters, metal stud case 15.5
  (0.7 dB)
• LOS, 0 to 4 meters, metal stud case 16.6 (0.7
  dB)
• NLOS, 4 to 10 meters, metal stud case 17.0 (0.8
  dB)

37

• Introduction
• Measurement Campaign
• Data Analysis
• Statistical Environmental Models
• Analytical Models
• Conclusions/Recommendations

38
Data Used for the Analysis

• 429 of the 471 channel records
• all vertically polarized measurements.
• duplicate measurements removed.

39
General Remarks on the Data

• Data collection SNRs varied from about 40 dB for
  1-meter boresight scans to about 15 dB for some
  10-meter NLOS scans.
• LOS and NLOS channels exhibit wide variations in
  path loss and RMS delay spread. Some NLOS
  channels have lower delay spreads than some LOS
  channels.
• The variations can be explained by grazing angles
  and destructive interference for LOS channels ,
  and low attenuation through materials for NLOS
  channels.

40
Scan 1 LOS 1m distance, Antenna Boresight1/r2
Path Loss
41
Scan 57 LOS 3.1m distance, office
environment, approximately 1/r5.28 Path Loss
Check this one!
42
Scan 6 NLOS 1.3m distance, office environment,
approximately 1/r26.5 path loss
43
Scan 15 NLOS2.7m distance, office
environmentapproximately 1/r2.07 Path Loss
0.02
0.015
0.01
0.005
Amplitude
0
-0.005
-0.01
-0.015
-0.02
0
2
4
6
8
10
12
14
16
18
20
Time (ns)
44
Descriptive Statistics of the Data

• CIRs and channel parameters extracted for all 429
  records.
• Statistical analysis and model fitting done only
  for metal stud measurements.
• 369 metal stud measurements.
• 60 wood stud measurements not enough for
  statistical breakdown.
• Three scenarios considered
• I. NLOS, 0 to 4 meters, metal stud.
• II. LOS, 0 to 4 meters, metal stud.
• III. NLOS, 4 to 10 meters, metal stud.
• Not enough LOS, 4 to 10 meter channels for
  analysis.

45
Explanation of Channel Statistics

• Channels characterized in terms of the following
  statistical parameters
• RMS delay as a function of distance.
• Mean excess delay as a function of distance.
• Number of multipath components per channel.
• Occupancy probabilities as a function of excess
  delay.
• Mean log relative magnitudes as a function of
  excess delay.

46
Channel Statistics
multipath component
amax
kth relative magnitude
a1
amplitudes
a0
ak
tk
t0
t1
time
delays
LOS delay
kth excess delay tk t0

• Mean excess delay is a weighted average of the
  excess delays in the CIR.
• CIR amplitudes are the weights
• RMS delay is the standard deviation of the excess
  delays.
• again using the CIR amplitudes as the weights.

47
Channel Statistics
48
Dependence of Channel Statistics on CLEAN
Algorithm Stopping Condition

• Channel statistics computed from channel impulse
  response as calculated by CLEAN algorithm.
• Dependence of channel statistics on stopping
  criteria assessed.
• The following energy capture stopping criteria
  were evaluated 80, 85, 90, 95

49
80 Energy Capture
(notional)
amplitudes
time
50
85 Energy Capture
(notional)
amplitudes
time
51
90 Energy Capture
(notional)
amplitudes
time
52
95 Energy Capture
(notional)
amplitudes
time
What is the effect on channel statistics?
53
Comparison of Statistics Across Energy Capture
Ratios
I. NLOS, 0 to 4 meters, metal stud
85 energy capture
95 energy capture
Avg. RMS Delay
11.57 ns
8.78 ns
12.41 ns
10.04 ns
Avg. Mean Excess Delay
Mean Number of Components per Channel
36.1
86.0
54
Comparison of Statistics Across Energy Capture
Ratios
II. LOS, 0 to 4 meters, metal stud
85 energy capture
95 energy capture
Avg. RMS Delay
6.36 ns
5.27 ns
5.17 ns
4.95 ns
Avg. Mean Excess Delay
Mean Number of Components per Channel
24.0
42.3
55
Comparison of Statistics Across Energy Capture
Ratios
III. NLOS, 4 to 10 meters, metal stud
85 energy capture
95 energy capture
Avg. RMS Delay
14.59 ns
16.80 ns
15.95 ns
14.24 ns
Avg. Mean Excess Delay
Mean Number of Components per Channel
61.6
117.7
56
85 Energy Capture Ratio Used for Statistical
Analysis

• Number of multipath components per channel is the
  statistic that is most sensitive to changes in
  the stopping criteria.
• Large change in number of multipath components
  causes only small changes in other statistics in
  going from 85 to 95 energy capture ratio.
• 85 stopping criteria also good from a least
  squares point of view.

57
Statistical Environmental Models

• Each environment characterized by statistical
  profile of channels collected from that
  environment.
• Statistical analysis and model fitting done only
  for metal stud measurements.
• 369 metal stud measurements.
• 60 wood stud measurements not enough for
  statistical breakdown.
• Three scenarios considered
• I. NLOS, 0 to 4 meters, metal stud (120
  channels).
• II. LOS, 0 to 4 meters, metal stud (xxx
  channels).
• III. NLOS, 4 to 10 meters, metal stud (xxx
  channels).
• Not enough LOS, 4 to 10 meter channels for
  analysis.

58
I. NLOS, 0 to 4 meters, metal stud
Histogram of Number of Measurements per Meter
Total Number of Measured Channels 120
59
I. NLOS, 0 to 4 meters, metal stud
Histogram of Number of Multipath Components Per
Channel
Mean Number of Components Per Channel 36.1
60
I. NLOS, 0 to 4 meters, metal stud
Multipath Arrival Time Distribution
Graph of the probability that an excess delay bin
contains a reflection.
61
I. NLOS, 0 to 4 meters, metal stud
Mean of Log Relative Magnitude vs. Excess Delay
Mean stdv.
Mean Log Relative Magnitude
Mean - stdv.
62
I. NLOS, 0 to 4 meters, metal stud
Mean RMS Delay vs. Distance
Mean stdv.
Mean RMS Delay
Mean - stdv.
Mean RMS Delay 8.78 ns
Standard Deviation of RMS Delay 4.34 ns
63
I. NLOS, 0 to 4 meters, metal stud
Average Mean Excess Delay vs. Distance
Mean stdv.
Avg. Mean Excess Delay
Mean - stdv.
Average Mean Excess Delay 10.04 ns
Standard Deviation of Mean Excess Delay 6.26 ns
64
II. LOS, 0 to 4 meters, metal stud
Histogram of Number of Measurements per Meter
Total Number of Measured Channels 79
65
II. LOS, 0 to 4 meters, metal stud
Histogram of Number of Multipath Components Per
Channel
Mean Number of Components Per Channel 24.0
66
II. LOS, 0 to 4 meters, metal stud
Multipath Arrival Time Distribution
67
II. LOS, 0 to 4 meters, metal stud
Mean of Log Relative Magnitude vs. Excess Delay
Mean stdv.
Mean Log Relative Magnitude
Mean - stdv.
68
II. LOS, 0 to 4 meters, metal stud
Mean RMS Delay vs. Distance
Mean stdv.
Mean RMS Delay
Mean - stdv.
Mean RMS Delay 5.27 ns
Standard Deviation of RMS Delay 3.37 ns
69
II. LOS, 0 to 4 meters, metal stud
Average Mean Excess Delay vs. Distance
Mean stdv.
Avg. Mean Excess Delay
Mean - stdv.
Average Mean Excess Delay 4.95 ns
Standard Deviation of Mean Excess Delay 4.14 ns
70
III. NLOS, 4 to 10 meters, metal stud
Histogram of Number of Measurements per Meter
Total Number of Measured Channels 119
71
III. NLOS, 4 to 10 meters, metal stud
Histogram of Number of Multipath Components Per
Channel
Mean Number of Components Per Channel 61.6
72
III. NLOS, 4 to 10 meters, metal stud
Multipath Arrival Time Distribution
73
III. NLOS, 4 to 10 meters, metal stud
Mean of Log Relative Magnitude vs. Excess Delay
Mean stdv.
Mean Log Relative Magnitude
Mean - stdv.
74
III. NLOS, 4 to 10 meters, metal stud
Mean RMS Delay vs. Distance
Mean stdv.
Mean RMS Delay
Mean - stdv.
Mean RMS Delay 14.59 ns
Standard Deviation of RMS Delay 3.41 ns
75
III. NLOS, 4 to 10 meters, metal stud
Average Mean Excess Delay vs. Distance
Mean stdv.
Avg. Mean Excess Delay
Mean - stdv.
Average Mean Excess Delay 14.24 ns
Standard Deviation of Mean Excess Delay 5.97 ns
76
Number of Components Per Channel comparison
across scenarios
NLOS
4 10 m
LOS
NLOS
0 4 m
0 4 m
77
Distribution of Multipath Arrival
Times comparison across scenarios
NLOS
4 10 m
LOS
NLOS
0 4 m
0 4 m
78
Mean of Log Relative Magnitude comparison across
scenarios
NLOS
4 10 m
LOS
NLOS
0 4 m
0 4 m
79
RMS Delay vs. Distance comparison across scenarios
NLOS
4 10 m
LOS
NLOS
0 4 m
0 4 m
80
RMS Delay vs. Distance comparison across scenarios
NLOS
4 10 m
LOS
NLOS
0 4 m
0 4 m
81

• Introduction
• Measurement Campaign
• Data Analysis
• Statistical Environmental Models
• Analytical Models
• Conclusions/Recommendations

82
Modeling Approach

• Attempted to fit two different models to the data
• A modified IEEE 802.11 channel model
• Modified D-K model
• Models evaluated on how well they reproduced the
  statistic distributions of the data
• Bhattacharyya distance calculated between
  simulated and measured distributions.

83
Modified IEEE 802.11 model

• Regularly spaced impulses
• modified for UWB to allow for random placement of
  impulses in each time bin
• Raleigh-distributed magnitudes
• input parameters
• TRMS RMS delay parameter
• TS time discretization unit
• Was not able to match both RMS delay and
  multipath intensity profile simultaneously.

84
I. NLOS, 0 to 4 meters, metal stud
Distribution of RMS Delay
Mean RMS Delay
measured 8.85 (ns)
simulated 8.58 (ns)
85
I. NLOS, 0 to 4 meters, metal stud
Mean of Log Relative Magnitude vs. Excess Delay
86
D-K Model

• Arrival time model
• Model clumping of multipath arrival times by
  making the probability of an arrival in a given
  excess delay bin dependent on whether there was
  an arrival in the previous bin.
• K value is the ratio of these conditional
  probabilities.
• Modeling assumption is that K is constant.
• D value is the time discretization unit.

positive conditional
negative conditional
87
D-K Model

• Amplitude model
• Log-normal model for multipath amplitudes
• Mean and standard deviation as functions of
  excess delay given by the statistics of the data.

88
Modified D-K Model

• Multipath arrival times governed by statistics of
  data
• Probability of a multipath arrival in a given
  time bin depends on whether previous bin was
  occupied.
• Positive and negative conditional probabilities
  derived from statistics of data.
• No assumption that ratio of conditional
  probabilities is constant.

89
Simulation Results

• time discretization unit D 0.1 ns for all
  cases.
• Empirical probabilities of occupancy and log
  relative magnitude data used as inputs to model.
• A D-K simulation would use approximations to
  these quantities as its inputs, and hence could
  perform no better.

90
II. LOS, 0 to 4 meters, metal stud
Multipath Arrival Time Distribution
95 Energy Capture data used.
91
II. LOS, 0 to 4 meters, metal stud
Mean of Log Relative Magnitude vs. Excess Delay
95 Energy Capture data used.
92
II. LOS, 0 to 4 meters, metal stud
Distribution of Number of Multipath Components
Per Channel
Mean Number of Components Per Channel
95 Energy Capture data used.
measured 42.3
simulated 43.9
93
II. LOS, 0 to 4 meters, metal stud
Distribution of RMS Delay
Mean RMS Delay
95 Energy Capture data used.
measured 6.36 (ns)
simulated 11.70 (ns)
94

• Introduction
• Measurement Campaign
• Data Analysis
• Statistical Environmental Models
• Analytical Models
• Conclusions/Recommendations

95
Conclusion

• Modeling channel response as a sum of
  scaled/delayed versions of channel input provides
  a good fit to data.
• Wide variety of channel characteristics, even
  within the same environment.
• Multipath arrival times and average power decay
  profiles follow linear or piece-wise linear
  trends.
• Exact parameter values for arrival times and
  decay profiles are dependent on the environment
  type.
• Occupancy probabilities and decay profiles do not
  completely characterize the channel data, since
  two models can have the same statistics for these
  quantities, and yet differ in the statistics of
  RMS delay.

96
Recommendations

• IEEE 802.11 and D-K model should not be used,
  because they do not provide good fits to the
  statistical models of the environments.
• Selected SG3A model should fit the collected
  data.
• Number of multipath components per channel
• Probability of occupancy
• Average power decay profile
• Distribution of RMS delay vs. distance
• Distribution of mean excess delay vs. distance

97
References

• R.A. Scholtz, Notes on CLEAN and Related
  Algorithms, Technical Report to Time Domain
  Corporation, April 20, 2001
• Homayoun Hashemi, Impulse Response Modeling of
  Indoor Radio Propagation Channels, IEEE Jornal
  on Slected Areas in Communications, VOL. 11, No.
  7, September 1993
• Theodore S. Rappaport, Wireless Communications
  Principles and Practice, 1996
• Intelligent Automation, Inc., Channel Impulse
  Response Modeling Comparison Analysis of CLEAN
  algorithm and FT-based Deconvolution Techniques,
  Technical Report to Time Domain Corporation,
  November 21, 2001
• Bob OHara and Al Petrick, IEEE 802.11 Handbook
  A Designers Companion, 1999

98
Definitions/Terminology
99
Terminology

• LOS
• Line of Sight (transmit and receive antenna have
  a clear visible field of view relative to each
  other)
• NLOS
• Non-Line of Sight
• CIR
• Channel Impulse Response
• Waveform Template
• correlation template used in the correlation
  process (CLEAN Algorithm)
• LTI
• Linear Time Invariant

100
Terminology

• CLEAN1
• Variant of a serial correlation algorithm

• Channel Modeled as LTI filter, with impulse
  response h(t) of the form

Where ak are the impulse amplitudes tk are the
impulse delays
101
Terminology

• RMS Delay Spread can be expressed as

102
Terminology

• Mean Excess Delay can be expressed as

103
Terminology

• Relative Magnitude can be expressed as

Where
104
Terminology

• Average Multipath Intensity Profile (MIP) (or
  Average Power Decay Profile (APDP) can be
  expressed as