Joint Modeling of the Multipath Radio Channel and User-Access Method

1
Joint Modeling of the Multipath Radio Channel and
User-Access Method
Zs. A. Polgar1), V. Bota1), M. Varga1), A.
Gameiro2)
1) Communications Department Technical University
of Cluj-Napoca, Romania 2)Electronics
Telecommunications Department University of
Aveiro, Portugal
2
Outline

• Mathematical modeling of the multipath Rayleigh
  faded radio channel.
• OFDMA type multiple access techniques short
  presentation.
• Theoretical computation of the received SINR
  p.d.f. for a specific OFDMA multiple access
  technique.
• Evaluation of the received SINR p.d.f. in the
  considered OFDMA multiple access based on
  measured/simulated data.
• Markov chain modeling of the channel SINR states
  in the considered OFDMA multiple access scheme.
• Further studies and developments.

3
Mathematical modeling of the multipath Rayleigh
faded radio channel.

• evaluation of the data transmission performances
  on a radio channel affected by frequency and time
  selectivity, requires the derivation of the
  channel state p.d.f., i.e. the p.d.f. of the
  instantaneous total SINR (total SINR background
  SNR SIR).
• the stated problem is a difficult one and pure
  analytical solutions are possible only for some
  special cases, e.g. the case of a radio channel
  characterized by multipath propagation with
  uniform power delay profile and Rayleigh fading
  Alouini - 3
• in this particular case the global SINR of the
  channel will be characterized by a ?
  distribution, while the SINR on each propagation
  path is characterized by an exponential
  distribution.
• another possible mathematical characterization of
  the entire transmission chain, very useful
  especially for packet based transmissions, can be
  made with Markov chains
• beside the probability distribution of the
  average received SINR, it allows the computation
  of instantaneous SINR for small time intervals.

4
Mathematical modeling of the multipath Rayleigh
faded radio channel.
(1)
(2)
(3)
(4)
(5)
(6)
Fig. 1 Multipath Rayleigh faded radio channel
model
(7)
5
Mathematical modeling of the multipath Rayleigh
faded radio channel.

• Possible solutions proposed in literature
• An infinite series for the computation of the
  c.d.f. and p.d.f. of sums of random variables
  (particular case of Rayleigh variables) Beaulieu
  - 14
• A number of several thousands of terms are
  required to obtain a good approximation of the
  p.d.f. of sums of random variables.
• It considers only the sum of real random
  variables the phase variations induced by
  multipath propagation and random phase variations
  induced by the small scale fading are not
  considered.
• Computation of the p.d.f of sums of random
  vectors, with identically and arbitrarily
  distributed lengths and uniformly distributed
  phases Abdi - 13
• Integration of the multipath propagation
  situation is required (phase distributed
  uniformly but with imposed mean value) .
• The p.d.f. obtained can be expressed as a
  definite integral including Bessel functions
  the obtained p.d.f. of the vector length can be
  expressed by Hankel transform H0a, where J0 is
  the zero order Bessel fct and Exy is the
  expectation of the combined vector
• The previous p.d.f. can be decomposed as a a
  series of Laguere polynomial, easier to compute
  than the definite integral.

(8)
6
Mathematical modeling of the multipath Rayleigh
faded radio channel.

• Closed-form upper-bound for the distribution of
  the weighted sum of Rayleigh variates
  Karagiannidis - 16
• An upper-bound is defined for the sum of Rayleigh
  distributed variables.
• The phase variations due to the multipath
  propagation and small scale fading are not
  considered.
• The proposed upper-bound relies on G Meijers
  functions very difficult to compute in the
  general case.
• Computing the distribution of sums of random sine
  waves and of Rayleigh-distributed random
  variables by saddle-point integration Helstrom -
  13
• The c.d.f. of a sum of independent random
  variables can be computed according to the
  following integral on a curve, where h(z) is the
  moment generating function of the v variable
• Saddle point integration is proposed to compute
  the previous integral the saddle point which has
  to be found as well the moment generating
  function.
• Random phases and Rayleigh distributed amplitudes
  are considered separately.
• Conclusion no simple closed form solutions are
  available for the
• stated problem.

(9)
7
Mathematical modeling of the multipath Rayleigh
faded radio channel.

• The proposed solution
• It considers a multipath channel with N
  propagation paths characterized by gains ai and
  delays ?i (i - index of the path). The level on
  each path is xiA?ai, where A is the amplitude of
  a test sine with frequency fl the level
  distribution on the propagation paths is given by
  distributions pi(x).
• The probability to obtain a level r at the output
  of the channel, if the phase distributions are
  not considered, is given by
• (10)
• (11)
• (12)
• The function Ql(r,x1,x2,xN-1, ?1,?2,, ?N-1),
  expresses the occurrence probability of a level
  xN on the last path, N, that would make the value
  of the entire
• received signal equal r, when the signals on the
  other N-1 paths have levels x1 xN-1

8
Mathematical modeling of the multipath Rayleigh
faded radio channel.

• The channel instantaneous SINR on frequency fl
  can be obtained by dividing the received signal
  power level on the considered frequency to the
  noise interferences power at receivers input.
• Since the noise power is constant for a short
  time interval and over a small frequency
  bandwidth, we may assume that the SINR p.d.f. is
  also expressed by (10), (11), (12).
• The channel instantaneous SINR on frequency fl
  can be obtained by dividing the received signal
  power level on the considered frequency to the
  noise interferences power at receivers input.
• If the phase distributions, p?i(?) on different
  propagation paths are considered, the probability
  of the level r at the output of the channel is
  expressed by
• (13)
• The presented relations are related to one
  discreet frequency (subcarrier)
• If the small scale fading is Rayleigh distributed
  then pi(x) are Rayleigh distributions and p?i(?)
  are uniform distributions.
• The integrals related to the amplitude
  distributions can be computed on a
• smaller, finite interval, according to the
  dynamic of the pi(x) distributions

9
OFDMA type multiple access techniques. Short
presentation.

• the OFDMA type multi-user access is based on a
  frequency-time signal pattern that can be
  adjusted to different propagation scenarios and
  which is able to cope with the frequency
  selectivity generated by the multipath
  propagation and the time variability generated by
  the user motion.
• the mentioned signal pattern, called bin, chunk
  or burst, is proposed by several existing and
  future high bit rate transmission systems -
  candidate systems for the future 4G mobile
  communication networks.
• Examples of existing or future mobile
  communication systems based on OFDMA multiple
  access
• the WINNER project - tends to propose radio
  interface technologies and system concepts for
  the future 4G mobile communication systems.
• the IEEE 802.16.x (xa,,e) standards (WiMAX)
  specify an OFDMA type multi-user access technique
  (called SOFDMA Scalable OFDMA)
• the FLASH-OFDM high speed mobile transmission
  technology proposed by Flarion Technologies.
• within these chunks, non-coded or coded QAM
  modulations are used adaptively, based on channel
  estimation or prediction, performed by the mobile
  station, using a number of scattered pilot
  sub-carriers.

10
OFDMA type multiple access techniques. Short
presentation.

• the chunk-allocation to different users in the
  downlink is performed based on the Best Frequency
  Position (BFP) method - each chunk is allocated
  to the user that predicts the best channel
  parameters (SINR) in the frequency band of that
  chunk or on a fast frequency hopping (FH)
  method.

Bchunk?channel coherence bandwidth
Tchunk?channel coherence time
Fig. 3 Chunk allocation according to BFP algorithm

• chunks are allocated by a scheduler using
  channel measure- ments performed by the mobile
  and channel prediction performed by the base
  station in the whole frequency band
• chunks are allocated in such a manner to
  increase the data amount transmitted by the BS
• one or several chunks are allocated to one user
  according to the channel and service
  characteristics

11
Theoretical computation of the received SINR
p.d.f. for the considered OFDMA multiple access
technique.

• Computation of the joint mobile channel-multiple
  access SINR p.d.f.
• Frequency Hopping user-chunk allocation
• The probability of the rated level to lie between
  a pair of thresholds Tk(dB) and Tk1(dB), wkFH,
  (Jk and Jk1 - corresponding the linear values)
  where modulation configuration k should be used
• The selection of a modulation configuration in a
  chunk m is made according to the average of the
  rated levels received on the Cu sub-carriers of
  that chunk, Pam.
• For FH bin-allocation method, the probability of
  a bin m to be assigned to one user is Pm1/Bu,Bu
  being the number of available chunks Nu is the
  no. of payload subcarriers

(14)
12
Theoretical computation of the received SINR
p.d.f. for the considered OFDMA multiple access
technique.

• Computation of the joint mobile channel-multiple
  access SINR p.d.f.
• Best Frequency Position user-chunk allocation
• The probability, wkBFP, that the maximum rated
  level lies between thresholds Jk, Jk1, where
  modulation configuration k should be used,
  requires the computation of the probability of
  the average rated level of chunk m to lie in this
  domain, Pkm, multiplied with the probability that
  the average rated levels of all other chunks, Qm
  to be smaller than the average rated level of bin
  m, it is expressed by

(15)
13
Theoretical computation of the received SINR
p.d.f. for the considered OFDMA multiple access
technique FH-allocation.
non-coded non-coded
nb SINR thr. (dB)
1 -?
2 8.3
3 13.2
4 16.2
5 20.2
6 23.6
7 26.6
8 29.8
Simulated
Computed
Fig 4 Measured and computed probabilities to
employ the specified non-coded configurations
(SINR of reference path16 dB) FH chunk
allocation method
Table 1 Non-coded configurations and
corresponding SINR thresholds
Const. No 1 2 3 4 5 6 7 8
PFH -computed 0.0745 0.1228 0.1341 0.2362 0.2066 0.1488 0.0663 0.0107
NFH /10000 - simulated 0.0734 0.1215 0.1338 0.2381 0.2062 0.1515 0.0581 0.0174
Table 2 Measured and computed probabilities to
employ the specified non-coded configurations
(SINR 16 dB, FH method)
14
Theoretical computation of the received SINR
p.d.f. for the considered OFDMA multiple access
technique BFP allocation.
Test parameters Test parameters
Carrier freq. fc 1.9GHz
OFDM symb. freq. fs 10kHz
OFDM guard Interv. G 11?s
Chunk size (no. subc.? no. OFDM symb.) 20?6120 108 payload symb.
Power delay profile 4 paths User speed a13dB?1200ns a23dB?2400ns a33dB?3600ns 120km/h
Simulated
Computed
Table 3 Test conditions related to fig. 4 and
fig. 5
Fig 5 Measured and computed probabilities to
employ the specified non-coded configurations
(SINR of reference path16 dB) BFP chunk
allocation method
Const. No 1 2 3 4 5 6 7 8
POA computed 0.0012 0.0013 0.0076 0.0756 0.2910 0.3936 0.2158 0.0139
NOA /10000 - simulated 0.0050 0.0025 0.0065 0.0675 0.2785 0.4165 0.2105 0.0130
Table 4 Measured and computed probabilities to
employ the specified non-coded configurations
(SINR 16 dB, BFP method)
15
Evaluation of the received SINR p.d.f. based on
measured/simulated data.

• the approach presented before requires a large
  amount of computation and should be performed for
  every SINR value for the BFP allocation, it
  should also be performed for every cell-carrier
  loading (Lc).
• an approximate method to compute the p.d.f. of
  the received signal level (and the SINR at the
  receiver), for any desired average SINR0 of the
  first arrived path consists of three steps
• compute, simulate or measure the probabilities
  of the receivers SINR to lay between an imposed
  set of thresholds Tk, for a given SINR0 of the
  first arrived path. Computation can be done using
  the method described above
• find an interpolation function f(x) that
  approximates the distribution of the SINR on the
  channel, fulfilling the conditions imposed by
  step a.
• translate and scale f(x) around the desired SINR
  of the first arrived wave, SINR0.
• this method requires a smaller amount of
  computation and ensures a good accuracy of the
  obtained p.d.f.
• as an example, the WP5 Macro channel model (18
  propagation paths) for a given average SINR0 16
  dB of the first path and Lc (cell load) 2,
  50, 75, 100 was considered both the BFP and
  FH chunk allocation methods were considered.

16
Evaluation of the received SINR p.d.f. based on
measured/simulated data.

• if f(x) is the function which interpolates the
  SINR p.d.f. and wk is the probability of the
  current SINR to lay within the k-th domain, the
  interpolation function should fulfill the
  conditions
• (Jk - linear values corresponding to logarithmic
  values Tk J1 min., JS max.)
• The interpolation function f(x) should also be
  positive and it should have only one maximum
  across the whole range of x variable considered
• To simplify the computation of f(x), the number
  of SINR domains is reduced by suppressing those
  who exhibit a wk probability below an imposed
  value, e.g. wk lt 110-3, and adjusting the
  lowest and highest thresholds, to Tkm and TkM.

(16)

• The f(x) was chosen to be a polynomial function
  of order SkM-km1, (17.a) Using the
  probabilities wk of the SINR to lay within the
  k-th interval, the coefficients of f(x) are
  computed using (17.b ) - (17.c).

(17)
17
Evaluation of the received SINR p.d.f. based on
measured/simulated data.
Tk dB T1-2 T28.3 T313.2 T416.2 T520.2 T623.6
Lc2 0 0 0 210-5 1.410-3 3.910-2
Lc50 0 0 0 510-5 3.210-3 5.510-2
Lc75 0 0 0 810-5 8.5 10-3 7.310-2
Lc100 0 0 3.110-4 6.810-3 2.610-2 8.810-2
Lcx 510-4 1.110-2 3.710-2 1.4710-1 2.3810-1 2.6710-1
Tk dB T726.6 T829.8 T933 T1036.2 T1139.4  
Lc2 3.410-1 5.310-1 8.710-2 7.810-4 0  
Lc50 3.510-1 5.010-1 8.810-2 7.910-4 0  
Lc75 3.410-1 4.810-1 8.710-2 8.210-4 0  
Lc100 3.110-1 4.810-1 8.910-2 1.010-3 0  
Lcx 2.2110-1 710-2 4.510-3 210-5 0  

• the SINR domains and the associated probabilities
  corresponding to BFP and FH chunk allocation are
  presented in in table 5

Tab. 5 SINR domains and associated probabilities
for BFP and FH chunk allocation SINR016dB Lcx
corresponds to the FH chunk allocation

• Scaling the f(x) function for other desired SINR0
  (SINR0-a) of the reference path, (fa(x))

(18)
18
Evaluation of the received SINR p.d.f. based on
measured/simulated data.

• both the analytical and the approximate method,
  ensure good accuracies of the wk SINR state
  probabilities.
• the wk SINR state probabilities delivered by the
  approximate method differ with less than 0.3
  from the ones obtained by computer simulations.
• the approximate method provides the probabilities
  wk, with a slightly greater error, but requires
  significantly less computation than the
  analytical approach.

19
Evaluation of the received SINR p.d.f. based on
measured/simulated data.

• the BFP allocation method ensures high state
  probabilities only for a limited number of
  channel states, 2 maximum 3, the rest of the
  states being employed quite seldom.
• the SINR average value of these states is 10-14
  dB higher than the SINR level of the firstly
  arrived path - this is because the BFP method
  ensures (with high probability) the chunk with
  the highest SINR for each user, taking advantage
  of the frequency diversity of the multipath
  Rayleigh channel in a very efficient way.
• with the FH method the cell-carrier load does no
  longer affect the p.d.f., because the user chunks
  are allocated independently according to a
  pseudorandom sequence.
• the FH allocation method employs with significant
  probabilities wk, more channel states, 5 or 6,
  because it does not place the user on the
  particular chunk that ensures the best SINR.
• the average SINRs of these states are smaller,
  with even 8 dB, or greater, with up to 14 dB,
  than the SINR of the firstly arrived wave, SINR0.
• the most employed states ensure an average SINR
  higher with 0-10 dB than SINR0.
• the FH chunk allocation method takes advantage of
  the channel frequency diversity in a less
  efficient manner - it performs an averaging over
  the
• whole available frequency bandwidth.

20
Markov chain modeling of the channel SINR states
in the considered OFDMA multiple access scheme.

• The p.d.f. of the total SINR is independent of
  the mobile speed.
• the total SINR p.d.f. shows only the average
  probabilities of the channels SINR to lie
  between each pair of thresholds.
• employment of the p.d.f. only allows the
  computation of average performances of a
  transmission scheme, over a relatively long time
  interval.
• If the transmission analysis requires the
  probability distributions of the studied
  parameters, distributions depending on the mobile
  speed, a Markov-chain modeling of the channel and
  multiuser-access method should be developed.
• Different chains should be derived for various
  cell-carrier loadings and mobile speeds.
• Some Markov-chains obtained by computer
  simulations, for some significant situations
  related to the considered OFDMA access scheme,
  are presented.
• The computer simulations required to derive these
  models with acceptable accuracy are far less time
  consuming than the complete evaluation by
  simulations of the desired performances of a
  transmission system.
• Out of the set of possible non-coded QAM
  modulations only the first 8 are used adaptively
  in each chunk.
• the complete Markov-chain assigned to the channel
  SINR states has 8 possible states, one assigned
  to each SINR domain.
• the Markov chains can be simplified by dropping
  some states with low or very low associated
  probabilities, i.e. smaller than 0.01, to allow a
  simpler and still accurate performance
  evaluation.

21
Markov chain modeling of the channel SINR states
in the considered OFDMA multiple access scheme.
Initial state
Initial state Final state speed Initial state Final state speed Tab.a BFP allocation, Lc2 Tab.a BFP allocation, Lc2 Tab.a BFP allocation, Lc2 State probab.
Initial state Final state speed Initial state Final state speed 6 7 8 State probab.
6 120 km/h 0.0839 0.0554 0.0269 0.03775
6 4 km/k 0.9713 0.0018 0.00077 0.03775
7 120 km/h 0.464 0.3972 0.1128 0.33888
7 4 km/k 0.0158 0.9807 0.0095 0.33888
8 120 km/h 0.4484 0.545 0.3563 0.62305
8 4 km/k 0.0127 0.0174 0.9897 0.62305
Tab.6 SINR state transition probabilities. BFP
chunk allocation, Lc2, speed 4km/h and 120km/h,
SINR016dB
22
Markov chain modeling of the channel SINR states
in the considered OFDMA multiple access scheme.

• The presented Markov chains can be used to
  compute two important parameters of a mobile
  transmission system, namely the packet error rate
  and the packet average length in bits, a packet
  of data being composed of an imposed number of
  chunks.
• each branch is labeled with a variable X used to
  count the number of chunks in the packet, and
  with a product of the state transition
  probability and the correct chunk probability of
  the branch originating from that state
• the correct chunk probability is computed based
  on the average SINR, QAM modulation corresponding
  to the considered state and chunk parameters
• the average SINR of a given state is computed
  based on the previously established SINRs
  p.d.f.
• for each possible combination of initial and
  final states i.e. the states corresponding to
  the first and last chunk in the packet - the
  transfer function of the graph (between the two
  considered states) is computed using the Masson
  formula 17.
• finally only the terms with an imposed power L
  (Lpacket length) are considered.
• To compute the average length of a packet, each
  branch in the graph will be labeled, beside the
  previously mentioned X variable, the state
  transition probability and with a term having the
  expression Yi, i being the number of bits/symbol
  of the QAM modulation used in the branch
  originating state.
• the obtained graph transfer functions are
  composed of terms having the following
  expression p?Yt?Xz. From each transfer function,
  the terms XL are extracted , and finally the
  average packet length is computed.

23
Markov chain modeling of the channel SINR states
in the considered OFDMA multiple access scheme.
Fig. 9 p.d.f. of the no. of bits/symbol in the
case of BFP allocation of the user chunk for
different loads of the cell carrier and different
speed values of the mobile
24
Further studies and developments.

• Study of methods which allow the accurate and
  fast computation of the p.d.f. of a sum of
  independent Rayleigh random variables e.g. an
  infinite-series approach for the computation of
  c.d.f. and p.d.f. or saddle-point integration
  approach.
• Alternative approach study of methods that give
  closed forms of upper/lower bounds of a sum of
  independent Rayleigh random variables.
• Study/computation of the c.d.f./p.d.f. of
  Rayleigh distributed vectors.
• Computation of the joint multipath-Rayleigh
  channel multiple access p.d.f.
• Theoretical Markov modeling of the
  multipath-Rayleigh channel
• Possible solution - mathematical modeling of the
  level transition rate and (based on this)
  computation of the SINR-state transition rate
• - time dimension is required - leading to
  a time domain modeling
• of the multipath-Rayleigh process should
  be performed
• - decomposition of the multipath-Rayleigh
  process as sum of weighted sine signals
  with p.d.f. restrictions, might be a possible
  solution
• Joint Markov modeling of the multipath-Rayleigh
  channel and multiple access technique.

25
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26
Annex demonstration of relations (10) (12)

• the transmitted and received signal, s(t) and
  r(t)

(17)

• alternative expression of received vector length

(18)
27
Annex demonstration of relations (10) (12)

• the probability of signal level on the N-th
  path, conditioned by the received signal and the
  signals on other levels

(19)