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Time Multiplexing and Power Allocation in Interference Limited Systems

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We wish to know whether the ARR-P may be extended by employing time multiplexing. ... coupling coefficient is high, time-multiplexing can buy us some extra rate. ... – PowerPoint PPT presentation

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Title: Time Multiplexing and Power Allocation in Interference Limited Systems


1
Time Multiplexing and Power Allocation in
Interference Limited Systems
  • Ashay Dhamdhere
  • Prof. Ramesh Rao

2
Motivation
  • We derive motivation from our study of the DSL
    system.
  • The DSL system is characterized by crosstalk
    between adjacent twisted pairs.
  • The capacity of a DSL system is interference
    limited.

3
Abstraction
  • We abstract out essential features of an
    interference limited system.
  • We study a problem that captures the essential
    features of an interference limited system.
  • We gain some insights about such a system.

4
System Model
  • We assume we have N transmit-receive pairs.
  • Transmitter Ti of user Ui transmits to receiver
    Ri of user Ui .
  • Only single direction traffic is considered.
  • No interference cancellation is performed.

5
System Model (Contd.)
  • All users see identical channel transfer
    functions. For the purpose of this analysis we
    assume,
  • The crosstalk functions are also flat,

6
System Model (contd.)
  • Interference from other users is modeled as AWGN.
  • The rate achieved by Ui is
  • Here W is the system bandwidth, and N is the
    noise PSD at each receiver.
  • Pi is the power transmitted by transmitter Ti

7
System Model (Contd.)
  • Each transmitter satisfies the power constraint
  • User Ui has weight wi. Users expect data rates
    in proportion to their weights.

8
Problem Formulation
  • Under the constraint
  • we want to maximize the sum rates of users.

9
Problem Formulation(Contd.)
  • We want to know how to achieve this maximum.
  • Which power allocation schemes should we use?
  • Do we need to multiplex between different power
    allocation schemes?

10
The Two-User Case
  • We wish to know the set of rates that may be
    achieved by using power allocation alone. We call
    this region the ARR-P.
  • We wish to know the nature of the ARR-P. We shall
    look at power allocations that yield boundary
    points of the ARR-P.
  • We wish to know whether the ARR-P may be extended
    by employing time multiplexing.
  • For this analysis we fix W1, and b12b21b

11
Nature of the ARR-P

R2
R2r2
R1
  • We may immediately fix two points on the two-user
    ARR-P. (Rmax,0) is obtained using power
    allocation (Pmax,0), and (0,Rmax) is obtained
    using power allocation (0, Pmax).
  • Consider a slice of the ARR-P at R2 r2. What
    is the maximum value of R1 subject to the above
    constraint?

12
Nature of the ARR-P (Contd.)
P2
R2r2
P1
  • Consider the two-user Shannon Capacity formula
    for U2.
  • Define K222R2-1. For a fixed R2, K2 is fixed.
  • From the rate equations, we get, P2K2bP1K2N
  • The constraint lines are shown in figure above.
  • Along the constraint line, R1 increases as
    P1increases
  • Hence the maximum R1 occurs when P1is at its
    maximum value subject to the constraints.


13
Mapping Powers to Rates
  • When K2 0, we are operating at (Pmax, 0)
    which corresponds to (Rmax, 0).
  • As K2 increases, we traverse the curve P1Pmax
    until we hit the point (Pmax, Pmax).
  • We then begin to traverse the curve P2Pmax.
  • Finally we arrive at (0, Pmax) which corresponds
    to (0, Rmax).

14
Shape of the ARR-P
  • The boundary is traced by moving along the curve
    P1Pmax or P2Pmax
  • Thus the ARR-P consists of two half-regions.

15
Shape of the ARR-P (Contd.)
  • We simulated the two-user ARR-P, fixing Pmax, N
    and W, and using different values of b.
  • When the coupling coefficient is low each
    half-region appears concave-in.
  • As the coupling coefficient increases the half
    regions begin to appear concave-out.

16
Extending the ARR-P
  • When the coupling coefficient is low,
    time-multiplexing does not buy us anything.
  • When the coupling coefficient is high,
    time-multiplexing can buy us some extra rate.

17
Formalizing the above concepts
  • The convex hull of the ARR-P is achievable.
  • We call the convex hull of the ARR-P, the ARR.
  • The convex hull of a set A in is the convex
    combination of finitely many points belonging to
    A.
  • A convex combination of
  • where

18
The Convex Hull
  • How many points do we actually need to get to a
    point on the convex hull of a set A in ?
  • Answer n1
  • The convex hull of A is precisely the set of all
    convex combinations of at most (n1) elements of
    A.

19
Drawing some Parallels
  • The ARR-P corresponds to the region A in
  • The ARR corresponds to the convex hull of A
  • A linear combination of points in A is equivalent
    to time-divisioning between different power
    allocation schemes.
  • We need only n1 points from A to reach a point
    on the convex hull of A. Thus we need to
    time-division between at most n1 power
    allocation schemes to reach a point on the ARR.

20
Introducing Fairness Constraints
  • Users have weights
  • We want to satisfy,
  • Our Goal Maximize the system capacity subject to
    the fairness constraints.

21
Putting it all together
  • Formulate as a maximization problem
  • The Unknowns
  • The n1 power allocation schemes. Each scheme
    corresponds to the powers allotted to each of the
    n transmitters.
  • The time fraction for which each scheme is in
    operation.
  • The constraints
  • Average power constraints on the transmitters
  • Fairness constraints on the achieved rates
  • The objective function r

1
22
Putting it all together (Contd.)
23
Putting it all together (Contd.)
  • The Constraints
  • Average Power Constraints on the Transmitters.
  • Fairness Constraints on the achieved rates.
  • The Objective Function r1

24
Simulations
  • We performed simulations for the three-user case.
  • We fixed Pmax 1
  • We fixed N 0.2
  • We fixed W 1
  • We assigned the following weights,
  • w11, w22, w33.
  • We then solve for the unknowns.

25
Some Results
  • In this case all three users are strongly
    coupled. It is best to use time multiplexing.
  • It is clear from the power allocation schemes
    returned that the system capacity is maximized
    when only one transmit-receive pair is in
    operation at any given time.

26
Some Results (Contd.)
  • Here user 1 is strongly coupled with the other
    two users.
  • It is prudent to let user 1 transmit by itself,
    and let the other two transmit together.

27
Some Results (Contd.)
  • Here users 12 are strongly coupled.
  • Note that users 12 never transmit together

28
Some Results (Contd.)
  • Here all users are weakly coupled.
  • Note that in all allocation schemes, all
    transmit-receive pairs are operating
    simultaneously.

29
A Four User Case
  • We set w1w2w3w41.
  • We see that U1 and U2 are heavily coupled same
    with U3 and U4.
  • Only two schemes have non-zero weights.
  • The solution is intuitive.

30
A Note about the Assumptions
  • We have assumed that users transmit with a flat
    input power spectral density in their respective
    bands.
  • This assumption is borne out by the work of
    Veeravalli et al. We shall talk about this in the
    next two slides.

31
A Note about the Assumptions
  • Two users in the system
  • Both users want equal data rates.
  • Flat channel and crosstalk functions.
  • Question What input Power Spectral Densities
    should be used?
  • The intuition Use symmetric Input Power Spectral
    Densities

32
A Note about the Assumptions
PSD1
PSD1
PSD1
PSD1
f
f
PSD2
PSD2
PSD2
PSD2
f
f
W
W
0.5W
  • Different classes of symmetric functions were
    considered.
  • It was found that only two input PSDs are
    optimal.
  • Either transmit together in the same band with
    flat input PSD, or split the band, and use a flat
    input PSD in each band.

33
A Note about the Assumptions
  • We have extended this to the case of multiple
    users.
  • We assume that users transmit with a flat input
    PSD in the bands assigned to them.

34
Conclusion
  • We have a formulation that allows us to analyze
    interference limited systems.
  • It presents certain insights regarding power
    allocation and time-multiplexing.

35
From Abstract to Concrete
  • We would like to see how our formulation relates
    to the existing DSL systems.

36
ADSL Physical Layer
  • Twisted copper pairs have a bandwidth larger than
    1MHz.
  • This bandwidth is divided into subchannels, each
    of which is 4Khz wide.
  • In each subchannel a Discrete Multitone
    Modulation is run.
  • DMT allows the input PSD to be tailored across
    the frequency spectrum.

37
Some Features of the Proposed Scheme
  • The Proposed Scheme handles a single flat
    channel.
  • We have a PSD constraint in place.
  • A (suboptimal) extension of our scheme to
    multiple subchannels is to require that users get
    data rates in proportion to their weights, in
    each subchannel. We shall explore this scheme
    further.

38
How the Proposed Scheme applies to a Real System
  • Recall that the Proposed Scheme advocates using
    Time/Frequency Divisioning and Power Control.
  • Now consider any subchannel of the system.
  • If frequency divisioning is used, it means
    splitting a subchannels further into
    sub-subchannels.
  • If time divisioning is used, it means that at any
    time t, some users are transmitting in that
    subchannel while other users are silent.

39
Effect on Higher Layers
  • We wish to know the effect of such a scheme on
    upper layers of the ADSL system.

40
Future Work
  • We would like to extend this scheme to multiple
    frequency bands.
  • We would like to apply this formulation to some
    real world system.
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