Title: Time Multiplexing and Power Allocation in Interference Limited Systems
1Time Multiplexing and Power Allocation in
Interference Limited Systems
- Ashay Dhamdhere
- Prof. Ramesh Rao
2Motivation
- 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.
3Abstraction
- 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.
4System 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.
5System Model (Contd.)
- All users see identical channel transfer
functions. For the purpose of this analysis we
assume, - The crosstalk functions are also flat,
-
6System 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
7System Model (Contd.)
- Each transmitter satisfies the power constraint
- User Ui has weight wi. Users expect data rates
in proportion to their weights.
8Problem Formulation
- Under the constraint
-
-
- we want to maximize the sum rates of users.
9Problem 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?
10The 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
11Nature 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?
12Nature 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.
13Mapping 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).
14Shape 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.
-
15Shape 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.
16Extending 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.
17Formalizing 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
-
18The 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.
19Drawing 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.
20Introducing Fairness Constraints
- Users have weights
- We want to satisfy,
-
- Our Goal Maximize the system capacity subject to
the fairness constraints.
21Putting 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
22Putting it all together (Contd.)
23Putting it all together (Contd.)
- The Constraints
- Average Power Constraints on the Transmitters.
- Fairness Constraints on the achieved rates.
- The Objective Function r1
24Simulations
- 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.
-
25Some 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. -
26Some 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. -
27Some Results (Contd.)
- Here users 12 are strongly coupled.
- Note that users 12 never transmit together
-
28Some Results (Contd.)
- Here all users are weakly coupled.
- Note that in all allocation schemes, all
transmit-receive pairs are operating
simultaneously. -
29A 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.
30A 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.
31A 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
32A 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.
33A 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.
34Conclusion
- We have a formulation that allows us to analyze
interference limited systems. - It presents certain insights regarding power
allocation and time-multiplexing.
35From Abstract to Concrete
- We would like to see how our formulation relates
to the existing DSL systems.
36ADSL 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.
37Some 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.
38How 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.
39Effect on Higher Layers
- We wish to know the effect of such a scheme on
upper layers of the ADSL system.
40Future Work
- We would like to extend this scheme to multiple
frequency bands. - We would like to apply this formulation to some
real world system.