The Power-Bandwidth Tradeoff in MIMO Systems

The Power-Bandwidth Tradeoff in

MIMO Systems

Marwan A. Hammouda

August 15, 2012

The Purpose of this Presentation ..� The purpose of this presentation is to highlight the concept of

power-bandwidth trade-off in MIMO systems.

� Methods/techniques might be used to optimize/deal with this trade-off are out of the scope.

Agenda� MIMO Systems

�Background

�System Structure

�Performance Improvements

• Power-Bandwidth Trade-off�The Concept�Example: SISO AWGN-Channel�EE-SE Trade-off for MIMO System�EE-SE Trade-off for MIMO System

• EE-ES Trade-off for MIMO Systems� MIMO Capacity �EE-SE Approximations – 1�EE-SE Approximations – 2�Simulation Results�Discussion

• Conclusions• References

MIMO SystemsMIMO SystemsMIMO SystemsMIMO SystemsBackground

� Four Basic Models:

User data stream

.

.

User data stream

.

.

.

.Channel

Matrix H

s1

s2

sm

y1

y2

yn

.

.

h11

h12

MIMO SystemsMIMO SystemsMIMO SystemsMIMO SystemsSystem Structure

y = Hs + n

Matrix Hsm

s

yn

yTransmitted vector Received vector

Where H =

h11 h21 …….. hm1

h12 h22 …….. hm2

h1n h2n …….. hmn

. . …….. .

m

n

hij is a Complex Gaussian random variable that models fading gain between the ith transmit and jth receive antenna

MIMO SystemsMIMO SystemsMIMO SystemsMIMO SystemsPerformance Improvements

1. Spatial multiplexing gain spectral efficiency

� Yields a linear (in the minimum of the number of transmit and receive antennas) increase in capacity for no additional power or bandwidth expenditure

� The corresponding gain is realized by simultaneously transmitting independent data streams in the same frequency band.

� In rich scattering environments, the receiver exploits differences in the spatial signatures of the multiplexed streams to separate the different signals, thereby realizing a capacity gain.


1. Spatial multiplexing gain


2. Diversity gain link reliability

� A powerful technique to mitigate fading and increase robustness to interference

� Principle: provide the receiver with multiple identical Principle: provide the receiver with multiple identical copies of a given signal over (ideally) independent fading paths.

� Intuitively, the more independently fading, identical copies of a given signal the receiver is provided with, the faster the bit error rate (BER) decreases as a function of the per signal SNR.


2. Diversity gain

� At high SNR values, it has been shown that.

where d represents the diversity gain and the coding gain. where d represents the diversity gain and the coding gain.

�Definition: For a given transmission rate R, the diversity gain is:

Where is the BER at the given rate and SNR.


3. Array gain power efficiency

�Achieved in MIMO systems through the enhancement of average signal-to-noise ratio (SNR) due to the transmission and reception by multiple antennas.

�Availability of channel state information (CSI) at the transmitter/receiver is necessary to realize transmit/receive array gains.

� Principle: To obtain the full array gain, one should transmit using the maximum eigenmode of the channel


3. Array gain

Hint: For maximum array gain, use only the maximum eigenchannel.

Where

Is the singular value decomposition (SVD) of H, and

PowerPowerPowerPower----Bandwidth TradeBandwidth TradeBandwidth TradeBandwidth Trade----offoffoffoffThe concept

� Two basic definitions:

� Spectral efficiency (SE): directly related to the channel capacity in bit/s. This metric indicates how efficiently a limited frequency spectrum is utilized. SE is quantified by: (in bit/s/Hz), where R is the data rate and B is the channel bandwidth.where R is the data rate and B is the channel bandwidth.

�Energy Efficiency (EE): closely related to the power consumption of the communication system. EE is usually quantified by the energy-per-bit to noise power spectral density ratio, , where (in Joules) and P is the signal power.

The efficiency of a communication system has traditionally been measured in terms of SE and EE.

� For any communication systems, it is desired to minimize the consumed power, and minimize the required bandwidth as well for a given R, or equivalently maximizing both SE and EE.

� However, this is not possible!� However, this is not possible!

� As a simple example, for AWGN channel, any achievable data rate is upper bounded by

The power-bandwidth trade-off is commonly known as EE-SE trade-off, where maximizing both EE and SE is Equivalent to maximizing one and minimizing the other.


To mathematically formulate the EE-SE trade-off, lets follow the following steps:

� Via the Shannon’s capacity theorem, as far as the maximumachievable SE, C, is concerned, it can be expressed as:

Where is the signal-to-noise ratio (SNR).

�Without loss of generality,

� Now, considering the achievable SE, , then can be expresses as:


�Then, by inserting (2) in (1), the EE-SE trade-off can be easily expressed as:

Where is the inverse Where is the inverse function of f.

So, as indicated in (3), the problem of defining a closed-form Expression for the EE-SE trade-off is generally equivalent to obtaining an explicit expression for the inverse function of the channel capacity per unit bandwidth,

PowerPowerPowerPower----Bandwidth TradeBandwidth TradeBandwidth TradeBandwidth Trade----offoffoffoffExample: SISO AWGN-Channel

� As a simple example, consider a simple additive white Gaussian noise (AWGN) channel. In this case,

And hence, is directly given by

Substituting this formula in (3), and using C instead of S, EE-SE trade-off for AWGN channel can be expressed as:

PowerPowerPowerPower----Bandwidth TradeBandwidth TradeBandwidth TradeBandwidth Trade----offoffoffoffExample: SISO AWGN-Channel

6

8

10

12

N0(dB)

Points above the curve satisfiesShannon’s limit R < B log2(1 + γ),while the points below don’t

10-3

10-2

10-1

100

101

-2

0

2

4

SpectralEff iciency(bit/s/Hz)

Eb/N

Figure 1: EE-SE Trade-off for AWGN Channel

PowerPowerPowerPower----Bandwidth TradeBandwidth TradeBandwidth TradeBandwidth Trade----offoffoffoffEE-SE Trade-off for MIMO System

� In MIMO systems, the closed form of EE-SE trade-off is more complicated since doesn’t have a straightforward formulation. In this case, approximations of can provide an acceptable solution.

� As known, capacity expression for MIMO differs according � As known, capacity expression for MIMO differs according to the channel model used. In this presentation, the channel is assumed to be a Rayleigh Fading channel with Gaussian noise, which is a general case and other cases can be considered as special cases.

EEEEEEEE----ES TradeES TradeES TradeES Trade----off for MIMO Systemsoff for MIMO Systemsoff for MIMO Systemsoff for MIMO SystemsMIMO Capacity

Consider the MIMO-system model

y = Hx + n

With:

o The signal is transmitted over M transmit antennas o The signal is transmitted over M transmit antennas and received over N received antennas,

o , is a random matrix having independent and identically distributed (i.i.d.) complex circular Gaussian entries with zero-mean and unit variance.

o n: zero–mean complex Gaussian noise. Independent and equal variance real and imaginary parts. E[nn†] = IN


o Also consider , where is the total transmitted power.

o Then, the ergodic channel capacity per unit bandwidth of the MIMO Rayleigh fading channel is accordingly expressed as:

� H matrix is assumed to be unknown at the transmitter. � Considering the special case when the channel is a deterministic

Gaussian and H still Unknown at the transmitter, (4) can bere-expressed as:


o Since it is not easy to find a closed-form for of (4), two main approaches are usually used to plot EE-SE trade-off curves,

� Numerically, where different values for are computed numerically for different SNR levels and corresponding Eb/No numerically for different SNR levels and corresponding Eb/No values are also computed for those SNR values.

�Approximated expressions for (4) are first introduced such that the inverse can be computed and obtained in a closed-form expression.

We will mention two of the research work done to obtainapproximated closed-form for the EE-SE.

EEEEEEEE----ES TradeES TradeES TradeES Trade----off for MIMO Systemsoff for MIMO Systemsoff for MIMO Systemsoff for MIMO SystemsEE-SE Approximations - 1

o In [1], they approximated the EE-SE trade-off for the situation of low SE and low EE.

oThe approximated EE-SE trade-off is expressed as:

(5)

� Where donates the minimum required for reliable

communication, and denotes the slope of spectral efficiency in b/s/Hz/(3 dB) at the point

(5)


Where are the first and second derivatives of

, respectively. , respectively.

For our assumed channel model,


o In [2], a closed-form approximation is provided for (4), as follow:

WhereWhere

And is the ratio between receive and transmit antennas

oThis approximation was approved to have an acceptable accuracy for M or N >2


oA simpler formulation can even be expressed as:

Where , and

o Solving this approximated expression to find its inverses, gave the following solutions:

� Case I: M=N, ,

Then: (6)


� Case II: M≠N, , then

Where: ,and

(7)

and

are values depending on the value of (see table I in [2])


In (6) and (7), is the real branch of the Lambert W function which is the inverse of the function and thus it satisfies , and then,

EEEEEEEE----ES TradeES TradeES TradeES Trade----off for MIMO Systemsoff for MIMO Systemsoff for MIMO Systemsoff for MIMO SystemsSimulation Results

The following simulation result presented in [2], where the EE-SE trade-offs is plotted for three methods:

� Numerical computation: as mentioned previously using Monte Carlo simulation to get the inverse of (4).

� Using approximation-1 as described in (5), and detailed in [1].

� Using approximation-2 as described in (6) and (7) and detailed in [2].

Results are simulated for different number of transmit (t) and receive (r) antennas.

EEEEEEEE----ES TradeES TradeES TradeES Trade----off for MIMO Systemsoff for MIMO Systemsoff for MIMO Systemsoff for MIMO SystemsSimulation Results

• (5)Monte-Carlo(6) and (7)

EEEEEEEE----ES TradeES TradeES TradeES Trade----off for MIMO Systemsoff for MIMO Systemsoff for MIMO Systemsoff for MIMO SystemsDiscussion

Simulation results show a good fit between the exact expression in (4) and the approximated ones in (6) and (7) for wide range of SE and different values of transmit and receive antennas, while the fit is not that accurate with the approximated expression in (5) when the SE is getting higher specially for small number of (5) when the SE is getting higher specially for small number of transmit and receive antennas.

ConclusionsConclusionsConclusionsConclusions

�The use of MIMO systems is a powerful performance enhancing technology.

� Energy efficiency (EE) and spectral efficiency (SE) are important metrics in evaluation the performance of the communication systems, where they are better to be maximized. communication systems, where they are better to be maximized.

�However, maximizing energy efficiency EE while maximizing the SE is a conflicting object, the concept of power-bandwidth trade-off!

� Some works have been done to formulate the SE-EE trade-off for MIMO systems.

ReferencesReferencesReferencesReferences[1] S. Verdu, "Spectral efficiency in the wideband regime", IEEE Trans. Inf. Theory,,

vol. 48, no. 6, pp. 13191343, June 2002

[2] F. Hliot, M. Imran, and R. Tafazolli, "On the Energy efficiency -Spectral efficiency

Trade-o over the MIMO Rayleigh Fading Channel“ ,IEEE Trans. Communications,

VOL. 60, NO. 5, MAY 2012.

[3] I. E. Telatar, "Capacity of multi-antenna Gaussian channels“ ,Europe. Trans. Telecomm. Related Techno, vol. 10, no. 6, pp. 585596, Nov. 1999.Telecomm. Related Techno, vol. 10, no. 6, pp. 585596, Nov. 1999.

[4] F. Hliot, M. Imran, and R. Tafazolli, "An accurate closed-form approximation of the energy efficiency -spectral efficiency trade-o over the MIMO Rayleigh fading channel“ ,in Proc. 2011 IEEE ICC, 4th Int. Workshop Green Comm..,

[5] O. Oyman and A. J. Paulraj, "Spectral efficiency of relay networks in the power limited regime",in Proc. 2004 Allerton Conf. Commun., Control Computing.

[6] S. de la Kethulle, "An Overview of MIMO Systems in Wireless communications," Lecture in Communication Theory for Wireless Channels, September 27, 2004.

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The Power-Bandwidth Tradeoff in MIMO Systems