Part IV MULTIPLE ACCESS AND ADVANCED TRANSCEIVER SCHEMES 363
19.8 Adaptive Modulation and Capacity
Adaptive modulation changes the coding scheme and/or modulation method depending on channel- state information – choosing it in such a way that it always “pushes the limit” of what the channel can transmit. In OFDM, modulation and/or coding can be chosen differently for each subcarrier, and it can also change with time. We thus not only accept the fact that the channel (and thus the SNR) shows strong variations but even exploit this fact. On subcarriers with good SNR, transmission is done at a higher rate than on subcarriers with low SNR. In other words, such an adaptive modulation selects a modulation scheme and code rate according to the channel quality of a specific subcarrier.
Let us compare adaptive modulation for OFDM with coded OFDM and multicarrier CDMA. Both of the latter systems try to “smear” the data symbols over many subcarriers, so that each symbol sees approximately the same average SNR. It can be shown that – at least theoretically – systems using adaptive modulation perform better than systems whose modulation and coding are fixed once and for all.
19.8.1 Channel Quality Estimation
Adaptive modulation requires that the transmitter knows channel-state information. This require- ment sounds trivial, but is quite difficult to realize in practice. It requires the channel to be reciprocal – e.g., the Base Station (BS) learns the channel state while it is in receiving mode;
when it then transmits, it relies on the fact that the channel is still in the same state. This can only be fulfilled in systems with Time Domain Duplex (TDD) in slowly time-varying channels. Alter- natively, feedback from the receiver to the transmitter can be used to inform the transmitter about the channel state (see also Sections 17.5 and 20.1.6). It is also noteworthy that the transmitter has to know the channel-state informationfor the time instant when it will transmit – in other words, it has to look into the future. Channel prediction is thus an important component for many adaptive modulation systems.
19.8.2 Parameter Adaptation
Once the channel state is known, the transmitter has to decide how to select the correct transmission parameter for each subcarrier – namely, coding rate and modulation alphabet. Furthermore, we also have to consider how much power should be assigned to each channel. In the following, we will assume that the channel is frequency selective, but time invariant. This will make the discussion easier, and the principles can be easily extended to doubly selective channels.
Let us first consider the question of how much power should be assigned to each channel. In order to find the answer, let us first reformulate the question in a more abstract way: “Given a number of parallel subchannels with different attenuations, what is the distribution of transmission power that maximizes capacity?” The answer to this latter question was given by Shannon in the 1940s, and is known as “waterfilling.” Power allocationPn of thenth subchannel is
Pn=max
0, ε− σn2
|αn|2
(19.31) where αn is the gain (inverse attenuation) of the nth subchannel, σn2 is noise variance, and the thresholdεis determined by the constraint of the total transmitted powerP as
P = N n=1
Pn (19.32)
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Waterfilling can be interpreted visually, according to Figure 19.16. Imagine a number of con- nected vessels. At the bottom of each vessel is a block of concrete with a height that is proportional to the inverse SNR of the subchannel that we are considering. Then take water, and pour it into the vessels; the amount of poured water is proportional to the total transmit power that is available.
Because the vessels are connected, the surface level of the water is guaranteed to be the same in all vessels. The amount of power assigned to each subchannel is then the amount of water in the vessel corresponding to this subchannel. Obviously, subchannel 1, which has the highest SNR, has the most water in it. It can also happen that some subchannels that have a poor SNR (like channel 5), do not get any power assigned to them at all (the concrete block of that vessel is sticking out of the water surface). Essentially, waterfilling makes sure that energy is not wasted on subchannels that have poor SNR: in the OFDM context, this means not wasting power on subcarriers that are in a deep fade.
sn2/⏐a1⏐2 sn2/⏐a2⏐2
sn2/⏐a5⏐2
Figure 19.16 Principle behind waterfilling.
With waterfilling, power is allocated preferably to subchannels that have a good SNR (“give to the rich” principle). This is optimum from the point of view of theoretical capacity; however, it requires that the transmitter can actually make use of the large capacity on good subchannels.
In each subchannel (subcarrier), signaling as close to capacity as possible should be performed.5 This means that the transmitter has to adapt the data rate according to the SNR that is available (note that waterfilling increases SNR differences between subcarriers). Consequently, the coding rate and the constellation size of the modulation alphabet have to be adjusted. For very high SNR, the constellation size, and thus the PAR, has to be very large. A Quadrature Amplitude Modulation (QAM) of alphabet size 64 currently seems to be the largest constellation size that can be used in practical systems. The capacity per subchannel is limited by log2(Na), where Na is the size of the symbol alphabet. It is thus wasteful to assign more energy to one stream than can be actually exploited by the alphabet. If the available alphabet is small, a “giving to the poor” principle for power allocation is preferable – i.e., assigning power that cannot be exploited by good subchannels to bad subchannels.
Example 19.3 Waterfilling: consider an OFDM system with three tones, withσn2=1,αn2=1, 0.4, 0.1, and total power
Pn=15. Compute the power assigned to different tones according to (i) waterfilling, (ii) equal power allocation, (iii) predistortion (inverting the channel attention), and compute the resulting capacity.
5We assume in the following that near-capacity-achieving codes are used. If this is not the case, we usually try to choose the data rate in such a way that a certain BER can be guaranteed.
From Eq. (19.31) we find thatε=9.25 gives the correct solution: in that case, the power in the different subchannels are
P1=8.25 (19.33)
P2=6.75 (19.34)
P3=0 (19.35)
that is, no power is assigned to the channel that suffers from the strongest attenuation. The total capacity can be computed as
Cwaterfill= N n=1
log2(1+αn2Pn/σn2)=5.1 bit/s/Hz (19.36) For equal power allocation:
P1=P2=P3=5 (19.37)
so that capacity becomes:
Cequal-power=4.8 bit/s/Hz (19.38)
For the predistortion case, the powers become
P1=1.1 (19.39)
P2=2.8 (19.40)
P3=11.1 (19.41)
from which we obtain a capacity of:
Cpredistort=3.2 bit/s/Hz (19.42)
In this example, equal power allocation gives almost as high a capacity as (optimum) waterfilling, while predistortion leads to significant capacity loss.
19.8.3 Signaling of Chosen Parameters
Most information-theoretic investigations assume a continuum of modulation alphabets that can realize any arbitrary transmission rate. In practice, the transmitter has a finite and discrete set of modulation alphabets available (BPSK, Quadrature-Phase Shift Keying (QPSK), 16-QAM, and 64- QAM). There is also a finite set of possible code rates: the different codes are usually obtained form a “mother” code by different amounts of puncturing. Thus, the available data rates form a discrete set.
After the transmitter has decided which transmission mode – i.e., combination of signal constel- lation and encoder – to use on each tone, it has to communicate that decision to the receiver. There are three possibilities to achieve that task:
• Explicit transmission: the transmitter can send, in a predefined and robust format, the index of the transmission mode it intends to use. Transmission of this information itself should always
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be done in the same mode, and care should be taken that the message is well protected against errors during transmission.
• Implicit transmission: implicit transmission is possible when the transmitter gets its channel- state information from the receiver via feedback. In such a case, the receiver knows exactly what channel-state information is available to the transmitter, and thus the basis on which the decision for a transmission mode is being made. Thus, the receiver just needs to know the decision rule on which the transmitter bases its choice of transmission mode. If the receiver feeds back the mode that the transmitter should use, the situation is even simpler.
The drawback to this method is that errors in channel-state feedback (from the receiver to the transmitter) not only lead to a wrong choice of transmission mode (which is bad, but usually not fatal) but also to detection and decoding using the wrong code, which leads to very high error rates.
• Blind detection: from the received signal, the receiver can try to determine the signal constella- tion. This can be achieved by considering different statistical properties of the received signal, including the PAR, autocorrelation functions, and higher order statistics of the signal.