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Figure 2. Above: Binary symmetric channel Below Binary erasure channel.
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7. Binary symmetric channel v.s. Binary Erasure channel Suppose the input X of the channel is equally probable to take values X0 and X1, (a = 0.5), whereas the output can take values of Y0 and Y1. The probability of receiving the right symbol is denoted by p. In other words, p(Y0|X0) = p(Y1|X1) = p. In the binary erasure channel, the output Y can take one more value e which is called the erasure. The probability of getting the erasure is 1 - p and p(Y0|X1) = p(Y1|X0) = 0. Both channels are shown as reference in gure 2. (a) Compute H(X|Y) in the binary symmetric channel (b) Compute H(X|Y) in the binary erasure channel (c) Compute H(Y) in both cases. (d) What does H(Y) tend to if p 1 How about p 0 Explain. 8. Venn Diagram Use Venn Diagram to illustrate the following (a) I(X; Y |Z) (b) I(X1, X2, X3; Y) 9. Channel Capacity of AWGN Channels (a) Given an AWGN channel with channel output given by Yn = Xn + Zn, show that the capacity is
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2 sx C = log 2 1 + 2 sz 2 2 where s x is the variance of X and s z is the variance of Z. Assume everything is complex. What is the unit of the capacity obtained. (b) In practice, we transmit continuous waveforms x(t) instead of discretetime symbols, Xn. Using (a) and Nyquist sampling theorem, show that the channel capacity of the continuous waveform AWGN channel with bandwidth W is given by
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P C = W log 2 1 + Wh0 where h0 is the single-sided noise power spectral density. What is the unit of the capacity obtained.
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In this chapter, we explore the design of channel-adaptive physical layer for wireless link with multiple antennas based on information theoretical framework. Speci cally, we consider a point-to-point communication link with nT transmit antennas and nR receive antennas. Notationwise, we denote such multiple-input multiple-output (MIMO) link as a nT nR wireless link. As we discussed in 1, wireless channels are characterized by time-varying multipath fading. For instance, the attenuation introduced by the channel uctuates over a large dynamic range over a relatively short time interval. At one time, the channel fading may be very good, facilitating reliable communications. At other times, the channel state may be very bad and the communication link may hardly be able to deliver any information reliably. In fact, the channel capacity1 depends on the availability of channel state information (CSI) at the receiver and at the transmitter. As we shall discuss, there are in general two functions of CSI. The availability of CSI at the receiver serves the purpose of detection. On the other hand, the availability of CSI at the transmitter serves the purpose of adaptation. The nT nR wireless MIMO channels, however, are characterized by timevarying fading nR nT channel matrices. As a result of the multiple antennas, the MIMO channel can be viewed as a vector channel with m* = min{nT, nR} spatial channels. In fact, the data rate of the MIMO link can be increased by spatial multiplexing of the m* spatial channels and therefore, the channel
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1 Capacity refers to the maximum data rate that can be delivered through the wireless link with arbitrarily low error probability.
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Channel-Adaptive Technologies and Cross-Layer Designs for Wireless Systems with Multiple Antennas: Theory and Applications. By V. K. N. Lau and Y.-K. R. Kwok ISBN 0-471-64865-5 2006 by John Wiley & Sons, Inc.
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capacity increases with m* in a linear manner asymptotically. Hence, by increasing m*, we can increase the channel capacity at the same power and bandwidth budget. This is a very signi cant gain compared to increasing the capacity by increasing the transmit power. Hence, it is the focus of this chapter to study the combined bene t of MIMO and channel adaptation. In this chapter, we shall focus on the ideal situation where the knowledge of CSI (at the transmitter and/or the receiver) is perfect. The effect of imperfect CSI knowledge on the channel capacity will be discussed in the next chapter. We begin by reviewing various discrete-time fading channels that serve as a common foundation for further discussion. Based on the channel models, we review the concept of ergodic capacity and outage capacity. The ergodic and outage capacity of the MIMO channels with various levels of CSI availability will be discussed in detail. For instance, we label the scenario as CSIR when the receiver has CSI and label it as CSIT when the transmitter has CSI. We shall discuss the channel capacities of the four cases, namely, without any CSI, with perfect CSIR, with perfect CSIT, as well as with both perfect CSIR and perfect CSIT. With perfect CSIR, channel-matched decoding can be done, resulting in enhanced channel capacity compared to the baseline case without any CSI knowledge. When the transmitter has perfect CSIT, adaptation (in both the spatial and temporal dimensions) can be performed at the transmitter, and this results in additional enhancement of channel capacity. For instance, the optimal MIMO transmission scheme with CSIR only is shown to be a bank of nT isolated channel encoders [126]. On the other hand, the optimal MIMO transmission scheme with perfect CSIT is shown to be a cascade of channel encoder bank, adaptive power control matrix, and an eigenbeamforming matrix. The adaptive power control matrix is to perform temporal and spatial (spatiotemporal) water lling on each individual spatial channel. The eigenbeamforming matrix is to decompose the nR nT channel matrix into m* = min[nR, nT] independent spatial channels so that data carried along each spatial channel will not interfere with each other at the receiver. In other words, the nR nT MIMO fading channel with CSIT and CSIR is decomposed into m* = min[nR, nT] independent parallel channels.
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