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Lecture9 - ECE 5670 Digital Communications Lecture 9...

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ECE 5670 : Digital Communications Lecture 9: Capacity of the Continuous time AWGN Channel 1 2/22/2011 Instructor: Salman Avestimehr Introduction In the penultimate lecture we saw the culmination of our study of reliable communication on the discrete time AWGN channel. We concluded that there is a threshold called capac- ity below which we are guaranteed arbitrarily reliable communication and above which all communication is hopelessly unreliable. But the real world is analog and in the last lecture we saw in detail the engineering way to connect the continuous time AWGN channel to the discrete time one. In this lecture we will connect these two story lines into a final statement: we will derive a formula for the capacity of the continuous time AWGN channel. This is the largest rate of reliable communication (as measured in bits/second) and depends only on the two key physical resources: bandwidth and power. We will see the utility of this formula by getting a feel for how the capacity changes as a function of the two physical resources has more impact on the capacity The Continuous Time AWGN Channel The channel is, naturally enough, y ( t ) = x ( t ) + w ( t ) , t > 0 . (1) The power constraint of ¯ P Watts on the transmit signal says that lim N →∞ 1 NT integraldisplay NT 0 ( x ( t )) 2 dt ¯ P. (2) The (two-sided) bandwidth constraint of W says that much of the energy in the transmit signal is contained within the spectral band bracketleftbig - W 2 , W 2 bracketrightbig . We would like to connect this to the discrete time AWGN channel: y [ m ] = x [ m ] + w [ m ] , m 1 . (3) This channel came with the discrete-time power constraint: lim N →∞ 1 N N summationdisplay m =1 ( x ( m ]) 2 P N. (4) We have already seen that there are W channel uses per second in the continuous time channel if we constrain the bandwidth of the analog transmit voltage waveform to W Hz. 1 Based on lecture notes of Professor Pramod Viswanath at UIUC. 1
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So, this fixes the sampling rate to be W and thus unit time in the discrete time channel corresponds to 1 W seconds. To complete the connection we need to: 1. connect the two power constraints ¯ P and P ; 2. find an appropriate model for the continuous time noise w ( t ) and connect it to the variance of the additive noise w [ m ].
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