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ch7_parts1and2 - Chapter 7 Optimal Demodulation From Intro...

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Chapter 7 Optimal Demodulation From Introduction to Communication Systems Copyright by Upamanyu Madhow, 2008-2010 As we saw in the last chapter, we can send bits over a channel by choosing one of a set of waveforms to send. For example, when sending a single 16QAM symbol, we are choosing one of 16 passband waveforms: s b c ,b s = b c p ( t ) cos 2 π f c t - b s p ( t ) sin 2 π f c t where b c , b s each take values in 1 , ± 3 } . We are thus able to transmit log 2 16 = 4 bits of information. In this chapter, we establish a framework for recovering these 4 bits when the received waveform is a noisy version of the transmitted waveform. More generally, we consider the fundamental problem of M-ary signaling in additive white Gaussian noise (AWGN): one of M signals, s 1 ( t ) , ..., s M ( t ) is sent. The received signal equals the transmitted signal plus white Gaussian noise (to be defined later in this chapter). At the receiver, we are faced with a hypothesis testing problem: we have M possible hypotheses about which signal was sent, and we have to make our “best” guess as to which one holds, based on our observation of the received signal. For our application, we are typically interested in finding a guessing strategy, more formally termed a decision rule, which minimizes the probability of error (i.e., the probability of making a wrong guess). We can now summarize the goals of this chapter as follows. Goals: We wish to design optimal receivers when the received signal is modeled as follows: H i : y ( t ) = s i ( t ) + n ( t ) , i = 1 , , , .M where H i is the i th hypothesis, corresponding to signal s i ( t ) being transmitted, and where n ( t ) is white Gaussian noise. We then wish to analyze the performance of such receivers, to see how performance measures such as the probability of error depend on system parameters. It turns out that, for the preceding AWGN model, the performance depends only on the received signal- to-noise ratio (SNR) and on the “shape” of the signal constellation { s 1 ( t ) , ..., s M ( t ) } . Underlying both the derivation of the optimal receiver and its analysis is a geometric view of signals and noise as vectors, which we term signal space concepts. Once we have this background, we are in a position to discuss elementary power-bandwidth tradeo ff s. For example, 16QAM has higher bandwidth e ffi ciency than QPSK, so it makes sense that it requires higher SNR, and hence higher transmit power, for the same probability of error. We will be able to quantify this intuition based on the material in this chapter. We will also be able to perform link budget calculations: for 1
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example, how much transmit power is needed to attain a given bit rate using a given constellation at a given range, given transmit and receive antenna directivities? Plan: We build up the background needed to attain the preceding goals in a step-by-step fashion, as follows.
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