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AWGN channel handout1

AWGN channel handout1 - 172 Capacity of wireless channels...

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172 Capacity of wireless channels transforming the channels back to the AWGN channel, or by using the type of heuristic sphere-packing arguments we have just seen. 5.2 Resources of the AWGN channel The AWGN capacity formula (5.8) can be used to identify the roles of the key resources of power and bandwidth . 5.2.1 Continuous-time AWGN channel Consider a continuous-time AWGN channel with bandwidth W Hz, power constraint ¯ P watts, and additive white Gaussian noise with power spectral density N 0 / 2. Following the passband–baseband conversion and sampling at rate 1 /W (as described in Chapter 2), this can be represented by a discrete- time complex baseband channel: y±m² = x±m² + w±m²³ (5.9) where w±m² is ±² ´ 0 ³N 0 µ and is i.i.d. over time. Note that since the noise is independent in the I and Q components, each use of the complex channel can be thought of as two independent uses of a real AWGN channel. The noise variance and the power constraint per real symbol are N 0 / 2 and ¯ P/´ 2 respectively. Hence, the capacity of the channel is 1 2 log ± 1 + ¯ P N 0 W ² bits per real dimension ³ (5.10) or log ± 1 + ¯ P N 0 W ² bits per complex dimension (5.11) This is the capacity in bits per complex dimension or degree of freedom. Since there are W complex samples per second, the capacity of the continuous- time AWGN channel is C awgn ´ ¯ P³Wµ = W log ± 1 + ¯ P N 0 W ² bits/s (5.12) Note that SNR · = ¯ P/´N 0 is the SNR per (complex) degree of freedom. Hence, AWGN capacity can be rewritten as C awgn = log ´ 1 + SNR µ bits / s / Hz (5.13) This formula measures the maximum achievable spectral efficiency through the AWGN channel as a function of the SNR.
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173 5.2 Resources of the AWGN channel 5.2.2 Power and bandwidth Let us ponder the significance of the capacity formula (5.12) to a communica- tion engineer. One way of using this formula is as a benchmark for evaluating the performance of channel codes. For a system engineer, however, the main significance of this formula is that it provides a high-level way of thinking about how the performance of a communication system depends on the basic resources available in the channel, without going into the details of specific modulation and coding schemes used. It will also help identify the bottleneck that limits performance. The basic resources of the AWGN channel are the received power ¯ P and the bandwidth W . Let us first see how the capacity depends on the received power. To this end, a key observation is that the function SNR ²³ = log ± 1 + SNR ² (5.14) is concave , i.e., f ±± ±x² 0 for all x 0 (Figure 5.4). This means that increasing the power ¯ P suffers from a law of diminishing marginal returns: the higher the SNR, the smaller the effect on capacity. In particular, let us look at the low and the high SNR regimes. Observe that log 2 ± 1 + x log 2 e when x 0 ´ (5.15) log 2 ± 1 + log 2 x when x ² 1 µ (5.16) Thus, when the SNR is low, the capacity increases linearly with the received power ¯ P : every 3 dB increase in (or, doubling) the power doubles the capacity.
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AWGN channel handout1 - 172 Capacity of wireless channels...

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