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Unformatted text preview: EE 568 Homework Solution 1 EE568 Homework Solution 3 Problem 2.2 The BSC capacity curve was obtained in problem 1.7, and the capacity curve of the BPSKAWGN channel is available from either from running the FEClimits program or using the precomputed curves. Note that the capacity plotted in problem 1.7 was in units of bits per BSC channel use. These two curves are plotted together in Fig. 1 along with their difference. Thus, information theory predicts that softin decoding provides and additional coding gain of nearly 2 dB at lower code rates and approximately 1 dB at very high code rates.21 1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 E b /N (dB) Capacity = Bits Per Channel Use Capacity BPSK vs BSC 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Capacity: Bits Per Channel Use ∆ E b /N o (dB) Difference in Required Eb/No (dB) for BPSK and BSC capacity BPSK (soft input) BSC (hard input) Numerical Precision Error SoftInput has more potential for greater capacity at a given Eb/No since it contains more "information" relative to the BSC (hardinput). Figure 1: Comparison of capacity for the BPSKAWGN channel with real observations and the BSC. For the case of 3bit quantization considered in problem 2.1, the SIR for the resulting DMC can be computed in terms of p ( j  i ) = Pr { output = j  input = i } and p ( i ) = Pr { input = i } : SIR = 1 2 +3 X j = 4 p ( j  0) log 2 p ( j  0) p ( j  0) + p ( j  1) + p ( j  1) log 2 p ( j  1) p ( j  0) + p ( j  1) (1) The transition probabilities p ( j  i ) are given as a function of E c /N in the solution to problem 8. Once this SIR value is computed for a specific E c /N value, we can convert to the minimum value of E b /N via E b /N = 1 SIR E c /N . Not that the units of SIR are bits per channel use (corresponding to a single 1D BPSK channel use). EE 568 Homework Solution 221 1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 E b /N (dB) Capacity = Bits Per Channel Use Capacity BPSK vs DMC (3bit Quant) vs BSC BPSK (soft input) DMC: Quant 3 bits BSC (hard input) Discrete Memoryless Channel (DMC) 3bit Quantization performs within 0.2 dB to BPSK Binary Symmetric Channel (DMC with 1bit quant) performs appx 2 dB degradation relative to BPSK (infinite quantization) Figure 2: Comparison of capacity for the BPSKAWGN channel with real observations, 3bit quantization, and 1bit quantization (the BSC)....
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This note was uploaded on 02/27/2008 for the course EE 568 taught by Professor Chugg during the Fall '07 term at USC.
 Fall '07
 CHUGG

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