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Challenge05_Quant_Arith0

# Challenge05_Quant_Arith0 - The worst case accumulation is 8...

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Lesson Title: Quantization Case Study Lesson Number: 05 Challenge: A dedicated multiply-accumulate unit is to be used to perform the 16-sample DSP SAXPY (S=AX+Y) operation: = = 16 1 i i i Y X Z The input data sets X i and Y i are 8-bit signed numbers with magnitudes bounded from above by 4. The implementation architecture is shown below for a full-precision multiplier, an extended precision accumulator (20-btis), and a 16-bit rounder. Q: What is the approximate output error variance, in bits, due to the multiply-accumulate unit? Rround your answer to the nearest bit value. Consider what would happen if the accumulator wordwidth were set to 16-bits. Consider what would happen if the multiplier only outputted the 8 MSBs of the full precision product? 8 8 X i Y i 16 Z i 20 Round 16 MAC Z

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The data format for the words X i and Y i is shown below. Since X i and Y i are assumed to be bounded between ± 4, the quantization step-size can be expressed as =2 -5 . This is also the weight of the LSB. The data format for the full precision product is shown below.
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Unformatted text preview: The worst case accumulation is: 8 16 1 16 1 16 2 256 16 * 16 4 * 4 = = = ≤ = ∑ ∑ = = i i i i Y X Z The extended precision accumulator wordwidth has 4-additional bits of precision (headroom) to permit 16 full precision products to the accumulated without overflowThe data format for the accumulator output word is shown below. Retaining the 16-most significant bits results in a output data word having the format shown below. The weight of the LSB of the SAXPY result is now ∆ =2-7 . The error statistics, in bits, is therefore (assuming rounding) E ( e )=0; σ 2 = ∆ /12 log 2 ( σ = ∆ /sqrt(12)) = log 2 (2-7 /sqrt(12)) = -7 – 1.79 ~ -9 or, approximately 9 bits of fractional precision is left in the accumulator’s output. Why did Motorola call its 24-DSP fixed-point product the DSP56xxx? S F=8-3=5 I=2 S F=16-5=11 I=2+2=4 S I=4+4=8 F=20-9=11 S I=8 F=16-9=7...
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Challenge05_Quant_Arith0 - The worst case accumulation is 8...

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