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HW10-solution - EE456 HW-10 SOLUTIONS Theory: i.) Using the...

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EE456 HW-10 SOLUTIONS Theory: i.) Using the given relations for propagate and generate signals we could write: For generate: ii.) Propagate i:j ( P i:j ) is the collective ‘propagate’ signal of the bits j to i. It determines whether the bit block propagates a carry-in that comes in from bit j all the way to bit i. To illustrate with an example, in a 16 bit ripple carry adder, if P 15:0 is 1, then an input carry of 1 will propagate through all the 1-bit adders to generate a carry-out in the end. Similarly, Generate i:j ( G i:j ) is the collective ‘generate’ signal of the bits ranging from j to i. It determines whether the bit block generates a carry-out or not. Using these generalized propagate and generate signals then, we could argue that a carry will be generated only under two conditions: i) There’s propagating carry in coming from the previous block ii) A carry is generated in the current block. To understand block propagate signal is even more straightforward: A carry will be propagated to the next bit blocks as long as each subsequent block propagates the incoming carry. We could observe that this is indeed the intended description of the propagate and generate notation: The idea that leads to high speed and parallelism in prefix-tree adders could be understood if we look at the answer to the first problem. We could see that as long as we have block generate (and propagate) signals, we do not have to worry about the previous stages because essentially all the information is contained within the block signals! Note that we could have
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stopped calculating at the second line of G 7:0 as long as we know what G 6 : 0 is. Parallel prefix-tree adders differ only in their ways to construct G 7:0. But the main point to understand here is that ‘all the information regarding the evolution of a carry inside an adder’ is contained in G 7:0 for an 8-bit adder. Although the details of parallelism or implementation could change, the basic idea is ‘looking ahead’ and ‘predicting’ the carry output to avoid naively waiting for a carry to propagate from the sums. iii.)
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This note was uploaded on 09/24/2009 for the course ECE 456 taught by Professor Mohammadi during the Spring '09 term at Purdue.

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HW10-solution - EE456 HW-10 SOLUTIONS Theory: i.) Using the...

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