Understanding output width in signed multipliers

Sep 21, 2004 3 Replies
U

using Xilinx DDS and MULTIPLIER cores, I simulated a system with scalable output. However after the scaler I notice a loss of amplitude of a factor two, even when the output scaler was at maximum positive value (0111..) or minimum negative value (1000..).



Thinking hard, I see that only one case ( 1000... * 1000..) delivers a result with both the top bits different. So if I clip the value range for the scale factor at the low end to exclude 1000..., the second top bit will never be different than the top bit and for an M bit * N bit multiplication using only (M+N-1) bits of the result will not drop any information.



Is that right?


Uwe Bonnes bon@elektron.ikp.physik.tu-darmstadt.de Institut fuer Kernphysik Schlossgartenstrasse 9 64289 Darmstadt --------- Tel. 06151 162516 -------- Fax. 06151 164321 ----------
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B

Yes.

In a two's-complement multiplication with inputs S+n and S+m, where S represents a sign bit and n, m represent the number of remaining bits, the product will have dimensions 2S+n+m in all cases except

10...0*10...0 (i.e., max neg times max neg). If you can avoid this case, you can throw away the redundant-sign-bit MSB.

How to avoid it? You can, as suggested, use a clipper to limit one of the inputs to a negative value of 10...01. Or if, as is often the case, one of the inputs comes from a coefficient table, you can select/scale your coefficients to avoid the max. negative number.

Obligatory clarification for those who care: In casual conversation (i.e., at parties, coronations, parole hearings) we often refer to the MSB of a two's-comp number as the sign bit. It's more accurate to call it a magnitude bit with a negative weight. For example, the bits in a 4-bit, two's-comp number have weights (from MSB to LSB) of -8, 4,

2, and 1. Looking at two's-comp this way removes a lot of the mystery from two's-comp arithmetic.

Bob Perlman Cambrian Design Works

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U

Bob Perlman wrote: : Obligatory clarification for those who care: In casual conversation : (i.e., at parties, coronations, parole hearings) we often refer to the : MSB of a two's-comp number as the sign bit. It's more accurate to : call it a magnitude bit with a negative weight. For example, the bits : in a 4-bit, two's-comp number have weights (from MSB to LSB) of -8, 4, : 2, and 1. Looking at two's-comp this way removes a lot of the mystery : from two's-comp arithmetic.

Nice explanation! Thanks

Uwe Bonnes bon@elektron.ikp.physik.tu-darmstadt.de Institut fuer Kernphysik Schlossgartenstrasse 9 64289 Darmstadt --------- Tel. 06151 162516 -------- Fax. 06151 164321 ----------
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S

Absolutely, Bob. Once my parole officer explained this to me, designing distributed arithmetic multipliers became a walk in the park! Cheers, Syms.

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