Question

Design a combinational circuit that compares two 4-bit unsigned numbers A and B to see whether B is greater than 2A. The circuit has one output X, so that:
X = 1 if 2A < B, and
X = 0 if 2A >= B.
Please de descriptive and detailed in drawing.


The book we are using in class is this: http://www.cramster.com/logic-and-computer-design-fundamentals-4th-solutions-3631 and we are on chapter 4.

Answer 1

<p>First start by writing all possibilities of A and B such that 2A &lt; B, where A and B both lie between o and 15 because it four bit, unsignednumber. Let first list how many possibilities of A are possible for each of value of B.<br /><br />B A<br />1 0<br />2 0<br/>3 0, 1<br />4 0, 1<br />5 0, 1, 2<br />6 0,1, 2<br />7 0, 1, 2, 3<br />8 0, 1, 2, 3<br />9 0, 1, 2, 3, 4<br/>10 0, 1, 2, 3, 4<br />11 0, 1, 2, 3, 4, 5<br />12 0, 1, 2, 3, 4, 5<br />13 0, 1, 2, 3, 4, 5, 6<br />14 0, 1, 2, 3, 4, 5,6<br />15 0, 1, 2, 3, 4, 5, 6, 7<br /><br />Now, lets denote A and B by four bits value (abcd)₂and(efgh)₂ respectively. So from the above table we get:</p>
<p>Y = a'b'c'de'f'g'h' + a'b'cd'e'f'g'h' + ... and this is freaking long.</p>
<p>&#160;</p>
<p>OR</p>
<p>&#160;</p>
<p>Assuming you've already got a full subtractor circuit designed, it'll have two outputs: answer, and borrow out, and 3 inputs: A, B, and borrowin.<br /> <br /> Use 4 subtractors, one for each bit, with the borrow output from the least significant subtractor wired to the borrow input ofthe second least significant subtractor, etc.<br /> <br /> If the borrow output of the most significant subtractor is set (1), then you knowthat A minus B is a negative number, therefore A &lt; B. If A=B or A&gt;B, the borrow output will be 0.</p>

Answer 2

<p>First start by writing all possibilities of A and B such that 2A &lt; B, where A and B both lie between o and 15 because it four bit, unsignednumber. Let first list how many possibilities of A are possible for each of value of B.<br /><br />B A<br />1 0<br />2 0<br/>3 0, 1<br />4 0, 1<br />5 0, 1, 2<br />6 0,1, 2<br />7 0, 1, 2, 3<br />8 0, 1, 2, 3<br />9 0, 1, 2, 3, 4<br/>10 0, 1, 2, 3, 4<br />11 0, 1, 2, 3, 4, 5<br />12 0, 1, 2, 3, 4, 5<br />13 0, 1, 2, 3, 4, 5, 6<br />14 0, 1, 2, 3, 4, 5,6<br />15 0, 1, 2, 3, 4, 5, 6, 7<br /><br />Now, lets denote A and B by four bits value (abcd)₂and(efgh)₂ respectively. So from the above table we get:</p>
<p>Y = a'b'c'de'f'g'h' + a'b'cd'e'f'g'h' + ... and this is freaking long.</p>
<p>&#160;</p>
<p>OR</p>
<p>&#160;</p>
<p>Assuming you've already got a full subtractor circuit designed, it'll have two outputs: answer, and borrow out, and 3 inputs: A, B, and borrowin.<br /> <br /> Use 4 subtractors, one for each bit, with the borrow output from the least significant subtractor wired to the borrow input ofthe second least significant subtractor, etc.<br /> <br /> If the borrow output of the most significant subtractor is set (1), then you knowthat A minus B is a negative number, therefore A &lt; B. If A=B or A&gt;B, the borrow output will be 0.</p>