A binary integer B in the range 0 <= B < 27 can be represented by 7 bits in "fixed-point" format, B = B6B5B4B3B2B1B0. We can represent numbers in the same range with less precision using 6 bits in a floating-point notation, F = M.2E, where M is a 4-bit mantissa M = M3M2M1M0 and E is a 2-bit exponent E = E1E0. In this system, the smallest integer is 0.2° and the largest is 15.23. We can write B = M•2E+T where the truncation error T satisfies 0 <= T < 2E. For example:
• if B = 1011010 we would choose M = 1011 and E = 11 and the truncation error would be T = 10;
• if B = 0110101 we would choose M = 1101 and E = 10 and the truncation error would be T =1.
(a) Start by expressing the following binary integers B in the form B = M•2E +T where M, E and T are as specified above:
(i) 1111111;
(ii) 1111000;
(iii) 0000000;
(iv) 0001100; and
(v) 1010101.
(b) Design a combinational circuit which takes as input the 7-bit unsigned binary integer B = B6B5B4B3B2B1B0 and which outputs the 4-bit mantissa M = M3M2M1M0 and the 2-bit exponent E = E1E0 of the floating-point representation of the number. You do not need to sketch the circuit, rather you should write down a sum-of-products expression for each of the six ouput variables in terms of the input variables. As always, you should explain the reasoning behind your design.