ECEN 248: Introduction to Digital System DesignDepartment of Electrical EngineeringTexas A&M UniversityAssignment #9Due Tuesday, December 1, 20091. [15 points.] Suppose I have a SOP expression for a funciton f on n-variables, x1, x2, · · · xn. Assumethat there are k cubes in the expression of f. I would like to represent the SOP expression as an k × nmatrix B, with each row representing a cube of f, and each column representing a variable. The rulesfor constructing the matrix B are:• If the literal xjis present in a cube i, then I write a ’1’ in the (i, j) entry of B.• If the literalxjis present in a cube i, then I write a ’0’ in the (i, j) entry of B.• If the literal xjis not present in a cube i, then I write a ’-’ in the (i, j) entry of B.For example, suppose I had a 4-variable function f = x1x3+ x2x4. The corresponding matrix Bwould be:B =1 − 0 −− 1 − 1(a) Write down the matrix B for the function x1x2x3+ x2x3x5+ x4+ x1.(b) Write down a rule for computing the cofactor of a function f, represented in the above fashion(with respect to a literal xi). The result should also be in the same format as above.(c) Write down a rule for computing the cofactor of a function f, represented in the above fashion(with respect to a literalxi). The result should also be in the same format as above.(d) Compute the cofactor of the matrix of part (a) with respect to x2andx3.2. [10 points.](a) Prove that x · fx= x · f.12Assignment #9(b) Suppose I have a function f on n variables x1, x2, · · · , xn. Suppose we have a cube c of thisfunction, written as a product of k literals l1· l2· l3· · · lk. Naturally k < n. What is the numberof minterms covered by this cube, as a function of k and n?3. [25 points.] Suppose I want to build a counter which counts from 0 up to 4 (skipping the value 2).When the count reaches 4, my counter then starts counting from 0 to 4 again (always skipping thevalue 2). In other words, the counter output sequence is 0 → 1 → 3 → 4 → 0 → 1 → 3 → 4 · · · .The count transitions are made on the rising edge of a CLK signal. Use DFFs for any state elementsrequired by the counter. Assume we want to implement the counter as a Moore machine. Encode thecounter states as per the following table:Counter state Encoding0 001 013 104 11(a) Can I implement the above with a combinational circuit? If so, state why. If not, state why not.(b) Write down the state transition diagram for the counter.(c) Write down the state transition table from the above.(d) Next write down the minimum SOP for all the NS and output functions.November 19, 2009 Sunil P Khatri ECEN