课程名称︰数位系统与实验 课程性质︰必修 课程教师︰廖世伟 开课学院：电机资讯学院 开课系所︰资讯工程学系 考试日期（年月日）︰104/5/11 考试时限（分钟）：180分钟 试题 : NTU CSIE Digital System and Labs 2015 MIDTERM May 11, 2015 1. (14%) Plot the following functions on a Karnaugh map and convert them into minimum sum of products. a. (4%) f(a,b,c) = Σm(0,1,3,6) b. (4%) g(w,x,y,z) = Σm(3,4,7,10,11,14) + Σd(2,13,15) c. (6%) Find all minimum sum of products expressions F(a,b,c,d) = Σm(0,2,3,7,8,9,13,15) + Σd(1,12) 2. (21%) For the three functions of four variables shown on the maps below a. (7%) Find a minimum cost two-level NAND circuit (where minimum cost is minimum number of gates, and among those with the same number of gates, minimum number of gates inputs). Assume all inputs are available both complemented and uncomplemented (minimum is 10 gates). b. (7%) Implement it with the ROM c. (7%) Implement it with the PLA with eight product terms available (You may use fewer.) http://imgur.com/SaeboLZ 3. the identity of each of the following Boolean equations, using algebraic manipulatoin. a. (3%) (x+y)(x'+y) = y b. (3%) x+y(x+z)+xz = x+yz c. (3%) (xy+z'(s+t'))' = (x'+y)(z+s't) 4. (6%) Simplify the logic diagram below. http://imgur.com/jJsVchZ 5. (12%) In crptography, the term "plaintext" is information a sender wishes to pass to receivers. while the term "ciphertext" is the result of encrypted plaintext. Here we introduce the "XOR cipher" where the ciphertext is generated by plaintext XOR the key: plaintext: 1011 1110 1110 1111 the key: 0110 0110 0110 0110 ──────────────── ciphertext: 1101 1000 1000 1001 Now we define that the 4-bit codes indicate the hexadecimal numbers, from 0000 = 0 to 1111 = F. So the example above encrypts the "BEEF" to "D889" with the key "0110". And if the key were "1101 0011", the ciphertext would be "6D3C". Answer the questions: i) What is the ciphertext if the plaintext is "BABEFACE" with the key "3D"? ii) We have just intercepted the enemy's ciphertext "3310AE86", and we have known that they encrypt the text "C" as "6". Please decrypt the intercepted message. iii) The intelligence agency just indicates that our enemy just changed their encryption method. They ANDand OR plaintext with key, respectively, then XOR these 2 results to generate the ciphertext. According to this rule, what is the plaintext for the question (ii)? 6. (12%) Given the Venn's Diagram as below, each number represents a set, use sum of sets to answer the following questions. For examples: A．B = 4 + 7, A + B = 1 + 2 + 4 + 5 + 6 + 7: http://imgur.com/lwaSsyA i) (ABC)' ii) (A NAND B) NOR C iii) F(A, B, C) = Σm(2, 3, 5, 7) iv) Minterm m_3 of F(A, B, C) v) A ⊕ B ⊕ C vi) A NXOR B NXOR C 7. (10%) Consider a 6-bit division checker D as the following: the first 4-bit indicates dividend, and the rest represents divisor. D returns True (= l) if the dividend is divisible by the divisor; otherwise it returns False 0). By the way, D returns X (don't care) if the divisor is zero. For example: 101010 means "Is 10 divisible by 2?" the answer is true, so D = l. 011000 means "Is 6 divisible by 0?" in this example, D = X due to zero divisor. i) Draw the K-map for D (6%) ii) Write the Boolean expression for D, your answer should include a 1-literal term and a 2-literal term at least (4%) 8. (16%) The following left figure shows 4 to 5 magnitude comparator, which takes 2 binary numbers as input and determine whether the first number is greater than (G), less than (L) the other number by 1 (X), by 2 (Y), orby 3 (Z). We regard inputs as A1A0 and B1B0 in unsigned integers, and assume that 1 means true in output, while 0 means false. For example, if the input is 0110, this comparator should output (0, l, l, 0, 0) since 1 is less than 2 by l. Answer the following questions: http://imgur.com/6iqlgwi i) Under condition the output would be (0, 0, 0, 0, 0), and how many possibilities are there. (2%) ii) Draw the K-maps for G and X (2%) iii) Draw the PLA implementation of this magnitude comparator (4%) iv) Now we combine two magnitude comparators to implement a bigger comparator as the right figure above. Let the function F means A - B = 5. Write the Boolean expression for F using ten outputs above (G2, L2, ..., Z1) only. (4%) http://imgur.com/8DypaKS 9. (Optional A, choose either A or B only) (10%) What is the function of the following circuit diagram? Explain your answer. http://imgur.com/xwwOD6C 9. (Optional B, choose either A or B only) (10%) (Verilog HDL) Take a look at the following Verilog program. a. (3%) Can a Verilog module name start with a number? b. (4%) Does "clk" below need to be declared in the parameter list? c. (3%) Should the module end with "endmodule" or "endmodules"? ┌───────────────────────────────┐ │module 16x16-MAC (out, rst, opl, op2) │ │ │ │input clk, rst; │ │ │ │input [15:0] op1, op2; │ │ │ │output [39:0] out; │ │ │ │reg [31:0] product; op1 * op2; │ │ │ │assign product │ │ │ │//40-bit accumulator after 16x16 multiplier │ │ │ │always @(posedge clk || negedge rst) │ │ │ │ if(rst) out 0; │ │ │ │ else out = out + product; │ │ │ │endmodules │ └───────────────────────────────┘ --

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