Exam for Physics 45, October, 8 5 points - closed book - calculators allowed - show your work Problem : (6 Points) The 4 bit shift register circuit shown in Figure has been initialized to contain the following values at t = : Q =, Q =, Q =, Q =. (Note that the input D has permanently been set to.) a) List the values of Q, Q, Q and Q for the next clock cycles after t =. Also write down the decimal equivalent that Q, Q, Q and Q represent. t Q (MSB) Q Q Q (LSB) D Decimal 6 8 b) The definition of a right or left shift register depends on how the shift operation affects its numerical value. (It does not depend on how it is drawn on the page!) Identical to the decimal system, a right shift operation results in a division and a left shift operation results in a multiplication. Based on this information, is the shift register shown above a left shift or right shift register? Decimal equivalent of register is:, 6,, 8 (last value would be 4 but next higher bit dropped off ), i.e., it multiplies; therefore, it is a left shift register. c) Name two specific circuit applications where shift registers are used. Paralled to Serial or Serial to Parallel converters, random number generators, mathematical operations.
Problem ) (6 Points) Shown on the next page is a 7 segment display similar to the one used in the Frequency Counter lab. It consists of 7 bar-shaped LEDs labeled A through G. By turning the appropriate combination of the LEDs on and off, all numerals and some letters can be displayed. For example, to display the number all LEDs except G are turned on; for the number, only B and C are turned on, etc. (See the diagram on the next page on the right.) You will design part of the logic driver circuit for such a 7 segment display. Your driver must be able to display the numerals through (and including) 9. Each numeral is represented by its 4 bit binary number representation, n n n n with n representing the LSB and n the MSB. For example, the numeral 5 is represented by. a) Complete the truth table shown below for the remaining numerals and indicate the LED segments that have to be turned on, or off. INPUTS OUTPUTS n n n n A B C D E F G b) From your complete truth table, write down the Boolean Logic expression that resents the LED segment labeled E in terms of its inputs n, n, n and n. (You do not have to simplify this expression.) E = n n n n + n n n n + n n n n + n n n n
Not required but simplifying previous result yields: E = n n n n + n n n n + n n n n + n n n n = n n n ( n n n n + n n n = n ( n n + n n ) + n ) + n n n ( n + n ) = Problem ) (7 Points) A decade counter counts in binary from (decimal ) to, etc. up to (decimal 9) before it repeats this sequence. An incomplete circuit diagram of a synchronous decade up-counter with 4 D-type Flip-Flops is shown on the next page. Note that Q represents the LSB and Q the MSB. a) Write down the complete truth table for such a synchronous decade up-counter, i.e., show all the Q i outputs (i.e., Q, Q, Q and Q ) and the D i inputs (i.e., D, D, D and D ) which will produce the sequence specified previously. D i takes on one cycle prior what Q i will become. Since Q i is specified, D i is defined too. Dec. Q Q Q Q D D D D Value 4 5 6 7 8 9 b) Work out the Boolean logic for the D input in terms of Q, Q, Q and Q, i.e., find the appropriate Boolean logic expression to implement D ( Q, Q, Q, Q ). D = Q Q Q Q + Q Q Q Q
c) Draw the logic circuit for the D input (from part b) using standard or inputs logic gates such as AND, NAND, OR, NOR, NOT and XOR. d) Some counters have a roll-over output that goes HI when the counter output is. Such a feature is very useful because it can be employed as a clock input to another decade counter to form a decade ripple counter. How would you implement such a rollover output circuit that goes HI whenever the counter is? Draw the roll-over output circuit using standard or inputs logic gates such as AND, NAND, OR, NOR, NOT and XOR. Roll Overout = Q Q Q Q Problem 4) (6 Points) a) Use a Venn diagram to display an XOR function. Specifically show Q, where Q = A B In your diagram, clearly indicate and label A, B and Q. A B = AB + AB, then shaded area is XOR.
Show that the two Boolean Expressions below are true: b) A + AB = A + B A + AB = A + AB = A ( + B) + AB = A + AB + AB = A + B( A + A) = A + B = A + B c) AB + AB = A B AB + AB = AB + AB = AB AB = ( A + B) ( A + B) = ( A + B) ( A + B) = AA + AB + BA + BB = AB + AB = ( AB + AB) = A B You will receive full points if you use Boolean logic to prove it. If you are unable to do so, you can use a Venn diagram to prove it but you will loose half a point for each proof. If you are still unable to do it with a Venn diagram then you may use a complete truth table but you will loose point for each proof. Problem 5) ( Points) Which 6 th century English writer and poet was the first to use the following Boolean expression (in a slightly different form) in one of his plays: B + B...? Have a Happy Halloween!