Chapter 2 Notes

Example: Suppose we are to design a circuit to control a single LIGHT from two switches, A and B. Whenever a single switch is UP the light should be ON, otherwise the LIGHT should be OFF. The state of the LIGHT is a function of the switches A and B.

A high-level specification of the system is:

    Input:        A an element of {UP, DOWN}, B an element of {UP, DOWN}
    Output:     LIGHT an element of {ON, OFF}

    Function:   Switch is described by the following table:
 

A, B	DOWN, DOWN	DOWN, UP	UP, DOWN	UP, UP
LIGHT = Switch(A,B)	OFF	ON	ON	OFF

One possible binary encoding for A and B, and decoding of LIGHT is:
 

A UP DOWN Ab=Encode(A) 1 0

B UP DOWN Bb=Encode(B) 1 0

LIGHTb 0 1 LIGHT=Decode(LIGHTb) OFF ON

The function Switch_b is obtained by replacing the high-level specification with binary values. The resulting binary specification is:

    Input:       A_b an element of {1, 0}, B_b an element of {1,0}
    Output:     LIGHT_b an element of {0,1}

    Function:   Switch_b is described by the following table:
 

AbBb	00	01	10	11
LIGHTb = Switchb(Ab,Bb)	0	1	1	0

Consists of three components, from the above example::
• Input set - A an element of {UP, DOWN}, B an element of {UP, DOWN}
• Output set - LIGHT an element of {ON, OFF}
• Input/Output function - LIGHT=Switch(A,B) or LIGHT_b = Switch_b(A_b,B_b)
Among other representations of the function are the following in table format.

     A    B   |LIGHT      A   B| LIGHT       Ab  Bb| LIGHTb
     Down Down| OFF       F   F|   F         0   0 | 0
     Down Up  | ON        F   T|   T         0   1 | 1
     Up   Down| ON        T   F|   T         1   0 | 1 
     Up   Up  | OFF       T   T|   F         1   1 | 0

Example: A 7-segment display is used to display the digits from 0-9 on calculators and many other displays. The 7-segment display appears as: where each segment is ON or OFF dependent upon the digit to be displayed. For example, to display the digit 0 would require segments a, b, c, d, e, and f to be ON with segment g OFF. To display the digit 2 would require segments a, b, g, e, and d to be ON and f and c segments OFF as in the above.

For segment a:

• Inputs - x an element of {0,1,2,3,4,5,6,7,8,9}
• Outputs - a an element of {ON, OFF}
• Function - a = ON when x = 0, 2, 3, 5, 6, 7, 8, 9 otherwise a = OFF

 
A tabular specification is:

 x | a=f(x) 
 0 | ON
 1 | OFF 
 2 | ON
 3 | ON
 4 | OFF
 5 | ON
 6 | ON
 7 | ON
 8 | ON
 9 | ON

Some standard representations are ASCII (Table 2.1, page 18), one's and two's complement, Gray code (Figure 2.6a, page 20), Binary Coded Decimal (Table 2.3, page 21), etc.

Example:    For the segment a example and most others studied in this course, the coding of input digits 0-9 will be in binary 0000₂-1001₂. Coding output values {ON, OFF} values to binary {1, 0}.

2.4 Binary Specification of Combinational Systems

Adding to the previous tabular the binary coding of input and output values gives the switching function:
 

 x  x3x2x1x0 | a=f(x) ab=f(x3,x2,x1,x0)
 0  0 0 0 0 | ON       1
 1  0 0 0 1 | OFF      0 
 2  0 0 1 0 | ON       1
 3  0 0 1 1 | ON       1
 4  0 1 0 0 | OFF      0
 5  0 1 0 1 | ON       1
 6  0 1 1 0 | ON       1
 7  0 1 1 1 | ON       1
 8  1 0 0 0 | ON       1
 9  1 0 0 1 | ON       1

f(x)=One-set(0,2,3,5,6,7,8,9)
f(x)=Zero set(1,4)

Incomplete Switching Functions -  Generally when some inputs that can be ignored are mapped to don't care outputs, that is the output could be any state. For the a segment only 0-9 values input values are valid, outputs for 10-15 are considered don't care. Adding that to the table yields:
 

 x  x3x2x1x0 | a=f(x) ab=f(x3,x2,x1,x0)
 0  0 0 0 0 | ON       1
 1  0 0 0 1 | OFF      0 
 2  0 0 1 0 | ON       1
 3  0 0 1 1 | ON       1
 4  0 1 0 0 | OFF      0
 5  0 1 0 1 | ON       1
 6  0 1 1 0 | ON       1
 7  0 1 1 1 | ON       1
 8  1 0 0 0 | ON       1
 9  1 0 0 1 | ON       1
10  1 0 1 0 |          -
11  1 0 1 1 |          -
12  1 1 0 0 |          -
13  1 1 0 1 |          -
14  1 1 1 0 |          -
15  1 1 1 1 |          -

The don't care set is dc(10,11,12,13,14,15)

The unsimplified, sum of products (Or is + or sum, And is product, Not is ') SE for the a LED segment is:

f(x₃,x₂,x₁,x₀) = x₃'x₂'x₁'x₀'+x₃'x₂'x₁x₀'+x₃'x₂'x₁x₀+x₃'x₂x₁'x₀+x₃'x₂x₁x₀'+x₃'x₂x₁x₀+x₃x₂'x₁'x₀'+x₃x₂'x₁'x₀

For the light switch table below, there are two equivalent switching expressions (LIGHT_b=Switch(A_b,B_b)) given as:

    Ab  Bb| LIGHTb=Switch(Ab,Bb)
    0   0 | 0
    0   1 | 1
    1   0 | 1 
    1   1 | 0

The gate representation for two corresponding switching expressions are:
 
 

Switch(Ab,Bb)= Ab'Bb+AbBb'

Switch(Ab,Bb)=Ab XOR Bb

Minterms - Shorthand representation. Where:

f(x₃,x₂,x₁,x₀) = x₃'x₂'x₁'x₀'+x₃'x₂'x₁x₀'+x₃'x₂'x₁x₀+x₃'x₂x₁'x₀+x₃'x₂x₁x₀'+x₃'x₂x₁x₀+x₃x₂'x₁'x₀'+x₃x₂'x₁'x₀
 
 

0-15 Minterm to Product Term Correspondence

m0 =  x3'x2'x1'x0'       m1 =  x3'x2'x1'x0 m2 =  x3'x2'x1x0' m3 =  x3'x2'x1x0 m4 =  x3'x2x1'x0' m5 =  x3'x2x1'x0 m6 =  x3'x2x1x0' m7 =  x3'x2x1x0

m8 =  x3x2'x1'x0' m9 =  x3x2'x1'x0 m10 =  x3x2'x1x0'  m11 =  x3x2'x1x0  m12 =  x3x2x1'x0'  m13 =  x3x2x1'x0  m14 =  x3x2x1x0'  m15 =  x3x2x1x0

f(x₃,x₂,x₁,x₀) = m₀ +  m₂ + m₃ + m₅ + m₆ + m₇ + m₈ + m₉ = Sm(0,2,3,5,6,7,8,9)

Maxterms - The unsimplified, product of sums (OR is + or sum, AND is product, NOT is ') Switching Expression for the a segment is:

f(x₃,x₂,x₁,x₀) = (x₃+x₂+x₁+x₀')(x₃+x₂'+x₁+x₀) = M₁*M₄ = PM(1,4)
 

0-15 Maxterm to Sum Term Correspondence

M0 =  x3+x2+x1+x0       M1 =  x3+x2+x1+x0'   M2 =  x3+x2+x1'+x0   M3 =  x3+x2+x1'+x0'   M4 =  x3+x2'+x1+x0   M5 =  x3+x2'+x1+x0'   M6 =  x3+x2'+x1'+x0   M7 =  x3+x2'+x1'+x0'

M8 =  x3'+x2+x1+x0   M9 =  x3'+x2+x1+x0'   M10 =  x3'+x2+x1'+x0   M11 =  x3'+x2+x1'+x0'   M12 =  x3'+x2'+x1+x0   M13 =  x3'+x2'+x1+x0'   M14 =  x3'+x2'+x1'+x0   M15 =  x3'+x2'+x1'+x0'

Don't cares - Output can be either state, 0 or 1, so can be chosen as best fits the design requirements. If all don't cares are chosen to be 1, then Sdc(10, 11, 12, 13, 14,15), if chosen to be 0, then  Pdc(10,11, 12, 13, 14, 15).

Equivalent - Minterm and maxterm representations are equivalent, so:
 

Sm(0,2,3,5,6,7,8,9) = PM(1,4)


For the LIGHT example the product and sum terms are:
 

Term A  B | LIGHT   Product  Sum
 0   0  0 | 0        A'B'    A+B
 1   0  1 | 1        A'B     A+B'
 2   1  0 | 1        AB'     A'+B
 3   1  1 | 0        AB      A'+B'
A'B+AB' = Sm(1,2) = PM(0,3) = (A+B)(A'+B')


For which a proof can be constructed by:

 
(A+B)(A'+B') = AA'+AB'+BA'+BB'
                      =  0    +AB'+A'B+0
                      = AB'+A'B

Using the simpler product of sums expression, the VHDL implementation of the module to light the a LED segment is:

ENTITY a_segment IS  PORT( x3, x2, x1, x0 : IN BIT;              a                     : OUT BIT); END a_segment; ARCHITECTURE behavioral of a_segment IS BEGIN   PROCESS (x3, x2, x1, x0)   BEGIN     a <= (x3 OR x2 OR x1 OR NOT x0) AND (x3 OR NOT x2 OR x1 OR x0);   END PROCESS; END behavioral;

• ENTITY - Serves the same function as a C++ function prototype, specifies the input and output signals of design in the PORT. BIT is the basic type for representing values of 0 or 1. Groups of bits can be represented using BIT_VECTOR, many other predefined and user defined data types are available.
• PORT - Defines the name, type, and number of inputs and outputs. Very much like a C++ function parameter definition.
• ARCHITECTURE - Defines the implementation details of module, much like the statements of a C++ function. Several levels of detail can be specified, the one given is often called the behavior or logical level since it specifies the behavior or logic of the design. A more detailed design would specify the physical devices used to implement the design.
• PROCESS - Remember in a combinational system the output is a function of the current value of the inputs. The PROCESS statement lists signals. When any of these signals change, the statements within the PROCESS block are executed. Unlike a C++ function which executes and returns, the module never returns or terminates, but recomputes the output for any input change. Being a combinational system, it produces an output that corresponds to its inputs. If the input changes, the output may also change. By listing in the PROCESS statement all signals that should be observed for changes, the output will be continually updated based upon all listed input signals. For example, if only PROCESS(x0) were used, the statements within the PROCESS block would be executed only when x0 changed, but the current values of all the signals would be used to compute the output. One important point of note is that statements within the PROCESS block are executed sequentially versus concurrently. For the PROCESS block
 

PROCESS (x, y, z)   BEGIN     a <= x AND y;     b <= a OR z;   END PROCESS;

signal a would be assigned first then signal b would be assigned.
• <= or signal assignment. In this example, can only be done to an output so x0 <= x1 would be erroneous.