Muxes and Encoders - Digital System Design - Lecture Slides, Slides of Digital Systems Design

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Shannon’s Expansion

Muxes and Encoders

 Tri-State Buffers

 A tri-state buffer has one input x, one output f and

one control line e

 Z means high impedance, i.e. no output at all

x f

e

x f

e = 0

e = 1

x f

Equivalent circuits normal tri-state buffer symbol e x f 0 0 z 0 1 z 1 0 0 1 1 1

 2x1 MUX Using Tri-State Buffers

 Tri-state buffers can be used in place of SOP form

 Note – this is the one case where the output lines can just

be “tied together”

 The control line s selects which buffer will pass its input

f w 0 w 1 s

 Crossbar Switch

 A crossbar switch connects any input to any output

 A 2x2 crossbar connects 2 inputs to 2 outputs

 You can route each input straight thru, or cross them

 When s = 0, y1 = x1 and y2 = x2 (no switching)

 When s = 1, y1 = x2 and y2 = x1 (switched)

x 1 x 2 y 1 y 2 s

 Using MUXes for Synthesis

 MUXes are very significant combinational devices

 MUXes are a key component of FPGAs (field

programmable gate arrays) – to be discussed soon

 MUXes can be used to synthesize any combinational

logic function

 Makes use to Shannon's expansion

 Shannon's Expansion

 Any Boolean function f(w

,w

,…,w

n

) can be written in

the form

f(w

,w

,…,w

n

) = w

' f(0,w

,…,w

n

) + w

f(1,w

,…,w

n

 w

acts as the selector control input and it selects between

f(0,w

,…,w

n

) and f(1,w

,…,w

n

 f(0,w

,…,w

n

) is called the cofactor of f with respect to w

 f(1,w

,…,w

n

) is called the cofactor of f with respect to w

 Example: f(w

,w

,w

) = w

w

+ w

w

+ w

w

 w 1 ' f(0,w 2 ,w 3 ) + w 1 f(1,w 2 ,w 3 ) = w 1 ' (w 2 w 3 ) + w 1 (w 2 + w 3 )

 More Shannon's Expansion Examples

 f(w

,w

,w

) = w

'w

' + w

w

+ w

w

 Expand on w

 f(w 1 ,w 2 ,w 3 ) = w 1 ' f(0, w 2 ,w 3 ) + w 1 f(1,w 2 ,w 3 ) =

w 1 ' (w 3 ') + w 1 (w 2 + w 3 )

 Use a 2x1 MUX

0 1

 Same Example ….

 f(w

,w

,w

) = w

'w

' + w

w

+ w

w

 Expand on both w

and w

 f(w 1 ,w 2 ,w 3 ) = w 1 'w 2 'f(0,0,w 3 ) + w 1 'w 2 f(0,1,w 3 ) +

w 1 w 2 'f(1,0,w 3 ) + w 1 w 2 f(1,1,w 3 ) =

w 1 'w 2 '(w 3 ') + w 1 'w 2 (w 3 ') + w 1 w 2 '(w 3 ) + w 1 w 2 (1)

 Use a 4x1 MUX

 Example Continued …

 f(w

,w

,w

) = w

'(g) + w

(h)

 where g = w 2 w 3

 where h = w 2 +w 3 +w 2 w 3

 Expanding g on w 2 gives

g = w

'(0) + w

(w

 Expanding h on w

gives

h = w

'(w

) + w

w 2 0 w 3 1 f w 1

 Last Example: Use Only a 4x1 MUX

 Try this one: f(w

,w

,w

) =  m (3,5,6,7)

 f = w

'w

w

+ w

w

'w

+ w

w

w

' + w

w

w

 Use a 4x1 MUX by expanding on both w

and w

 f = w 1 'w 2 '(0) + w 1 'w 2 (w 3 ) + w 1 w 2 '(w 3 ) + w 1 w 2 (1)

f w 1 0 w 2 1 w 3

 4 to 2 Encoder

 A 4 to 2 encoder encodes a 4 bit one-hot input into a

2 bit binary encoded output

w 3 w 2 w 1 w 0 y 1 y 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 y^ y 0 1 w 1 w 0 w 2 w 3

 Building Encoders

 Notice that since the input is guaranteed to be one-

hot, the design of an encoder is simple

 y 1 = w 3 + w 2

 y 0 = w 3 + w 1

 Each output signal y is just the sum of the input

signals that are asserted for the “on set” of y

 These encoder types are called binary encoders

w 3 w 2 w 1 w 0 y 1 y 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1

 4 to 2 Priority Encoder

 4 input lines (w

:w

 w

has highest priority; w

has lowest priority

 2 output lines (y

:y

 1 “valid” (z) output to determine if any line is active at

all

w 3 w 2 w 1 w 0 y 1 y 0 z 0 0 0 0 x x 0 0 0 0 1 0 0 1 0 0 1 x 0 1 1 0 1 x x 1 0 1 1 x x x 1 1 1

 Logic for Priority Encoders

 Let i

= w

'w

'w

'w

i 1 = w 3 'w 2 'w 1

i

= w

'w

i

= w

 Then y

= i

+ i

y

= i

+ i

 z = i

+ i

+ i

+ i

same structure as one-hot encoder: sum of terms in the on set w 3 w 2 w 1 w 0 y 1 y 0 z 0 0 0 0 x x 0 0 0 0 1 0 0 1 0 0 1 x 0 1 1 0 1 x x 1 0 1 1 x x x 1 1 1