Muxes and Encoders
Shannon’s Expansion

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
e
x
f
0
0
z
0
1
z
1
0
0
1
1
1
e= 0
x
f
Equivalent circuits
e
e= 1
x
x
f
normal tri-state buffer symbol
f

Other Types of Tri-State Buffers
e
e
f
x
f
x
active high, not inverted
active high, inverted
output is enabled (connected) when e = 1
e
e
f
x
active low, not inverted
f
x
active low, inverted
output is enabled (connected) when e = 0

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
w0
s
w1
f

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)

s
x1
y1
x2
y2

2x2 Crossbar Switch Using MUXes

A 2x2 crossbar can be constructed from 2x1 MUXes
x1
0
1
y1
s
x2
0
1
y2

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(w1,w2,…,wn) can be written in
the form
f(w1,w2,…,wn) = w1' f(0,w2,…,wn) + w1 f(1,w2,…,wn)

w1 acts as the selector control input and it selects between
f(0,w2,…,wn) and f(1,w2,…,wn)

f(0,w2,…,wn) is called the cofactor of f with respect to w1'
f(1,w2,…,wn) is called the cofactor of f with respect to w1


Example: f(w1,w2,w3) = w1w2 + w1w3 + w2w3

w1' f(0,w2,w3) + w1 f(1,w2,w3) = w1' (w2w3) + w1 (w2 + w3)

MUXes Based on Shannon's Expansion

f(w1,w2,w3) = w1' (w2w3) + w1 (w2 + w3) is
implemented using a 2x1 MUX as follows …
w2
w3
w1
0
1
f

More Shannon's Expansion Examples

f(w1,w2,w3) = w1'w3' + w1w2 + w1w3

Expand on w1:


f(w1,w2,w3) = w1' f(0, w2,w3) + w1f(1,w2,w3) =
w1' (w3') + w1(w2 + w3)
Use a 2x1 MUX
0
1

Same Example ….

f(w1,w2,w3) = w1'w3' + w1w2 + w1w3

Expand on both w1 and w2:


f(w1,w2,w3) = w1'w2'f(0,0,w3) + w1'w2f(0,1,w3) +
w1w2'f(1,0,w3) + w1w2f(1,1,w3) =
w1'w2'(w3') + w1'w2(w3') + w1w2'(w3) + w1w2(1)
Use a 4x1 MUX

Implementing Using Only MUXes

Shannon's expansion can be used to implement
functions using MUXes exclusively

Example: f(w1,w2,w3) = w1w2 + w1w3 + w2w3

Use 2x1 MUXes only to implement this

Expanding on w1

f(w1,w2,w3) = w1'(w2w3) + w1(w2+w3+w2w3)

Let g = w2w3
Let h = w2+w3+w2w3


Example Continued …

f(w1,w2,w3) = w1'(g) + w1(h)




where g = w2w3
where h = w2+w3+w2w3
Expanding g on w2 gives
g = w2'(0) + w2(w3)
w2
w1
0
w3
Expanding h on w2 gives
h = w2'(w3) + w2(1)
f
1

Last Example: Use Only a 4x1 MUX

Try this one: f(w1,w2,w3) = m(3,5,6,7)

f = w1'w2w3 + w1w2'w3 + w1w2w3' + w1w2w3

Use a 4x1 MUX by expanding on both w1 and w2

f = w1'w2'(0) + w1'w2(w3) + w1w2'(w3) + w1w2(1)
w2
w1
0
w3
1
f

Encoders

The opposite of decoding is encoding

Encoder encodes one-asserted information



An 2n bit one-hot value is presented as input
A binary encoded output of size n is the result
Example: input is 01000000
 output is 110 (i.e. 6)
7 6 5 4 3 2 1 0
w0
y0
n
n
outputs
2
inputs
w
2n – 1
yn – 1

4 to 2 Encoder

A 4 to 2 encoder encodes a 4 bit one-hot input into a
2 bit binary encoded output
w3 w2 w1 w0 y1
y0
w0
0
0
0
1
0
0
w1
0
0
1
0
0
1
w2
0
1
0
0
1
0
w3
1
0
0
0
1
1
y1
y0

Building Encoders

Notice that since the input is guaranteed to be onehot, the design of an encoder is simple


y1 = w3 + w2
y0 = w3 + w1
w3 w2 w1 w0 y1
y0
0
0
0
1
0
0
0
0
1
0
0
1
0
1
0
0
1
0
1
0
0
0
1
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

Priority Encoders

What if you can’t guarantee, or don’t want to
guarantee, that the input is one-hot?

You can encode the input with the highest priority


Priority can be assigned in many ways
One priority scheme is to pick the input lines in order
according to position
7 6 5 4 3 2 1 0

Example: input is 01100100  output is 110 since
line 6 is “more significant” than line 5 or line 2

What if input is 00000000? Compare to 00000001?

4 to 2 Priority Encoder

4 input lines (w3:w0)

w3 has highest priority; w0 has lowest priority

2 output lines (y1:y0)

1 “valid” (z) output to determine if any line is active at
all
w3 w2 w1 w0 y1
y0
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
i0 = w3'w2'w1'w0
i1 = w3'w2'w1
i2 = w3'w2
i3 = w3
Then y0 = i1 + i3
y1 = i2 + i3
z = i0 + i1 + i2 + i3
w3 w2 w1 w0 y1
y0
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
same structure as one-hot encoder:
sum of terms in the on set

74147 Priority Encoder

All inputs are active low

All outputs are active low


0110 is "9"
1110 is "1"