1. More on Decoders and Muxes
More on Decoders and Muxes
ECGR2181
Lecture Notes 2A
Logic System Design I

2. Making Larger Components
Consider destination then use component given to work toward solution.
Think about the functionality of the destination component
Think about what the given parts can do
Then, bring it all together
Finally, once design is done make a couple of test cases to see if the design is valid. If not, repeat from start.
Logic System Design I

3. i.e., construct a 16-input mux from any number of 4-input muxes
Making Large Muxes
i.e., construct a 16-input mux from any number of 4-input muxes
16-input Multiplexer i0 i2 i1 i3 i4 i6 i5 i7 i8
i10 i9
i11
i12
i14
i13
i15 S0 S2 S1 S3 Y
4-input Mux i0 i2 i1 i3 S0 S1 Y
4-input Mux i0 i2 i1 i3 S0 S1 Y
4-input Mux i0 i2 i1 i3 S0 S1 Y
???
4-input Mux i0 i2 i1 i3 S0 S1 Y
4-input Mux i0 i2 i1 i3 S0 S1 Y
Logic System Design I

4. Example Continued T.T. Goal: Logic System Design I S3 S2 S1 S0 Y Z? i0
4-input Mux i0 i2 i1 i3 S0 S1 Y S3 S2 S1 S0 Y Z? i0 1 i1 i2 i3 i4 i5 i6 i7 i8 i9
i10
i11
i12
i13
i14
i15 Z0 Z0 Z1 Z2 Z3 Z0 Z1 Z2 Z3 S0 S1 Z0 Z1 Z2 Z3 i4 i6 i5 i7
4-input Mux i0 i2 i1 i3 S0 S1 Y Z0 Z1 Z2 Z3 Z1 Z3 Z2 Z1 Z0
4-input Mux i0 i2 i1 i3 S0 S1 Y Y S2 S3 i8
i10 i9
i11
4-input Mux i0 i2 i1 i3 S0 S1 Y Z2
i12
i14
i13
i15
4-input Mux i0 i2 i1 i3 S0 S1 Y Z3
Logic System Design I

5. Signal Flow Illustrated
If S=7, then Y will equal the value on i7 S3 S2 S1 S0 Y i0 1 i1 i2 i3 i4 i5 i6 i7 i8 i9
i10
i11
i12
i13
i14
i15 i0 i2 i1 i3
4-input Mux i0 i2 i1 i3 S0 S1 Y Z0
= i3
= i11
= i15 S0 S1 i4 i6 i5 i7 i0 i2 i1 i3 S0 S1 Y
4-input Mux Z1 Z3 Z2 Z1 Z0
4-input Mux i0 i2 i1 i3 S0 S1 Y S0 S1 Y S2 S3 i8
i10 i9
i11
4-input Mux i0 i2 i1 i3 S0 S1 Y Z2 S0 S1
i12
i14
i13
i15
4-input Mux i0 i2 i1 i3 S0 S1 Y Z3 S0 S1
Logic System Design I

6. Signal Flow Illustrated, again
If S=9, then Y will equal the value on i9 S3 S2 S1 S0 Y i0 1 i1 i2 i3 i4 i5 i6 i7 i8 i9
i10
i11
i12
i13
i14
i15 i0 i2 i1 i3
4-input Mux i0 i2 i1 i3 S0 S1 Y Z0
= i1
= i5
= i13 S0 S1 i4 i6 i5 i7
4-input Mux i0 i2 i1 i3 S0 S1 Y Z1 Z3 Z1 Z0 Z2
4-input Mux i0 i2 i1 i3 S0 S1 Y S0 S1 Y S2 S3 i8
i10 i9
i11
4-input Mux i0 i2 i1 i3 S0 S1 Y Z2 S0 S1
i12
i14
i13
i15
4-input Mux i0 i2 i1 i3 S0 S1 Y Z3 S0 S1
Logic System Design I

7. Enable Lines T.T. Goal: Logic System Design I G S3 S2 S1 S0 Y x 1 i0x 1 i0 i1 i2 i3 i0 i2 i1 i3
4-input Mux i0 i2 i1 i3 S0 S1 Y G G S3 S2 S1 S0 Y x 1 i0 i1 i2 i3 i4 i5 i6 i7 i8 i9
i10
i11
i12
i13
i14
i15 Z0 G S0 S1 i4 i6 i5 i7
4-input Mux i0 i2 i1 i3 S0 S1 Y G Z1 Z3 Z2 Z1 Z0
4-input Mux i0 i2 i1 i3 S0 S1 Y G S0 S1 Y S2 S3 i8
i10 i9
i11
4-input Mux i0 i2 i1 i3 S0 S1 Y G Z2 S0 S1
i12
i14
i13
i15
4-input Mux i0 i2 i1 i3 S0 S1 Y G Z3 S0 S1
Logic System Design I

8. Enable Lines, Continued
16-input Mux, w/ 2 enable inputs {1 active-high (G0) & 1 active-low (G1)} i0 i2 i1 i3
4-input Mux i0 i2 i1 i3 S0 S1 Y G
For devices with multiple Enable inputs, all Enables must be asserted for the device operate as per it’s definition. G1 G0 S3 S2 S1 S0 Y x 1 i0 i1 i2 i3 i4 i5 i6 i7 i8 i9
i10
i11
i12
i13
i14
i15 Z0 S0 S1 i4 i6 i5 i7
4-input Mux i0 i2 i1 i3 S0 S1 Y G Z1 Z3 Z2 Z1 Z0
4-input Mux i0 i2 i1 i3 S0 S1 Y G S0 S1 Y
= 0 S2 S3 i8
i10 i9
i11
4-input Mux i0 i2 i1 i3 S0 S1 Y G Z2 S0 S1
i12
i14
i13
i15
4-input Mux i0 i2 i1 i3 S0 S1 Y G
Disabled State Z3 S0 S1
Enabled State
Logic System Design I

9. Enable Lines, Continued
16-input Mux, w/ 2 enable inputs {1 active-high (G0) & 1 active-low (G1)} i0 i2 i1 i3
4-input Mux i0 i2 i1 i3 S0 S1 Y G G1 G0 S3 S2 S1 S0 Y x 1 i0 i1 i2 i3 i4 i5 i6 i7 i8 i9
i10
i11
i12
i13
i14
i15 Z0 G0 S0 S1 i4 i6 i5 i7
4-input Mux i0 i2 i1 i3 S0 S1 Y G Z1 Z3 Z2 Z1 Z0
4-input Mux i0 i2 i1 i3 S0 S1 Y G S0 S1 Y G1 S2 S3 i8
i10 i9
i11
4-input Mux i0 i2 i1 i3 S0 S1 Y G Z2 S0 S1
i12
i14
i13
i15
4-input Mux i0 i2 i1 i3 S0 S1 Y G Z3 S0 S1
Logic System Design I

10. The New Mux
Here is a the TT & symbol for our shinny new 16-input Mux, with 2 enable lines G1 G0 S3 S2 S1 S0 Y x 1 i0 i1 i2 i3 i4 i5 i6 i7 i8 i9
i10
i11
i12
i13
i14
i15
16-input Multiplexer i0 i2 i1 i3 i4 i6 i5 i7 i8
i10 i9
i11
i12
i14
i13
i15 S0 S2 S1 S3 Y G0 G1
Logic System Design I

11. Mux Implementation of Combinational Logic
Make an inverter out of a 4-input Mux G1
G1’
Vcc G S1 S0 Y x 1 i0 i1 i2 i3
4-input Mux i0 i2 i1 i3 S0 S1 Y G G1
G1’
Start with the Mux, keep an eye on the TT.
Configure the inputs such that …
when G1 = ‘0’ the output will equal ‘1’
and, when G1 = ‘1’ the output will equal ‘0’
Enable the Mux always.
Never leave unused inputs unconnected.
Logic System Design I

12. Mux Implementation of Boolean Expression
More complex design … F(a,b,c)
For a 8-input mux the output can be described as
Y = i0(S2’S1’S0’) + i1(S2’S1’S0) + i2(S2’S1S0’) + i3(S2’S1S0) + … … + i4(S2S1’S0’) + i5(S2S1’S0) + i6(S2S1S0’) + i7(S2S1S0)
Connect the inputs of the function (a,b,c) to the select lines of the mux … with a connected to S3
Y = i0(a’b’c’) + i1(a’b’c) + i2(a’bc’) + i3(a’bc) + i4(ab’c’) + i5(ab’c) + i6(abc’) + i7(abc)
Continued 
Logic System Design I

13. Mux Implementation of Boolean Expression (cont.)
Connect the inputs to the mux (i7 – i0) to one or zero depending upon the value in the truth table
For F(a,b,c) = m(0,2,5,7) ... i0=1; i1=0; i2=1; i3=0; i4=0; i5=1; i6=0; i7=1
F(a,b,c) = 1(a’b’c’) + 0(a’b’c) + 1(a’bc’) + 0(a’bc) + 0(ab’c’) + 1(ab’c) + 0(abc’) + 1(abc)
F(a,b,c) = a’b’c’ + a’bc’ + ab’c + abc
F(a,b,c) = m0 + m2 + m5 + m7
Vcc
8-input Multiplexer i0 i2 i1 i3 i4 i6 i5 i7 S0 S2 S1 Y G0 G1
Begin by connecting the system inputs (a,b,c) to the select lines as described above
Connect the minterms where the function equals ‘1’ to Vcc F
Connect the remaining mux inputs to gnd
Connect the enable lines to the appropriate Vcc or gnd to always enable the mux c a b
Connect the output of the mux to the output of the system (F)
Logic System Design I

14. More Mux Implementation of Comb. Logic
Implement F(x,y,z) = Σm(3,4,6,7) using a 4-input mux
Vcc x y z F 1
4-input Mux i0 i2 i1 i3 S0 S1 Y G z z F x y
Connect the most significant two inputs to the select lines
Pairs of minterms (where x & y remain constant) are then considered
For xy = 00, the output F is independent of z … i0 should be connected to gnd.
For xy = 01, the output F is dependent on z … when z = 0, F = 0 and when z = 1 F = 1; thus i1 = z
For xy = 10, the output F is dependent on z … when z = 0, F = 1 and when z = 1 F = 0; thus i2 = z’
For xy = 11, the output F is independent of z … F = 1 for both cases of z … i3 should be connected to Vcc
Enable mux and connect the output of the mux to F
Logic System Design I

15. More on Decoders Recap:
When enabled: the input combination value (i) is the subscript of the output that is asserted. Otherwise, the output is zero. Where i is i1 & i0 concatenated.
When not enabled all outputs are zero.
Making larger decoders:
A decoder is used to select the appropriate output decoder G i1 i0 y3 y2 y1 y0 x 1
2x4 decoder y0 y2 y1 y3 G i0 i1
Logic System Design I

16. 3x8 Decoder using only 2x4 decoders
Expanding Decoders
3x8 Decoder using only 2x4 decoders G i2 i1 i0 y7 y6 y5 y4 y3 y2 y1 y0 x 1
2x4 decoder y0 y2 y1 y3 G i0 i1
???
3x8 decoder y0 y2 y1 y3 G i0 i1 i2 y4 y6 y5 y7
2x4 decoder y0 y2 y1 y3 G i0 i1
2x4 decoder y0 y2 y1 y3 G i0 i1
2x4 decoder y0 y2 y1 y3 G i0 i1
2x4 decoder y0 y2 y1 y3 G i0 i1
Logic System Design I

17. 3x8 Decoder Logic System Design I Connect the outputs as shownx 1
Connect the least significant inputs to the inputs of the output decoders
Each combination of i1 & i0 will produce two asserted outputs; thus, we need to enable the decoder with the desired output and disable the others …
Connect the first level decoder shown based on the remaining inputs. In this case, i2 of the new decoder When i2 = 0, the top decoder is enabled. When i2 = 1 the bottom decoder is enabled.
Connect the enable for the new decoder to the enable of the first level mux. i1 i0
2x4 decoder y0 y2 y1 y3 G i0 i1 y0 y1 y2 y3 i2
2x4 decoder y0 y2 y1 y3 G i0 i1 z0 z1 G i1 i0
2x4 decoder y0 y2 y1 y3 G i0 i1 y4 y5 y6 y7
NOTE: Unused inputs must connect to something … unused outputs are left hanging.
Logic System Design I

18. Decode binary word to 1-of-12 code (12 outputs)
4x12 Decoder (non-std.)
Decode binary word to 1-of-12 code (12 outputs) G
G0_L i3 i2 i1 i0
y11
y10 y9 y8 y7 y6 Y5 y4 y3 y2 y1 y0 x 1 X
Logic System Design I

19. 4-to-12 Decoder (non-std.)i1 i0
2x4 decoder y0 y2 y1 y3 G i0 i1 y0 y1 y2 y3 z0 i2
2x4 decoder y0 y2 y1 y3 G i0 i1 z0 i1 i0
2x4 decoder y0 y2 y1 y3 G i0 i1 y4 y5 y6 y7 i3 z1 z2
G0_L
2x4 decoder y0 y2 y1 y3 G i0 i1 G1
Vcc
Gint z1 i1 i0
2x4 decoder y0 y2 y1 y3 G i0 i1 y8 y9
y10
y11 z2
Logic System Design I

20. 4x16 Decoder with an active-high enable
4-to-16 Decoder
4x16 Decoder with an active-high enable i1 i0
2x4 decoder y0 y2 y1 y3 G i0 i1 y0 y1 y2 y3 z0
4x16 decoder y0 y2 y1 y3 G i0 i1 y4 y6 y5 y7 y8
y10 y9
y11
y12
y14
y13
y15 i2 i3 i1 i0
2x4 decoder y0 y2 y1 y3 G i0 i1 y4 y5 y6 y7 i2 i3
2x4 decoder y0 y2 y1 y3 G i0 i1 z0 z1 z2 z3 z1 G i1 i0
2x4 decoder y0 y2 y1 y3 G i0 i1 y8 y9
y10
y11 z2 i1 i0
2x4 decoder y0 y2 y1 y3 G i0 i1
Y12
y13
y14
y15 z3
Logic System Design I

21. Demultiplexer (demux)
A demux performs the reverse operation from the mux
The demux routes one input to 1-of-n outputs based on the select-line input combination
System description for 4 output demux (2 select-lines)
y0 = i·s1’·s0’
y1 = i·s1’·s0
y2 = i·s1·s0’
y3 = i·s1·s0
Recall from the decoder a description could be given as follows:
y0 = G·s1’·s0’
y1 = G·s1’·s0
y2 = G·s1·s0’
y3 = G·s1·s0
Logic System Design I

22. Demux (concluded)
You can see, from the previous expressions, that the demux could be implemented with a decoder that has an enable
Just connect I to G of the decoder and you have a demux.
Warning: Most CAD tools rely on the user knowing this fact … … the libraries do not contain any demuxes, just decoders.
2x4 decoder y0 y2 y1 y3 G i0 i1 i
2x4 decoder y0 y2 y1 y3 G i0 i1 1 i
2x4 decoder y0 y2 y1 y3 G i0 i1 s0 s1 i
2x4 decoder y0 y2 y1 y3 G i0 i1 1 i
2x4 decoder y0 y2 y1 y3 G i0 i1 1 i
Logic System Design I