Understanding Decoders: Building Blocks for Digital Circuits | Papers Computer Architecture and Organization

September 22, 2003 Decoders 1

Decoders

•Next, we’ll look at some commonly used circuits: decoders and

multiplexers.

–These serve as examples of the circuit analysis and design

techniques from last lecture.

–They can be used to implement arbitrary functi ons.

–We are introduced to abstraction and modularity as hardware

design principles.

•Throughout the semester, we’ll often use decoders and multiplexers as

building blocks in designing more complex hardware.

September 22, 2003 Decoders 2

What is a decoder

•In older days, the (good) printers used be like typewriters:

–To print “A”, a wheel turned, brought the “A” k ey up, which then was

struck on the paper.

•Letters are encoded as 8 bit codes inside the co mputer.

–When the particular combination of bits that encodes “A” is

detected, we want to activate the output line corresponding to A

–(Not actually how the wheels worked)

•How to do this “detection” :

decoder

•General idea: given a k bit input,

–Detect which of the 2^k combinations is represented

–Produce 2^k outputs, only one of which is “1”.

September 22, 2003 Decoders 3

What a decoder does

•A n-to-2ndecoder takes an n-bit input and produces 2noutpu ts. The n

inputs represent a binary number that deter mines which of the 2n

outputs is

uniquely

true.

•A 2-to-4 decoder operates according t o the following truth table.

–The 2-bit input is called S1S0, and the four outputs are Q0-Q3.

–If the input is the binary number i, then output Q i is uniquely true.

•For instance, if the input S1 S0 = 10 (decimal 2), then output Q2 is

true, and Q0, Q1, Q3 are all false.

•This circuit “decodes” a binary number into a “ one-of-four” code.

S1 S0 Q0 Q1 Q2 Q3

00 1 0 0 0

010 10 0

100 0 1 0

11000 1

September 22, 2003 Decoders 4

How can you build a 2-to-4 decoder?

•Follow the design procedures from last tim e! We have a truth table, so

we can write equations for each of the four ou tputs (Q0-Q3), based on

the two inputs (S0-S1).

•In this case there’s not much to be simplified. Here are the equations:

S1 S0 Q0 Q1 Q2 Q3

00 1 0 0 0

010 10 0

100 0 1 0

11000 1

Q0 = S1’ S0’

Q1 = S1’ S0

Q2 = S1 S0’

Q3 = S1 S0

September 22, 2003 Decoders 5

A picture of a 2-to-4 decoder

S1 S0 Q0 Q1 Q2 Q3

00 1 0 0 0

010100

1000 1 0

11000 1

September 22, 2003 Decoders 6

Enable inputs

•Many devices have an additional enable input, which is used to “activate”

or “deactivate” the device.

•For a decoder,

–EN=1 activates the decoder, so it behaves as specified earlier.

Exactly one of the outputs will be 1.

–EN=0 “deactivates” the decoder. By conven tion, that means

all

the decoder’s outputs are 0.

•We can include this additional input in the decoder’s truth table:

EN S1 S0 Q0 Q1 Q2 Q3

0000 000

001 0 000

0100000

0110 000

1001000

1010100

1100010

1110001

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September 22, 2003 Decoders 1

Decoders

• Next, we’ll look at some commonly used circuits: decoders and

multiplexers.

– These serve as examples of the circuit analysis and design

techniques from last lecture.

– They can be used to implement arbitrary functions.

– We are introduced to abstraction and modularity as hardware

design principles.

• Throughout the semester, we’ll often use decoders and multiplexers as

building blocks in designing more complex hardware.

September 22, 2003 Decoders 2

What is a decoder

• In older days, the (good) printers used be like typewriters:

– To print “A”, a wheel turned, brought the “A” key up, which then was

struck on the paper.

• Letters are encoded as 8 bit codes inside the computer.

– When the particular combination of bits that encodes “A” is

detected, we want to activate the output line corresponding to A

– (Not actually how the wheels worked)

• How to do this “detection” :decoder

• General idea: given a k bit input,

– Detect which of the 2^k combinations is represented

– Produce 2^k outputs, only one of which is “1”.

September 22, 2003 Decoders 3

What a decoder does

• A n-to-2n^ decoder takes an n-bit input and produces 2 n^ outputs. The n

inputs represent a binary number that determines which of the 2n

outputs isuniquely true.

• A 2-to-4 decoder operates according to the following truth table.

– The 2-bit input is called S1S0, and the four outputs are Q0-Q3.

– If the input is the binary number i, then output Qi is uniquely true.

• For instance, if the input S1 S0 = 10 (decimal 2), then output Q2 is

true, and Q0, Q1, Q3 are all false.

• This circuit “decodes” a binary number into a “one-of-four” code.

S1 S0 Q0 Q1 Q2 Q

September 22, 2003 Decoders 4

How can you build a 2-to-4 decoder?

• Follow the design procedures from last time! We have a truth table, so

we can write equations for each of the four outputs (Q0-Q3), based on

the two inputs (S0-S1).

• In this case there’s not much to be simplified. Here are the equations:

S1 S0 Q0 Q1 Q2 Q

Q0 = S1’ S0’

Q1 = S1’ S

Q2 = S1 S0’

Q3 = S1 S

A picture of a 2-to-4 decoder

S1 S0 Q0 Q1 Q2 Q

Enable inputs

• Many devices have an additional enable input, which is used to “activate”

or “deactivate” the device.

• For a decoder,

– EN=1 activates the decoder, so it behaves as specified earlier.

Exactly one of the outputs will be 1.

– EN=0 “deactivates” the decoder. By convention, that meansall of

the decoder’s outputs are 0.

• We can include this additional input in the decoder’s truth table:

EN S1 S0 Q0 Q1 Q2 Q

September 22, 2003 Decoders 7

An aside: abbreviated truth tables

• In this table, note that whenever

EN=0, the outputs are always 0,

regardless of inputs S1 and S0.

• We can abbreviate the table by

writing x’s in the input columns

for S1 and S0.

EN S1 S0 Q0 Q1 Q2 Q

0 x x 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 1

September 22, 2003 Decoders 8

• Decoders are common enough that we want to encapsulate them and

treat them as an individual entity.

• Block diagrams for 2-to-4 decoders are shown here. Thenames of the

inputs and outputs, not their order, is what matters.

• A decoder block provides abstraction:

– You can use the decoder as long as you know its truth table or

equations, without knowing exactly what’s inside.

– It makes diagrams simpler by hiding the internal circuitry.

– It simplifies hardware reuse. You don’t have to keep rebuilding the

decoder from scratch every time you need it.

• These blocks are like functions in programming!

Blocks and abstraction

Q0 = S1’ S0’

Q1 = S1’ S

Q2 = S1 S0’

Q3 = S1 S

September 22, 2003 Decoders 9

A 3-to-8 decoder

• Larger decoders are similar. Here is a 3-to-8 decoder.

– The block symbol is on the right.

– A truth table (without EN) is below.

– Output equations are at the bottom right.

• Again, only one output is true for any input combination.

S2 S1 S0 Q0 Q1 Q2 Q3 Q4 Q5 Q6 Q

Q0 = S2’ S1’ S0’

Q1 = S2’ S1’ S

Q2 = S2’ S1 S0’

Q3 = S2’ S1 S

Q4 = S2 S1’ S0’

Q5 = S2 S1’ S

Q6 = S2 S1 S0’

Q7 = S2 S1 S

September 22, 2003 Decoders 10

Understanding Decoders: Building Blocks for Digital Circuits, Papers of Computer Architecture and Organization

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Partial preview of the text

Download Understanding Decoders: Building Blocks for Digital Circuits and more Papers Computer Architecture and Organization in PDF only on Docsity!

Decoders

• Next, we’ll look at some commonly used circuits: decoders and

multiplexers.

– These serve as examples of the circuit analysis and design

techniques from last lecture.

– They can be used to implement arbitrary functions.

– We are introduced to abstraction and modularity as hardware

design principles.

• Throughout the semester, we’ll often use decoders and multiplexers as

building blocks in designing more complex hardware.

What is a decoder

• In older days, the (good) printers used be like typewriters:

– To print “A”, a wheel turned, brought the “A” key up, which then was

struck on the paper.

• Letters are encoded as 8 bit codes inside the computer.

– When the particular combination of bits that encodes “A” is

detected, we want to activate the output line corresponding to A

– (Not actually how the wheels worked)

• How to do this “detection” :decoder

• General idea: given a k bit input,

– Detect which of the 2^k combinations is represented

– Produce 2^k outputs, only one of which is “1”.

What a decoder does

• A n-to-2n^ decoder takes an n-bit input and produces 2 n^ outputs. The n

inputs represent a binary number that determines which of the 2n

outputs isuniquely true.

• A 2-to-4 decoder operates according to the following truth table.

– The 2-bit input is called S1S0, and the four outputs are Q0-Q3.

– If the input is the binary number i, then output Qi is uniquely true.

• For instance, if the input S1 S0 = 10 (decimal 2), then output Q2 is

true, and Q0, Q1, Q3 are all false.

• This circuit “decodes” a binary number into a “one-of-four” code.

S1 S0 Q0 Q1 Q2 Q

How can you build a 2-to-4 decoder?

• Follow the design procedures from last time! We have a truth table, so

we can write equations for each of the four outputs (Q0-Q3), based on

the two inputs (S0-S1).

• In this case there’s not much to be simplified. Here are the equations:

S1 S0 Q0 Q1 Q2 Q

Q0 = S1’ S0’

Q1 = S1’ S

Q2 = S1 S0’

Q3 = S1 S

A picture of a 2-to-4 decoder

S1 S0 Q0 Q1 Q2 Q

Enable inputs

• Many devices have an additional enable input, which is used to “activate”

or “deactivate” the device.

• For a decoder,

– EN=1 activates the decoder, so it behaves as specified earlier.

Exactly one of the outputs will be 1.

– EN=0 “deactivates” the decoder. By convention, that meansall of

the decoder’s outputs are 0.

• We can include this additional input in the decoder’s truth table:

EN S1 S0 Q0 Q1 Q2 Q

An aside: abbreviated truth tables

• In this table, note that whenever

EN=0, the outputs are always 0,

regardless of inputs S1 and S0.

• We can abbreviate the table by

writing x’s in the input columns

for S1 and S0.

EN S1 S0 Q0 Q1 Q2 Q

EN S1 S0 Q0 Q1 Q2 Q

• Decoders are common enough that we want to encapsulate them and

treat them as an individual entity.

• Block diagrams for 2-to-4 decoders are shown here. Thenames of the

inputs and outputs, not their order, is what matters.

• A decoder block provides abstraction:

– You can use the decoder as long as you know its truth table or

equations, without knowing exactly what’s inside.

– It makes diagrams simpler by hiding the internal circuitry.

– It simplifies hardware reuse. You don’t have to keep rebuilding the

decoder from scratch every time you need it.

• These blocks are like functions in programming!

Blocks and abstraction