Decoders (2 - 4, 3 - 8) — Hinglish notes

3.3.8 · Hardware › Combinational Circuits

The Core Idea: outputs = minterms

KYUN har output ek minterm hota hai. Variables $A, B$ ka minterm ek product term hota hai jo exactly ek combination ke liye $1$ hota hai. Jaise $\overset{ˉ}{A} B$ sirf tab $1$ hota hai jab $A = 0, B = 1$ . Kyunki hum chahte hain ki har output exactly ek input combination ke liye high ho, isliye har output ek minterm hi hota hai.

Scratch se derivation (2:4): Truth table likho. Hum chahte hain ki $D_{k} = 1$ sirf row $k$ mein ho.

$A_{1}$	$A_{0}$	$D_{3}$	$D_{2}$	$D_{1}$	$D_{0}$
0	0	0	0	0	1
0	1	0	0	1	0
1	0	0	1	0	0
1	1	1	0	0	0

Har column mein ek hi 1 hai → har output ek single minterm hai → seedha padh lo. K-map simplification ki zaroorat nahi; ek decoder literally "ek gate per minterm" hota hai.

Enable input (practically zaroori cheez)

Active-high enable ke saath, har output ban jaata hai $D_{k} = E \cdot (minterm)$ . Example: $D_{2} = E \cdot A_{1} \cdot \overline{A_{0}}$ .

3:8 Decoder

3:8 ko do 2:4 decoders + Enable se banana (Feynman-style construction):

MSB $A_{2}$ decide karta hai ki kaun sa 2:4 decoder active hoga.
$A_{1} A_{0}$ ko dono 2:4 decoders ko feed karo.
Top decoder $A_{2} = 0$ handle karta hai (outputs $D_{0} .. D_{3}$ ): ise $E = \overline{A_{2}}$ se enable karo.
Bottom decoder $A_{2} = 1$ handle karta hai (outputs $D_{4} .. D_{7}$ ): ise $E = A_{2}$ se enable karo.

Yeh kyun kaam karta hai: Kisi bhi input ke liye, exactly ek hi decoder enabled hoga (dono ek saath on nahi ho sakte kyunki $A_{2}$ aur $\overline{A_{2}}$ opposite hote hain). Jo enabled hoga woh apne chaar mein se ek line pick karega. Result: poore 8 mein one-hot. Yahi hierarchical decoder trick hai aur kisi bhi size tak generalize hoti hai.

Worked Examples

Common Mistakes

Recall Feynman: 12-saal ke bacche ko samjhao

Ek light switchboard socho jisme 4 lamps hain. Tumhare paas 4 switches nahi hain — tumhare paas 2 switches hain jo ek number "on/off" code (00,01,10,11) mein dikhaate hain. Decoder ek chhoti machine hai jo woh 2-switch number padhti hai aur sirf matching lamp jalati hai. Do switches → 4 lamps. Teen switches → 8 lamps. "Enable" button ek master power button hai: ise off karo aur saare lamps bujh jaate hain chahe tum koi bhi number set karo.

Flashcards

What does an

n : 2^{n}

decoder do?

n

-bit binary input ko one-hot output mein convert karta hai —

2^{n}

output lines mein se exactly ek activate karta hai (woh wali jo input ki decimal value ke equal ho).

Why is each decoder output a minterm?

Kyunki har output exactly ek input combination ke liye 1 hona chahiye, jo ki precisely minterm ki definition hai.

2:4 decoder: write

D_{2}

D_{2} = A_{1} \overline{A_{0}}

(fire karta hai jab

A_{1} A_{0} = 10

ho).

3:8 decoder: which output fires for input 101?

D_{5}

, kyunki

10 1_{2} = 5

; equals

A_{2} \overline{A_{1}} A_{0}

What is the role of the Enable input?

Jab disabled ho, saare outputs inactive hote hain; jab enabled ho decoder normally kaam karta hai. Expansion aur demux use bhi allow karta hai.

How do you build a 3:8 decoder from two 2:4 decoders?

A_{1} A_{0}

ko dono ko feed karo; top ko

\overline{A_{2}}

se enable karo (lines 0–3), bottom ko

A_{2}

se (lines 4–7).

Decoder vs encoder?

Decoder: binary→one-hot (few→many). Encoder: one-hot→binary (many→few). Yeh dono inverses hain.

How to implement

F = \sum m (1, 4, 7)

with a 3:8 decoder?

Outputs

D_{1}, D_{4}, D_{7}

ko OR karo.

A decoder with data on the Enable pin acts as what?

Ek 1-to-

2^{n}

demultiplexer (data bit ko addressed output par route karta hai).

For active-low outputs, what does the selected line look like?

Woh 0 ho jaati hai jabki baaki saari 1 rehti hain.

Connections

Minterms and Maxterms — decoder outputs hi minterms hote hain.
Multiplexers — dual/complementary block; MUX + decoder ek common pair hai.
Demultiplexers — ek decoder + enable = demux.
Encoders — inverse operation.
Memory Address Decoding — real-world use: RAM rows/chips select karna.
Sum of Products (SOP) — decoders kisi bhi SOP ko OR gate ke saath realize karte hain.
7-Segment Display Driver — numbers ko display segments mein decode karna.