Decoders, Encoders, Priority Encoders & Code Converters
“Active-low minterm generators, priority resolution truth tables, and BCD-to-seven-segment logic.”
1.Decoders and Priority Encoders in Hardware Architecture
§ 3.1.A Decoder is a combinational circuit that converts an n-bit binary input code into 2^n unique output lines. Exactly one output line is activated for each input combination.
Minterm Generator: Because each output corresponds to a specific minterm, any Boolean function expressed in canonical SOP form f = \sum m(1, 3, 7) can be synthesized simply by feeding the corresponding decoder outputs into an OR gate.
Active-Low Decoders: Most real TTL/CMOS decoders use active-low outputs (Y_i = 0 when selected, 1 otherwise). By De Morgan's laws: \sum m_i = (\prod m_i')' = (\prod Y_i)'. Therefore, connecting active-low decoder outputs to a NAND gate directly produces the OR of minterms!
Priority Encoders: Standard encoders assume exactly one input is active at any time. When multiple inputs are asserted simultaneously (as in hardware interrupt controllers), a standard encoder produces corrupted outputs. A Priority Encoder guarantees that the output reflects the index of the highest-priority asserted request.
2. Worked Analysis: Logic Function Synthesis via Active-Low Decoder
Active-low outputs combined into a NOR or AND gate perform different dual functions.
Problem Statement (Modeled on GATE Pattern):
A 3-to-8 decoder has active-low outputs Y0' through Y7' where Y_k' = 0 iff input (A, B, C) represents binary k.
The outputs Y1', Y3', Y5', and Y7' are connected to the inputs of an active-low input NAND gate (which is equivalent to an OR gate).
What Boolean function f(A, B, C) is realized?
Step 1: Write output expression of the NAND gate:
f = (Y1' Y3' Y5' * Y7')'.
Step 2: Apply De Morgan's theorem:
f = (Y1')' + (Y3')' + (Y5')' + (Y7')'.
Step 3: Relate to minterms:
Since Y_k' = (m_k)', we have (Y_k')' = m_k.
Therefore, f = m1 + m3 + m5 + m7.
Step 4: Minimize the minterm sum:
m1 = A'B'C, m3 = A'BC, m5 = AB'C, m7 = ABC.
Factor out C: f = (A'B' + A'B + AB' + AB) C = 1 * C = C.
Result: The circuit computes the simple identity f(A, B, C) = C.
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