What is an Encoder?
An encoder is a combinational logic circuit that performs the inverse function of a decoder. It takes \(\mathsf{2^n}\) input lines, where only one input is active (logic high) at any given time—this is known as the one-hot format—and converts that active input into an n-bit binary code representing its position.
Clarifications
- “One-hot” means exactly one of the \(\mathsf{2^n}\) inputs is set to 1, while all others are 0.
- The encoder outputs the binary index of the active input line.
- For example, in a 4-to-2 encoder, if the third input line (corresponding to the index 2) is high, the output will be binary 10.
- An Encoder has:
- \(\mathsf{2^n}\) inputs presented as 0s except for 1 signal.
- \(\mathsf{n}\) outputs: exhibiting the binary equivalent index value.
Symbolic Representation of an Encoder
An encoder is typically represented by a vertical rectangular symbol, with the input lines shown on the left side and the output lines on the right. An encoder symbol may be annotated as \(\mathsf{2_n\times{}n}\) or \(\mathsf{2^n-\textsf{to}-n}\) to denote its role in compressing \(\mathsf{2^n}\) one-hot inputs into their corresponding \(\mathsf{n}\)-bit binary representation.
Truth Tables of an Encoder
Truth tables of \(\mathsf{4\times{}2}\) and \(\mathsf{8\times{}3,}\) are given here. It must be mentioned that only the allowed combinations are shown, otherwise 4-input function can have 16 unique combinations while those an 8-input function go as high as 256. The inputs and outputs are shown with MSB first (on the left-hand side). For clarity, the LSB has been marked with the index 0.
Truth Table of a \(\mathsf{4\times{}2}\) Encoder
Complete Truth Table of a \(\mathsf{4\times{}2}\) Encoder
In this complete truth table, the entries in pink show the forbidden input combinations and that the output against those inputs is not valid.
Truth Table of a \(\mathsf{8\times{}3}\) Encoder
Internal Circuit of a \(\mathsf{4\times{}2}\) Encoder
We can solve for the logic diagram using the truth tables or K-maps. However, with large number of inputs, it may become impractical to solve the k-map manually. A simpler workaround exists; from the truth table of an encoder, it’s evident that each unique one-hot input pattern corresponds to a distinct binary output. This means the encoder compresses positional information from \(\mathsf{2^n}\) inputs into an \(\mathsf{n}\)-bit binary code. Internally, the encoder uses OR gates to detect which input line is active and generate the corresponding binary representation. Each output bit is formed by logically combining specific input lines that contribute to that bit’s high state.
\[
\begin{align}
\mathsf{Z_0}&\mathsf{=A_1+A_3}\\
\mathsf{Z_1}&\mathsf{=A_2+A_3}
\end{align}
\]
Problems with a Regular Encoder
A standard encoder assumes that its inputs are in one-hot format, meaning exactly one of the \(\mathsf{2^n}\) inputs is active (logic high) at any given time. If this condition is violated—such as when multiple inputs are high simultaneously—the encoder may produce an incorrect or undefined binary output that does not correspond to the intended input index. To ensure reliable operation, the encoder must be constrained to receive strictly one-hot input patterns.
Why Priority Encoders Are Needed?
When multiple inputs may be active at once (i.e., the input is not guaranteed to be one-hot), a regular encoder fails to resolve ambiguity. This is where priority encoders come in:
- Priority Resolution: They assign precedence to higher-order inputs. If multiple inputs are high, the one with the highest priority (usually the highest index) determines the output.
- Additional Logic: Priority encoders include extra circuitry to detect and resolve conflicts, ensuring a deterministic output even under invalid input conditions.
- Valid Output Flag: Many priority encoders also include a “valid” output signal to indicate whether any input is active at all, which helps downstream logic avoid misinterpretation.
Symbolic Representation of a Priority Encoder
A priority encoder is an encoder, and hence, displayed in the same way. A Valid (V) signal, may be added to the outputs, if present.
Truth Tables of a Priority Encoder
Truth tables of \(\mathsf{4\times{}2}\) and \(\mathsf{8\times{}3,}\) priority encoders are given here. All combinations are shown by means of using \(\mathsf{X}\), don’t care, to encode priority. The inputs and outputs are shown with MSB first (on the left-hand side). For clarity, the LSB has been marked with the index 0. The Valid (\(\mathsf{V}\)) output is invalid (0) when all inputs are 0, and 1 otherwise.
Truth Table of a \(\mathsf{4\times{}2}\) Priority Encoder
The placement of X, don’t care, values in the table show that in this configuration, \(\mathsf{A_3}\) has the highest priority as if it is detected as 1, all the other input signals are ignored and the corresponding output is produced.
Truth Table of a \(\mathsf{8\times{}3}\) Priority Encoder
The placement of X, don’t care, values in the table show that in this configuration, \(\mathsf{A_7}\) has the highest priority as if it is detected as 1, all the other input signals are ignored and the corresponding output is produced.
Internal Circuit of a \(\mathsf{4\times{}2}\) Priority Encoder
Solving Output Functions \(\mathsf{Z_0,Z_1,}\) and \(\mathsf{V}\) Using a Condensed Truth Table
We aim to derive the output expressions for \(\mathsf{Z_0,Z_1,}\) and \(\mathsf{V}\) by analyzing the corresponding Karnaugh maps (K-maps), populated directly from a condensed truth table. Although the truth table is logically complete, it does not list all 16 rows explicitly for the 4-input combinations. Instead, it uses wildcards (X) to represent multiple input combinations compactly.
Interpreting the Condensed Truth Table
- Rows 1 & 2 represent the binary combinations 0000 (\(\mathsf{m_0}\)) and 0001 (\(\mathsf{m_1}\)).
- Row 3 contains an X in \(\mathsf{A_0}\), denoting two combinations: 0010 (\(\mathsf{m_2}\)) and 0011 (\(\mathsf{m_3}\)). These are highlighted in pink in both the truth table and K-map.
- Row 4 shows 01XX, representing four combinations: 0100 to 0111 (\(\mathsf{m_4}\) to \(\mathsf{m_7}\)), marked in blue.
- Row 5 uses 1XXX, covering eight combinations: 1000 to 1111 (\(\mathsf{m_8}\) to \(\mathsf{m_{15}}\)), shown in green.
Deriving Output \(\mathsf{Z_1}\)
- Two prime implicants are identified:
- \(\mathsf{A_3}\): Covers \(\mathsf{m_8}\) to \(\mathsf{m_{15}}\).
- \(\mathsf{A_2}\): Covers \(\mathsf{m_4}\) to \(\mathsf{m_7}\) and \(\mathsf{m_{12}}\) to \(\mathsf{m_{15}}\).
- The don’t-care condition at m₀ is treated as 0 to simplify the expression.
Final Expression:
\[\mathsf{Z_1=A_2+A_3}\]
Deriving Output \(\mathsf{Z_0}\)
- Again, two prime implicants emerge:
- \(\mathsf{A_3}\): Covers \(\mathsf{m_0}\) to \(\mathsf{m_{15}}\).
- \(\mathsf{A_1A_2′}\): Covers \(\mathsf{m_2,m_3,m_{10},m_{11}}\).
- The don’t-care at m₀ is treated as 0 for minimal form.
Final Expression:
\[\mathsf{Z_0=A_3+A_1A_2′}\]
Deriving Output V (Valid Output Indicator)
- The V output is essentially the logical OR of all inputs.
- Using the K-map, we identify four prime implicants, each grouping eight minterms:
- \(\mathsf{A_3}\): Covers \(\mathsf{m_{8}}\) to \(\mathsf{m_{15}}\)
- \(\mathsf{A_2}\): Covers \(\mathsf{m_4}\) to \(\mathsf{m_7}\) and \(\mathsf{m_{12}}\) to \(\mathsf{m_{15}}\)
- \(\mathsf{A_1}\): Covers \(\mathsf{m_2,m_3,m_6,m_7,m_{10},m_{11},m_{14},m_{14}}\)
- \(\mathsf{A_0}\): Covers \(\mathsf{m_1,m_3,m_5,m_7,m_9,m_{11},m_{13},m_{15}}\)
Final Expression:
\[\mathsf{V=A_3+A_2+A_1+A_0}\]
Final Internal Circuit of \(\mathsf{4\times{}2}\) Priority Encoder
