Understanding the Demultiplexer Implementation Problem

The question asks for the number of smaller $1 \text{-} \text{to} \text{-} 4$ demultiplexers needed to construct a larger $1 \text{-} \text{to} \text{-} 256$ demultiplexer. This involves understanding how demultiplexers function and how they can be combined in a hierarchical structure.

Demultiplexer Basics

A demultiplexer (DEMUX) is a fundamental digital logic circuit. It takes a single data input line and routes it to one of several output lines. The selection of the output line is determined by the signals applied to its control inputs, known as select lines.

• A $1 \text{-} \text{to} \text{-} N$ demultiplexer is characterized by having 1 data input, $k$ select lines, and $N$ output lines, where $N$ is related to $k$ by the equation $N = 2^k$.
• The specific demultiplexer we aim to implement is a $1 \text{-} \text{to} \text{-} 256$ type. To select one of the 256 possible outputs, it requires $k = \log_2(256) = 8$ select lines. Let's denote these lines as $S_7, S_6, S_5, S_4, S_3, S_2, S_1, S_0$, ordered from most significant to least significant.
• The building block available is a $1 \text{-} \text{to} \text{-} 4$ demultiplexer. This type requires $k = \log_2(4) = 2$ select lines and provides 4 distinct output lines.

Hierarchical Implementation Strategy

Implementing a larger demultiplexer using smaller ones is typically done using a hierarchical approach. Since the target $1 \text{-} \text{to} \text{-} 256$ demultiplexer requires 8 select lines and the available $1 \text{-} \text{to} \text{-} 4$ units use 2 select lines each, we can group the 8 select lines into 4 pairs: $(S_7, S_6)$, $(S_5, S_4)$, $(S_3, S_2)$, and $(S_1, S_0)$. Each pair will control a stage in the hierarchy.

Calculating the Number of Demultiplexers per Stage

We can determine the number of $1 \text{-} \text{to} \text{-} 4$ demultiplexers required at each stage of the hierarchy:

• Stage 1: Control by ($S_7, S_6$)

  These two select lines determine which of the four main blocks (or groups) of the final 256 outputs will be active. To achieve this selection, we need one $1 \text{-} \text{to} \text{-} 4$ demultiplexer whose select lines are $S_7$ and $S_6$. The single data input of this stage serves as the main data input for the entire $1 \text{-} \text{to} \text{-} 256$ demultiplexer.

  Number of units = $1$
• Stage 2: Control by ($S_5, S_4$)

  The output selected in Stage 1 needs to be further routed to one of four sub-paths. Since there was 1 active path from Stage 1, we need 4 separate $1 \text{-} \text{to} \text{-} 4$ demultiplexers for this stage. Each of these 4 units receives the data signal passed from Stage 1 and uses $S_5, S_4$ as its select lines to choose one of its 4 outputs.

  Number of units = $4$
• Stage 3: Control by ($S_3, S_2$)

  Similarly, the 4 active paths from Stage 2 must be expanded. Each of the 4 Stage 2 outputs controls a group of $1 \text{-} \text{to} \text{-} 4$ demultiplexers. Thus, we need $4 \times 4 = 16$ units of $1 \text{-} \text{to} \text{-} 4$ demultiplexers in this stage. Each uses $S_3, S_2$ as select lines.

  Number of units = $16$
• Stage 4: Control by ($S_1, S_0$)

  This is the final stage. The 16 active paths from Stage 3 need to be routed to the final 256 outputs. Each Stage 3 output controls a group of $1 \text{-} \text{to} \text{-} 4$ demultiplexers. Therefore, we require $16 \times 4 = 64$ units of $1 \text{-} \text{to} \text{-} 4$ demultiplexers. These units use the least significant select lines ($S_1, S_0$) and produce the final 256 distinct output paths.

  Number of units = $64$

Total Demultiplexer Count

To find the total number of $1 \text{-} \text{to} \text{-} 4$ demultiplexers needed for the entire $1 \text{-} \text{to} \text{-} 256$ implementation, we sum the number of units required at each stage:

$ \text{Total Units} = (\text{Units Stage 1}) + (\text{Units Stage 2}) + (\text{Units Stage 3}) + (\text{Units Stage 4}) $

$ \text{Total Units} = 1 + 4 + 16 + 64 $

$ \text{Total Units} = 85 $

Thus, 85 units of $1 \text{-} \text{to} \text{-} 4$ demultiplexers are required.