Electrical – Building a 4×16 decoder with only four 2×4 enabled decoder

A \$2\$-by-\$4\$ decoder has two input lines and four output lines, only one of which is logical \$1\$ at any time. Which line is \$1\$ depends on the input bit pair which can be \$00, 01, 10, 11\$.

So take two such \$2\$-by-\$4\$ decoders which give you four input lines. Let the output lines be \$a_0, a_1, a_2, a_3\$ for one decoder and \$b_0, b_1, b_2, b_3\$ for the other. Use the \$16\$ AND gates to compute the \$16\$ functions \$ a_i \wedge b_j, 0 \leq i \leq 3, 0 \leq j \leq 3\$. We now have a \$4\$-by-\$16\$ circuit with the property that only one output is a logical \$1\$ at any time: which one depends on the values of $i$ and $j$ which in turn depend on the \$4\$ input bits. In other words, we have a \$4\$-by-\$16\$ decoder constructed from two \$2\$-by-\$4\$ decoders and \$16\$ AND gates.

1. You have 64 memory elements, and you need them 4 wide (because 16b/4b=4). You therefore have 16 banks to switch between.

2. You have a 24-bit address space, and devices with 20 address bits. 24-20=4, and 2**4=16.