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I am trying to solve the following question.

How many 3 to 8 decoders with an enable input are needed to construct a 6 to 64 line decoder without using any other logic gate.

I have a Digital Electronics textbook in which the author gives a simple cookbook method :

If n is the number of input lines in the available decoder and N is the number of input lines in the desired decoder, then the number of individual decoders required to construct the desired decoder circuit would be \$2^{N−n}\$.

With above explanation, the answer would be 8. Then I checked the answer & it was given as 9. I have only answer & no solution & even answer also may be given wrong.

After bit of struggling, I come up with this design. Let there be 8 decoders cascaded. The 3 inputs of each of decoder would serve as inputs for target decoder. And one more 3 x 8 decoder connected to each of these 8 decoders enable line. So that makes 9 decoders.

Here is my design:

enter image description here

I would like to know:

  1. Is my approach correct?
  2. Where did I do mistake in understanding from textbook?
  3. Most importantly, how to tackle problems like these. If inputs are in large number, it would be very difficult for me to come up with a design. Is there any standard formulae or any method?
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  • \$\begingroup\$ Please edit your question, hit Ctrl-M and draw a circuit diagram of your design. \$\endgroup\$ – jippie Apr 14 '13 at 10:15
  • \$\begingroup\$ Sure, please give me a minute. \$\endgroup\$ – avi Apr 14 '13 at 10:18
  • \$\begingroup\$ @jippie - I have added my design. I drawn by hand since I couldn't find decoder option in Circuit Labs. \$\endgroup\$ – avi Apr 14 '13 at 13:13
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  1. Your approach is correct.

  2. This is probably an error, or oversight, in the textbook: the authors forgot to allow for selecting the final decoder, which is what your 9th decoder does.

  3. There's not a standard formula. First, you have to stack up enough decoders to give you the number of output pins you need. Then, you have to "tree up" enough additional decoders to select the output-layer decoders.

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  • \$\begingroup\$ Thank you for confirming my design. Your last point is really good & more like a formulae to figure out. I have one more doubt. In question posted, it specifically says not to use any inverters. If I were to use inverters, how to design it ? Can you give me a hint ? (no solution needed) \$\endgroup\$ – avi Apr 15 '13 at 12:16
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    \$\begingroup\$ It would depend on the exact decoder(s) you used. Some of them will have more than one enable line, and you can then use inverters and extra gates to substitute for some of the "select tree" decoders. With the "no other devices" rule, you have to think about cascading the decoders, which is what the problem wanted you to do. \$\endgroup\$ – John R. Strohm Apr 18 '13 at 2:49

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