RAM Module Design Calculation

This problem involves determining the number of smaller RAM modules needed to construct a larger RAM module with specific capacity and bit width requirements. We need to figure out how many 128 x 8 bit RAM units are required to build a 32K x 32 bit RAM.

Understanding RAM Specifications

Let's break down the specifications:

• Small RAM Module:
  • Number of words: 128
  • Bits per word: 8
  • Total capacity: $128 \times 8$ bits
• Target Large RAM Module:
  • Number of words: 32K (where K = 1024) = $32 \times 1024$ words
  • Bits per word: 32
  • Total capacity: $32K \times 32$ bits = $(32 \times 1024) \times 32$ bits

Calculating Total Capacity

First, let's calculate the total storage capacity in bits for both the small module and the target large module.

• Capacity of one small RAM module = $128 \times 8$ bits = 1024 bits.
• Capacity of the target large RAM module = $32K \times 32$ bits = $(32 \times 1024) \times 32$ bits = $32768 \times 32$ bits = 1,048,576 bits.

Determining the Number of Small RAMs

To find the number of small RAM modules needed, we divide the total capacity of the target large RAM by the capacity of a single small RAM module.

Number of small RAMs =
$ \frac{\text{Capacity of Large RAM}}{\text{Capacity of Small RAM}} $

Number of small RAMs = $ \frac{(32 \times 1024) \times 32 \text{ bits}}{128 \times 8 \text{ bits}} $

We know that $128 \times 8 = 1024$. So the expression becomes:

Number of small RAMs = $ \frac{32 \times 1024 \times 32}{1024} $

We can cancel out the 1024 from the numerator and denominator:

Number of small RAMs = $ 32 \times 32 $

Number of small RAMs = 1024

Alternative Calculation (Word and Bit Perspective)

We can also think about this in terms of how many smaller units are needed to meet the word count and the bit width requirements.

• Word Requirement: The target RAM needs 32K words ($32 \times 1024$ words). Each small RAM provides 128 words. Number of sets of 128 words needed = $ \frac{32 \times 1024}{128} = \frac{32768}{128} = 256 $ So, we need 256 groups, where each group can handle 128 words.
• Bit Width Requirement: The target RAM needs 32 bits per word. Each small RAM provides 8 bits per word. Number of 8-bit units needed for 32 bits = $ \frac{32}{8} = 4 $ So, for each word address, we need 4 modules connected in parallel to achieve the 32-bit width.
• Total Modules: To get the total number of 128x8 RAM modules, multiply the number of word groups by the number of parallel bit units. Total Modules = (Number of word groups) $\times$ (Number of bit units per word) Total Modules = $256 \times 4 = 1024$

Both methods confirm that 1024 modules of 128 x 8 bits are required.

Final Answer Summary

To design a 32K x 32 bit RAM using 128 x 8 bit RAM modules, you need 1024 such modules.

Calculation: $ \frac{1,048,576 \text{ bits}}{1024 \text{ bits/module}} = 1024 \text{ modules} $