Duplicate Error Detection Problem The sender sends an 8-bit word { x 1 , x 2 , …

Question

Duplicate Error Detection Problem

The sender sends an 8-bit word { x1, x2, … , x8 } in duplicate. The receiver receives the following two words meant to be identical copies of the word sent. Namely the receiver receives Y = { y1, y2, … , y8 } and Z = { z1, z2, … , z8 } . The receiver creates two parity calculator circuits. The first one, PC1, receives 8 input bits, namely the word Y, the second one, PC2, receives 8 input bits, namely the word Z. The output of PC1, is py , the parity of the word Y and the output of PC2, is pz , the parity of the word Z. That is:

py = y1 y2 y3 y4 y5 y6 y7 y8

pz = z1 z2 z3 z4 z5 z6 z7 z8

Then the receiver creates a final circuit for each pair of input bits of the Y and Z words. Namely for example one of these eight final circuits takes as input bits y1 , py , z1 , pz . The output bit w1 is to actually equal the original bit sent x1 .

Using the Boolean operators, +, ·, ~, and create the output expression w1 involving the four input bits y1 , py , z1 , pz .  

w1 = _____________________________

As partial support for your answer consider the following example.

Y = 1 1 0 1 1 10 0            Z= 1 1 0 0 1 1 0 0  

Calculate py = ___________________ calculate pz = _________________

In this example since y1 = 1 and z1 = 0 , then using your expression for w1 above

the value of w1 = _________________________________ which must have been the original value of x1 .

Explanation / Answer

Using the Boolean operators, +, ·, ~, and created the output expression w1 involving the four input bits y1 , Py , z1 , Pz .

SHOW ALL WORK

w1 = y1 Py Z1 Pz

answer consider the following example.

Y = 1 1 0 1 1 1 0 0 1 Z = 1 1 0 0 1 1 0 0 1

Calculate Py = 0 calculate Pz = 1

In this example since y1 = 1 and z1 = 1 , then using your expression for w1 above

the value of w1 = 1 which must have been the original value of x1 .

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