Always remember the expression in terms of exponentials and the identity equations when dealing with trigonometric and hyperbolic functions!
The hyperbolic identity equation:
Coshx squre minus sinhx squre equals 1.
Coshx = 1/2 times the sum of e to the x and e to the negative x.
Sinhx = 1/2 times the result of e to the x minus e to the negative x.
The trigonometric identity equation:
Cosx squre plus sinx squre is equal to 1.
Sinx = 1/2 times the result of e to the power of ix take away e to the power of -ix.
Cosx = minus i/2 times the sum of e to the ix and e to the -ix.
I won't forget them hopefully.
Monday, 28 May 2007
Thursday, 24 May 2007
look-ahead carry
When we use parallel adders to do addition operations, the main problem would be that the propagation delay could become quite long when the number of bits is increasing. This is mainly because the result in the current position depends on the carry which has been calculated in the previous position. Although modern super deluxe CPUs seem really powerful when doing a simple addition, we gotta keep in mind that small delays would reault in astronomically big delays when doing complex calculations especially in electronics and computer science.
Fortunately engineers came up with the idea of look-ahead carry, which uses a certain combinational circuit to calculate all the carries within a constant time and then does the addition for each bit synchronously. Each carry could be represented in terms of the inputs of the least significant bit, so each carry could have a similar expression and these expresstions only need a two-level AND-OR gates to be implemented. Hence the time to be consumed to calculate all the carries will become the time which does the two-level basic AND-OR operations instead of taking a multiple full adder calculation time. As a result, the total time consumed has been reduced from a linearly increasing number to a constant number, which has increased the calculation speed dramatically.
This is what I learned from the subject digital systems, especially after I've done question 1 in assignment 8. The reason why I wanted to spend 15 minutes on writing this is that I have gained some wisdom from the idea of look-ahead carry. We know that the carry is calculated by the full adder for each bit, but the time to be used for a whole addition operation depends on the number of bits to be handled. If engineers had spent plenty of time only on how to modify the full adder to reduce the whole propagation deley, they wouldn't have got look-ahead carry. So we should try not to stick to the object which we are supposed to put emphasis on when tackling a certain problem. I reckon this is also the essence of brainstorming to some extent.
Fortunately engineers came up with the idea of look-ahead carry, which uses a certain combinational circuit to calculate all the carries within a constant time and then does the addition for each bit synchronously. Each carry could be represented in terms of the inputs of the least significant bit, so each carry could have a similar expression and these expresstions only need a two-level AND-OR gates to be implemented. Hence the time to be consumed to calculate all the carries will become the time which does the two-level basic AND-OR operations instead of taking a multiple full adder calculation time. As a result, the total time consumed has been reduced from a linearly increasing number to a constant number, which has increased the calculation speed dramatically.
This is what I learned from the subject digital systems, especially after I've done question 1 in assignment 8. The reason why I wanted to spend 15 minutes on writing this is that I have gained some wisdom from the idea of look-ahead carry. We know that the carry is calculated by the full adder for each bit, but the time to be used for a whole addition operation depends on the number of bits to be handled. If engineers had spent plenty of time only on how to modify the full adder to reduce the whole propagation deley, they wouldn't have got look-ahead carry. So we should try not to stick to the object which we are supposed to put emphasis on when tackling a certain problem. I reckon this is also the essence of brainstorming to some extent.
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