Authors: Mircea R. Stan, Alexandre F. Tenca and Milos D. Ercegovac

1. Long and Fast Up/Down Counters Authors:Mircea R. Stan, Alexandre F. Tenca and Milos D. Ercegovac Reader: Pushpinder Kaur Chouhan

2. Long and Fast Up/Down Counters • Introduction • Basic Counters • Counters Classification • Prescaled Counters • Constant-Time Up/down Counters • Alternative Design • Conclusion • References

3. Long and Fast Up/Down Counters • Introduction • Definition of counter • What is the article about • Basic Counters • Counters Classification • Prescaled Counters • Constant-Time Up/down Counters • Alternative Design • Conclusion • References

4. Introduction • Counter is a special type of adder in which one operand is kept constant. • Mod number – number of states 2n in a counter (n is number of flip-flops) • Generally counter can be written as Binary modulo 2n n-bit counter, where the value s(t) of the counter is incremented by one in each clock cycle: s(t+1) = s(t) mod 2n In Out Up/Down Load CNT Reset CLK TC N N Black-box generic counter model with the most common control signals

5. Introduction • Problem- When the up/down counter changes direction from counting up to counting down or vise versa. • Solution- • use Shadow register for storing the previous counter state. • restore the previous state in constant time, by storing the carry bits in a Carry/Borrow register.

6. Long and Fast Up/Down Counters • Introduction • Basic Counters • Asynchronous Counter • Synchronous Counter • Synchronous Decade Counter • Ring Counter • Twisted-ring Counter • Counters Classification • Prescaled Counters • Constant-Time Up/down Counters • Alternative Design • Conclusion • References

7. Basic Counters Asynchronous Counter –

8. Basic Counters Asynchronous up/down Counter – Sequence Table -

9. Basic Counters Synchronous Counter – Synchronous up/down Counter –

10. Basic Counters Asynchronous Decade Counter –

11. Basic Counters D Q3 CLK Q3 D Q2 CLK Q2 D Q1 CLK Q1 D Q0 CLK Q0 Ring Counter – Q3 Q2 Q1 Q0 1000 0100 0010 0001

12. Basic Counters D Q2 CLK Q2 D Q1 CLK Q1 D Q0 CLK Q0 Twisted-ring Counter – Q2 Q1 Q0 000 100 110 111 011 001

13. Long and Fast Up/Down Counters • Introduction • Basic Counters • Counters Classification • Based on initialization of the state • Based on counting sequence • Based on the way of supplying the clock signal • Based on the state diagram • Based on the number of states • Based on the state encoding • Prescaled Counters • Constant-Time Up/down Counters • Alternative Design • Conclusion • References

14. Counters Classification • Based on initialization of the state – • Noninitalizable • Resettable • Loadable • Based on counting sequence – • Up-only • Down-only • Up/down • Based on the way of supplying the clock signal – • Asynchronous • Semisynchronous • Synchronous • Based on the state diagram – • Periodic • Aperiodic

15. Counters Classification • Based on initialization of the state – • Based on counting sequence – • Based on the way of supplying the clock signal – • Based on the state diagram – • Based on the number of states – • Modulo-2N • Modulo-P • Based on the state encoding – • Binary • Quasi-Binary • Non-Binary

16. Long and Fast Up/Down Counters • Introduction • Basic Counters • Counters Classification • Prescaled Counters • Characteristics • Basic idea • Block Diagram • Partitioning • Constant-Time Up/down Counters • Alternative Design • Conclusion

17. Prescaled Counter • Characteristics – • Binary counting sequence • Clock period independent of counter size • Readable on the fly • Space complexity linear in the number of bits i.e., (O(N)) • Count up, down, or up/down • Resettable

18. Prescaled Counter • Basic Idea – Characteristic of binary number system. i.e., The higher order bits are stable for long periods of time and the terminal count (TC) output from the two least significant bits, which becomes a CARRY-in into the most significant block, is periodic with a lower frequency than the clock signal. Binary Sequence Counting Up

19. Prescaled Counter Prescaling long counters requires partitioning them into a series of sub-blocks of increasing sizes in order to take advantage of the reduced frequency required by high order bits.

20. Prescaled Counter • Partitioning methods - • Top-down manner – First determine the size of most significant block, which is chosen as large as possible and then recursively determine the size of the lower order blocks. N-bit counter is first partitioned into an (N - log2N ) most significant block and into another log2N block. Eg:- 64-bit counter is partitioned in block sizes 58, 3, 2, 1 and 128-bit counter in 121, 4, 2, 1. • Bottom-up manner -First determine the size of least significant block, then choose the second block as large as possible and then the third and so on. Least significant block with n0 = 1-bit the second block with n1 = 2 to the power n0 = 2-bits and the third block with n2 = 2 the power (n0 + n1) = 8 bits and so on.Eg:- 64-bit counter is partitioned in block sizes 51, 8, 2, 1 and 128-bit counter in 117, 8, 2, 1.

21. Long and Fast Up/Down Counters • Introduction • Basic Counters • Counters Classification • Prescaled Counters • Constant-Time Up/down Counters • Main idea • Block Diagram • Least-Significant Bit Counter • Configuration Bit • Clock Period • Up/Down Ring Counter • Partitioning • Incrementer/Decrementer • Alternative Design • Conclusion

22. Constant-Time Up/down Counters • Main idea - Store the previous block value in the Shadow register whenever the block is loaded with a new value and simply load this value when necessary, instead of trying to compute it. • Issues that determine the structure of the new counter • The prescaled TC generation must itself be up/down, for that up/down ring counter is used. • Each block needs to be configurable for counting either up or down. A separate configuration bit for each block is needed to keep track of the block configuration. • Each sub-block has a shadow register that stores the previous block value. When the block configuration is “up”, the shadow register stores the present value minus one LSB and when the configuration is “down”, it stores the present value plus one LSB.

23. Constant-Time Up/down Counters

24. Partitioning Determining the partition sizes for the proposed up/down counter proceeds top-down, the minimum clock period (Tclk) is larger than the combinational unit delay due to the extra complexity. If we consider Tclk = pd, where d is the unit delay, the partitioning first divides the N-bit counter into a most significant N - (log2 N/P) block and into another (log2 N/P) block which is recursively divided in the same manner until the smaller block is a 1-bit counter. Example: For p = 4 and N = 64, the partitioning leads to the sizes: 60, 3, 1.

25. Up/Down Ring Counter • The ring counter inside each block is used to generate the TC in constant time for the block. • When counting up, TC=1 when the state of the enable counter is s(t)=1000…0 • When counting down, TC=1 when the state of the enable counter is s(t)=0000…0

26. Incrementer/decrementer Incrementer/decrementer can be easily implemented as a ripple carry chain. For an n-bit block, the delay through the ripple chain will be n times the unit logic delay.

27. Least-Significant Bit Counter • A 1- bit counter counts in the same sequence, so it does not effect whether it count up or down. • It act as both 1-bit least significant bit and as a ring counter for the second block. • There is no need for a shadow register or configuration bit for the first block.

28. Configuration Bit A configuration bit for each higher order block keeps track of how the block is configured (up/down). There are four cases: • Block is configured up and a carry-in comes from ring counter. • Block is configured down and a borrow-in comes from ring counter. • Block is configured up and a borrow-in comes from ring counter. • Block is configured down and a carry-in comes from ring counter.

29. Example- 64bit ( 60,3,1)

30. Constant-Time Up/down Counters

31. Long and Fast Up/Down Counters • Introduction • Basic Counters • Counters Classification • Prescaled Counters • Constant-Time Up/down Counters • Alternative Design • Use of Carry/Borrow Register • Conclusion • Comment • References

32. Alternative Design • Store the bit-wise XOR between the previous state and the current state in a Carry/Borrow register. With this information available, it is possible to restore the desired previous state in one gate delay.

33. Long and Fast Up/Down Counters • Introduction • Basic Counters • Counters Classification • Prescaled Counters • Constant-Time Up/down Counters • Alternative Design • Conclusion • Comment • References

34. Conclusion • Use Shadow register to store the previous value of the counter and just swap the value when counter change the direction. • Use the Carry/Borrow register to store the bit-wise XOR between the previous state and current state. • Use of shadow register is better than the carry/borrow register, as CBreg restore the value in one gate delay.

35. Long and Fast Up/Down Counters • Introduction • Basic Counters • Counters Classification • Prescaled Counters • Constant-Time Up/down Counters • Alternative Design • Conclusion • Comment • References

36. Comment • Use twisted-ring counter instead of ring counter in constant up/down counter with shadow register will be fast and may use less clock period. • Mod-N Twisted-ring counter can be constructed by connecting N/2 flip-flops, where as in Mod-N ring counter N flip-flops are required. • We can use Bottom-up manner partitioning, as in top-down manner partitioning the counters of different sizes will require different partition sizes. • The state bits in the twisted-ring counter are such that the s(t)=(100…0) state can be detected by the two most significant bits and the s(t)=(000…0) state can be detected by testing the most and least significant bits.

37. Up/Down Twisted-ring Counter 000 100 110 111 011 001

38. Long and Fast Up/Down Counters • Introduction • Basic Counters • Counters Classification • Prescaled Counters • Constant-Time Up/down Counters • Alternative Design • Conclusion • Comment • References

39. References • Digital Systems Principles and Application By Ronald J. Tocci. • Computer Arithmetic Algorithms and Hardware Designs By Behrooz Parhami • http://www.play-hookey.com/digital/synchronous counter.html • http://www.eelab.usyd.edu.au/digital_tutorial/part2/counter01.html • http://www.hcc.hawaii.edu/~richardi/113/index.htm

40. Questions ?

41. Basic Counters Synchronous Decade Counter – • Q0 toggles on each clock pulse. • Q1 changes on the next clock pulse each time Q0=1 and Q3=0. • Q2 changes on the next clock pulse each time Q0=Q1=1. • Q3 changes on the next clock pulse each time Q0=1, Q1=1 and Q2=1 (count 7), or when Q0=1 and Q3=1 (count 9).

42. Basic Counters Differential Counter - ∆ = k (d1-d2)