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Implement Synthesizable Square Root Algorithm On Fpga Engineering Essay


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The main objective of this paper is to implement synthesizable square root algorithm on FPGA. As square root function is not synthesizable on Silicon, this paper proposes optimized non restoring square root algorithm for unsigned 8 bit number on ED2C20F484C7 device in Cyclone II family. This algorithm is implemented in gate level abstraction of Verilog HDL. The basic building block of the design is CSM (Controlled Subtract Multiplex) block. It makes use of only subtract operation and append 01 which is an improvement over restoring algorithm.

Keyword: FPGA,CSM,Verilog HDL,fixed point


The square root function is a basic operation in computer graphic and scientific calculation application. Due to its algorithm complexity, the square root operation is hard to be designed in digital system. As known, digital system has been used in daily life or industrial purpose that may have been in need of square root operation to fully execute its functions. Scientists have developed various algorithms for square root calculation. But the implementation of algorithms is difficult because of their complexities and thus results into long delays for its completion.

There are two main families of algorithms that can be used to extract square roots. The first family is that of digit recurrence, which provides one digit (often one bit) of the result at each iteration[6]. Each iteration consists of additions and digit-by-number multiplications (which have comparable cost)

Such algorithms have been widely used in microprocessors that didn't include hardware multipliers. Most of the FPGA implementations in vendor tools or in the literature use this approach. Second family of algorithms uses multiplications. It includes quadratic convergence recurrences derived from the Newton-Raphson iteration [5]. The digit recurrence approaches allow one to build minimal hardware, while multiplicative approaches allow one to make the best use of available resources when these include multipliers. Also there are estimation method and digit-by-digit method. Digit-by-digit method is classified into two distinct classes: restoring and non- restoring algorithm [1].

In restoring algorithm, remainder is restored in the regular flow. So its implementation needs more hardware. Compared to the restoring algorithm, the non restoring algorithm does not restore the remainder, which can be implemented with fewest hardware resource and the result is hardware simple implementation. It is most suitable for FPGA implementation.

Restoring and non restoring square root calculation

Restoring Algorithm

Step 1: If it is a 2n bit number then divide it in a group of 2 bits

Step2: Subtract 1 from the first 2 digits (starting from MSB)

Step3: Whenever the result of the subtraction is positive then the developed root is '1' otherwise '0'

Step4: Whenever the result is negative, write it as it is. We have to restore the wrong guess by appending

'01' and guessed square root.

Step5: Now take the next two digits

Step6: Append '01' (to be subtracted from next two digits of dividend) and guessed square root to

subtract from the remainder.

Step7: If the result of subtraction is negative then restore previous remainder by adding wrong guess by

appending '01' and guessed square root.

Step8: Every time guessed square root has to be updated while appending '01'.

Step9: Continue the steps until the group of two digits end

1 0 0 1.1 0 1 0

01 01 11 01.00 00 00 00

- 01

00 01 take next two digits from dividend

- 1 01 Append 01

Negative value 11 00

+ 1 01

0 0 01 11

-10 01

11 10 Negative value

+ 10 01

01 11 01

- 10 00 01

11 00 00

- 10 01 01

01 01 1 00

10011 01


+ 1001101

010110 00

- 010011001

000010111 00

- 100110101


Figure 1: The example of restoring algorithm to solve square root

B. Proposed Modified Non Restoring Algorithm

A little modification in non restoring algorithm makes calculation faster. It uses only subtract operation and appends "01". It uses n stage pipelining to find square root of 2n bit number. The following algorithm describes the modified non restoring square root algorithm.

Step1: Start

Step2: Initialize the radicand (p) which is 2n bit number. Divide the radicand in two bits beginning at

binary point in both directions.

Step3: Beginning on the left (most significant), select the first group of one or two digit (If n is odd

then first group is one digit ,else two bits)

Step4: Select the first group of bits and subtract' 01' from it. If borrow is zero, result is positive and

quotient is 1 else it is 0.

Step5: Append 01(to be subtracted next two digits of dividend) and guessed square root to subtract

from remainder of previous stage

Step6: If result of subtraction is negative, write previous remainder as it is and quotient is considered as

0, else write the difference as remainder and quotient as 1.

Step7: Repeat step 5 and step 6 until end group of two digits.

Step8: End

1 0 0 1.1 0 1 0

01 01 11 01.00 00 00 00 00

- 01

00 01 take next two digits from dividend

- 1 01 Append 01


- 10 01

01 11 01

100 01

11 00 00

1001 01

001011 00


00101100 00

- 10011001

0000010111 00



Figure 2: The example of modified non restoring algorithm to solve square root

Basic Building Block for Non restoring algorithm

Inputs of the building block are x,y,b and u while d and b0(borrow) are outputs.

If b0=0, then d<=x-y-b else d<=x;

b0=( ~ x .y)+(b.~x)+(by);

d= (~x.y.~b.~u)+(~x.~y.b.~u)+(x~y.~b)+(x.u)+(x.y.b);


Figure 3: RTL schematic of CSM block

The generalization of simple implementation of non restoring digit by digit algorithm for unsigned 6 bit square root by array structure is shown in Fig.4. Each row of the circuit executes one iteration of non restoring digit by digit square algorithm, where it only uses subtract operation and appends '01'.

Figure 4: Pipelined structure of 6 bit unsigned square root number

The design can be optimized by minimizing the logic expressions and can be implemented by modifying CSM block. The specialized entities A,B,C,D,E,F,G and H are derived from CSM block and are defined as follows:

For csmA, ybu = 100

b0 = ~x

d = ~x

For csmB, yu = 00

b0 = ~x.b

d = ~x.b + ~b.x

For csmC, u = 0

b0 = ~x.y + ~x.b + y.b

d = ~x.y.~b + ~x.~y.b + x.~y.~b + x.y.b

For csmD, yb = 10

b0 = ~x

d = ~x.~u + x.u

For csmE, y = 0

b0 = ~x.b

d = ~x.b.~u + ~b.x + x.u

For csmF, xy = 00

b0 = b

d = b.~u

For csmG, xyb = 010

b0 = ~x

d = ~u

For csmH, xyu = 000

b0 = b

Figure 5: Optimized Pipelined structure of 8 bit unsigned square root number

Results and analysis

The Non Restoring algorithm can be implemented with least hardware resources and the result will be the faster than restoring square rooting techniques. The source code is implemented in such a way that it can be extended according to user's requirement to calculate complicated square root in FPGA.

Figure 6: Simulation result of 8 bit square root using non restoring algorithm

The DE1 kit has 4 seven segment displays only so the maximum number which can be displayed is 9999d and also it doesn't have a decimal point. Hence output obtained is less precise if one of the displays is considered as a decimal point.

Table 1 shows the list of Logic Elements usage for 8 bit implementation. This indicates the size of the

implemented circuit "hardware resource".

Table 1: Comparison of LE's usage in 8 bit implementation


Implementation of non restoring algorithm for 8 bit



8 bit (with seven segment)



8 bit (without seven segment)



optimized 8 bit (with seven segment)



optimized 8 bit (without seven segment)


Table 2: PowerPlay Power Analyzer Status


PowerPlay Power Analyzer Status

8 bit with optimization (mW)

8 bit without optimization (mW)





Total Thermal Power Dissipation





Core Dynamic Thermal Power Dissipation





Core Static Thermal Power Dissipation





I/O Thermal Power Dissipation





This implementation and analysis shows that proposed method is most efficient of hardware resource. This is reasonable, because it only uses subtract operation and append 01. The result shows that the proposed algorithm is easy to implement and also uses less resources. The result is extended for square root implementation of 8 bit floating point number and also it can be expanded to larger numbers to solve complicated square root problem in FPGA implementation.

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