 Open Access
 Total Downloads : 136
 Authors : Usman Ali, Tariq Jamil
 Paper ID : IJERTV2IS100253
 Volume & Issue : Volume 02, Issue 10 (October 2013)
 Published (First Online): 09102013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Bitwise Bytewide Analysis of ShiftRight Operations on Binary Representation of Complex Numbers
AUTHORS AND AFFILIATIONS
1Laboratory of Signals and Systems L2S, CNRSSupÃ©lec, University of Paris 11, Plateau de Moulon 91192, GifsurYvette, FRANCE
2Department of Electrical and Computer Engineering, Sultan Qaboos University, AlKhod 123, Muscat, SULTANATE OF OMAN
CORRESPONDING AUTHOR
Complex numbers play very important role in electrical and computer engineering. Arithmetic operations involving complex numbers represented in binary number system are handled in todays computers by the application of divideandconquer technique wherein complex numbers are partitioned into real parts and imaginary parts and then operations between each pair (real with real, imaginary with imaginary) are carried out independent of one another. This results in multiple suboperations within an operation and delay in the production of the final result. Complex Binary Number System provides a oneunit binary representation of a complex number and allows for arithmetic operations on complex numbers to be carried out in exactly the same way as the real numbers. In this paper, we have investigated the effect of bitwise bytewide shift right operations on the complex binary representation of complex numbers and analyzed these results using mathematical equations.
Complex number, complex binary number, multipleshift, shiftright

Introduction
Complex numbers play very important role in electrical and computer engineering. These days, arithmetic operations involving complex numbers represented in binary number system are handled by the application of divideandconquer technique wherein complex numbers are partitioned into real parts and imaginary parts and then operations between each pair (real with real, imaginary with imaginary) are carried out independent of one another. This results in multiple suboperations within an operation and delay in the production of the final result. For example, lets consider arithmetic operations involving two complex numbers ( + ) and ( + ). Their addition involves two individual additions, one for the real parts( + )and one for the imaginary parts( + ). Their subtraction involves two individual subtractions, one for the real parts( )and one for the imaginary parts( ). Multiplication involves four individual multiplications
, , , , one subtraction 2 = , and one addition + . Finally, division
involves six individual multiplications , , , , 2, 2, two additions + and
2 +2
2 +2
2 + 2, one subtraction , and then two individual divisions +
and 2 +2 . If we
assume that each individual addition/subtraction, involving complex numbers, takes nsec to complete and each individual multiplication/division takes nsec to execute, such that
(multiplication can be assumed to be repeatedaddition and division can be assumed to be repeated subtraction), then each complex addition/subtraction will take 2 nsec, each complex multiplication will take 4 + + = (2 + 4) nsec, and each complex division will take 6 + 2 + + 2 = (3 + 8) nsec. Now lets imagine a number system in which complex arithmetic does not involve any combination of individual arithmetic sub operations as described previously. That is, addition, subtraction, multiplication, or division of complex numbers is just one pure addition, one puresubtraction, one puremultiplication, or one puredivision operation respectively and not a combination of various individual sub operations within a given arithmetic operation. This will effectively reduce the complex addition/subtraction time to nsec and complex multiplication/division time to nsec. Mathematically, such a complex number system will yield reduction in execution time of addition/subtraction operation roughly by a factor of 2 Ã— 100 = 50%, for multiplication
(2+4) Ã— 100 = 4 Ã— 100 = 400% (since is very small compared to ), and for division
(3+8) Ã— 100 = 8 Ã— 100 = 800%. With the reduction in execution times of complex
arithmetic operations roughly by factors of 50% to 800% in digital signal and image processing applications of engineering where complex numbers are most frequently utilized, it is possible to achieve tremendous improvement in the overall performance of systems, provided that a technique exists which treats a complex number as a single entity (rather than two entities comprising of real and imaginary parts) and facilitates a singleunit representation of complex numbers in binary format within a microprocessor environment (rather than two individual representations for real and imaginary parts respectively, as in
todays computers). Such a unique number system is referred to as Complex Binary Number System (CBNS) [1].
There have been several efforts in the past to define a binary number system (0 or 1) with bases other than 2 which would facilitate a oneunit representation of complex numbers. In 1960, Donald Knuth described a number system with base 2 and analyzed the arithmetic operations of numbers based on this imaginary base [2]. He was unsuccessful in providing a division algorithm for binary numbers based on this imaginary base and considered it as a main obstacle towards hardware implementation of this number system.Four years later, Walter Penney defined a binary number system, first by using a negative base of 4 [3], and then by using a complex number (1 + ) as the base [4]. Like Donald Knuth before him, Walter Penney was unable to formulate an efficient division process using these bases and, consequently, these number systems remained dormant for more than thirty years. V.
Stepanenko, in 1996, defined a number system with the base 2 and generated real and imaginary parts of complex numbers by taking even and odd powers of this base respectively [5]. He succeeded somewhat in resolving the division problem as an allinone operation but, in his algorithm, everythingreduces to good choice of an initial approximation in a NewtonRaphson iteration which may or may not converge.Jamil et al, in 2000 [6], revisited Penneys number system with base (1 + ) and presented a detailed analysis of this number system, now famously called Complex Binary Number System (CBNS) [7]. Extensive details of the algorithms to be followed in CBNS for binary representation of a complex number can be found in [1] and the implementation statistics of the arithmetic circuits designed based on this number system can be found in [8,9,10,11]. In this paper, we haverestricted ourselves to investigating the effects of multiplebit (from 1 bit to 8 bits) shiftright operations only on a complex number represented in singleunit binary notation of complex binary number system.
This paper is organized as follows: In section 2, well present basic information about CBNS and how to represent an integeronly complex number into this new number system. Then well take the CBNS representation of complex numbers and, in section 3,give a comprehensive analysis of the effect of multiplebit shift right operations on the complex numbers. Conclusions are presented in section 4, which are followed by acknowledgments and references.

( + )base Complex Binary Number System
Mathematically, the value of any nbit binary number (123 210)in base 2 can be represented in the form of a power series as 1(2)n1 + 2(2)n2 + 3(2)n3 +
+ 2(22 +1(2)1 +0(2)0. Following the same procedure, the value of an nbit binary number with base (1 + )can be written in the form of a power series as 1(1 + )n1
+ 2(1 + )n2 + 3(1 + )n3 + + 2(1 + )2 +1(1 + )1 +0(1 + )0 where the coeffecients 1, 2, 3, , 2, 1, 0 are binary in nature (0 or 1) and belong to complex binary number system. Using the conversion algorithms given in [1], we are able to
obtain a oneunit binary representation of any given complex number, whether it is made from integers, fractions, or floating point numbers, in Complex Binary Number System (CBNS). In this paper only the conversion algorithm for integershas been excerpted from [1].
The reader is referred to [7] for a thorough coverage of the conversion algorithms for fractions and floating point numbers.
A positive integer N can be converted into CBNS representation by following these steps:

Express N in terms of powers of 4 by repeatedly dividing by 4 and keeping track of the remainders. This will result in a base 4 number ( 5,4,3,2,1,0,) where {0,1,2,3}

Convert Base 4 number ( 5,4,3,2,1,0,)to Base 4 by replacing each digit in the odd location (1,3,5 ,) with its negative to get ( 5,4,3,2,1,0,). Next, we normalize the new number, i.e., get each digit in the range 0 to 3, by repeatedly adding 4 to the negative digits and adding
a 1 to the digit on its left. This operation will get rid of negative numbers but may create some digits with a value of 4 after the addition of a 1. To normalizethis, we replace 4 by a 0 and subtract a 1 from the digit on its left. This subtraction might once again introduce negative digits which will be normalized using the previous method but this process will definitely terminate.

Lastly, we replace each digit in the normalized representation by its equivalent
binary representation in CBNS, i.e., 0 0000, 1 0001, 2 1100, and 3
1101.
For example,
201210 = 1,3,3,1,3,0 4 = 1,3, 3,1, 3,0 4
= 1,2,0,1,2,1,0 = 0001 1100 0000 0001 1100 0001 0000
= 1110000000001110000010000 (1+ )
To convert a negative integer into CBNS format, we simply multiply the representation of the corresponding positive integer with 11101 (equivalent to (1) (1+ )) according to the multiplication algorithm for CBNS[1].
For example,
201210 = 111000000000111000001 Ã— 11101
= 110000000000110111010000 (1+)
To obtain binary representation of a positive or negative imaginary number in CBNS, we multiply the corresponding CBNS representation of positive or negative integer with 11 (equivalent to (+) (1+ )) or 111 (equivalent to () (1+)) according to the multiplication algorithm for CBNS [1]. Thus
+201210 = 111000000000111000001 Ã— 11
= 10000000000010000110000 (1+ )
201210 = 111000000000111000001 Ã— 111
= 111010000000111010001110000 (1+)
After obtaining CBNS representations for all types of integers (real and imaginary), it is now possible for us to represent an integer complex number (both real and imaginary parts of the complex number are integers) simply by adding the real and imaginary CBNS representations according to the addition algorithm for CBNS [1]. Thus
201210 +201210
= 1110000000001110000010000 1+ + 10000000000010000110000 1+
= 1110100000001110100011100000 1+


MultipleBit Shift Right Operations in CBNS
To analyze the effects of shiftright operations on a complex number represented in CBNS format, we wrote a computer program in C++ language which allowed for variations in magnitude and sign of both real and imaginary components of a complex number to be generated automatically in a linear fashion, and then decomposed the complex binary number after the shiftright operation into its real and imaginary components. We restricted the length of the original binary bit array to 800 bits and 0s were padded on the leftside of the binary data when the given complex number required less than maximum allowable bits for representation in CBNS format.
To illustrate these restrictions, lets consider the following complex number: Original complex number represented in CBNS before padding:
9010 +9010 = 110100010001000 1+
Padded complex binary array such that the total size of the array is 800 bits.
9010 +9010 = 0 0110100010001000 1+
Shifting this binary array by 1bit to the right will yield 00 011010001000100 1+ ensuring that total arraysize remains 800 bits. This is done by removing one 0 from the rightside and inserting one 0 on the leftside of the number. Similarly, shifting of the original binary array by 2,3,4,5,6,7,8bits to the right will yield respectively:
000 01101000100010 1+
0000 0110100010001 1+
00000 011010001000 1+
000000 01101000100 1+
0000000 0110100010 1+
00000000 011010001 1+
000000000 01101000 1+
All the time we make sure that the total arraysize remains 800 bits by removing 2,3,4,5,6,7,8 bits, respectively from the rightside of the original array.
Table 1 presents an overall summary of the effect on the signs of the complex numbers, represented in CBNS format, because of multiplebit shiftright operations (1 to 8 bits).
Table 1. Effect on signs of complex numbers in CBNS format after shiftright operations
Before ShiftRight
After ShiftRight by 1bit
After ShiftRight by 2bits
Real
Imaginary
Real
Imaginary
Real
Imaginary
+
0
0
+
0
+
+
0
0
+
+
0
0
+
+
0
+
+
0
+
+
0
+
+
+
+
0
0
+
+
Before ShiftRight After ShiftRight After ShiftRight
by 3bits by 4bits
Real
Imaginary
Real
Imaginary
Real
Imaginary
+
0
+
0
0
+
+
0
0
+
+
+
0
0
0
+
+
+
+
0
+
0
+
+
0
+
+
0
+
+
+
Table 1 (continued). Effect on signs of complex numbers in CBNS format after shiftright operations
Before ShiftRight
After ShiftRight by 5bits
After ShiftRight by 6bits
Real
Imaginary
Real
Imaginary
Real
Imaginary
+
0
+
+
0
0
0
+
0
+
+
+
0
0
+
0
+
+
0
+
+
+
+
0
+
0
+
+
0
+
Before ShiftRight After ShiftRight After ShiftRight
by 7bits by 8bits
Real
Imaginary
Real
Imaginary
Real
Imaginary
+
0
+
+
0
0
+
0
0
+
0
+
0
+
+
0
+
+
0
+
+
+
0
+
+
+
0
+
+
0
Shiftright operations on complex binary numbers affect not only the signs of the given complex numbers (as shown in Table 1) but also have impact on the magnitudes of the complex numbers according to different mathematical relationships. To find out the effects of shiftright operations on the magnitudes of the complex numbers, we varied the magnitude of real and imaginary components of the original complex numbers in a linear fashion (Fig. 1). The complex numbers obtained after shiftright operations were analysed by obtaining mathematical equations describing their behavior, as given in Figs. 29.
Fig.1. Mathematical equations describing variations in magnitudes of complex numbers in the Cartesian plane (before shiftright)
Fig. 2. Mathematical equations describing variations in magnitudes of complex numbers in the Cartesian plane (after shiftright by 1bit)
Fig.3. Mathematical equations describing variations in magnitudes of complex numbers in the Cartesian plane (after shiftright by 2bits)
Fig. 4. Mathematical equations describing variations in magnitudes of complex numbers in the Cartesian plane (after shiftright by 3bits)
Fig.5. Mathematical equations describing variations in magnitudes of complex numbers in the Cartesian plane (after shiftright by 4bits)
Fig. 6. Mathematical equations describing variations in magnitudes of complex numbers in the Cartesian plane (after shiftright by 5bits)
Fig.7. Mathematical equations describing variations in magnitudes of complex numbers in the Cartesian plane (after shiftright by 6bits)
Fig. 8. Mathematical equations describing variations in magnitudes of complex numbers in the Cartesian plane (after shiftright by 7bits)
Fig. 9. Mathematical equations describing variations in magnitudes of complex numbers in the Cartesian plane (after shiftright by 8bits)
To fully understand the variations in the sign and magnitude of the complex numbers before and after the shiftright operation, we used Microsoft Excel to draw graphs as shown in the Figs.1017.
Fig. 10 and Fig. 11 illustrate the effect on the real and imaginary parts of the complex numbers after 18bits shiftright operations for positive and negative real only complex numbers (no imaginary part), respectively.
Fig. 12 and Fig. 13 present the effect on the real and imaginary parts of the complex numbers after 18bits shiftright operations for positive and negative only imaginary complex numbers (no real part), respectively.
The four cases of Â±RealÂ±Imaginary complex numbers represented in CBNS format before the shift, and effects of 18 bits shiftright operations on the sign and magnitude of complex numbers are presented in Figs. 14,15,16, and 17 respectively.
25
20
15
10
5
0
20
Real Imag
Real shift 2bit Imag shift 2bit
Real Imag
Real shift 2bit Imag shift 2bit
15
10
5
5
10
15
1 4 7 10 13 16 19
0
1 4 7 10 13 16 19
Real
Imag
Real shift 1bit
Imag shift 1bit
Real
Imag
Real shift 1bit
Imag shift 1bit
Real Imag
Real shift 3 bits
Imag shift 3 bits
Real Imag
Real shift 3 bits
Imag shift 3 bits
Real Imag
Real shift 4 bits
Imag shift 4 bits
Real Imag
Real shift 4 bits
Imag shift 4 bits
25 25
20 20
15 15
10 10
5 5
0 0
5
10
35
30
25
20
15
10
1 3 5 7 9 11 13 15 17 19
Real Imag
Real Shift Right 5bit
5
10
35
30
25
20
15
10
5
1 3 5 7 9 11 13 15 17 19
Real Imag Real Shift
Right 6bit
5 Imag Shift 0
Imag Shift
0
1 4 7 10 13 16 19 22 25 28
Right 5bit
5
10
Right 6bit
35 35
30 Real 30
25 25
20 Imag 20
Real Imag
15 Real Shift 15
Real Shift
10 Right 7bit
10 Right 8bit
5 Imag Shift 5
Imag Shift
Right 8bit
0
5 1 4 7 10 13 16 19 22 25 28
Right 7bit 0
1 4 7 10 13 16 19 22 25 28
Fig. 10. Effects of shiftright operation on sign and magnitude of a positive realonly complex number (18 bits)
15
10 Real
5
0 Imag
5 1 4 7 10 13 16 19
Real shift 1
10 bit
15 Imag shift 1
20 bit
25
0
1 4 7 10 13 16 19 Real
5
Imag
10
Real shift 2
bits
15 Imag shift 2
bits
20
10
5 Real 0
5 1 4 7 10 13 16 19 Imag
10 Real shift 3
bits
15 Imag shift 3
20 bits
25
10
5 Real 0
5 1 4 7 10 13 16 19 Imag
10 Real shift 4
bits
15 Imag shift 4
20 bits
25
0
10
5 1 4 7 10 13 16 19 22 25 28
10
15
20
25
30
Real Imag
Real Shift Right 5 bit
Imag Shift Right 5 bit
5
0
5
10
15
20
25
30
Real Imag
Real Shift Right 6bit
Imag Shift Right 6bit
35
35
0
5
0
5 1 5 9 13 17 21 25 29
10
15
20
25
30
35
Real
Imag
Real Shift Right 7bit
Imag Shift
Right 7bit
5 1 4 7 10 13 16 19 22 25
10
15
20
25
30
Real Imag
Real Shift Right 8bit
Imag Shift Right 8bit
35
Fig. 11. Effects of shiftright operation on sign and magnitude of a negative realonly complex number (18 bits)
25
20
15
10
5
0
25
1
4
7
10
13 16
1
1
4
7
10
13 16
1
Real Imag
Real shift 2 bits
Imag shift 2 bits
Real Imag
Real shift 2 bits
Imag shift 2 bits
20
15
10
5
0
5
10
Real Imag
Real shift 3 bits
Imag shift 3 bits
Real Imag
Real shift 3 bits
Imag shift 3 bits
15
1 4 7 10 13 16 19
5 9
10
15
Real Imag
Real shift 1 bit
Imag shift 1 bit
Real Imag
Real shift 1 bit
Imag shift 1 bit
20
15
10
5
25
Real Imag
Real shift 4
bits
Imag shift 4 bits
Real Imag
Real shift 4
bits
Imag shift 4 bits
20
15
10
5
0
0
1 4 7 10 13 16 19
40
30
20
10
Real Imag
Real Shift Right 5bit
5
10
35
30
25
20
15
10
1 4 7 10 13 16 19
Real Imag
Real Shift Right 6bit
0 Imag Shift Right
5 Imag Shift
10
1 4 7 10 13 16 19 22 25 28
5bit 0
1 4 7 10 13 16 19 22 25 28
Right 6bit
35 35
30 Real 30
25 25
20 Imag 20
Real Imag
15 15
Real Shift
Real Shift
10 Right 6bit
10 Right 8bit
5 Imag Shift 5
Imag Shift
0
1 4 7 10 13 16 19 22 25 28
Right 6bit 0
1 5 9 13 17 21 25 29
Right 8bit
Fig. 12. Effects of shiftright operation on sign and magnitude of a positive imaginaryonly complex number (18 bits)
15
10
5
0
15
Real
Imag
Real shift 2 bits
Imag shift 2 bits
Real
Imag
Real shift 2 bits
Imag shift 2 bits
10
5
0
5
10
15
20
25
1 4 7 10 13 16 19
5
10
15
20
25
1 4 7 10 13 16 19
Real
Imag
Real shift 1 bit
Imag shift 1 bit
Real
Imag
Real shift 1 bit
Imag shift 1 bit
Real
Imag
Real shift 3
bits
Imag shift 3 bits
Real
Imag
Real shift 3
bits
Imag shift 3 bits
Real
Imag
Real shift 4 bits
Imag shift 4 bits
Real
Imag
Real shift 4 bits
Imag shift 4 bits
0 10
2 1 4 7 10 13 16 19
4
6
5
0
8
10
12
14
16
18
20
5
10
15
20
25
1 4 7 10 13 16 19
10 0
5
0
Real
5
10
1 4 7 10 13 16 19 22 25 28 Real
5
10
15
20
25
30
35
1 4 7 10 13 16 19 22 25 28
Imag
Real Shift Right 5bit
Imag Shift Right 5bit
15
20
25
30
35
Imag
Real Shift Right 6bit
Imag Shift Right 6bit
1 4 7 10 13 16 19 22 25 28
1 4 7 10 13 16 19 22 25 28
5
0 Real
0
5
10
Real
Imag
5
10
15
1 4 7 10 13 16 19 22 25 28
Imag
15 Real Shift
20
Real Shift
20
Right 7bit
25
Right 8bit
25 Imag Shift Imag Shift
30
35
Right 7bit
30
35
Right 8bit
Fig. 13. Effects of shiftright operation on sign and magnitude of a negativeimaginaryonly complex number (18 bits)
25
20
15
10
5
0
5 1 4 7 10 13 16 19
25
Real Imag
Real shift 2 bits
Imag shift 2 bits
Real Imag
Real shift 2 bits
Imag shift 2 bits
20
15
10
5
0
10
15
20
25
5
10
15
1 4 7 10 13 16 19
Real Imag
Real shift 1 bit
Imag shift 1 bit
Real Imag
Real shift 1 bit
Imag shift 1 bit
20
18
16
14
12
10
8
6
4
25
Real
Imag
Real shift 4 bits
Imag shift 4 bits
Real
Imag
Real shift 4 bits
Imag shift 4 bits
20
15
10
5
0
Real
Imag
Real shift 3 bits
Imag shift 3 bits
Real
Imag
Real shift 3 bits
Imag shift 3 bits
2 5
0
1 4 7 10 13 16 19
1 4 7 10 13 16 19 10
35 35
30 Real 30
25 25
20 Imag 20
15 15
Real Shift Right
10 5bit 10
Real Imag
Real Shift Right 6bit
5 Imag Shift Right
5 Imag Shift
0
1 4 7 10 13 16 19 22 25 28
5bit
0
5 1 4 7 10 13 16 19 22 25 28
Right 6bit
35 35
30 Real 30
25 25
20 Imag 20
15 15
10 10
10 10
Real Shift
Right 7bit
5 Imag Shift 5
Real Imag
Real Shift Right 8bit
Imag Shift Right 8bit
0
5 1 4 7 10 13 16 19 22 25 28
Right 7bit 0
1 4 7 10 13 16 19 22 25 28
Fig. 14. Effects of shiftright operation on sign and magnitude of a +Real+Imaginary complex number (18 bits)
1 4 7 10 13 16 19
1 4 7 10 13 16 19
25
20
15
10
5
0
5
10
15
20
25
25
1 4 7 10 13 16 19
1 4 7 10 13 16 19
Real Imag
Real shift 2 bits
Imag shift 2 bits
Real Imag
Real shift 2 bits
Imag shift 2 bits
20
15
10
5
0
5
10
15
20
25
Real Imag
Real shift 1bit Imag shift 1bit
Real Imag
Real shift 1bit Imag shift 1bit
25
20
15
10
5
0
5
10
15
20
25
1 4 7 10 13 16 19
25
1 4 7 10 13 16 19
1 4 7 10 13 16 19
Real
Imag
Real shift 4 bits
Imag shift 4 bits
Real
Imag
Real shift 4 bits
Imag shift 4 bits
20
15
10
5
0
5
10
15
20
25
Real Imag
Real shift 3 bits
Imag shift 3 bits
Real Imag
Real shift 3 bits
Imag shift 3 bits
40 40
30 Real 30
20 20
10 Imag 10
Real Imag
0
Real Shift
0
Real Shift
10
20
30
40
1 4 7 10 13 16 19 22 25 28
Right 5bit Imag Shift Right 5bit
10
20
30
40
1 5 9 13 17 21 25 29
Right 6bit Imag Shift Right 6bit
40
30 Real
20
40
30 Real
20
10
0
10
1 4 7 10 13 16 19 22 25 28
Imag
Real Shift Right 7bit
10
0
10
1 4 7 10 13 16 19 22 25 28
Imag
Real Shift Right 8bit
20 Imag Shift
20 Imag Shift
30
40
Right 7bit
30
40
Right 8bit
Fig. 15. Effects of shiftright operation on sign and magnitude of a +RealImaginary complex number (18 bits)
1 3 5 7 9 11 13 15 17 19
1 3 5 7 9 11 13 15 17 19
25
20
15
10
5
0
5
10
15
20
25
1 3 5 7 9 11 13 15 17 19
1 3 5 7 9 11 13 15 17 19
25
20
15
10
5
0
5
10
15
20
25
25
1 3 5 7 9 11 13 15 17 19
1 3 5 7 9 11 13 15 17 19
Real Imag
Real shift 1 bit
Imag shift 1 bit
Real Imag
Real shift 1 bit
Imag shift 1 bit
Real Imag
Real shift 2 bits
Imag shift 2 bits
Real Imag
Real shift 2 bits
Imag shift 2 bits
20
15
10
5
0
5
10
15
20
25
1 3 5 7 9 11 13 15 17 19
1 3 5 7 9 11 13 15 17 19
Real
Imag
Real shift 4 bits
Imag shift 4 bits
Real
Imag
Real shift 4 bits
Imag shift 4 bits
25
20
15
10
5
0
5
10
15
20
25
Real
Imag
Real shift 3 bits
Imag shift 3 bits
Real
Imag
Real shift 3 bits
Imag shift 3 bits
40
30
20
10
0
10
1 4 7 10 13 16 19 22 25 28
Real Imag
Real Shift Right 5bit
40
30
20
10
0
10
1 5 9 13 17 21 25 29
Real Imag Real Shift
Right 6bit
20 Imag Shift 20 Imag Shift
30
40
Right 5bit
30
40
Right 6bit
40 40
30 Real 30
20 20
10 Imag 10
Real Imag
0
Real Shift
0 Real Shift
10
1 5 9 13 17 21 25 29
Right 7bit
10
1 4 7 10 13 16 19 22 25 28
Right 8bit
20 Imag Shift
20
Imag Shift
30
40
Right 7bit
30
40
Right 8bit
Fig.16. Effects of shiftright operation on sign and magnitude of a Real+Imaginary complex number (18 bits)
25
20
15
10
5
0
5
10
15
20
25
15
1 4 7 10 13 16 1
1 4 7 10 13 16 1
1
4
7
10
13 16
19
1
4
7
10
13 16
19
Real Imag
Real shift 1 bit
Imag shift 1 bit
Real Imag
Real shift 1 bit
Imag shift 1 bit
Real Imag
Real shift 2 bits
Imag shift 2 bits
Real Imag
Real shift 2 bits
Imag shift 2 bits
10
5
0
5
9 10
15
20
25
1
4
7
10
13
16
19
1
4
7
10
13
16
19
Real
Imag
Real shift 3 bits
Imag shift 3 bits
Real
Imag
Real shift 3 bits
Imag shift 3 bits
Real Imag
Real shift 4 bits
Imag shift 4 bits
Real Imag
Real shift 4 bits
Imag shift 4 bits
0 10
1 4 7 10 13 16 19 5
5 0
5
10
15
20
10
15
20
25
0
5
10
15
20
25
30
35
1 4 7 10 13 16 19 22 25 28 Real
Imag
Real Shift Right 5bit
Imag Shift Right 5bit
10
5
0
5
10
15
20
25
30
35
1 4 7 10 13 16 19 22 25 28
Real Imag
Real Shift Right 6bit
Imag Shift Right 6bit
10
5
0
5
10
15
20
25
30
35
Real Imag
Real Shift Right 7bit
Imag Shift Right 7bit
0
5
10
15
20
25
30
35
1 4 7 10 13 16 19 22 25 28 Real
Imag
Real Shift Right 8bit
Imag Shift Right 8bit
Fig. 17. Effects of shiftright operation on sign and magnitude of a RealImaginary complex number (18 bits)
After analyzing Figs. 117, we are able to obtain the characteristic equations describing complex numbers in CBNS format after shiftright operations. These equations are given in Table 2.
Table 2. Characteristic equations describing complex numbers in CBNS format after shiftright operations
Type Number
of
Complex
After ShiftRight by 1bit
After ShiftRight by 2bits
Realold
Imaginaryold
Realnew
Imaginarynew
Realnew
Imaginarynew
+
0
Â½Realold+ Â¼
Â½Realold+ Â¼
0
+Â½Realold Â¼
0
Â½Realold+ Â¼
Â½Realold+ Â¼
0
+Â½Realold Â¼
0
+
Â½Imagold+ Â¼
Â½Imagold+ Â¼
Â½Imagold + Â¼
0
0
Â½Imagold+ Â¼
Â½Imagold+ Â¼
Â½Imagold + Â¼
0
+
+
0
Imagold
Â½Realold + Â¼
Â½Imagold + Â¼
+
Realold
0
Â½Realold + Â¼
Â½Imagold + Â¼
+
Realold
0
Â½Realold + Â¼
Â½Imagold + Â¼
0
Imagold
Â½Realold + Â¼
Â½Imagold + Â¼
Type
of
Complex
After ShiftRight
After ShiftRight
Number
by 3bits
by 4bits
Realold
Imaginaryold
Realnew Imaginarynew
Realnew Imaginarynew
+
0
Â¼ Realold Â¼ Realold
Â¼Realold 0
0
0
+
Â¼ Realold Â¼ Realold
Â¼ Imagold Â¼ Imagold
Â¼Realold 0
0 Â¼Imagold
0
+
+
+
+
Â¼ Imagold Â¼ Imagold
Â½Realold + Â¼ 0
0 Â½Imagold Â¼
0 Â½Imagold Â¼
Â½Realold + Â¼ 0
0 Â¼Imagold
Â¼Realold Â¼Imagold
Â¼Realold Â¼Imagold
Â¼Realold Â¼Imagold
Â¼Realold Â¼Imagold
Table 2 (continued) Characteristic equations describing complex numbers in CBNS format after shiftright operations
Type Number
of
Complex
After ShiftRight by 5bits
After ShiftRight by 6bits
Realold
Imaginaryold
Realnew
Imaginarynew
Realnew
Imaginarynew
+
0
Realold
Realold
0
Realold
0
Realold
Realold
0
Realold
0
+
Imagold +
Imagold +
Imagold +
0
0
Imagold +
Imagold +
Imagold +
0
+
+
0
Â¼Imagold
Realold +
Realold +
+
Â¼ Realold +
0
Realold
Imagold
+
Â¼ Realold +
0
Realold
Imagold
0
Â¼ Imagold
Realold +
Imagold +
Type
of
Complex
After ShiftRight
After ShiftRight
Number
by 7bits
by 8bits
Realold
Imaginaryold
Realnew Imaginarynew
Realnew Imaginarynew
+
0
Â¼ Â¼
Â¼ 0
0
Â¼ Â¼
Â¼ 0
0
+
0 0
0 0
0
0 0
0 0
+
+
Realold + 0
0 0
+
0 Imagold
Â¼ Â¼
+
0 Imagold
Â¼ Â¼
Realold + 0
0 0

Conclusions
The role of complex numbers is vital in all types of engineering applications and the importance in these numbers in the professional realm of an engineer cannot be ignored. However, ever since the birth of computer, these numbers have been treated as distant relatives of the real world and nothing substantial has ever been done in the field of computer architecture and arithmetic to improve the performance of arithmetic operations involving these type of numbers. CBNS provides a viable alternative for a singleunit binary representation of complex numbers with the premise of substantial enhancement in the speed of arithmetic operations dealing with these types of numbers. In this paper, we have looked in detail on how shiftright operations of 18 bits on a complex number represented in CBNS affect the signs and magnitudes of these numbers. Continuing with this work, we intend to do similar analysis for shiftleft operations on complex numbers represented in CBNS so that a wholesome picture about the true benefits of CBNS are established within the engineering community.
Acknowledgments
The work presented in this paper has been the result of a research grant: IG/ENG/ECED/06/02 provided by Sultan Qaboos University (Oman) and we gratefully acknowledge the encouragement rendered to us for this project. Preliminary versions of this paper, related to only nibblesize (up to 4bits) shiftright operations have appeared previously in the Proceedings of the IEEE SoutheastCon 2007 and the Journal of Computers (Academy Publisher) in 2008. The positive feedback received from reviewers of these previous publications prompted us to engage in more thorough and extended analysis (up to 8bits) of shiftright operationsinvolving complex binary numbers and we are thankful to these reviewers for their valuable input.
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