VHDL Design and Simulation
Dr. Anthony D. Johnson
radix_conversions7.fm - 1
p30 1.3.Converting
Positional
Between two Bases
Number
Representations
An often encountered practical situation is that a number N is available as represented in one of the
positional representations, but some processing step that is to be performed needs a different
positional representation of N. This creates a conversion task that can be formally specified as,
Problem: Conversion from One to Another Positional Number Representation
Problem description:
Given is:
NrA
representation of a positive mixed number N in radix rA representation,
rA, rB
two radixes (bases),
(rB)rA
representation of the radix rB in radix rA,
(rA)rB
representation of the radix rA in radix rB.
Determine:
NrB
p31
representation of the number N radix rB representation.
1.3.1Conversion Methods
To improve the clarity and simplicity of presentation, thefollowing descriptions of conversion
techniques will assume that conversions are done:
-
from a given radix rA representation NrA,
to a desired radix rB representation NrB.
Three basic techniques applied in conversions between positional number representations are listed
here together with their areas of application where they are preferable for providing a “one step”
conversion using decimal arithmetic:
(a) expanding positional notation technique is preferable for:
- conversion from number representations in any radix rA to decimal representation,
(b) division by (rB)rA (radix rB in rA representation) technique is preferable for:
-
conversion of integer numbers from decimal to any radix rB representation,
(c) multiplication by (rB)rA technique is preferable for.
-
conversion of fractional numbers from decimal to any radix rB representation.
This preference list shows that when one of the radixes is ten, any representation conversion can be
accomplished using only decimal arithmetic in at most two conversion steps:
- from any radix rA to decimal representation in a single application of technique (a),
-
from decimal to any radix rB representation:
-
for integer and fractional numbers, in a single application of technique (b) or (c)
respectively,
for mixed numbers, in one application of each of the techniques (b) and (c) to the
integer and fractional part separately,
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VHDL Design and Simulation
Dr. Anthony D. Johnson
radix_conversions7.fm - 2
One more special conversion technique exists that avoids using non decimal arithmetic when
conversions between radixes 210, 810, and 1610 representations are needed:
(d) grouping of digits when rA = (rB)k technique is applicable for:
-
conversions between representations in radixes 210, 810, and 1610:
- in one step between representations in radixes 210 and 810, and 210 and 1610,
- in two steps between representations in radixes 810 and 1610.
For all other combinations of radixes, two variations are available of the general case method:
(e) general conversion method technique is applicable for any combination od radixes:
- indirect, two step method uses only decimal arithmetic:
- first step: from radix rA to radix 10,
- second step: from radix 10 to radix rB,
-
direct, one step, method uses non decimal arithmetic in the application of basic
techniques (a), (b), and (c).
p31 1.3.1_1 Conversion Method: Series Substitution
Series Substitution conversion method is also called Expanding Positional Notation conversion
method.
Positional Representation of Numbers has been introduced by Equation (1.3).
Nr = (an-1an-2... a2a1ao.a-1a-2 ... a-m)r =
(1-3)
Nr= an-1⋅rn-1+an-2⋅rn-2+... a2⋅r2+a1⋅r1+a0⋅r0+a-1⋅r-1+a-2⋅r-2+ ... a-m⋅r-m
Expanding Positional Notation expands the NrA representation by suming up the products, of the
powers of base rA and their corresponding digits ai; which is always conveniently doable in radix 10
arithmetic.
This is practical for hand calculations, but its computer implementation would be slow due to the
involved exponentiations.
p32 1.3.1_2 Conversion Method: Division by Radix (rB)rA
When Division by Radix technique is used, conversion of integer numbers from decimal to any other
radix representation applies only decimal arithmetic .
As described by the algorithm DRM, division by radix method applies a succession of divisions by
the radix (rB)rA, :
DRM-1: with counter i set to i=0, the initial division is applied to the fraction NrA/(rB)rA, and
the quotient Q0 and remainder R0 of this division are written on the same line to the
right of the fractional expression;
DRM-2: incrementing the counter i by one at each successive division, divisions is applied to
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VHDL Design and Simulation
Dr. Anthony D. Johnson
radix_conversions7.fm - 3
the fraction Qi-1/(rB)rA, followed by the new quotient Qi and remainder Ri written again
on the same line to the right of the fractional expression;
DRM-3: the last division by (rB)rA will be the one that results in a quotient whose value equals
zero; the remainder Ri of this last division then becomes the MSD of NrB, and the other
remainders are appended to Ri in the reverse chronological order so that:
NrB = (Ri, Ri-1,..., R1, R0)rB
is the result of the conversion.
Examples of the application of division by radix method, from decimal representation of 2310 and
1910 to octal and binary representation are,
quot. remain.
NrA = 2310 23/8
2/8
(rB)rA = 810
2
0
7
2
NrB = 2 78
quot. remain.
NrA = 1910 19/2 9
1
(rB)rA = 210 9/2 4
1
4/2 2
0
2/2 1
0
1/2 0
1 0 0 1 12= 1910
The apparent “magic” supporting the division by radix (rB)rA technique is more easily understood
when one rewrites the explicit positional number representation of equation (1-3) as,
NrA = ((( an-1⋅rA+ an-2)⋅rA+...+ a2)⋅rA+ a1)⋅rA + ao.
(1-4)
The form (1-4), being equivalent to radix rA expression of the number N, shows why the remainders
of the successive divisions by radix (rB)rA (starting with the division of N itself, and continuing by
the divisions of resulting quotients of preceding divisions) constitute the series of digits of the
representation NrA.
The initial division by rA is applied to the right hand side of the equation (1-4), and yields the quotient
and remainder as,
Q0 = (( an-1⋅rA + an-2)⋅rA+...+ a2)⋅rA+ a1
R0=ao.
The second division by radix rA is applied to the quotient Q0, and yields the quotient and the
remainder as,
Q1 = (( an-1⋅r A+ an-2)⋅rA+...+ a3)⋅rA+ a2
R 1= a 1.
And so on, until the n-th division yields the quotient dn-1, which is not divisible by rA (an-1<rA), so
that the (n+1)-st division yields the quotient 0 (which is the condition for ending the division process)
and the remainder an-1.
8/25/14
VHDL Design and Simulation
Dr. Anthony D. Johnson
radix_conversions7.fm - 4
Look up the textbook also.!
p33 1.3.1_3Conversion Method: Multiplication by Radix (rB)rA
When Multiplication by Radix technique is used, conversion of fractional numbers from decimal
representation to any radix rB representation applies only decimal arithmetic.
The information needed for the execution of MRM technique consists of:
- a fractional number in radix rA representation, NrA = (a-1a-2 ...a-m)rA
-
radix (rB)rA of the representation NrB which should be created,
-
number of digits nd to be calculated for NrB.
Multiplication by radix technique applies a succession of nd multiplications by the radix (rB)rA, as
described by the algorithm MRM:
MRM-1: Execute two initialization settings:
- set counter i to i = 0,
- set parameter P0 to P0 = NrA.
MRM-2: Execute the following steps nd times, for i = 0 to i = nd -1:
-
increment i to i = i+1,
calculate the product Pi-1⋅(rB)rA, and assign its value to the new parameter Pi,
-
set a-i = In{Pi},
-
set Pi = Fr{Pi},
MRM-3: after the step MRM-2 has been applied nd times, all nd digits of the representation NrB
have been calculated, and the result of conversion is:
NrB = (.a-1a-2a-3 ... a-(nd-1)a-nd)rB
The following two conversions are examples of the application of multiplication by radix technique.
to conversions of fractional numbers 0.351610 and 0.7350510 to octal and ternary representation
respectively.
Conversion info:
NrA = 0.351610
(rB)rA = 810
ndB = 6
Conversion process
i = 1: 0.3516 ⋅8 = 2.8128 b-1 = 2
Conversion result
i = 2: 0.8128 ⋅8 = 6.5024 b-2 = 6
i = 3: 0.5024 ⋅8 = 4.0192 b-3 = 4
i = 4: 0.0192 ⋅8 = 0.1536 b-4 = 0
i = 5: 0.1536 ⋅8 = 1.2288 b-5 = 1
i = 6: 0.2288 ⋅8 = 1.8304 b-6 = 1
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NrB = 0.2640118 = 0.3515610
VHDL Design and Simulation
Dr. Anthony D. Johnson
radix_conversions7.fm - 5
Conversion info:
Conversion process
NrA = 0.7350510
i = 1: 0.73505 ⋅3 = 2.20515 b-1 = 2
(rB)rA = 310
i = 2: 0.20515 ⋅3 = 0.61545 b-2 = 0
ndB = 8
i = 3: 0.61545 ⋅3 = 1.84635 b-3 = 1
Conversion result
i = 4: 0.84635 ⋅3 = 2.53905 b-4 = 2
i = 5: 0.53905 ⋅3 = 1.61715 b-5 = 1
i = 6: 0.61715 ⋅3 = 1.85145 b-6 = 1
i = 7: 0.85145 = 2.55435 b-7 = 2
i = 8: 0.55435 ⋅3 = 1.66305 b-8 = 1
NrB = 0.201211213 = 0.73494810
Understanding how the MRM technique works takes a different kind of consideration than the one
used for DRM technique.
p35 1.3.2 General Conversion Algorithms
Algorithm 1.1 Works directly between any two bases, but may require performing arithmetic in
an unfamiliar base.
Algorithm 1.2Uses only decimal arithmetic, but works indirectly, in two conversions, the first to
base 10, and then the second from base 10.
p36 1.3.3 Conversion between radixes rA and rB when (rA)k= rB, or rA= (rB)k
Bit grouping (more precisely, digit grouping) conversion method is applicable to conversion
between representations in radixes rA and rB, iff rA= (rB)k or (rA)k= rB, where k is a positive integer.
Practical cases:
rB
k
rA
2
2
3
4
8
16
When rA= (rB)k and k > 0 is an integer:
1o r A > r B,
2o when the digits of rA and rB representations are denoted by DA and DB, then each DA
is representable by k DB digits,
3o the following conversions are not possible by this method:
- conversions involving base 10 representation,
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VHDL Design and Simulation
Dr. Anthony D. Johnson
-
radix_conversions7.fm - 6
direct conversions between base 8 and base 16 because,
generally: lg(rA) = k⋅ lg(rB)
⇐
specifically: for rA=16 and rB=8,
k=
k=
lg(rA)
lg(rB)
lg(rA)
lg(rB)
≠ integer
lb16
=
lb8
=
4
3
≠ integer
4o indirect conversion between base 8 and base 16 by this method involves two successive
conversions:
- the first to a representation in base 2, and
- the second from the representation in base 2.
Bit grouping conversion method is performed separately on the integer and fractional parts.
When (rA)k = rB, the digits of radix rA representation of a number N are separated into groups of k
consecutive digits to the left and to the right of the fractional point. This is then followed by expansion
of each separate group into one radix rB digit.
When rA = (rB)k, the digits of radix rA representation of a number N are converted individually into
rB representation, followed by a concatenation of obtained groups of k rB digits.
Algorithm 1.3 calculates only in base 10 arithmetic, but requires a longer procedure.
p36 Algorithm 1.3(a) from base rB to base rA for rA= (rB)k
Example 1.24 Binary to Octal
p36 Algorithm 1.3(b) from base rA to base rB for rA= (rB)k
Example 1.25 Hexadecimal to Octal
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VHDL Design and Simulation
Dr. Anthony D. Johnson
radix_conversions7.fm - 7
CONVERSION BETWEEN PRACTICALLY
SIGNIFICANT NUMBER REPRESENTATIONS
CONVERSION TECHNIQUES
Division &
Multiplication
by Radix
Bit
Grouping
⇔
OCTAL
⇔
BINARY
⇐
⇐
⇐
⇐
⇐
DECIMAL
Expanding
Positional Notation
⇐
HEXADECIMAL
Calculation using
Decimal Arithmetic
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DECIMAL