fixed point representation and signed numbers

[blackpadme]

Hi, i'm working with fixed point numbers which have two bits for no-
fractional part (in two complement) and eight bits for fractional
(i.e. pi/2 = 0.5703 => 00.10010010). But, what happen if i need
signed representation for the fractional part ? 11.10010010 = -1.5703.
What can i do if a need to represent -pi/2 and i need to work across
division, add, multiplicacion etc. ?

Thanks.

 

[Jonathan Bromley]

On Fri, 5 Sep 2008 07:50:48 -0700 (PDT), blackpadme wrote:

Hi, i'm working with fixed point numbers which have two bits for no-
fractional part (in two complement) and eight bits for fractional
(i.e. pi/2 = 0.5703 => 00.10010010). But, what happen if i need
signed representation for the fractional part ? 11.10010010 = -1.5703.
What can i do if a need to represent -pi/2 and i need to work across
division, add, multiplicacion etc. ?

It would be very unusual to have both the integer part and
the fractional part represented as 2s complement. It makes much
more sense to keep the usual fixed-point representation in which
the fraction is invariably unsigned:

10.00000000 -2 + .00 = -2.00
10.01000000 -2 + .25 = -1.75
10.10000000 -2 + .50 = -1.5
10.11000000 -2 + .75 = -1.25
11.00000000 -1 + .00 = -1.00
11.01000000 -1 + .25 = -0.75
11.10000000 -1 + .50 = -0.5
11.11000000 -1 + .75 = -0.25
00.00000000 0 + .00 = 0.00

You can then treat the whole 10-bit fixed-point number as if
it were a regular 2s complement integer, except that it is
scaled by 1/256.
--
Jonathan Bromley, Consultant

DOULOS - Developing Design Know-how
VHDL * Verilog * SystemC * e * Perl * Tcl/Tk * Project Services

Doulos Ltd., 22 Market Place, Ringwood, BH24 1AW, UK
jonathan.bromley@MYCOMPANY.com
http://www.MYCOMPANY.com

The contents of this message may contain personal views which
are not the views of Doulos Ltd., unless specifically stated.

 

[KJ]

On Sep 5, 10:50 am, blackpadme <rando...@gmail.com> wrote:

Hi, i'm working with fixed point numbers which have two bits for no-
fractional part (in two complement) and eight bits for fractional
(i.e. pi/2 =  0.5703 => 00.10010010). But, what happen if i need
signed representation for the fractional part ? 11.10010010 = -1.5703.
What can i do if a need to represent -pi/2 and i need to work across
division, add, multiplicacion etc. ?

Google for "Fixed_pkg_c.vhdl" (I think it's at vhdl.org somewhere).
This relatively new package nicely handles all the bit fiddlings
required to work with fixed point numbers...saves you the trouble of
re-inventing the wheel.

KJ

 

[blackpadme]

On 6 sep, 01:26, KJ <kkjenni...@sbcglobal.net> wrote:

On Sep 5, 10:50 am, blackpadme <rando...@gmail.com> wrote:

Hi, i'm working with fixed point numbers which have two bits for no-
fractional part (in two complement) and eight bits for fractional
(i.e. pi/2 =  0.5703 => 00.10010010). But, what happen if i need
signed representation for the fractional part ? 11.10010010 = -1.5703..
What can i do if a need to represent -pi/2 and i need to work across
division, add, multiplicacion etc. ?

Google for "Fixed_pkg_c.vhdl" (I think it's at vhdl.org somewhere).
This relatively new package nicely handles all the bit fiddlings
required to work with fixed point numbers...saves you the trouble of
re-inventing the wheel.

KJ

Thanks Jonathan and KJ.

 

[rickman]

On Sep 5, 11:23 am, Jonathan Bromley <jonathan.brom...@MYCOMPANY.com>
wrote:

On Fri, 5 Sep 2008 07:50:48 -0700 (PDT), blackpadme wrote:
Hi, i'm working with fixed point numbers which have two bits for no-
fractional part (in two complement) and eight bits for fractional
(i.e. pi/2 = 0.5703 => 00.10010010). But, what happen if i need
signed representation for the fractional part ? 11.10010010 = -1.5703.
What can i do if a need to represent -pi/2 and i need to work across
division, add, multiplicacion etc. ?

It would be very unusual to have both the integer part and
the fractional part represented as 2s complement. It makes much
more sense to keep the usual fixed-point representation in which
the fraction is invariably unsigned:

10.00000000 -2 + .00 = -2.00
10.01000000 -2 + .25 = -1.75
10.10000000 -2 + .50 = -1.5
10.11000000 -2 + .75 = -1.25
11.00000000 -1 + .00 = -1.00
11.01000000 -1 + .25 = -0.75
11.10000000 -1 + .50 = -0.5
11.11000000 -1 + .75 = -0.25
00.00000000 0 + .00 = 0.00

You can then treat the whole 10-bit fixed-point number as if
it were a regular 2s complement integer, except that it is
scaled by 1/256.

I don't understand what you are saying. Or maybe you are just using
some sort of a mnemonic for 2's complement. But what you have written
above is no different than just treating the entire number as a 2's
complement value...

Or are you just messing with our heads?

Rick

 

[Jonathan Bromley]

On Sat, 6 Sep 2008 15:59:01 -0700 (PDT), rickman <gnuarm@gmail.com>
wrote:

what you have written above is no different
than just treating the entire number as a 2's
complement value...

Precisely. The OP was asking about using a signed
representation for the *fraction* as well as the integer
part of a fixed-point number, and I was trying to show
why that doesn't make a lot of sense.

Or are you just messing with our heads?

Not intentionally. I'm a little hard-pressed
to understand why my attempt to enumerate
a few values in conventional 2.8 fixed-point
signed representation is upsetting you so

Perhaps this Number Representations 101
express my intent more clearly, at least
if you view it with a monospaced font:

Ordinary binary integers work like this
(using a 4-bit example)

1011 = 8 + 2 + 1 = 11
||||_ 1
|||__ 2
||___ no 4
|____ 8

Twos complement works by making the most significant
bit have the same value it would usually have, but
negated:

1011 = -8 + 2 + 1 = -5
||||_ 1
|||__ 2
||___ no 4
|____ -8

0011 = 2 + 1 = 3
||||_ 1
|||__ 2
||___ no 4
|____ no -8

In particular, note that every bit EXCEPT the MSB
works in exactly the same way as it did for straight
binary; the only difference is that the MSB is negative.

Now let's move to fixed-point, using 2 integer and
2 fraction bits for the example. Basically it's
just an integer scaled by 1/4:

1011 = 2 + 0.5 + 0.25 = 2.75 = 11/4
||||_ 0.25
|||__ 0.50
||___ no 1
|____ 2

And in twos complement, it's exactly the same
except that the MSB is negative:

1011 = -2 + 0.5 + 0.25 = -1.25 = -5/4
||||_ 0.25
|||__ 0.50
||___ no 1
|____ -2

0011 = 0.5 + 0.25 = 0.75 = 3/4
||||_ 0.25
|||__ 0.50
||___ no 1
|____ no -2

The fraction bits remain positive; there is no
need to introduce a special signed representation
for the fraction. As rickman said, it's precisely
a twos-complement integer - except you think of it
as being right-shifted by the number of fraction bits.
--
Jonathan Bromley, Consultant

DOULOS - Developing Design Know-how
VHDL * Verilog * SystemC * e * Perl * Tcl/Tk * Project Services

Doulos Ltd., 22 Market Place, Ringwood, BH24 1AW, UK
jonathan.bromley@MYCOMPANY.com
http://www.MYCOMPANY.com

The contents of this message may contain personal views which
are not the views of Doulos Ltd., unless specifically stated.

 

[David Bishop]

KJ wrote:

On Sep 5, 10:50 am, blackpadme <rando...@gmail.com> wrote:
Hi, i'm working with fixed point numbers which have two bits for no-
fractional part (in two complement) and eight bits for fractional
(i.e. pi/2 = 0.5703 => 00.10010010). But, what happen if i need
signed representation for the fractional part ? 11.10010010 = -1.5703.
What can i do if a need to represent -pi/2 and i need to work across
division, add, multiplicacion etc. ?


Google for "Fixed_pkg_c.vhdl" (I think it's at vhdl.org somewhere).
This relatively new package nicely handles all the bit fiddlings
required to work with fixed point numbers...saves you the trouble of
re-inventing the wheel.

You will find versions of this package tailored for several EDA tools at:
http://www.vhdl.org/fphdl/vhdl.html

 

[rickman]

On Sep 7, 4:46 am, Jonathan Bromley <jonathan.brom...@MYCOMPANY.com>
wrote:

On Sat, 6 Sep 2008 15:59:01 -0700 (PDT), rickman <gnu...@gmail.com
wrote:

what you have written above is no different
than just treating the entire number as a 2's
complement value...

Precisely.  The OP was asking about using a signed
representation for the *fraction* as well as the integer
part of a fixed-point number, and I was trying to show
why that doesn't make a lot of sense.

Or are you just messing with our heads?

Not intentionally.  I'm a little hard-pressed
to understand why my attempt to enumerate
a few values in conventional 2.8 fixed-point
signed representation is upsetting you so

The problem is that you seem to be saying that there is nothing
different about fixed point integer vs. fraction and yet, you describe
the integer as signed and the fraction as not. I can describe
integers in the exact same terms you are describing fixed point by
talking about the integer part above 8 and the integer part below 8.
It is just simple math...

10 11 = -8 + (2 + 1) = 5
|| ||_ 1
|| |__ 2
||
||____ no 4
|_____ 8

The way you are looking at it, the separation really is not at the
fixed point, it is at the ***sign bit*** -2**(n-1) + ...

The definition of 2's comp of k is 2**n - k. You talk about
interpreting the bits with odd weights, i.e. -2**(n-1) * bit (n-1)
instead. Yes, this works, but this is not the definition of 2's
complement.

1011 = -16 + 8 + 2 + 1 = 5
||||_ 1
|||__ 2
||___ no 4
|____ 8

By using this altered notation, you make the integer and fraction
*appear* different.

10.11 = -2 + 0.5 + 0.25 = -1.25 = -5/4
|| ||_ .25
|| |__ .5
||____ no 1
|_____ 2

or

10.11 = -4 + 2 + (0.5 + 0.25) = -1.25
|| ||_ .25
|| |__ .5
||____ no 1
|_____ 2

Notice that I treat *all* the bits as positive values to be added to
the -2**n value. So the full number is a ***single*** 2's complement
entity. It makes no *sense* to talk about the integer and fraction as
separate notations.

Obviously we are saying the same conclusion, that there is no need to
introduce any special handling of the fraction vs. the integer. But
in the explanation of this conclusion, you *do* exactly that, treat
the integer and fraction differently!

Rick

Perhaps this Number Representations 101
express my intent more clearly, at least
if you view it with a monospaced font:

Ordinary binary integers work like this
(using a 4-bit example)

   1011       = 8 + 2 + 1 = 11
   ||||_ 1
   |||__ 2
   ||___ no 4
   |____ 8

Twos complement works by making the most significant
bit have the same value it would usually have, but
negated:

   1011       = -8 + 2 + 1 = -5
   ||||_ 1
   |||__ 2
   ||___ no 4
   |____ -8

   0011       = 2 + 1 = 3
   ||||_ 1
   |||__ 2
   ||___ no 4
   |____ no -8

In particular, note that every bit EXCEPT the MSB
works in exactly the same way as it did for straight
binary; the only difference is that the MSB is negative.

Now let's move to fixed-point, using 2 integer and
2 fraction bits for the example.  Basically it's
just an integer scaled by 1/4:

   1011       = 2 + 0.5 + 0.25 = 2.75 = 11/4
   ||||_ 0.25
   |||__ 0.50
   ||___ no 1
   |____ 2

And in twos complement, it's exactly the same
except that the MSB is negative:

   1011       = -2 + 0.5 + 0.25 = -1.25 = -5/4
   ||||_ 0.25
   |||__ 0.50
   ||___ no 1
   |____ -2

   0011       = 0.5 + 0.25 = 0.75 = 3/4
   ||||_ 0.25
   |||__ 0.50
   ||___ no 1
   |____ no -2

The fraction bits remain positive; there is no
need to introduce a special signed representation
for the fraction.  As rickman said, it's precisely
a twos-complement integer - except you think of it
as being right-shifted by the number of fraction bits.