ELEN 33 Introduction to Digital Signal Processing Systems
Lecture 6: Floating Point Notes for TI formats
10 April 2000
Floating Point Representation
Values are written in scientific notation and saved as two numbers. One is the
exponent. The other is the magnitude and associated sign of the number.
Normalization to guarantee uniqueness
3.25*104=32.5*103=325000*10-1=0.00325*107
Increase dynamic range
Companding=Compressing and expanding
Use small amplitude steps for low amplitude and larger steps for high amplitude.
Floating Point Formats for TMS320C31:
TMS320C3x - Short
Internal
P4-5 User's Guide
exp s
15
fraction
11
10
0
The exponent is 4 bit 2's complement.
The mantissa is 12 bit 2's complement with an implied most significant nonsign bit. The
implied binary point is between bits 11 and 10. The value is:
0
if e=-8
01.f x 2e if s=0
10.f x 2e if s=1
TMS320C3x - Single
Precision
P4-6 User's Guide
exp
31
s
fraction
24 23 22
0
The exponent is 8 bit 2's complement.
The mantissa is 24 bit 2's complement with an implied most significant nonsign bit. The
implied binary point is between bits 23 and 22. The value is:
0
if e=-128
01.f x 2e if s=0
10.f x 2e if s=1
Copyright March 2000, Sally L. Wood
Page 1
ELEN 033 Lecture 6 Notes
TMS320C3x Extended Precision
P4-7 User's Guide
exp
39
s
fraction
32 31 30
0
The exponent is 8 bit 2's complement.
The mantissa is 32 bit 2's complement with an implied most significant nonsign bit. The
implied binary point is between bits 31 and 30. The value is:
0
if e=-128
01.f x 2e if s=0
10.f x 2e if s=1
Examples of 32 bit single precision
Normalized
Exp bits Sign Fraction Bits 22-0
X*2exp
31-24
bit 23
Decimal
1
2
3
4
5
6
7
8
9
Binary
00001.000
00010.000
00011.000
00100.000
00101.000
00110.000
00111.000
01000.000
01001.000
X
00001.000
00001.000
00001.100
00001.000
00001.010
00001.100
00001.110
00001.000
00001.001
exp
0
1
1
2
2
2
2
3
3
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
0000 0000
0000 0001
0000 0001
0000 0010
0000 0010
0000 0010
0000 0010
0000 0011
0000 0011
0
0
0
0
0
0
0
0
0
000 0000 0000 0000 0000 0000
000 0000 0000 0000 0000 0000
100 0000 0000 0000 0000 0000
000 0000 0000 0000 0000 0000
010 0000 0000 0000 0000 0000
100 0000 0000 0000 0000 0000
110 0000 0000 0000 0000 0000
000 0000 0000 0000 0000 0000
001 0000 0000 0000 0000 0000
1000 0000
11111.000
11110.000
11101.000
11100.000
11011.000
11010.000
11001.000
11000.000
10111.000
11110.000
11110.000
11110.100
11110.000
11110.110
11110.100
11110.010
11110.000
11110.111
-1
0
1
1
2
2
2
2
3
Copyright March 2000, Sally L. Wood
1111 1111
0000 0000
0000 0001
0000 0001
0000 0010
0000 0010
0000 0010
0000 0010
0000 0011
Page 2
Hexadecimal
00 00 00 00
01 00 00 00
01 40 00 00
02 00 00 00
02 20 00 00
02 40 00 00
02 60 00 00
03 00 00 00
03 10 00 00
80 00 00 00
1
1
1
1
1
1
1
1
1
000 0000 0000 0000 0000 0000
000 0000 0000 0000 0000 0000
100 0000 0000 0000 0000 0000
000 0000 0000 0000 0000 0000
110 0000 0000 0000 0000 0000
100 0000 0000 0000 0000 0000
010 0000 0000 0000 0000 0000
000 0000 0000 0000 0000 0000
111 0000 0000 0000 0000 0000
ELEN 033 Lecture 6 Notes
FF 80 00 00
00 80 00 00
01 C0 00 00
01 80 00 00
02 E0 00 00
02 C0 00 00
02 A0 00 00
02 80 00 00
03 F0 00 00
mu-law and A-law:
Companding using 8 bits to represent values in a manner similar to floating point formats
described above. Input values over the range -1 to 1are mapped to 8-bit values with a step
size of approximately 0.00025 at the lowest amplitude and a step size of approximately
0.032 at the largest amplitude. There is an algorithm that converts from linear to mu law
encoding, and there is a Matlab function that will implement this. This representation is
not suitable for computation because it uses so few bits, but it is appropriate for
increasing the dynamic range of an audio signal using only 8 bits.
The most significant bit of the mu-law representation is the sign bit, and the exponent is 3
bits. The representation looks more like the other formats if it is bitwise complemented.
xmu=0:255;
xlin=mu2lin(xmu);
xmuc=255-xmu;
The table shows positive values (for the MSB of xmuc=0) represented by 128 of themulaw codes. Xmuc=16*xmucH+xmucL. Negative values have the same magnitudes.
xmucH
xmucL
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
.0002
.0005
.0007
.0040
.0045
.0050
.0121
.0131
.0140
.0282
.0302
.0604
.1249
.2538
.5116
.5428
.5741
.6053
.0032
.0034
.0037
.0114
Copyright March 2000, Sally L. Wood
.4882
Page 3
.9491
.9803
ELEN 033 Lecture 6 Notes