Analog to Digital and Digital to Analog Conversion - Digital Signal Processing Laboratory - page 58

4.1.1.3 Binary Encoding
The third and last stage in the A/D process is binary encoding, where the
quantized signal , n = 0,1,2 … N - 1 is encoded to yield the final digital
signal , n = 0,1,2 … N - 1. As an illustration, if the number of quantizer
levels L = 8 = 2 3 , the number of binary bits required to encode all the L levels
is 3. Table 4.1 illustrates the encoding procedure, using two's complement
coding, which is very convenient in the decoding process at the receiver.
The two's complement code (TCC) is quite easily obtained from the offset
binary code (OBC), by complementing the left-most bit of the OBC . Some of the
advantages of the two's complement code are as follows:

ˆ ()

xn

ˆ

xn
B

()

• The decimal form of the TCC includes both positive and negative

numbers and is given by the following equation:

Decimal number = - a 0 2 0 + a 1 2 -1 a 2 2 -2 + … a B 2 - B (4.9)

where the original binary number is [ a 0 a 1 a 2 a B … a B ].

• The decoding process at the receiver is more efficient, as illustrated

in Table 4.2.

T ABLE 4.1
Binary Encoding Process
Quantizer Level
(for L= 8)

Offset Binary Code
(3-Bit)

Two's Complement Code
( 3 -Bit)

-4 000 100

-3 001 101

-2 010 110

-1 011 111

0 100 000

1 101 001

2 110 010

3 111 011

T ABLE 4.2
Binary Decoding Process
Two's Complement Code
(3-Bit)

Decimal Value
(from Equation 4.9)

Actual
Quantizer Level

100 -1 -4

101 -3/4 -3

110 -1/2 -2

111 -1/4 -1

000 0 0

001 1/4 1

010 1/2 2

011 3/4 3