248 8. ANALOG PERIPHERALS

We can now combine both sampling and quantization processes together to identify a quan-

tized level of a sampled value. Suppose a sampled analog signal value is 3.4V and the input signal

range is 0V to 5V. Using a converter with 8 bits, we can ﬁnd the quantized level of the sampled value

using the following equation.

Quantized level =

s a mp l e d i np ut v a l ue – lowest possible input value

δ

Thus, given the input sample value of 3.4V, the quantized level becomes

3.4V −0 V

19.53 mV

∼

=

174.09.

Since we can only have integer levels, the quantized level becomes 174. So the sampling error is the

difference between the true analog value and the sampled value. It is the amount of approximation

the converter had to make. See Fig. 8.4 for a pictorial view of this concept. For the example, the

input sampled value 3.4 V is represented as the quantized level 174 and the quantized error is

0.09 × δ = 1.76 mV . Note that the maximum quantization error is the resolution of the converter.

Example: Given a sampled signal value of 7.21 V, using a 10 bit ADC converter with input

range of 0V and 10V, ﬁnd the corresponding quantization level and the associated quantization error.

Answer: First, we ﬁnd the quantized level.

Quantized level =

7.21 − 0

δ

where δ =

10

2

10

= 9.77 mV . Thus, the quantized level is 738.3059. Since we always round

down, the quantized level is 738 and the associated quantization error is 0.3059 × 9.77 mV

∼

=

2.987 mV .

8.3.3 ENCODING

The last step of the ADC conversion process is the encoding. The encoding process converts the

quantized level of a sampled analog signal value into a binary number representation. Consider the

following simple case, ﬁrst. Suppose we have a converter with four bits. The available quantization

levels for this converter are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. Using four bits, we can

represent the quantization levels as 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001,

1010, 1011, 1100, 1101, 1110, and 1111. Once we identify a quantization level, we can uniquely

represent the quantization level as a binary number. This process is called encoding. Similar to a

decimal number, the position of each bit in a binary number represents a different value. For example,

binary number 1100 is decimal number (1 × 2

3

) + (1 × 2

2

) + (0 × 2

1

) + (0 × 2

0

) = 12.

Knowing the weight of each bit, it is a straight forward process to represent a decimal number as a

binary number.

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