1. Quantization
Prof. Siripong Potisuk

2. A Block Diagram of a DSP System2

3. Analog-to-Digital Conversion (ADC)
Discretize the independent variable or time of an analog signal (Sampling)
Discretize the dependent variable or amplitude of an analog signal by rounding off to the nearest integer (Quantization)
Each quantization level represented using binary encoding scheme (Encoding)
Flash, Successive approximation, Sigma-delta

4. A Typical ADC Process The process of converting analog
voltage with infinite precision to finite
precision is called the quantization process.

5. Analog-to-Digital Conversion (ADC)

6. Quantizer Input-output Characteristics
Similar to passing a discrete-time signal through a piecewise constant staircase type function
2 types: mid-tread and mid-rise

7. 7

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9. Quantization Level
Suppose the value of x[n] ranges over the interval [xmin, xmax]. The spacing between adjacent quantization level or step size (ADC resolution) is
L = # of quantization levels
N = # of binary bits used to represent the value of x[n]
The resulting quantization level, xq , is
i is an index corresponding to the binary code

10. Example
A speech signal has a maximum frequency of 4 kHz. We want to digitize it and send it in a file using 2 bytes (i.e., 16 bits) per sample. What would be the minimum length of the file occupied by the signal for each minute of recording? Assume the signal is not compressed. 10

11. Quantization Error Also known as quantization noise
Modeled as a random variable uniformly distributed over the interval [-D/2, D/2] with probability density p(eq) = 1/D.
The average power of the quantization noise is

13. Example 2.9 & 2.10 (text)
Assuming that a 3-bit ADC accepts an analog input
ranging from 0 to 5 volts, determine
(a) the number of quantization levels;
(b) the step size or resolution of the quantizer;
(c) the quantization level corresponding to the
analog value of 3.2 volts;
(d) the binary code produced by the encoder.
(e) the quantization error corresponding to the 3.2-V
analog input. 13

14. Problem 2.27 (text) Assuming that a 3-bit ADC accepts an analog input
ranging from -2.5 to 2.5 volts, determine
(a) the number of quantization levels;
(b) the step size or resolution of the quantizer;
(c) the quantization level corresponding to the
analog value of -1.2 volts;
(d) the binary code produced by the encoder.
(e) the quantization error corresponding to the
analog input. 14

15. Example
Suppose the amplitude of a discrete-time signal x[n] is constrained to lie in the interval [-10, 10]. If the average power of the quantization noise is to be less than 0.001, what is the minimum number of bits that are needed to represent the value of x[n]?

16. Signal-to-quantization Noise Ratio (SNRq)
A figure of merit expressed in terms of the ratio between signal power and the quantization noise power
Usually expressed in decibels (dB)

17. For a full-scale sinusoidal signal with amplitude A,
Special Case: The signal has a full-scale dynamic range
Thus,
For a full-scale sinusoidal signal with amplitude A,
Increasing 1 bit of the ADC quantizer can improve SNRq by 6 dB  The 6-dB rule

18. Example In a DSP system, the output SNRq is to be held to a
minimum of 40 dB. Determine the number of required quantization levels, and the corresponding
SNRq assuming a full-scale sinusoidal input. 18

19. Flash ADC Unit One of several ways to implement ADC
Consists of a series of reference voltages created by equal resistors
A set of comparators is used to compare the input voltage with the reference voltages
An encoding logic unit outputs the binary sequence
Offers high conversion speed
Impractical for high-resolution applications

20. An Example of a Simple 2-bit Flash ADC

21. Non-uniform Quantization
Needed for signals whose smaller amplitudes predominate and larger ones are rare (i.e.,
speech)
Difficult to design

22. Companding
Pre- and post-processing applied to a uniform quantizer to achieve non-uniform quantization
The combination of compression and expansion
The signal samples first compressed before passing through a uniform quantizer
To restore the signal samples to their correct relative level, an expander with a characteristic complementary to that of the compressor is used in the receiver

23. Characteristics of a compressor

24. Compression Law
 -law (used in digital telephony in North America & Japan with  = 255)
A-law (used in Europe with A = 87.6)