1. CEN352 Digital Signal Processing Lecture No. 6 Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Kingdom of Saudi Arabia September 24 th, 2012 الدكتور / انور مجيد ميرزا

2. Quantization A digital signal is a sequence of numbers (samples) in which each number is represented by a finite number of digits (finite precision i.e. finite number of digits). quantization The process of converting a discrete-time continuous-amplitude signal into a digital signal by expressing each sample value as a finite (instead of an infinite) number of digits is called quantization. quantization error quantization noise The error introduced in representing the continuous-valued signal by a finite set of discrete value levels is called quantization error or quantization noise. 2

3. Quantization: Example 3 Analog signal

4. Quantization: Example 4 Analog signal Discrete-time signal (Truncation)(Rounding) 01 10.9 20.81 30.729 40.6561 50.59049 60.531441 70.4782969 80.43046721 90.387420489 Table: Table: Numerical Illustration of Quantization with One Significant Digit using Truncation or Rounding

5. Quantization: Example 5 Analog signal Discrete-time signal

6. Quantization: Example 6 Analog signal Discrete-time signal

7. Quantization: Example 7 Analog signal Discrete-time signal (Truncation)(Rounding) 011.0 10.9 20.810.8 30.7290.7 40.65610.60.7 50.590490.50.6 60.5314410.5 70.47829690.40.5 80.430467210.4 90.3874204890.30.4 Table: Table: Numerical Illustration of Quantization with One Significant Digit using Truncation or Rounding

8. Quantization: Example 8 Analog signal Discrete-time signal

9. Quantization: Example 9 Analog signal Discrete-time signal (Truncation)(Rounding) 011.0 0.0 10.9 0.0 20.810.8 -0.01 30.7290.7 -0.029 40.65610.60.7 0.0439 50.590490.50.6 0.00951 60.5314410.5 -0.031441 70.47829690.40.5 0.0217031 80.430467210.4 -0.03046721 90.3874204890.30.4 0.012579511 Table: Table: Numerical Illustration of Quantization with One Significant Digit using Truncation or Rounding

10. Quantization: How it done? 10 Given an analog signal x(t) and the number of bits m in an ADC Find out the number of quantization levels: L=2 m Find out the step size of the quantizer: Δ = (xmax ‒ xmin)/L The index corresponding to the binary code is: i = round(x ‒ xmin)/ Δ Values of the quantization levels: xq = xmin + i Δ, where i=0,1,2,…,L

11. Assuming that a 3-bit ADC channel accepts analog input ranging from 0 to 5 volts, determine the following: a. Number of quantization levels b. Step size of the quantizer or resolution c. Quantization level when the analog voltage is 3.2 volts d. Binary code produced by the ADC Solution In this case x min = 0 volts, x max = 5 volts and m = 3 bits. Part(a): Part(a): The number of quantization levels are given by L=2 m = 2 3 = 8. Part (b): Part (b): Step size is given by Δ = ( x max – x min )/L = (5 – 0)/8 = 0.625 volts Part(c): Part(c): The index to the quantization level is given by i = round((x – x min )/ Δ ) = round((3.2 – 0)/0.625) = round(5.12) = 5 Therefore, the quantization level for analog voltage 3.2 volts is x q = x min + i* Δ = 0 + 5 * 0.625 = 3.125 Part (d): Part (d): The 3-bit binary code corresponding to level 5 is 101. 11 Quantization: Example 2.9

12. Signal to Quantization Noise The error introduced by the quantization process is given by The signal to quantization noise ratio (SQNR or simply SNR) is given by The signal to quantization noise ratio (SQNR or simply SNR) in decibels (dB) is given by 12 Quantization

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