XỬ LÝ TÍN HIỆU SỐ Dsp chapter2 student 22062015

Slide 1 Click to edit Master subtitle style Nguyen Thanh Tuan, M Eng Department of Telecommunications (113B3) Ho Chi Minh City University of Technology Email nttbk97@yahoo com Quantization Chapter 2 D[.]

Chapter Quantization Nguyen Thanh Tuan, Click M.Eng to edit Master subtitle style Department of Telecommunications (113B3) Ho Chi Minh City University of Technology Email: nttbk97@yahoo.com Quantization process Fig: Analog to digital conversion  The quantized sample xQ(nT) is represented by B bit, which can take 2B possible values  An A/D is characterized by a full-scale range R which is divided into 2B quantization levels Typical values of R in practice are between 1-10 volts Digital Signal Processing Quantization Quantization process Fig: Signal quantization  Quantizer resolution or quantization width (step) Q  R R  A bipolar ADC   xQ (nT )  2 R 2B  A unipolar ADC  xQ (nT )  R Digital Signal Processing Quantization Quantization process  Quantization by rounding: replace each value x(nT) by the nearest quantization level  Quantization by truncation: replace each value x(nT) by its below nearest quantization level  Quantization error: e(nT )  xQ (nT )  x(nT )  Consider rounding quantization:  Q Q e 2 Fig: Uniform probability density of quantization error Digital Signal Processing Quantization Quantization process  The mean value of quantization error e  Q /2  Q /2 ep(e)de   Q /2  e  Q /2 Q /2 de 0 Q Q /2 Q2  The mean-square error (power)   e   e p(e)de   e de  Q 12  Q /2  Q /2 2  Root-mean-square (rms) error: erms    e2  Q 12  R and Q are the ranges of the signal and quantization noise, then the signal to noise ratio (SNR) or dynamic range of the quantizer is defined as R SNR dB  20log10    20log10 (2 B )  B log10 (2)  B dB Q which is referred to as dB bit rule Digital Signal Processing Quantization Example  In a digital audio application, the signal is sampled at a rate of 44 KHz and each sample quantized using an A/D converter having a full-scale range of 10 volts Determine the number of bits B if the rms quantization error mush be kept below 50 microvolts Then, determine the actual rms error and the bit rate in bits per second Digital Signal Processing Quantization Digital to Analog Converters (DACs)  We begin with A/D converters, because they are used as the building blocks of successive approximation ADCs Fig: B-bit D/A converter  Vector B input bits : b=[b1, b2,…,bB] Note that bB is the least significant bit (LSB) while b1 is the most significant bit (MSB)  For unipolar signal, xQ є [0, R); for bipolar xQ є [-R/2, R/2) Digital Signal Processing Quantization DACs Rf  Full scale R=VREF, B=4 bit 2Rf 4Rf I 8Rf MSB i xQ=Vout 16Rf bB b1 LSB -VREF Fig: DAC using binary weighted resistor  b1 b3 b2 b4 I  V      REF  2R 4R 8R 16R f f f  f     b1 b2 b3 b4  xQ  VOUT   I  R f  VREF       16  xQ  R24  b1 23  b2 22  b3 21  b4 20   Q  b1 23  b2 22  b3 21  b4 20  Digital Signal Processing Quantization DACs  Unipolar natural binary xQ  R(b1 21  b2 22   bB 2 B )  Qm where m is the integer whose binary representation is b=[b1, b2,…,bB] m  b1 2B1  b2 2B2   bB 20  Bipolar offset binary: obtained by shifting the xQ of unipolar natural binary converter by half-scale R/2: R R xQ  R(b1  b2   bB )   Qm  2 1 2 B  Two’s complement code: obtained from the offset binary code by complementing the most significant bit, i.e., replacing b1 by b1   b1 R xQ  R(b1  b2   bB )  1 Digital Signal Processing 2 B Quantization Example  A 4-bit D/A converter has a full-scale R=10 volts Find the quantized analog values for the following cases ? a) Natural binary with the input bits b=[1001] ? b) Offset binary with the input bits b=[1011] ? c) Two’s complement binary with the input bits b=[1101] ? Digital Signal Processing 10 Quantization A/D converter  For rounding quantization, we shift x by Q/2: Digital Signal Processing 15  For the two’s complement code, the sign bit b1 is treated separately Quantization Example  Consider a 4-bit ADC with the full-scale R=10 volts Using the successive approximation algorithm to find offset and two’s complement of rounding quantization for the analog values x=3.5 volts Digital Signal Processing 16 Quantization Oversampling noise shaping  e2 fs Pee(f)  e'2 f s' e(n) -f’s/2 -fs/2 fs/2 f’s/2 '2  e2  e'2   '   e2  f s e' fs fs fs Digital Signal Processing HNS(f) f x(n) 17 ε(n) xQ(n) Quantization Oversampling noise shaping Digital Signal Processing 18 Quantization Dither Digital Signal Processing 19 Quantization Uniform and non-uniform quantization Digital Signal Processing 20 Quantization