1. ECE 4371, 2009 Class 9 Zhu Han Department of Electrical and Computer Engineering Class 9 Sep. 22 nd, 2009

2. Quantization Scalar Quantizer Block Diagram Mid-tread Mid-rise

3. Equations

4. Quantization Noise

5. Quantization Noise Level

6. Quantization SNR, 6dB per bit

7. Example SNR for varying number of representation levels for sinusoidal modulation 1.8+6 X dB, example 3.1 Number of representation level L Number of Bits per Sample, R SNR (dB) 32531.8 64637.8 128743.8 256849.8

8. Conditions for Optimality of Scalar Quantizers Let m(t) be a message signal drawn from a stationary process M(t) -A  m  A m 1 = -A m L+1 =A m k  m k+1 for k=1,2,…., L The kth partition cell is defined as J k : m k < m  m k+1 for k=1,2,…., L d(m, v k ): distortion measure for using v k to represent values inside J k.

9. Condition for Optimal Quantizer

10. Condition One

11. Condition Two

12. Vector Quantization

13. Vector Quantization image and voice compression, voice recognition statistical pattern recognition volume rendering

14. Rate Distortion Curve Rate: How many codewords (bits) are used? –Example: 16-bit audio vs. 8- bit PCM speech Distortion: How much distortion is introduced? –Example: mean absolute difference(L 1 ), mean square error (L 2 ) Vector Quantizer often performs better than Scalar Quantizer with the cost of complexity Rate (bps) Distortion SQ VQ

15. Non-uniform Quantization Motivation –Speech signals have the characteristic that small- amplitude samples occur more frequently than large-amplitude ones –Human auditory system exhibits a logarithmic sensitivity u More sensitive at small- amplitude range (e.g., 0 might sound different from 0.1) u Less sensitive at large- amplitude range (e.g., 0.7 might not sound different much from 0.8) histogram of typical speech signals

16. Non-uniform Quantizer x Q x ^ FF -1 Example F: y=log(x)F -1 : x=exp(x) y y ^ F: nonlinear compressing function F -1 : nonlinear expanding function F and F -1 : nonlinear compander We will study nonuniform quantization by PCM example next A law and  law

17.  Law/A Law The  -law algorithm (μ-law) is a companding algorithm, primarily used in the digital telecommunication systems of North America and Japan. Its purpose is to reduce the dynamic range of an audio signal. In the analog domain, this can increase the signal to noise ratio achieved during transmission, and in the digital domain, it can reduce the quantization error (hence increasing signal to quantization noise ratio).compandingdigitaltelecommunication North AmericaJapandynamic rangesignal A-law algorithm used in the rest of worlds. A-law algorithm A-law algorithm provides a slightly larger dynamic range than the mu-law at the cost of worse proportional distortion for small signals. By convention, A-law is used for an international connection if at least one country uses it.

18.  Law

19. A Law

20.  Law/A Law

21. Analog to Digital Converter Main characteristics –Resolution and Dynamic range : how many bits –Conversion time and Bandwidth: sampling rate Linearity –Integral –Differential Different types –Successive approximation –Slope integration –Flash ADC –Sigma Delta

22. Successive approximation Compare the signal with an n-bit DAC output Change the code until –DAC output = ADC input An n-bit conversion requires n steps Requires a Start and an End signals Typical conversion time –1 to 50  s Typical resolution –8 to 12 bits Cost –15 to 600 CHF

23. Single slope integration Start to charge a capacitor at constant current Count clock ticks during this time Stop when the capacitor voltage reaches the input Cannot reach high resolution –capacitor –comparator - + IN C R S Enable N-bit Output Q Oscillator Clk Counter Start Conversion Vin Counting time

24. Flash ADC Direct measurement with 2n-1 comparators Typical performance: –4 to 10-12 bits –15 to 300 MHz –High power Half-Flash ADC –2-step technique u 1st flash conversion with 1/2 the precision u Subtracted with a DAC u New flash conversion Waveform digitizing applications

25. Sigma-Delta ADC

26. Over-sampling ADC Hence it is possible to increase the resolution by increasing the sampling frequency and filtering Reason is the noise level reduce by over sampling. Example : –an 8-bit ADC becomes a 9-bit ADC with an over-sampling factor of 4 –But the 8-bit ADC must meet the linearity requirement of a 9-bit

27. Resolution/Throughput Rate

28. Digital to Analog conversion DAC DAC Input code = n 0110001 0100010 0100100 0101011 : Output voltage = V out (n) V +ref V -ref

29. Digital to Analog Converter Pulse Width Modulator DAC Pulse Width Modulator Delta-Sigma DAC Binary Weighted DAC R-2R Ladder DAC R-2R Ladder Thermometer coded DAC Segmented DAC Hybrid DAC

30. Pulse Code Modulation (PCM) Pulse code modulation (PCM) is produced by analog-to-digital conversion process. Quantized PAM As in the case of other pulse modulation techniques, the rate at which samples are taken and encoded must conform to the Nyquist sampling rate. The sampling rate must be greater than, twice the highest frequency in the analog signal, f s > 2f A (max) Telegraph time-division multiplex (TDM) was conveyed as early as 1853, by the American inventor M.B. Farmer. The electrical engineer W.M. Miner, in 1903. PCM was invented by the British engineer Alec Reeves in 1937 in France.Alec Reeves1937 It was not until about the middle of 1943 that the Bell Labs people became aware of the use of PCM binary coding as already proposed by Alec Reeves.Bell Labs

31. Figure The basic elements of a PCM system. Pulse Code Modulation

32. Encoding

33. Advantages of PCM 1. Robustness to noise and interference 2. Efficient regeneration 3. Efficient SNR and bandwidth trade-off 4. Uniform format 5. Ease add and drop 6. Secure DS0: a basic digital signaling rate of 64 kbit/s. To carry a typical phone call, the audio sound is digitized at an 8 kHz sample rate using 8-bit pulse-code modulation. 4K baseband, 8*6+1.8 dBdigitalsignalingkbit/skHzpulse-code modulation Virtues, Limitations and Modifications of PCM

34. 0000 1111 1110 1101 1100 1011 1010 1001 0001 0010 0011 0100 0101 0110 0111 0000011001110011110010011011 Numbers passed from ADC to computer to represent analogue voltage Resolution= 1 part in 2 n PCM