Chapter 2 : Formatting and Baseband Modulation

1. Chapter 2: Formatting and Baseband Modulation

2. Contents • Formatting • Sampling Theorem • Pulse Code Modulation (PCM) • Quantization • Baseband Modulation

3. Introduction

4. Introduction • Formatting • To insure that the message is compatible with digital signal • Sampling : Continuous time signal x(t)  Discrete time pulse signal x[n] • Quantization : Continuous amplitude  Discrete amplitude • Pulse coding : Map the quantized signal to binary digits • When data compression is employed in addition to formatting, the process is termed as a source coding. • Baseband Signaling: Pulse modulation • Convert binary digits to pulse waveforms • These waveforms can be transmitted over cable.

5. Baseband Systems

6. Formatting Textual Data

7. Formatting Textual Data (Cont.) • A system using a symbol set with a size of M is referred to as an M-ary system. • For k=1, the system is termed binary. • For k=2, the system is termed quaternary or 4-ary. M=2k • The value of k or M represents an important initial choice in the design of any digital communication system.

8. Formatting Analog Information • Baseband analog signal • Continuous waveform of which the spectrum extends from dc to some finite value (e.g. few MHz) • Analog waveform  Sampled version • PAM  Analog waveform FT Sampling process ((e.g.) sample-and-hold) Low pass filtering

9. Sampling Theorem • Uniform sampling theorem A bandlimited signal having no spectral components above fm hertz can be determined uniquely by values sampled at uniform intervals of

10. Nyquist Criterion • A theoretically sufficient condition to allow an analog signal to be reconstructed completely from a set of uniformly spaced discrete-time samples • Nyquist rate e.g.) speech 8kHz audio 44.1kHz

11. Impulse Sampling • In the time domain, • In the frequency domain,

12. Impulse Sampling (Cont.) FT FT FT

13. Impulse Sampling (Cont.) • The analog waveform can theoretically be completely recovered from the samples by the use of filtering (see next figure). • Aliasing If , some information will be lost. • Cf) Practical consideration • Perfectly bandlimited signals do not occur in nature. • A bandwidth can be determined beyond which the spectral components are attenuated to a level that is considered negligible.

14. Impulse Sampling (Cont.)

15. Aliasing • Due to undersampling • Appear in the frequency band between

16. Sampling and aliasing • Helicopter 100Hz • Sampling at 220Hz  100Hz • Sampling at 80Hz  20Hz • How about sampling at 120Hz? -100 0 100 f -100 120 -320 340 0 100 f 320 -340 -120 540 -20 60 -100 -180 -140 -60 20 100 f

17. Aliasing (Cont.) • Effect in the time domain

18. Natural Sampling • More practical method • A replication of X(f), periodically repeated in frequency every fs Hz. • Weighted by the Fourier series coefficients of the pulse train, compared with a constant value in the impulse-sampled case.

19. Natural Sampling (Cont.) -1/T 1/T -1/T 1/T

20. Sample-and-Hold • The simplest and most popular sampling method • Significant attenuation of the higher-frequency spectral replicates • The effect of the nonuniform spectral gain P(f) applied to the desired baseband spectrum can be reduced by postfiltering operation.

21. Oversampling • the most economic solution • Analogprocessing is much more costly than digital one.

22. Quantization • Analog  quantized pulse with “Quantization noise” • L-level uniform quantizer for a signal with a peak-to-peak range of Vpp= Vp-(-Vp) =2Vp • The quantization step • The sample value on is approximate to • The quantization error is o

23. Quantization Errors

24. Quantization Errors (Cont’d) • SNR due to quantization errors

25. Pulse Code Modulation (PCM) • Each quantization level is expressed as a codeword • L-level quantizer consists of L codewords • A codeword of L-level quantizer is represented by l-bit binary digits. • PCM • Encoding of each quantized sample into a digital word

26. Pulse Code Modulation (PCM)

27. PCM Word Size • How many bits shall we assign to each analog sample? • The choice of the number of levels, or bits per sample, depends on how much quantization distortion we are willing to tolerate with the PCM format. • The quantization distortion error p = 1/(2 # of levels)

28. Uniform Quantization • The steps are uniform in size. • The quantization noise is the same for all signal magnitudes. • SNR is worse for low-level signals than for high-level signals.

29. Uniform Quantization (Cont.)

30. Nonuniform Quantization • For speech signals, many of the quantizing steps are rarely used. • Provide fine quantization to the weak signals and coarse quantization to the strong signals. • Improve the overall SNR by • The more frequent  the more accurate • The less frequent  the less accurate • But, we only have uniform quantizer. Probability Volt

31. Nonuniform Quantization (Cont.) • Companding = compress + expand • Achieved by first distorting the original signal with a logarithmic compression characteristic and then using a uniform quantizer. • At the receiver, an inverse compression characteristic, called expansion, is applied so that the overall transmission is not distorted. Probability Probability o o Volt Volt x[n] x[n]’ Compression Quantization Expansion

32. Nonuniform Quantization (Cont.) Ξ

33. Baseband Transmission • Digits are just abstractions: a way to describe the message information. • Need something physical that will represent or carry the digits. • Represent the binary digits with electrical pulses in order to transmit them through a (baseband) channel. • Binary pulse modulation: PCM waveforms • Ex) RZ, NRZ, Phase-encoded, Multi-level binary • M-ary pulse modulation : M possible symbols • Ex) PAM, PPM (Pulse position modulation), PDM (pulse duration modulation)

34. PCM Waveform Types • Nonreturn-to-zero (NRZ) • NRZ-L, NRZ-M (differential encoding), NRZ-S • Return-to-zero (RZ) • Unipolar-RZ, bipolar-RZ, RZ-AMI • Phase encoded • bi--L (Manchester coding), bi--M, bi--S • Multilevel binary • Many binary waveforms use three levels, instead of two.

35. PCM Waveforms (Fig. 2.22)

36. Choosing a PCM Waveform • DC component • Magnetic recording prefers no dc (Delay, Manchester..) • Self-clocking • Manchester • Error detection • Duobinary (dicode) • Bandwidth compression • Duobinary • Differential encoding • Noise immunity

37. Spectral Densities of PCM Waveforms • Normalized bandwidth • Can be expressed as W/RS [Hz/(symbol/s)] or WT • Describes how efficiently the transmission bandwidth is being utilized • Any waveform type that requires less than 1.0Hz for sending 1 symbol/s is relatively bandwidth efficient. • Bandwidth efficiency R/W [bits/s/Hz]

38. Spectral Densities of PCM Waveforms Bandwidth efficiency Mean = 0  DC=0 R/W [bits/s/Hz] Transition period↑ W ↓ FT

39. Autocorrelation of Bipolar Note that where xi+1 xi xi+2 xi-1 x(t) 0 To 2To xi xi+1 xi+2 xi-1 x(t+Ʈ) -Ʈ To-Ʈ 2To-Ʈ A2 A2 Rx(Ʈ) -Ʈ 0 To To-Ʈ 2To 2To-Ʈ

40. Autocorrelation of Bipolar Note that where xi+1 xi xi+2 xi-1 x(t) 2To 0 To xi xi+1 xi+2 xi-1 x(t+Ʈ) -Ʈ To-Ʈ 2To-Ʈ A2 A2 A2 Rx(Ʈ) -Ʈ 0 To To-Ʈ 2To 2To-Ʈ

41. Autocorrelation and power spectrum density of Bipolar Gx(f) 1 FT 0 0 0

42. Rectangular function Rect(t) FT 0

43. M-ary Pulse-Modulation • M=2k-levels : k data bits per symbol  longer transition period  narrower W  more channel • Kinds of M-ary pulse modulation • Pulse-amplitude modulation (PAM) • Pulse-position modulation (PPM) • Pulse-duration modulation (PDM) • M-ary versus Binary • Bandwidth and transmission delay tradeoff • Performance and complexity tradeoff

44. Example. Quantization Levels and Multilevel Signaling • Analog waveform fm=3kHz, 16-ary PAM, quantization distortion  1% (a) Minimum number of bits/sample? |e|= 1/2L < 0.01  L>50  L=64  6bits/sample (b) Minimum required sampling rate? fs > 2*3kHz = 6kHz (c) Symbol transmission rate? 9103symbols/sec 6*6 kbps = 36kbps  36[kbps]/4[bits/symbol] (d) If W=12kHz, the bandwidth efficiency? R/W = 36kbps / 12kHz = 3bps/Hz

45. HW#3 • P2.2 • P2.8 • P2.9 • P2.14 • P2.18