Chapter 2: Formatting and Baseband Modulation

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

Introduction

Introduction Formatting Baseband Signaling: Pulse modulation 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.

Baseband Systems

Formatting Textual Data

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.

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

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

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 fm fs Speech 3.2kHz (사람의 한계) 8 kHz (휴대폰) Audio 20kHz (가청주파수) 44.1kHz (mp3) e.g.) speech 8kHz audio 44.1kHz

Impulse Sampling In the time domain, In the frequency domain,

Impulse Sampling (Cont.) FT FT FT

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.

Impulse Sampling (Cont.)

Aliasing Due to undersampling Appear in the frequency band between

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

Aliasing (Cont.) Effect in the time domain

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.

Natural Sampling (Cont.)

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.

Oversampling the most economic solution Analog processing is much more costly than digital one.

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

Quantization Errors

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

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

Pulse Code Modulation (PCM)

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)

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.

Uniform Quantization (Cont.)

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

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

Nonuniform Quantization (Cont.) Ξ

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)

PCM Waveform Types Nonreturn-to-zero (NRZ) Return-to-zero (RZ) 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.

PCM Waveforms (Fig. 2.22)

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

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]

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

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

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

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

Rectangular function Rect(t)   FT  

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

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

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