1Dept. of EE, NDHU

Chapter Two

Formatting and Baseband Modulation

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Digital Communication Transformation

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Formatting and Transmission of Baseband Signals

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Message, Characters, and Symbols

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Formatting Analog Information

• Formatting process

– Transform an analog waveform into a form that is compatible with a

digital communication system

• Sampling theorem

– A bandlimited signal having no spectral components above hertz

can be determined uniquely by values sampled at

, where is also called the

Nyquist rate

(2.1) sec 2

1

ms fT mf2

mf

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Impulse Sampling (Ideal Case)

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Spectra for Various Sampling Rate

Sampled spectrum (fs > 2fm)

Sampled spectrum (fs < 2fm)

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Natural Sampling

)/(csin)/1( , ssn TnTTC

n

nsnp nffCfX )()(

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Comparison of Impulse Sampling and Natural Sampling

• Impulse sampling (Ideal case)

• Natural sampling (A practical way)

n

ns

ss nffX

TX )(

11

sn

n

nss

s

n

nsns

TC

TnffXTnTcT

nffXCX

1

0 when ,)()/(sin1

)(1

10Dept. of EE, NDHU

Sample-and-Hold Operation

• Transfer function

where is the hold-operation and is the form of

• Two effects of hold-operation

– The significant attenuation of the higher frequency components

– The non-uniform spectral gain

• Post-filtering operation can compensate the effects of hold-

operation

(2.16) 1

)()(

ns

ss )X(f-nf

TfpfX

)( fp ss cfTT sin

)( fp

11Dept. of EE, NDHU

Aliasing for Sampling

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Eliminate Aliasing for Higher Sampling

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Aliasing Elimination

• Higher sampling rate

• Pre-filtering the original spectrum so that the new maximum frequency i

s reduced to fs/2 or less

• Post-filtering removes the aliased components

• Both the pre-filtering and the post-filtering will result a loss of signal inf

ormation

• Trade-off is required between the sampling rate and cutoff bandwidth

• Engineer’s version of the Nyquist sampling rate is ms ff 2.2

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Pre-filter Eliminates Alias

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Post-filter Eliminates Alias

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Alias Frequency by Sub-Nyquist Sampling Rate

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Sampling Process (I)

• Without oversampling (sampling rate is the Nyquist rate)

– The analog signal passes through a high performance analog low-pas

s filter

– Sampling rate is the Nyquist rate for the band-limited signal

– The samples are mapped to a finite list of discrete output levels and p

rocessed by the following digital signal process

18Dept. of EE, NDHU

Sampling Process (II)

• With over-sampling (sampling rate is higher than the Nyquis

t rate)

– The analog signal passes through a low performance analog low-pass

filter

– The pre-filtered signal is sampled at the higher Nyquist rate for the b

and-limited signal

– The samples are mapped to a finite list of discrete output levels and p

rocessed by a high performance digital filter to reduce the bandwidth

of the digital samples

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Analog Source Description

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Source of Corruption

• Sampling and quantizing effects

– Quantization noise due to round-off or truncation error

+ Increase the number of levels employed in the quantization process

– Quantizer saturation

+ AGC can be used to avoid the saturation

– Timing jitter

+ Stable clock

• Channel effects

– Channel noise (thermal noise, interference from other users)

– Intersymbol interference (ISI)

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Quantization Level

22Dept. of EE, NDHU

Signal to Noise Ratio for Quantized Pulse

• Assume the quantization error ,e, is uniformly distributed over a single i

nterval q-wide, the quantizer error variance is

• The peak power is

• The ratio of signal peak power to average quantization error power

12

1)(

2

2/

2/

2/

2/

222

q

deq

edeepeq

q

q

q

222 )2

(]2

)1([

LqLqVp

22

22

312/

4/)( L

q

qL

N

Sq

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Quantization Samples

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Pulse Code Modulation (PCM)

• Quantize PAM signal into a digital word

• Increase the number of levels

– Reduce the quantization noise

– Increase the number of bits per PCM sequence

– The data rate is thus increased, and the cost is a greater transmission

bandwidth

• Some communication systems can be tolerable to the time delay so that

the more quantization levels need not more bandwidth (ex: outer space

communication)

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Statistics of Speech Amplitudes

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Uniform and Non-uniform Quantization

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Quantizer Characteristics

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Compression Characteristics

Figure 2.20 Compression characteristics. (a) μ-law characteristic. (b) A-law characteristic.

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Compression Functions

• -law compression

• A-law

xxx

yy sgn)1log(

)]/(1log[ maxmax

11

sgnlog1

)]/(log[1

10 sgn

log1

)/(

max

maxmax

max

maxmax

x

x

Ax

A

xxAy

Ax

xx

A

xxAy

y

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Baseband Transmission

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Waveform Representation of Binary Digits

• Binary digits needs to be represented by physical waveform

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PCM Waveform Considerations

• DC component

– Eliminate DC energy to enable the system to be ac coupled

• Self-clocking

– Some PCM coding schemes aid in the recovery of the clock signal

• Error detection

• Bandwidth compression

– Such as multi-level codes

• Differential encoding

• Noise immunity

– Some PCM schemes have better error performance

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Spectral Densities of Various PCM Waveform

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Bits per PCM Word and Bits per Symbol

• PCM word size

– Required number of bits per analog sample for the allowable

quantization distortion

– For example, we specified the quantization error is specified not to

exceed a fraction of the peak-to-peak analog voltage ,

• Bits per symbol is decided by M-level signal transmission

p ppV

e

pppppppp pVL

V

L

V

L

Ve

2

2)1(2max

bits 2

1log levles

2

12 2 p

lp

Ll

36Dept. of EE, NDHU

Quantization Levels and Multi-level Signaling

• Example 2.3

– The information in an analog waveform, with the maximum frequency fm=3 kHz, is to

be transmitted over an M-ary PAM system, where the number of pulse levels is M=16.

The quantization distortion is specified not to exceed of the peak-to-peak analog

signal

(a) What is the minimum number of bits/sample, or bits/PCM word that should be used i

n digitizing the analog waveform?

(b) What is the minimum required sampling rate, and what is the resulting bit transmissio

n rate?

(c) What is the PAM pulse or symbol transmission rate?

(d) If the transmission bandwidth equals 12 KHz, determine the bandwidth efficiency for

this system

%1

37Dept. of EE, NDHU

Correlative Coding

• Transmit 2W symbols/s with zero ISI, using the theoretical minimum ba

ndwidth of W Hz, without infinitely sharp filters.

• Correlative coding (or duobinary signaling or partial response signaling)

introduces some controlled amount of ISI into the data stream rather than

trying to eliminate ISI completely

• Doubinary signaling

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Duobinary Decoding

• Example

– Binary digit sequence xk: 0 0 1 0 1 1 0

– Bipolar amplitudes xk : -1 -1 +1 -1 +1 +1 -1

– Coding rule yk=xk+xk-1 -2 0 0 0 2 0

– Decoding decision rule

+ If , decide that

+ If , decide that

+ If , decide opposite of the previous decision

• Error propagation could cause further errors

1ˆ kx

0ˆ kx

2ˆ ky

2ˆ ky

0ˆ ky

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Precoded Doubinary Signaling

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Duobinary Precoding

• Example

– Binary digit sequence 0 0 1 0 1 1 0

– Precoded sequence 0 0 1 1 0 1 1

– Bipolar sequence -1 -1 +1 +1 -1 +1 +1

– Coding rule -2 0 +2 0 0 +2

– Decoding decision rule

+ If , decide that

+ If , decide that

+ Decoded binary sequence 0 1 0 1 1 0

1 kkk wxw

}{ kx

}{ kw

1 kkk wwy

2ˆ ky

0ˆ ky

0ˆ kx

1ˆ kx

}ˆ{ kx

41Dept. of EE, NDHU

Duobinary Equivalent Transfer Function

fTjefH 21 1)(

elsewhere 02

1for cos2)(

that so

)(

)1(

2

1for )()()(

2

21

TfTfH

eeeT

Te

TffHfHfH

e

fTjfTjfTj

fTj

e

elsewhere 02

1for )(2 T

fTfH

)(sin)(sin)( T

Ttc

T

tcthe

42Dept. of EE, NDHU

Duobinary Transfer Function

43Dept. of EE, NDHU

Comparison of Binary with Duobinary Signaling

• Binary signaling assumes the transmitted pulse amplitude are independe

nt of one another

• Duobinary signaling introduces correlation between pulse amplitudes

• Duobinary technique achieve zero ISI signal transmission using a smalle

r system bandwidth

• Duobinary coding requires three levels, compared with the usual two lev

els for binary coding

• Duobinary signaling requires more power than binary signaling (~2.5 dB

greater SNR than binary signaling)