DC Chapter2b

7/28/2019 DC Chapter2b

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Practical Sampling Rates Speech

- Telephone quality speech has a bandwidth of 4 kHz(actually 300 to 3300Hz)

- Most digital telephone systems are sampled at 8000samples/sec

Audio:

- The highest frequency the human ear can hear isapproximately 15kHz

- CD quality audio are sampled at rate of 44,000

samples/sec Video

- The human eye requires samples at a rate of at

least 20 frames/sec to achieve smooth motion


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2.6 Pulse Code Modulation (PCM)

Pulse Code Modulation refers to a digital baseband signal that is

generated directly from the quantizer output

Sometimes the term PCM is used interchangeably with quantization


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See Figure 2.16 (Page 80)


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Advantages of PCM:

Relatively inexpensive

Easily multiplexed: PCM waveforms from different

sources can be transmitted over a common digital

channel (TDM)

Easily regenerated: useful for long-distance

communication, e.g. telephone

Better noise performance than analog system

Signals may be stored and time-scaled efficiently (e.g.,

satellite communication)

Efficient codes are readily available

Disadvantage:

Requires wider bandwidth than analog signals


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2.5 Sources of Corruption in the sampled,quantized and transmitted pulses Sampling and Quantization Effects

Quantization (Granularity) Noise: Results when

quantization levels are not finely spaced apart enoughto accurately approximate input signal resulting intruncation or rounding error.

Quantizer Saturation or Overload Noise: Results when

input signal is larger in magnitude than highestquantization level resulting in clipping of the signal.

Timing Jitter: Error caused by a shift in the sampler

position. Can be isolated with stable clock reference. Channel Effects

Channel Noise

Intersymbol Interference (ISI)


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The level of quantization noise is dependent on how close any

particular sample is to one of the L levels in the converter

For a speech input, this quantization error resembles a noise-like disturbance at the output of a DAC converter

Signal to Quantization Noise Ratio


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2.7 Uniform and Nonuniform Quantization A quantizer with equal quantization level is a Uniform Quantizer Each sample is approximated within a quantile interval

Uniform quantizers are optimal when the input distribution isuniform

i.e. when all values within the range areequally likely

Most ADCs are implemented using uniform quantizers

Error of a uniform quantizer is bounded by

2 2

q qe <


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The mean-squared value (noise variance) of the quantization error is

given by:

/ 2 / 2 / 22 2 2

/ 2 / 2 / 2

1 1( )2

q q q

q q q

e p e de e de e deq q

= =

=

3 / 2

/ 2

21 3 12

q

qqeq = =

Signal to Quantization Noise Ratio


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The peak power of the analog signal (normalized to 1Ohms )can beexpressed as:

Therefore the Signal to Quatization Noise Ratio is given by:

22 2 2

2 41

pppVV L q

P

= = =

2 2

2

/ 4

/ 12

23q

L q

qS N R L==


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where L = 2n

is the number of quantization levels for the converter.(n is the number of bits).

Since L = 2n

, SNR = 22n

or in decibels

p pVq =

210log (2 ) 6

10

nS n dB

N dB

= =

Ifq is the step size, then the maximum quantization error that canoccur in the sampled output of an A/D converter is q


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Nonuniform Quantization Nonuniform quantizers have unequally spaced levels

The spacing can be chosen to optimize the Signal-to-Noise Ratiofor a particular type of signal

It is characterized by:

Variable step size Quantizer size depend on signal size


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Many signals such as speech have a nonuniform distribution

See Figure on next page (Fig. 2.17)

Basic principle is to use more levels at regions with large probability

density function (pdf)

Concentrate quantization levels in areas of largest pdf

Or use fine quantization (small step size) for weak signals and

coarse quantization (large step size) for strong signals


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Statistics of speech Signal Amplitudes

Figure 2.17: Statistical distribution of single talker speech signal

magnitudes (Page 81)


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Nonuniform quantization using companding

Companding is a method of reducing the number of bits required inADC while achieving an equivalent dynamic range or SQNR

In order to improve the resolution of weak signals within a converter,

and hence enhance the SQNR, the weak signals need to be

enlarged, or the quantization step size decreased, but only for theweak signals

But strong signals can potentially be reducedwithout significantly

degrading the SQNR or alternatively increasing quantization step size

The compression process at the transmitter must be matched with anequivalent expansion process at the receiver


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The signal below shows the effect of compression, where theamplitude of one of the signals is compressed

After compression, input to the quantizer will have a more uniform

distribution after sampling

At the receiver, the signal isexpanded by an inverseoperation

The process of COMpressingand exPANDING the signal iscalled companding

Companding is a techniqueused to reduce the number of bitsrequired in ADC or DAC whileachieving comparable SQNR


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Basically, companding introduces a nonlinearity into the signal This maps a nonuniform distribution into something that more

closely resembles a uniform distribution

A standard ADC with uniform spacing between levels can be used

after the compandor (or compander)

The companding operation is inverted at the receiver

There are in fact two standard logarithm based compandingtechniques

US standard called-law companding

European standard calledA-law companding


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Input/Output Relationship of Compander

Logarithmic expression Y = log Xis the most commonly

used compander This reduces the dynamic range ofY


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Types of Companding -Law Companding Standard (North & SouthAmerica, and Japan)

where x and y represent the input and output voltages

is a constant number determined by experiment In the U.S., telephone lines uses companding with = 255

Samples 4 kHz speech waveform at 8,000 sample/sec

Encodes each sample with 8 bits, L = 256quantizer levels

Hence data rate R = 64 kbit/sec

= 0corresponds to uniform quantization

[ ]maxmax

log 1 (| | /

sgn( )log (1 )

e

e

x x

y y x

+

= +


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A-Law Companding Standard (Europe, China, Russia,

Asia, Africa)

where

x and y represent the input and output voltages A = 87.6

A is a constant number determined by experiment

maxmax

max

max

max

max

| |

| | 1sgn( ), 0

(1 )( )| |

1 log1 | |

sgn( ), 1(1 log )

e

e

xA

x xy x

x Ay xx

Ax x

y xA A x

<

+= +

< +


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2.8 Baseband Modulation Recall that analog signals can be represented by a sequence of discrete

samples (output of sampler) Samples are converted into bits. But bits are just abstract entities that

have no physical definition

We use pulses to convey a bit of information, e.g.,

In order to transmit the bits over a physical channel they must betransformed into a physical waveform

A line coder or baseband binary transmittertransforms a stream of bitsinto a physical waveform suitable for transmission over a channel

Line coders use the terminology markfor 1 and space to mean 0

In baseband systems, binary data can be transmitted using many kinds of

pulses


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There are many types of waveforms. Why? performance criteria! Each line code type have merits and demerits

The choice of waveform depends on operating characteristics of a

system such as:

Modulation-demodulation requirements

Bandwidth requirement

Synchronization requirement

Receiver complexity, etc.,


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Goals of Line Coding (qualities to look for)

A line code is designed to meet one or more of the following goals: Self-synchronization

The ability to recover timing from the signal itself

That is, self-clocking (self-synchronization) - ease of clock lock

or signal recovery for symbol synchronization

Long series of ones and zeros could cause a problem

Low probability of bit error

Receiver needs to be able to distinguish the waveform associatedwith a markfrom the waveform associated with a space

BER performance

relative immunity to noise

Error detection capability

enhances low probability of error


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Spectrum Suitable for the channel

Spectrum matching of the channel

e.g. presence or absence of DC level

In some cases DC components should be avoided

The transmission bandwidth should be minimized

Power Spectral Density

Particularly its value at zero

PSD of code should be negligible at the frequency near zero

Transmission Bandwidth Should be as small as possible

Transparency

The property that any arbitrary symbol or bit pattern can be

transmitted and received, i.e., all possible data sequence should

be faithfully reproducible


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Line Coder The input to the line encoder is

the output of the A/D converter

or a sequence of values an that

is a function of the data bit

The output of the line encoderis a waveform:

where f(t) is the pulse shape and Tb is the bit period (Tb=Ts/n fornbit quantizer)

This means that each line code is described by a symbol mappingfunction an and pulse shape f(t)

Details of this operation are set by the type of line code that isbeing used

( ) ( )n bn

s t a f t nT

=

=


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Commonly Used Line Codes

Polar line codes use the antipodal mapping

Polar NRZ uses NRZ pulse shape Polar RZ uses RZ pulse shape

, 1

, 0

n

n

n

A w h e n Xa

A w h e n X

+ ==

=


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Unipolar NRZ Line Code

Unipolar non-return-to-zero (NRZ) line code is defined byunipolar mapping

In addition, the pulse shape for unipolar NRZ is:

where Tb is the bit period

, 1

0, 0

n

n

n

A when Xa

when X

+ ==

=Where Xn is the n

th data bit

( ) , NRZ Pulse Shapeb

tf t

T

=


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Bipolar Line Codes

With bipolar line codes a space is mapped to zero anda mark is alternately mapped to -A and +A

It is also called pseudoternary signaling oralternate mark inversion

(AMI)

Either RZ or NRZ pulse shape can be used

, when 1 and last mark

, when 1 and last mark

0, when 0

n

n n

n

X A

a A X A

X

+ =

= = + =


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Manchester Line Codes

Manchester line codes use the antipodal mappingand the following split-phasepulse shape:

4 4( )

2 2

b b

b b

T Tt t

f t T T

+ =


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Summary of Line Codes


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Comparison of Line Codes Self-synchronization

Manchester codes have built in timing information because they

always have a zero crossing in the center of the pulse

Polar RZ codes tend to be good because the signal level alwaysgoes to zero for the second half of the pulse

NRZ signals are not good for self-synchronization

Error probability

Polar codes perform better (are more energy efficient) than

Unipolar or Bipolar codes

Channel characteristics

We need to find the power spectral density (PSD) of the linecodes to compare the line codes in terms of the channel

characteristics


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Comparisons of Line Codes Different pulse shapes are used

to control the spectrum of the transmitted signal (no DC value,

bandwidth, etc.)

guarantee transitions every symbol interval to assist in symbol timing

recovery1. Power Spectral Density of Line Codes (see Fig. 2.23, Page 90)

After line coding, the pulses may be filtered or shaped to further

improve there properties such as

Spectral efficiency

Immunity to Intersymbol Interference

Distinction between Line Coding and Pulse Shaping is not easy

2. DC Component and Bandwidth DC Components

Unipolar NRZ, polar NRZ, and unipolar RZ all have DC components

Bipolar RZ and Manchester NRZ do not have DC components


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First Null Bandwidth

Unipolar NRZ, polar NRZ, and bipolar all have 1st null bandwidths

ofRb = 1/Tb Unipolar RZ has 1st null BW of2Rb

Manchester NRZ also has 1st null BW of2Rb, although thespectrum becomes very low at 1.6Rb


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Generation of Line Codes

The FIR filter realizes the different pulse shapes

Baseband modulation with arbitrary pulse shapes can bedetected by

correlation detector

matched filter detector (this is the most common detector)


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Section 2.8.4: Bits per PCM Word and Bits per Symbol L=2l

Section 2.8.5: M-ary Pulse Modulation Waveforms

M = 2k

Problem 2.14: The information in an analog waveform, whosemaximum frequency fm=4000Hz, is to be transmitted using a 16-level

PAM system. The quantization must not exceed 1% of the peak-to-peak analog signal.

(a) What is the minimum number of bits per sample or bits per PCMword that should be used in this system?

(b) What is the minimum required sampling rate, and what is theresulting bit rate?

(c) What is the 16-ary PAM symbol Transmission rate?

Bits per PCM word and M-ary Modulation


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max

2 2

2

2

| | | 0.01 | | |2

122

1

log log (50) 62

8000 48000 16

4800012000 / sec

log ( ) 4

pp

pp lpp

qe pV p e

VV Lq q L

L p

l lp

fs Rs M

RR symbols

M

= =

= = =

=

= = =

= = =

Solution to Problem 2.14

DC Chapter2b

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