Digital Telephony1. 2 Analog/digital systems Analog signal -voltage -speech -pressure SP Analog Sampler Discrete signal F s F max Quantiz- er Error.

Digital Telephony 1

Digital Telephony

Digital Telephony 2

Analog/digital systems

Analog signal-voltage-speech-pressure

SPAnalog

Sampler

Discrete signal

FsFmax

Quantiz-er

Error isintroduced

Digital signal

A/D converter

ordata from

-tape-simulations-digital devices

DSP

Digital Signal Processor-digital computer-dedicated dig. hw-programmable hw

Digital signal

D/A

Analog signal

Digital Telephony 3

Issues

Reconstruction accuracyConditions for perfect reconstruction

Digital signal is not just an approx. representation of an analog signalCould be generated digitallyThe processing being performed may not be

realizable in analog The theory of discrete time signal processing is

independent of continuous

Digital Telephony 4

Digital vs. analog processing

DSP implementations are flexible, programmable and modular

More precise and repeatable Performance and cost effectiveness (riding

the microelectronics wave) Direct mapping of mathematical expressions

with less approximation possible (enables sophisticated algorithms)

Digital Telephony 5

Digital vs. analog ...

Digital hardware can be multiplexed better than analog. Allows integration of multiple operations and services on a h/w platform

Digital storage is more reliable, cheaper and more compact

Digital Telephony 6

On the other hand

Analog SP still offers higher bandwidth Higher dynamic range Can be very low power

Digital Telephony 7

Analog to Digital Conversion

To convert “real-world” analog signals to digital signals for processing

Sampling Quantizing and coding

Xa(t) X [n] Xq[n]Sampler

Quantizerand Coder

Analog signal Discrete signal Digital signal

Digital Telephony 8

Sampling Uniform

One sample every T seconds (ideally)x[n] = xa(nT), nSampling period: TSampling frequency: Fs=1/T

Assume: xa(t) = Acos( 2Ft+) = Acos(t+)

Then: x[n] = Acos[2FnT+] = Acos[Tn+]

= Acos[ n+], where T is called the normalized or discrete domain frequency

Digital Telephony 9

f = F/Fs must be rational in order for x[n] to be

periodic If f = k/N, then x[n] is periodic with period N Now, xa(nT) = Acos(Tn+)

= Acos((+2k/TTn+)

is periodic in with period 2/T Also, x[n] = Acos[ n+] = Acos[(+2k) n +]

is periodic in with period 2

Digital Telephony 10

xa(t)

n=

S(t) = (tnT)

xs(t) = xa(nT)(tnT)n=

convert todiscretesequence x[n] = xa(nT)


Let us look at the continuous time Fourier transform of xs(t)

Xs(j) = Xa(j) * S(j)

S(j) = ks

Xs(j) = Xa(jkjs)

2T

k=

12

1T

k=


Thus, Xa(j) must be bandwidth limited

If the max frequency in Xa(j) is N, then the sampling rate s2Nensures no information is lost due to aliasing

This sampling rate is known as Nyquist rate A lower sampling rate causes a distortion of

the signal due to Aliasing If no Aliasing occurs, the signal can be

perfectly reconstructed by passing through an ideal low pass filter with


Reconstruction

Xr(j) = Hr(j) Xs(j)

if Nc(sN)

then Xr(j) = Xc(j)

Hr(j)

c c s>2

Xs(j)


Reconstruction

Frequency response of ideal reconstruction filter

c c

T

Impulse response of ideal reconstruction filter

Hr(j) = {T, cc0, otherwise

hr(t)=sin t/Tt/T


Reconstruction

Xr(j) = Hr(j) Xs(j)

xr(t) = xs(t) * hr(t)

= [kxa(kT) t-kThr(t)

= kxa(kT) hr(t-kT)

kxa(kT)

sin t-nT)/Tt-nT)/T


xa(t) xs(t) hr(t)

xr(t)


Sampling theorem If the highest frequency contained in a signal

xa(t) is 0 and the signal is uniformly sampled at a rate s0, then xa(t) can be exactly recovered from its sample values using the interpolation function

and then xa(t) = kxa(kT) hr(t-kT), where {xa(kT) } are the samples of xa(t), and T=2s

hr(t)=sin t/Tt/T


Quantization and coding

Quantization:Converting discrete time signal to digitalxq(n) =Q [x(n)]

Quantization step


x

Q(x)


Quantization

Rounding: Assign x[n] to the closest quantization level

Quantization error eq[n] = xq[n] - x[n]

eq[n]

Uniformly distributed mean = 0 variance =


Quantization

Range of quantizer: xmax-xmin

Quantization levels: m Assuming uniform quantization

=Xm/ (m-1)

where Xm = (xmax-xmin)/2 is called the full-scale level of the A/D converter

m-1xmax-xmin


Coding

Coding is the process of assigning a unique binary number to each quantization level

Number of bits required log2m

Alternatively, given b+1 bits

xmax-xmin)/2b+1 =Xm /2b

For A/D devices, the higher Fs and m, the less the error (and the more the cost of the device)


Assuming dynamic range of A/D converter is larger than signal amplitudeSNR = 10 log10(xe) = 10 log10(x)

= 10 log10(12.22bx/Xm)

=6.02b +10.8 + 20 log10(Xm/x)

Quantizer

+

x(n) xq(n)

x(n) xq(n)

eq(n)

2 2 2

2 2


Uniformly Encoded PCM

X/Xm

20

40

60

80

-40 -30 -20 -10 0

Number of bitsper sample

13

1211109

8

dB

Sig

nal t

o Q

uant

iiatio

n N

oise

Rat

io (

dB)


Example

What is the minimum bit rate that a uniform PCM encoder must provide to encode a high fidelity audio signal with a dynamic range of 40 dB? Assume the fidelity requirements dictate passage of a 20-kHz bandwidth with a minimum signal-to-noise ratio of 50 dB. For simplicity, assume sinusoidal input signals.


Companding

Companded PCM with analog compression and expansion

A/D

CompressionLinear PCM

Encoder

InputSignal

D/A

Linear PCMDecoder Expansion

OutputSignal

CompressedDigital

Codewords

11 ])1[(1

sgn()(

11 )1ln(

)1ln(sgn()(

1||1

yy)yF

xx

x)xF

y


Segment Approximation

Input Sample Values

000

001

010

011

100

101

110

111

Uniform quantization


T1 Channel Bank

A/D

D/A

T1 transmissionLine

AnalogInputs

1

2

24•Eigth bits per PCM code word•companding functions with mu=255


Performance of a Encoder

10

20

30

40

-70 -60 -50 -40 -30dB

-20 -10 0 3

Signal Power of sinewave (dBm0)

Signal-to-quantization noise ratio (dB)

8 bit 2557 bit 100

Piecewise linear 8 bit 255

22

2733

7

0

2

12

1power noise

iiiqp


Total Noise Power

Signal Power relative to full-load signal (dBm0)-70

15 dB at which persons find communication difficult

Signal-to-total noise noise ratio

10

20

30

40

-60 -50 -40 -30dB

-20 -10 0 3

30 dB required for good communication

40 d

B r

ange

of

poss

ible

sig

nals

310eP

0eP

410eP

510eP

610eP

810eP


Error Performance

Fewer than 10% of 1 min intervals to have BER worse than 10E-6

Fewer than 0.2% of 1 sec intervals to have BER worse than 10E-3

92% error free sec


DS1 Signal Format

(8x24)+1=193 bits in 125 s 193 x 8000 = 1.544 Mbs Bit “robbing” technique used on each sixth

frame to provide signaling information

6101258000

1


Plesiochronous Transmission Rates

64 kbits/s

Japanese Standard North AmericaStandard

European Standard

1544 kbits/s 2048 kbits/s

8448 kbits/s

34368 kbits/s

139264 kbits/s

564992 kbits/s

6312 kbits/s

44736 kbits/s

274176 kbits/s

32064 kbits/s

97728 kbits/s

97728 kbits/s

x24x30

x4x3

x4

x4

x3

x4

x5 x7

x6

x4

x4x3


Plesiochronous Digital Hierarchy

MULTIPLEXINGLEVELS(DS)

# OF VOICECHANNELS

NORTHAMERICA

EUROPE JAPAN

0 1 0.064 0.064 0.064

1 24 1.544 1.544

30 2.048

48 3.152 3.152

2(4xDS1)

96 6.312 6.312

120 8.448


MultiplexingLevels

# OF VOICECHANNELS

NORTHAMERICA

EUROPE JAPAN

3 (7xDS2) 480 34.368 32.064

672 44.376

1344 91.053

1440 97.728

4 (6xDS3) 1920 139.264

4032 274.176

5760 397.200

7680 565.148


Plesiochronous Digital Hierarchy

The output of the M12 multiplexer is operating 136 kbs faster than the agragate rate of four DS1 6.312 vs 4x1.544=6.176

M12 frame has 1176 bits, i.e. 294-bit subframes ; each subframe is made of up of 49-bits blocks; each block starts with a control bit followed by a 4x12 info bits from four DS1 channels


Makeup of a DS2 Frame

M1 01 02 03 04 C1 01 02 03 04 F0 01 02 03 04 C2 01 02 03 04 C3 01 02 03 04 F1 01 02 03 04

M1 01 02 03 04 C1 01 02 03 04 F0 01 02 03 04 C2 01 02 03 04 C3 01 02 03 04 F1 01 02 03 04

Bit stuffing

4 M bits (O11X X=0 alarm) C=000,111 bit stuffing present/absent nominal stuffing rate 1796 bps, max 5367


Regenerative Repeaters

Amplifier Equalizer

Input Timingrecovery

Regenerator

Output

Spacing between adjacent repeatersPairdiameter(mm)

Loopattenuationat 1 MGHz(dB/km)

Loopresistance(/km)

Maximumdistance(km)

TotalRepeaters

Maxdistancesystem(km)

0.9 12 60 3 18 54

0.8 16 100 2.25 16 36


Digital Transmission Systems

Designation

Administration

BitRate

Line Code Media RepeaterSpacing

T1 AT&T 1.544 AMI/B8ZS Twisted pair 6000 ft

CEPT1 CCITT 2.048 B4ZS Twisted pair 2000 m

T1C AT&T 3.152 Bipolar Twisted pair 6000 ft

T148 ITT 2.37ternary

4B3T Twisted pair 6000 ft

9148A GTE 3.152 1-DDduobinary

Twisted pair 6000 ft

T1D AT&T 3.152 1+Dduobinary

Twisted pair 6000 ft

T1G AT&T 6.443 4-level Twisted pair 6000 ft

LD-4 Canada 274.176 B3ZS Coax 1900 m

T4M AT&T 274.176 Polar Coax 5700 ft


PCM System Enhancements

North AmericaSuperframe of 12 DS0’s has a sync sequence

101010 for odd (001110 for even frames) Extended superframe

24 frames - (4 S bits for frame allignment signal); 6 S bits for CRC-6 check; the rest 12 constitute 4 kbs data link

Digital Telephony1. 2 Analog/digital systems Analog signal -voltage -speech -pressure SP Analog Sampler Discrete signal F s F max Quantiz- er Error.

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