digital telephony1. 2 analog/digital systems analog signal -voltage -speech -pressure sp analog...
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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
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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
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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)
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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
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On the other hand
Analog SP still offers higher bandwidth Higher dynamic range Can be very low power
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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
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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
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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
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xa(t)
n=
S(t) = (tnT)
xs(t) = xa(nT)(tnT)n=
convert todiscretesequence x[n] = xa(nT)
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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=
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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
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Reconstruction
Xr(j) = Hr(j) Xs(j)
if Nc(sN)
then Xr(j) = Xc(j)
Hr(j)
c c s>2
Xs(j)
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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
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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
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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
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Quantization and coding
Quantization:Converting discrete time signal to digitalxq(n) =Q [x(n)]
Quantization step
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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 =
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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
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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)
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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
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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)
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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.
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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
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Segment Approximation
Input Sample Values
000
001
010
011
100
101
110
111
Uniform quantization
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T1 Channel Bank
A/D
D/A
T1 transmissionLine
AnalogInputs
1
2
24•Eigth bits per PCM code word•companding functions with mu=255
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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