Lý thuyết điều khiển nâng cao - Chapter 5+6

n function of ak
.
The transmitted signal spectrum must be matched with the channel
 properties. In baseband
systems, e.g., in cables, the degradation is not 
constant within the used frequency band. Typically, the channel degradation 
is increased in high frequencies. 
Therefore, a higher signal power should be put into lower frequencies, 
where the cable degradation is the smallest. This reduces cross talks and 
radio distortions.
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Line Coding vs. Nyquist-Pulse Shaping (1)
Two different approaches to shape spectrum:
(1) Line coding
ƒ The pulse waveform is a square pulse.
ƒ The spectrum is a sinc-type wide spectrum.
ƒ The DC-component can be removed by constructing the signal 
properly.
ƒ Usually, the symbol train is generated to have some correlation, in 
order to modify (or “shape”) the transmitted spectrum.
ƒ Mostly used for binary signaling.
ƒ In a pure line coding, the bandwidth consumption is not a limiting 
factor.
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Line Coding vs. Nyquist-Pulse Shaping (2)
(2) Nyquist-pulse shaping
ƒ Used when we want to reduce the bandwidth consumption. It is 
assumed that transmitted symbols are uncorrelated ⇒ the 
transmitted spectrum has the shape of Fourier transform of the 
pulse waveform.
ƒ The pulse waveform is optimized so that the needed bandwidth is 
small ⇒ adjacent pulses are overlapped in time domain.
ƒ The methods can also be combined. In practical systems, one of 
them is chosen.
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Line Coding (1)
The goal of line coding:
ƒ Spectrum management and spectrum shaping.
ƒ To remove the variation of DC-component in AC-coupled systems.
ƒ To avoid synchronization problems when the transmitted symbol 
train consists of long sequences with constant 0 or 1.
ƒ System monitoring during the normal operation is possible by 
using proper line codes.
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Line Coding (2)
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Line Coding (3)
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Transmission Limitations (1)
Digital baseband
transmission model:
The signal-plus-noise-and-interference waveform:
where td
is transmission delay and stands for pulse shape with 
transmission distortion (See a possibility waveform of y(t) in next slide).
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Transmission Limitations (2)
The task of the regenerator is to recover the digital message from y(t). 
The synchronization signal may help the regenerator by identifying the 
optimum sampling times:
If then
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Transmission Limitations (3)
When rectangular pulses are passed through a bandlimited channel, the 
pulses will spread in time and the pulse for each symbol will smear into the 
time intervals of succeeding symbols. This leads to an increased
 probability of the receiver making an error in detecting a symbol ⇒
 intersymbol interference – ISI.
The combined effects of noise and ISI may result in errors in the 
regenerated message.
If n(t) is white noise, then the noise power can be reduced by reducing the 
bandwidth of the LPF at receiver. However, the low pass filtering causes 
pulses to spread out which would increase the ISI. Consequently, the 
fundamental limitations of digital transmission is the relationship 
between ISI, bandwidth and signaling rate.
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Transmission Limitations (4)
The Nyquist statement:
Given an ideal low-pass channel with bandwidth B,
it is possible to 
transmit independent symbols at a rate r ≤
2B baud without ISI. It is not 
possible to transmit independent symbols at rate r > 2B.
Signaling at the maximum rate
r = 2B requires a special pulse shape, that 
is
sinc pulse:
having the
bandlimited spectrum:
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Transmission Limitations (5)
Eye diagram:
An experimental display to know the channel characteristics, it further 
clarifies digital transmission limitations.
Distorted polar binary signal and eye diagram:
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Transmission Limitations (6)
General binary eye diagram:
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Bandlimited Digital PAM Systems (1)
Consider digital baseband
transmission with bandlimited channel. 
Consequently, the rectangular signaling pulses would be severely
distorted 
(resulting in intersymbol
interference -
ISI). Instead, we must use 
bandlimited pulses specially shaped to avoid ISI.
‰ Nyquist- Pulse Shaping:
Assumed that noise is absent, the signal at the input of the
regenerator is:
As before, the condition for p(t) is:
which eliminates ISI, but now we impose additional requirement that the 
pulse spectrum be bandlimited:
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Bandlimited Digital PAM Systems (2)
where
This means that the signaling rate is:
in which B
may be considered as the minimum required transmission 
bandwidth, so that BT
≥
B.
The Nyquist
theorem states that the above bandlimited
spectrum is 
satisfied if the p(t) has the form:
With a cosine rolloff spectrum:
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Bandlimited Digital PAM Systems (3)
Then, the spectrum of p(t) is: 
and the corresponding pulse shape is:
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Bandlimited Digital PAM Systems (4)
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Bandlimited Digital PAM Systems (5)
When β = r/2 (100% rolloff), the pulse spectrum has the raised cosine 
shape:
and
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Noise and Errors (1)
We assumed that the channel is distortionless
so the receiver signal is free 
of ISI. Assumed that, the additive white noise with zero mean, independent 
of the signal.
Binary Error Probability:
Baseband
binary receiver:
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Noise and Errors (2)
A sample-and-hold (S/H) extracts from y(t) the sample values:
These sample values are compared with a fixed threshold level V:
If y(tk
) > V, the output of the comparator gets high level
(bit 1). If y(tk
) < V, 
the comparator goes low level
(bit 0).
Considering x(t) to be unipolar signal (ak
= 1 for bit 1, and ak
= 0 for bit 0).
Let variable Y
represents y(tk
) at an arbitrary sampling time, and n
represents 
n(tk
).
If H0
denotes hypothesis that ak
= 0 and Y
= n, then the pdf:
where pN
(n) is the pdf
of noise alone. Similar, H1
denotes hypothesis that 
ak
= A
and Y
= A
+ n, then:
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Noise and Errors (3)
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Noise and Errors (4)
The comparator implements the decision rule:
ƒ Choose H0 (ak = 0), if Y < V
ƒ Choose H1 (ak = A), if Y > V
The corresponding regeneration error probabilities are then given by:
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Noise and Errors (5)
The threshold value is adjusted to minimize the average error probability:
where
Normally,
Then, for optimum threshold Vopt
, we have:
Assumed that the noise is with Gaussian distribution with zero mean and 
variance σ2, so:
Then, we obtain:
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Noise and Errors (6)
Since pN
(n) is even function and Vopt
= A/2, then:
For the polar signal, ak
= ±
A/2, we have Vopt
= 0.
From [1], we can write:
The Q
function is then obtained from the Table.
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Noise and Errors (7)
Regenerative Repeater:
Long-haul transmission requires repeaters. For analog repeaters, we obtain:
where (S/N)1
is signal to noise ratio after one hop and m
is number of hops. 
The transmitted power per repeater must be increased linearly
with m. The 
contaminating noise progressively builds up from repeater to repeater.
In contrast, a digital repeater is a regenerator, regenerating new digital 
signal to next repeater. For m
is not too large, we obtain:
It requires much smaller transmitted power per repeater than analog repeater.
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Noise and Errors (8)
Matched Filtering:
Every baseband
digital receiver (including repeaters) should include a 
LPF designed to remove excess noise without introducing ISI. The
 optimum LPF for timelimited
pulses in white noise
is a matched filter.
Let the received signal with duration τ (τ ≤
D) as:
The matched filter is designed to maximize the signal to noise ratio (that 
means minimizing the error probability) at time: 
From [1], the impulse response of the matched filter is:
with
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Noise and Errors (9)
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Noise and Errors (10)
When the x(t) is in white noise, the output noise power from the matched 
filter is:
Considering binary transmission systems with bit rate rb
, average received 
power SR
and noise density N0
. We can characterize this system in terms 
of two parameters:
where Eb
corresponds to average energy per bit, while γb
represents the 
ratio of bit energy to noise density.
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Noise and Errors (11)
If the signal consists of timelimited
pulse p(t) with amplitude sequence ak
, 
then:
where for unipolar
signal and
for polar signal. 
Thus, we obtain:
Therefore,
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Noise and Errors (12)
M-ary Error Probability:
The bit error probability (or bit error rate - BER) is:
in which:
where r is the M-ary
signaling rate (symbol rate), rb
is the bit rate, and SR
 is average received power.