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Sampler Implementation
The implementation of a sample is most commonly done with a
SAMPLE AND HOLD circuit . In this
operations, a switch and storage mechanism (for example, a
transistor and a capacitor) is used to form a
sequence of the analog input waveform. These samples look like a
PAM ( Pulse Amplitude Modulation )
waveform as the amplitude of the sampled pulse can vary
continuously. The original analog waveformcan be recovered from
these PAM type samples simply by low pass filtering them. We know
from our
knowledge of sampling theorem, that if we under sample (i.e.
fs-2.2 fm
The anti aliasing filters described so far are analog filters;
there is an alternative to them . We can
oversample the signal thereby removing anti aliasing . The
larger number of samples can be filtered
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further by digital filters, instead of analog filters. It turns
out this third method is the most economic
solution for sampling. This is so because signal processing
preformed with high performance analog
equipment is typically more expensive than using digital signal
processing equipment to perform the
sample task. We enumerate the steps required in A/D conversion
for both the cases of under sampling
and oversampling:
Concept of Error Free Communication
In every communication system , the error probability of the
detected signal at the receiver is inversely
related to the transmitted signals power. Specifically, for
digital communication, the relationship
between error probability pe and bit energy Eb can be expressed
as,
Equation
By increasing the bit energy Eb , the error probability Pe can
be decreased. Since, the signal power
S=EbRb, Increase of Eb can be achieved either by increasing
signal power S or by decreasing the bit
transmission rate Rb . Increase of signal power ultimately gets
restricted by the signal handling capability
of the source. So , in Communication systems, reduction of error
rate can be achieved only by reducing
the transmission rate .
An optimum source code has no redundancy and a compact code has
a very very low redundancy.
When these optimum codes or compact codes are transmitted, in
transit, if noise causes some
information loss, these is no way to detect that . if we
increase redundancy, the immunity of the signal
against nose increases. A common way to increase redundancy is
to repeat a digit a number of times.
The receiver uses majority rule to decipher the message. So ,
ever if one out of there repeated digits is in
error, the communication is considered error free, because the
receiver can detect the transmitted
symbol correctly. This scheme fails, if noise corrupts two
digits to be erroneous. In that case five digits
should be repeated , i.e. redundancy should be increased.
Figure 4.13 shows all the eight possible sequences that can be
received by the receiver when a single bit
is repeated three times and transmitted. The sequences are
represented sas the vertices of a 3-
dimensional cube. We already know that if two binary sequences
of same length differ in j digits, then
the Hamming distance between them is j. In our case transmitted
sequences are either 000 or 111 and
they are occupying only two out of eight vertices. The majority
decision rule basically takes a decision in
favor of the message whose Hamming distance is closest to the
received sequence . Sequence .
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Sequences 000001,010 and 100 are within 1 unit of Hamming
distance form 000 but are at least two
units away from 111. So. If any of these are received, decision
would be 0 , Similarly , for 110,111,011
and 101 , the decision would be in favour of 1.
So, of the eight possible vertices, we used only two , separated
by 3 Hamming units. If we draw a
Hamming sphere of unit radius around each of these two vertices,
the two Hamming spheres becomenon-overlapping. The channel noise
can cause a shift in the received sequence compared to the
transmitted sequence and as long as this shift is equal to or
less than I unit, we can still detect the
message withour error.
