Digital

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Digital Communication Vector Space concept
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Signal space

• Signal Space
• Inner Product
• Norm
• Orthogonality
• Equal Energy Signals
• Distance
• Orthonormal Basis
• Vector Representation
• Signal Space Summary

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Signal Space
S(t)
S(s1,s2,)

• Inner Product (Correlation)
• Norm (Energy)
• Orthogonality
• Distance (Euclidean Distance)
• Orthogonal Basis

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ONLY CONSIDER SIGNALS, s(t)
T
t
Energy
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Inner Product - (x(t), y(t))
Similar to Vector Dot Product
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Example
A
T
t
-A
2A
A/2
t
T
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Norm - x(t)
Similar to norm of vector
A
T
-A
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Orthogonality
A
T
-A
Y(t)
B
Similar to orthogonal vectors
T
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X(t)

• ORTHONORMAL FUNCTIONS

T
Y(t)
T
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Correlation Coefficient
1 ? ? ? -1 ??1 when x(t)?ky(t) (kgt0)

• In vector presentation

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Example
Y(t)
X(t)
10A
A
t
t
-A
T
T/2
7T/8
Now,
? shows the real correlation
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Distance, d

• For equal energy signals

• ?-1 (antipodal)

• ?0 (orthogonal)

• 3dB better then orthogonal signals

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Equal Energy Signals

• To maximize d

(antipodal signals)

• PSK (phase Shift Keying)

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• EQUAL ENERGY SIGNALS
• ORTHOGONAL SIGNALS (?0)

PSK (Orthogonal Phase Shift Keying)
(Orthogonal if
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Signal Space summary

• Inner Product

• Norm x(t)

• Orthogonality

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• Corrolation Coefficient, ?

• Distance, d

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Modulation
QAM
BPSK
QPSK
BFSK
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Modulation

• Modulation
• BPSK
• QPSK
• MPSK
• QAM
• Orthogonal FSK
• Orthogonal MFSK
• Noise
• Probability of Error

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Binary Phase Shift Keying (BPSK)
-
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Binary antipodal signals vector presentation

• Consider the two signals

The equivalent low pass waveforms are
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The vector representation is Signal
constellation.
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The cross-correlation coefficient is
The Euclidean distance is
Two signals with cross-correlation coefficient
of -1 are called antipodal
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Multiphase signals

• Consider the M-ary PSK signals

The equivalent low pass waveforms are
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The vector representation is
Or in complex-valued form as
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Their complex-valued correlation coefficients are

and the real-valued cross-correlation
coefficients are
The Euclidean distance between pairs of signals
is
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The minimum distance dmin corresponds to the case
which m-k 1
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Quaternary PSK - QPSK
(00)
(10)
(01)
(11)

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X(t)
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(00)
(10)
(01)
(11)
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Exrecise
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MPSK
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Multi-amplitude Signal
Consider the M-ary PAM signals
m1,2,.,M
Where this signal amplitude takes the
discrete values (levels)
m1,2,.,M
The signal pulse u(t) , as defined is rectangular
U(t)
But other pulse shapes may be used to obtain a
narrower signal spectrum .
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Clearly , this signals are one dimensional (N1)
and , hence, are represented by the scalar
components
M1,2,.,M
The distance between any pair of signal is
M2
0
M4
0
Signal-space diagram for M-ary PAM signals .
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The minimum distance between a pair signals
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Multi-Amplitude MultiPhase signalsQAM Signals
A quadrature amplitude-modulated (QAM) signal or
a quadrature-amplitude-shift-keying (QASK) is
represented as
Where and are the
information bearing signal amplitudes of the
quadrature carriers and u(t)
.
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QAM signals are two dimensional signals and,
hence, they are represented by the vectors
The distance between a pair of signal vectors is
k,m1,2,,M
When the signal amplitudes take the discrete
values
In this
case the minimum distance is
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QAM (Quadrature Amplitude Modulation)
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QAMQASKAM-PM
Exrecise
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M256 M128 M64 M32 M16 M4

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For an M - ary QAM Square Constellation
In general for large M - adding one bit requires
6dB more energy to maintain same d .
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Binary orthogonal signals
Consider the two signals
Where either fc1/T or fcgtgt1/T, so that
Since Re(p12)0, the two signals are orthogonal.
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The equivalent lowpass waveforms
The vector presentation
Which correspond to the signal space diagram
Note that
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We observe that the vector representation for the
equivalent lowpass signals is
Where
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M-ary Orthogonal Signal
Let us consider the set of M FSK signals
m1,2,.,M
This waveform are characterized as having equal
energy and cross-correlation coefficients
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The real part of is
0
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First, we observe that 0 when
and .
Since m-k1 corresponds to adjacent frequency
slots , represent the
minimum frequency separation between adjacent
signals for orthogonality of the M signals.
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For the case in which ,the FSK
signals are equivalent to the N-dimensional
vectors
( ,0,0,,0)
(0, ,0,,0)
Orthogonal signals for MN3 signal space diagram
(0,0,,0, )
Where NM. The distance between pairs of signals
is
all m,k
Which is also the minimum distance.
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Biorthogonal Signal
A set of M bi-orthogonal signals can be
constructed from M/2 orthogonal signals by simply
including the negatives of the orthogonal signals
. Thus, we require NM/2 dimensions for the
construction of M bi-ortogonal signals .
M4
M6
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We note that the correlation between any pair of
waveforms is either or 0. The
corresponding distances are or
, with the latter being the minimum
distance.
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Orthogonal FSK(Orthogonal Frequency Shift Keying)
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0
1
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ORTHOGONAL MFSK
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All signals are orthogonal to each other
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How togeneratesignals
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0 T 2T
3T 4T 5T
6T

0 T 2T
3T 4T 5T
6T
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0 T 2T
3T 4T 5T
6T

0 T 2T
3T 4T 5T
6T
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0 T 2T
3T 4T 5T
6T

0 T 2T
3T 4T 5T
6T
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IQ Modulator

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IQ Modulator
Pulse shaping filter

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NOISE
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What about Noise

• White Gaussian Noise

T
T

• The coefficients are random variables !

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WHITE GAUSSIAN NOISE (WGN)
We write

• All are gaussian variables
• All are independent

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• All have same probability distribution

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• White Gaussian Noise has energy in every dimension

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Probability of Error for Binary Signaling
Exrecise
The two signal waveforms are given as These
waveforms are assumed to have equal energy E and
their equivalent lowpass um(t), m1,2 are
characterized by the complex-valued correlation
coefficient ?12 .
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The optimum demodulator forms the decision
variables Or,equivalently And decides in favor
of the signal corresponding to the larger
decision variable .
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Lets see that the two expressions yields the same
probability of error . Suppose the signal s1(t)
is transmitted in the interval 0?t?T . The
equivalent low-pass received signal
is Substituting it into Um expression
obtain Where Nm, m1,2, represent the noise
components in the decision variables,given by
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And . The probability
of error is just the probability that the
decision variable U2 exceeds the decision
variable u1 . But Lets define variable V as N1r
and N2r are gaussian, so N1r-N2r is also
gaussian-distributed and, hence, V is
gaussian-distributed with mean value
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And variance Where N0 is the power spectral
density of z(t) . The probability of error is now
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Where erfc(x) is the complementary error
function, defined as It can be easily shown
that
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Distance, d

• For equal energy signals

• ?-1 (antipodal)

• ?0 (orthogonal)

• 3dB better then orthogonal signals

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It is interesting to note that the probability of
error P2 is expressed as Where d12 is the
distance of the two signals . Hence,we observe
that an increase in the distance between the two
signals reduces the probability of error .
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