Wave Form Geometry
Transcript of Wave Form Geometry
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Lecture #11 Overview
p Vector representation of signal waveforms
p Two-dimensional signal waveforms
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Geometric Representation of Signals
We shall develop a geometric representation of signal waveformsas points in a signal space.
Such representation provides a compact characterization ofsignal sets for transmitting information over a channel andsimplifies the analysis of their performance.
We use vector representation which allows us to representwaveform communication channels by vector channels.
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Geometric Representation of Signals
Suppose we have a set of M signal waveforms sm(t), 1 m Mwhere we wish to use these waveforms to transmit over acommunications channel (recall QAM, QPSK).
We find a set of N M orthonormal basis waveforms for oursignal space from which we can construct all of our M signal
waveforms.
Orthonormal in this case implies that the set of basis signals areorthogonal (inner product
si(t)sj(t)dt = 0) and each has unit
energy.
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Orthonormal Basis
Recall that
i,
j,
k formed a set of orthonormal basis vectors for
3-dimensional vector space, R3, as any possible vector in 3-D
can be formed from a linear combination of them:v = vii + vjj + vkk Having found a set of waveforms, we can express the M signals
{sm(t)} as exact linear combinations of the {j(t)}
sm(t) = N
j=1
smjj(t) m = 1, 2, . . . , M
where
smj =
sm(t)j(t)dt
and
Em =
s2m(t)dt =N
j=1
s2mj
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Vector Representation
We can therefore represent each signal waveform by its vectorof coefficients smj, knowing what the basis functions are towhich they correspond.
sm = [sm1, sm2, . . . , smN]
We can similarly think of this as a point in N-dimensional space
In this context the energy of the signal waveform is equivalentto the square of the length of the representative vector
Em = |sm|2 = s2m1 + s2m2 + . . . + s2mN
That is, the energy is the square of the Euclidean distance ofthe point sm from the origin.
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Vector Representation (cont.)
The inner product of any two signals is equal to the dot
product of their vector representations
sm sn =
sm(t)sn(t)dt
Thus any N-dimensional signal can be represented geometricallyas a point in the signal space spanned by the N orthonormal
functions {j(t)}
From the example we can represent the waveformss1(t), . . . , s4(t) as
s1 = [2, 0, 0], s2 = [0,2, 0], s3 = [0,2, 1], s4 = [2, 0, 1]
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Pulse Amplitude Modulation (PAM)
In PAM the information is conveyed by the amplitude of the
transmitted (signal) pulse
Pulse
Amplitude
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Baseband PAM (cont.)
The pulses are transmitted at a bit-rate of Rb = 1/Tb bits/swhere Tb is the bit interval (width of each pulse).
We tend to show the pulse as rectangular ( infinitebandwidth) but in practical systems they are more rounded (finite bandwidth)
We can generalize PAM to M-ary pulse transmission (M 2)
In this case the binary information is subdivided into k-bit blockswhere M = 2k. Each k-bit block is referred to as a symbol.
Each of the M k-bit symbols is represented by one of M pulseamplitude values.
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Baseband PAM (cont.)
A
-A
-3A
3As4(t)
s1(t) s2(t)
s3(t)
e.g., for M = 4, k = 2 bits per block, as we need 4 differentamplitudes. The figure shows a rectangular pulse shape with
amplitudes {3A,A,A,3A} representing the bit blocks{01, 00, 10, 11} respectively.
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Two Dimensional Signals
Recall that PAM signal waveforms are one-dimensional.
That is, we could represent them as points on the real line, R.
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PAM points on the real line
We can represent signals of more than one dimension
We begin by looking at two-dimensional signal waveforms
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Orthogonal Two Dimensional Signals
t t
tt
A 2A
2AA
-A
0
0 T/2 T
T 0 T/2
0 T/2 T
S1(t)
S2(t)
S1(t)
S2(t)
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Two Dimensional Signals (cont.)
Recall that two signals are orthogonal over the interval (0, T) iftheir inner productT
0s1(t)s2(t)dt = 0
Can verify orthogonality for the previous (vertical) pairs of
signals by observation
Note that all of these signals have identical energy, e.g. energyfor signal s
2(t)
E=
T
0
[s2(t)]
2dt = T
T /2
[
2A]2dt = 2A2[t]T
T /2 = A
2T
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Two Dimensional Signals (cont.)
We could use either signal pair to transmit binary information
One signal (in each pair) would represent a binary 1 and theother a binary 0
We can represent these signal waveforms as signal vectors intwo-dimensional space, R2
For example, choose the unit energy square wave functions asthe basis functions 1(t) and 2(t)
1(t) =
2/T , 0 t T /2
0, otherwise
2(t) = 2/T , T /2 t T0, otherwise
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Two Dimensional Signal Waveforms
(cont.)
The waveforms s1(t) and s2(t) can be written as linearcombinations of the basis functions
s1(t) = s111(t) + s122(t)
s
1 = (s11, s12) = (AT /2, AT /2) Similarly, s2(t) s2 = (A
T /2,A
T /2) 45
45o
o
s1
s2
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Two Dimensional Signal Waveforms
(cont.)
We can see that the previous two vectors are orthogonal in 2-Dspace
Recall that their lengths give the energy
E1 = s12 = s211 + s212 = A2T
The euclidean distance between the two signals is
d12 =s1 s22 =
(s11 s21, s12 s22)2 =
(0, A
2T)2
= A2T = A22T = 2E16 ENGN3226: Digital Communications L#11 00101011
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Two Dimensional Signal Waveforms
(cont.)
Can similarly show that the other two waveforms are orthogonaland can be represented using the same basis functions 1(t) and
2(t)
Their representative vectors turn out to be a 45 rotation of theprevious two vectors.
s1
s2
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Representation of > 2 bits in 2-D
Simply add more vector points
The total number of points that we have, M, tells us how manybits k we can represent with each symbol, M = 2k, e.g.,
M = 8, k = 3
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Representation of > 2 bits in 2-D (cont.)
Note that the previous set of signals (vector representation) hadidentical energies
Can also choose signal waveforms/points with unequal energies
The constellation on the right gives an advantage in noisyenvironments (Can you tell why?)
1 2
2
1
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2-D Bandpass Signals
Simply multiply by a carrierum(t) = sm(t)cos2fct m = 1, 2, . . . , M 0 t T
For M = 4, k = 2 and signal points with equal energies, we canhave four biorthogonal waveforms
These signal points/vectors are
equivalent to phasors, where
each is shifted by /2 from each
adjacent point/waveform
For a rectangular pulse
um(t) =2Es
Tcos
2fct +
2m
M
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Carrier with Square Pulse
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2-D Bandpass Signals
This type of signalling is also referred to as phase-shift keying(PSK)
Can also be written as
um(t) = gT(t)Amc cos2fct gT(t)Ams sin2fctwhere g
T(t) is a square wave with amplitude 2Es/T and widthT, so that we are using a pair of quadrature carriers
Note that binary phase modulation is identical to binary PAM
A value of interest is the minimum Euclidean distance which
plays an important role in determining bit error rate
performance in the presence of AWGN.
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Quadrature Amplitude Modulation (QAM)
For MPSK, signals were constrained to have equal energies.
The representative signal points therefore lay on a circle in 2-Dspace
In quadrature amplitude modulation (QAM) we allow differentenergies.
QAM can be considered as a combination of digital amplitudemodulation and digital phase modulation
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QAM
Each bandpass waveform is represented according to a distinctamplitude/phase combination
umn(t) = AmgT(t) cos(2fct + n)
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