wave form geometry

8/3/2019 Wave Form Geometry

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Lecture #11 Overview

p Vector representation of signal waveforms

p Two-dimensional signal waveforms

1 ENGN3226: Digital Communications L#11 00101011


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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.

1 3 5 7 -1-3-5-7

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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(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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(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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