4.2-21

4.2.4 Orthogonal, Biorthogonal and Simplex Signals

• In PAM, QAM and PSK, we had only one basis function. For orthogonal,

biorthogonal and simplex signals, however, we use more than one

orthogonal basis function, so N-dimensional Examples: the Fourier basis;

time-translated pulses; the Walsh-Hadamard basis.

o We’ve become used to SER getting worse quickly as we add bits to the

symbol – but with these orthogonal signals it actually gets better.

o The drawback is bandwidth occupancy; the number of dimensions in

bandwidth W and symbol time Ts is

2 sN W T≈

o So we use these sets when the power budget is tight, but there’s plenty

of bandwidth.

Orthogonal Signals

• With orthogonal signals, we select only one of the orthogonal basis

functions for transmission:

The number of signals M equals the number of dimensions N.

4.2-22

• Examples of orthogonal signals are frequency-shift keying (FSK), pulse

position modulation (PPM), and choice of Walsh-Hadamard functions

(note that with Fourier basis, it’s FSK, not OFDM).

• Energy and distance:

o The signals are equidistant, as can be seen from the sketch or from

location

location

s

i j

s

E i

jE

⎡ ⎤⎢ ⎥

←⎢ ⎥⎢ ⎥− =⎢ ⎥

←⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

s s

so 2 m2 2log ( ) 2 ,i j s b bE M E kE d− = = = = ∀s s in ,i j

4.2-23

• Effect of adding bits:

o For a fixed energy per bit, adding more bits increases the minimum

distance – strong contrast to PAM, QAM, PSK.

o But adding each bit doubles the number of signals M, which equals the

number of dimensions N – and that doubles the bandwidth!

• Error analysis for orthogonal signals. Equal energy, equiprobable signals,

so receiver is

o Error probability is same for all signals. If 1s was sent, then the

correlation vector is

1

2

s

M

E nn

n

⎡ ⎤+⎢ ⎥⎢=⎢⎢ ⎥⎢ ⎥⎣ ⎦

c ⎥⎥

with real components.

4.2-24

o Assume 1s was sent. The probability of a correct symbol decision,

conditioned on the value of the received , is 1c

( ) ( ) ( )1 2 1 3 1

1

1

0

( )

12

cs M

M

1P c P n c n c n c

cQN

−

⎡ ⎤= < ∧ < ∧ ∧ <⎣ ⎦

⎛ ⎞⎛ ⎞⎜ ⎟= − ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

…

o Then the unconditional probability of correct symbol detection is

( )

( )

( )( ) ( )

1

1

11 1

0

12

11 1

00 0

21

12

1 1 1 21 exp22 2 2

1 11 exp 222

M

cs c

M

s

Ms

cP Q p c dcN

cQ cNN N

Q u u du

−∞

−∞

−∞

−∞

∞−

−∞

⎛ ⎞⎛ ⎞⎜ ⎟= − ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟= − − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ π ⎝ ⎠⎝ ⎠⎝ ⎠

⎛ ⎞= − − − γ⎜ ⎟π ⎝ ⎠∫

∫

∫ E dc

Needs a numerical evaluation.

o The unconditional probability of symbol error (SER) is

1es csP P= −

4.2-25

• Now for the bit error rate.

o All errors equally likely, at 1 2 1

es esk

P PM

=− −

. No point in Gray coding.

o Can get i bit errors in equiprobable ways, so ki

⎛ ⎞⎜ ⎟⎝ ⎠

1 1

1

112 1 2 1

222 1

k ks s

eb k ki i

k

s sk

k kP PP i ki i

kk P P

= =

−

−⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟−− −⎝ ⎠ ⎝ ⎠

= ≈−

∑ ∑

About half the bits are in error in the average symbol error.

• The SER for orthogonal signals is striking:

2 0 2 4 6 8 10 12 14 161 .10 6

1 .10 5

1 .10 4

1 .10 3

0.01

0.1

1M = 2M = 4M = 8M = 16

Symbol Error Prob, Orthogonal Signals

SNR per bit, gamma_b (dB)

prob

of s

ymbo

l err

or

The SER improves as we

add bits!

But the bandwidth

doubles with each

additional bit.

4.2-26

• Why does SER drop as M increases?

o All points are separated by the same 22log ( ) bd M= E

M

. Since

increases linearly with

2d

2log ( )k = , the pairwise error probability

(between two specific points) decreases exponentially with k, the

number of bits per symbol.

o Offsetting this is the number M-1 of neighbours of any point: M-1

ways of getting it wrong, and M increases exponentially with k.

o Which effect wins? Can’t get much analytical traction from the

integral expression two pages back. So fall back to a union bound.

• Union bound analysis:

o Without loss of generality, assume was transmitted. 1s

o Define as the event that r is closer to than to . , 2, ,mE m M= … ms 1s

4.2-27

o Note that is a pairwise error. It does not imply that the receiver’s

decision is m. In fact, we have events

mE

2 , , ME E… and they are not

mutually exclusive, since r could lie closer to two or more points than

to : 1s

o The overall error event 2 3 ME E E E= ∪ ∪ ∪… , so the SER is bounded

by the sum of the pairwise event probabilities1

[ ] [ ] [ ]

( )

2 32

20

ln(2)2 2

2( 1) ( 1) log ( )2

122

b

b

M

es M mm

b

kkk

P P E P E E E P E

dM Q M Q MN

e e

=

γ⎛ ⎞− −⎜ ⎟− γ ⎝ ⎠

= = ∪ ∪ ∪ ≤

⎛ ⎞= − = − γ⎜ ⎟⎜ ⎟

⎝ ⎠

≤ <

∑…

1 A general expansion is

[ ] [ ]1 21 1

1 1 1 etc.

M M M

M i ii i i j

j i

M M M

i j ki j k

j i k jk i

P A A A P A P A A

P A A A

= = =≠

= = =≠ ≠

≠

j⎡ ⎤∪ ∪ ∪ = − ∩⎣ ⎦

⎡ ⎤+ ∩ ∩ −⎣ ⎦

∑ ∑∑

∑∑∑

…

4.2-28

o From the bound, if ( )2ln 2 1.386bγ > = , then loading more bits onto a

symbol causes the upper bound on SER to drop exponentially to zero.

o Conversely, if ( )2ln 2bγ < , increasing k causes the upper bound on

SER to rise exponentially (SER itself saturates at 1).

o Too bad the dimensionality, the number of correlators and the

bandwidth also increase exponentially with k.

• This scheme is called “block orthogonal coding.” Thresholds are

characteristic of coded systems.

• Remember that this analysis is based on bounds…

2 0 2 4 6 8 10 12 14 161 .10 6

1 .10 5

1 .10 4

1 .10 3

0.01

0.1

1

10

M = 2M = 2M = 4M = 4M = 8M = 8M = 16M = 16

SER and Bound, Orthogonal Signals

SNR per bit, gamma_b (dB)

prob

of s

ymbo

l err

or

True SERs are solid lines,

bounds are dashed.

Upper bound is useful to

show decreasing SER, but is

too loose for a good

approximation.

4.2-29

Biorthogonal Signals

• If you have coherent detection, why waste the other side of the axis?

Double the number of signal points (add one bit) with sE± . Now

2M N= , with no increase in bandwidth. Or keep the same M and cut the

bandwidth in half.

• Energy and distance:

All signal points are equidistant from is

min2i k sE d− = =s s (the same as orthogonal signals)

except one – the reflection through the origin – which is farther away

2i i E−− =s s s

4.2-30

• Symbol error rate:

The probability of error is messy, but

the union bound is easy. A signal is

equidistant from all other signals but its

own complement.

o So the union bound on SER is

( ) ( )( )( )2

( 2) 2

( 2) second term redundant (Sec'n 4.5.4)

( 2) log ( )

s s s

s

b

P M Q Q

M Q

M Q M

≤ − γ + γ

≤ − γ

= − γ

which is slightly less than orthogonal signaling, for the same M and

same . The major benefit is that biorthogonal needs only half the

bandwidth of orthogonal, since it has half the number of dimensions.

bγ

o And the bit error probability is

2log ( )2b s

MP P≈

4.2-31

Simplex Signals

• The orthogonal signals can be seen as a mean values, shared by all, and

signal-dependent increments:

The mean signal (the centroid) is 1

1 M

mmM =

= ∑s s .

o The mean does not contribute to distinguishability of signals – so why

not subtract it?

m m′ = −s s s

11

1 1 location

1

m s

MM

EM m

M

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥

′ = ⎢ ⎥− ←⎢ ⎥⎢ ⎥⎢ ⎥−⎣ ⎦

s

where sE is the original signal energy.

4.2-32

o Removing the mean lowers the dimensionality by one. After rotation

into tidier coordinates:

• They still have equal energy (though no longer orthogonal). That energy is

22

1 21 1s m sE E M E 1M MM

⎛ ⎞ ⎛′ ′= = + − = −⎜ ⎟ ⎜⎝ ⎠ ⎝

s ⎞⎟⎠

The energy saving is small for larger M.

4.2-33

• The correlation among signals is:

[ ] ( )1

, 1Re 1 11

m nmn

m n

MM

M

−′ ′ρ = = = −

′ ′⋅ −−

s ss s

for m n≠

A uniform negative correlation.

• The SER is easy. Translation doesn’t change the error rate, so the SER is

that of orthogonal signals with a (M M 1)− SNR boost.

4.2.5 Vertices of a Hypercube and Generalizations

• More signals defined on a multidimensional space. Unlike orthogonal, we

will now allow more than one dimension to be used in a symbol.

• Vertices of a hypercube is straightforward: two-level PAM (binary

antipodal) on each of the N dimensions:

o With the Fourier basis, it is BPSK on each frequency at once – a

simple OFDM

o With the time translate basis, it is a classical NRZ transmission:

4.2-34

o Signal vectors:

1

2

m

mm

mN

ss

s

⎡ ⎤⎢ ⎥⎢=⎢⎢ ⎥⎣ ⎦

s ⎥⎥

with mn ss E= ± N

for and 1,m M= … 2NM = .

o In space, it looks like this

o Every point has the same distance from the origin, hence the same

energy

2

2log ( )m s bE M E N= = =s bE

4.2-35

o Minimum distance occurs for differences in a single coordinate:

11111

sm

EN

⎡ ⎤⎢ ⎥⎢ ⎥

= ⎢⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

s ⎥

111

11

sn

EN

⎡ ⎤⎢ ⎥⎢ ⎥

= ⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

s

min 2 2sm n

Ed EN

= − = =s s b

More generally, for Hd disagreements (Hamming distance),

2 2H sm n H

d E d EN

− = =s s b

o What are the effects on and bandwidth if we increase the number

of bits, keeping fixed?

mind

bE

• It’s easy to generalize from binary to PAM, PSK, QAM, etc. on each of the

dimensions:

o With the Fourier basis, it is OFDM

o With time translates, it is the usual serial transmission.

4.2-36

• Error analysis:

All points have same energy

s bE N E=

Euclidean distance for Hamming

distance h is 2 bd h= E

o If labeling is done independently on different dimensions (see sketch),

then the independence of the noise causes it to decompose to N

independent detectors. So the following structure

( )( )2

2

log ( ) 2

s b

b

P N Q

M Q

≤ γ

= γ

bP depends on labels

becomes

sP is meaningless

( )2b bP Q= γ , just

binary antipodal

4.2-37

• Generalization: user multilevel signals (PAM) in each dimension. Again, if

labeling is independent by dimension, then it’s just independent and

parallel use, like QAM. Not too exciting. Yet. These signals form a finite

lattice.

• Generalization: use a subset of the cube vertices, with points selected for

greater minimum distance. This is a binary block code.

o Advantage: greater Euclidean distance

o Disadvantage: lower data rate