204

Lecture 12: Tue Feb 23, 2021

Reminder:

• HW 3 postedLecture:

• signal space• signal vectors• signal-space (constellation) diagram• generalized Parseval• signal space diagrams for M-PSK

205

Today: M-ary CommunicationsTransmitter sends one of {s1( t ), sM( t )} every T seconds:

• bit rate Rb = bits/s

• “signal space” S = Span{s1( t ), sM( t )}

• Associate with x( t ) ∈ S the vector x of expansion coeffs in ON expansion:

• Signals in S obey Parseval:

⇒ Signal energy = squared length of corresponding vector. ⇒ Energy in the error = distance2 between vectors.

• signal-space diagram = constellation: sketch of the M signal vectors

T

------------log2 M

t

xx( t ) =X

i

N

=1xii( t ) x1

xN

x =

“signal vector”

...

–

x( t )y( t )*dt = x, y .

206

Signal SpaceThe most famous linear space.

In context of M-ary communications, the signal space is:

S = Span{s1( t ), sM( t )}.

207

Signal VectorsLet {1( t ) N( t )} be an ON basis for a signal space S

Associate with each signal in S the N-dim vector of expansion coefficients:

t

s

s( t ) =X

i

N

=1sii( t )

s1

s2

s3

sN

s = “signal vector”

208

Parseval Still Applies

For any x( t ), y( t ) ∈ S :

Inner product in one domain = inner product in another!

Special Cases

• |x( t )|2 dt =x 2

⇒ Signal energy = squared length of corresponding vector.

• |x( t ) – y( t )|2 dt = x – y 2

⇒ Energy in the error = distance2 between vectors.

–

x( t )y( t )*dt = x, y .

–

–

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M-ary CommunicationsThe m-th signal sm( t ) in the signal set {s1( t ), s2( t ), sM( t )} has a corresponding signal vector:

sm( t ) = X

ism,ii( t) sm = .

sm,1sm,2

sm,N

210

Signal-Space Diagram = ConstellationIn the context of M-ary communication, the signal-space diagram, orconstellation, is a sketch of the M signal vectors in N-dimensional space:

Key properties:

• Parseval ⇒ geometry (lengths, angles, distances) invariant to basis• squared length of signal vector = signal energy

Coming soon: Signal-space geometry is all we need to quantify achievable

• bandwidth efficiency• power efficiency

s1

s2s3

s4

211

Pop Quiz1

12

3

1

23

2

3

3

1

12

3

t

t

t

t

t

t

t

t

s1( t )

s8( t )

s2( t )

s3( t )

s4( t )

s5( t )

s6( t )

s7( t )

(a) Bit rate?(b) dimension of signal space?

(c) signal-space diagram?

0 2

212

Pop Quiz: Sketch Constellation

1

2

3

1

3

0

0

0

t

t

t

t

s1( t )

s2( t )

s3( t )

s4( t )

213

Gram-Schmidt Basis

2

1

3

0 t

t

t1

2

1( t )

2( t )

3( t )

1

1

1

214

Signal Space Diagram

s1

s3

1( t )

2( t )

3( t )

s2

s4

unit cube

Verify:

• Are waveform energies the same as squared vector length?• Is energy in error waveform the same as squared distance?• Do orthogonal signals have orthogonal vectors?

215

Example: A Binary Signal Set (M = 2)

A binary signal set:

The signal space is S = Span{s1( t ), s2( t )}.

Is {s1( t ), s2( t )} an ON basis for S?

s1( t )

s2( t )

0 1

2 t

t

1

1

1

216

Signal Space Diagram

Verify:

• Are waveform energies the same as squared vector length?• Is energy in error waveform the same as squared distance?

s1

s2

t

1( t )=s1( t )

2( t )=s2( t )

217

M=6 Example

The signal space is

S = Span{s1( t ), s2( t ) ... s6( t )}.

02

s3( t )

s4( t )

s5( t )

s6( t )t

s1( t )

s2( t )

0 1

2 t

t

1

1

1

218

Signal Space Diagram

s1

s2

s3

s4s5

s6

02

s3( t )

s4( t )

s5( t )

s6( t )t

1( t )=s1( t )

2( t )=s2( t )

219

M=6 Example

The signal space is

S = Span{s1( t ), s2( t ) ... s6( t )}.

02

s3( t )

s4( t )

s5( t )

s6( t )t

s1( t )

s2( t )

0 1

2 t

t

1

1

1

220

3 Different Bases

1( t )

2( t )

1( t )

2( t )

1( t )

2( t )

s1

s2

s3

s4s5

s6

s2

s3s4

s5s1

s6

GS{1, 2} GS{?, ?} GS{?, ?}

s1s6

s4s5

s3

s2

221

3 Different Bases

1( t )

2( t )

1( t )

2( t )

1( t )

2( t )

s1

s2

s3

s4s5

s6 s2

s3s4

s5s1

s6

GS{1, 2} GS{3, 4} GS{?, ?}

s6

s1

s4s5

s3

s2

222

3 Different Bases

1( t )

2( t )

1( t )

2( t )

1( t )

2( t )

s1

s2

s3

s4s5

s6 s2

s3s4

s5s1

s6

GS{1, 2} GS{3, 4} GS{4, 3}

s6

s1

s4s5

s3

s2

223

Signal Space Diagram for 8-ary PSK?sm( t ) = g( t )cos(2f0t + (m – 1) )2E

Eg

------- 2M------

s1( t )

s2( t )

s3( t )

s8( t )

T

224

Solution via Gram-SchmidtFnd basis for M-PSK’s S = span{ g( t )cos(2f0t + (m – 1) )}.

Step 1.

s1( t ) = g( t )cos(2f0t)

1( t ) = s1( t )/s1( t )= g( t )cos(2f0t) (when BW of g( t ) < f0)

2EEg

------- 2M------

2EEg

-------

2Eg

------

225

Step 2a

s21 = s2( t ), 1( t ) = g2( t ) cos(2f0t + /4)cos(2f0t)dt

= cos(/4) g2( t )dt + g2( t )cos(4f0t + /4)dt

= + 0 ( when 2{BW of g( t )} < 2f0, i.e. BW < f0)

=

⇒ s2( t ) = s211( t ) = g( t )cos(2f0t)

2Eg

------–

2EEg

-------

EEg

--------–

EEg

--------–

E2----

E2----

EEg

------

226

Step 2b e2( t ) = s2( t ) – s2( t )

= g( t )cos(2f0t + /4) – g( t )cos(2f0t)

= g( t ){cos(2f0t) – sin(2f0t) – cos(2f0t)}

= – g( t ) sin(2f0t)

2( t ) = s2( t )/s2( t )=– g( t )sin(2f0t)

2EEg

------- EEg

------

EEg

------

EEg

------

2Eg

-------

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Step 3

s3( t ) = g( t )cos(2f0t + )

= g( t )sin(2f0t)

= – 2( t )

already span{1( t ), 2( t )}

⇒ s3( t ) = s3( t )

⇒ ˆ e3( t ) = 0

⇒ skip this waveform

2EEg

------- 2---

2EEg

-------

E

228

Conclusion: ON Basis for PSKGiven passband M-ary PSK, sm( t ) = g( t )cos(2f0t + (m – 1) ),

the basis for the signal space S = Span{s1( t ), sM( t )} is:

⇒dimension is N = 2

2EEg-------

2M------

1( t ) = g( t )cos(2f0t)

2( t ) = – g( t )sin(2f0t)

2Eg------

2Eg------

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A By-Inspection Solution

sm( t ) = g( t )cos(2f0t + (m – 1) )

= g( t ) cos((m – 1) )cos(2f0t) – sin((m – 1) )sin(2f0t)

= sm,11( t ) + sm,22( t )

are orthogonal when either of following conditions are met:

• BW of g( t ) is less than f0 (i.e., sm( t ) is passband)• g( t ) = u( t ) – u(t – T ) and f02T is an integer, or f0 >> 1/T

2EEg-------

2M------

2EEg-------

2M------ 2

M------

1( t ) = g( t )cos(2f0t)

2( t ) = – g( t )sin(2f0t)

2Eg------

2Eg------

where

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M-PSK Signal Vectors

sm = =

For example, M = 8:

sm,1

sm,2E

(m 1– )2M------

cos

(m 1– )2M------

sin

E

s1

s2

s3s4

s8

231

M-ary PSK Signal Space Diagrams

E

M = 32

E E

E

M = 2 M = 4

M = 16

232

ExampleAn inefficient 8-ary PSK transmitter:

An efficient 8-ary PSK transmitter:

WAVEFORM GEN #1

WAVEFORM GEN #2

WAVEFORM GEN #8

WAVEFORM GEN #1

WAVEFORM GEN #2

sm1

sm2