Digital Modulation Saravanan Vijayakumaran sarva@ee.iitb.ac.in - - PowerPoint PPT Presentation

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Digital Modulation Saravanan Vijayakumaran sarva@ee.iitb.ac.in - - PowerPoint PPT Presentation

Dec 19, 2022 462 likes •692 views

Digital Modulation Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay August 19, 2013 1 / 21 Digital Modulation Definition The process of mapping a bit sequence to signals

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SLIDE 1

Digital Modulation

Saravanan Vijayakumaran sarva@ee.iitb.ac.in

Department of Electrical Engineering Indian Institute of Technology Bombay

August 19, 2013

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SLIDE 2

Digital Modulation

Definition

The process of mapping a bit sequence to signals for transmission over a channel.

Information Source Source Encoder Channel Encoder Modulator Channel Demodulator Channel Decoder Source Decoder Information Destination

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SLIDE 3

Digital Modulation

Example (Binary Baseband PAM)

1 → p(t) and 0 → −p(t)

t A p(t) t −A −p(t)

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SLIDE 4

Classification of Modulation Schemes

  • Memoryless
  • Divide bit sequence into k-bit blocks
  • Map each block to a signal sm(t), 1 ≤ m ≤ 2k
  • Mapping depends only on current k-bit block
  • Having Memory
  • Mapping depends on current k-bit block and L − 1 previous blocks
  • L is called the constraint length
  • Linear
  • Complex baseband representation of transmitted signal has the

form u(t) =

  • n

bng(t − nT) where bn’s are the transmitted symbols and g is a fixed baseband waveform

  • Nonlinear

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SLIDE 5

Signal Space Representation

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SLIDE 6

Signal Space Representation of Waveforms

  • Given M finite energy waveforms, construct an orthonormal basis

s1(t), . . . , sM(t) → φ1(t), . . . , φN(t)

  • Orthonormal basis

φi, φj = ∞

−∞

φi(t)φ∗

j (t) dt =

1 if i = j

  • therwise
  • Each si(t) is a linear combination of the basis vectors

si(t) =

N

  • n=1

si,nφn(t), i = 1, . . . , M

  • si(t) is represented by the vector si =
  • si,1

· · · si,N T

  • The set {si : 1 ≤ i ≤ M} is called the signal space representation or

constellation

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SLIDE 7

Constellation Point to Waveform

si,N si,N−1 . . . si,2 si,1 si(t) × × . . . × × φ1(t) φ2(t) φN−1(t) φN(t) +

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SLIDE 8

Waveform to Constellation Point

si(t) si,N si,N−1 . . . si,2 si,1 × × . . . × × φ∗

1(t)

φ∗

2(t)

φ∗

N−1(t)

φ∗

N(t)

  • .

. .

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SLIDE 9

Gram-Schmidt Orthogonalization Procedure

  • Algorithm for calculating orthonormal basis for s1(t), . . . , sM(t)
  • Consider M = 1

φ1(t) = s1(t) s1 where s12 = s1, s1

  • Consider M = 2

φ1(t) = s1(t) s1 , φ2(t) = γ(t) γ where γ(t) = s2(t) − s2, φ1φ1(t)

  • Consider M = 3

φ1(t) = s1(t) s1 , φ2(t) = γ1(t) γ1 , φ3(t) = γ2(t) γ2 where γ1(t) = s2(t) − s2, φ1φ1(t) γ2(t) = s3(t) − s3, φ1φ1(t) − s3, φ2φ2(t)

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Gram-Schmidt Orthogonalization Procedure

  • In general, given s1(t), . . . , sM(t) the kth basis function is

φk(t) = γk(t) γk where γk(t) = sk(t) −

k−1

  • i=1

sk, φiφi(t) is not the zero function

  • If γk(t) is zero, sk(t) is a linear combination of φ1(t), . . . , φk−1(t). It does

not contribute to the basis.

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SLIDE 11

Gram-Schmidt Procedure Example

2 t 1 s1(t) 2 t 1

  • 1

s2(t) 3 t

  • 1

1 s3(t) 3 t

  • 1

1 s4(t)

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SLIDE 12

Gram-Schmidt Procedure Example

2 t

1 √ 2

φ1(t) 2 t

1 √ 2

− 1

√ 2

φ2(t) 2 3 t

  • 1

1 φ3(t)

s1 = √ 2 T s2 =

  • √

2 T s3 = √ 2 1 T s4 =

  • −

√ 2 1 T

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Properties of Signal Space Representation

  • Energy

Em = ∞

−∞

|sm(t)|2 dt =

N

  • n=1

|sm,n|2 = sm2

  • Inner product

si(t), sj(t) = si, sj

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SLIDE 14

Modulation Schemes

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SLIDE 15

Pulse Amplitude Modulation

  • Signal Waveforms

sm(t) = Amp(t), 1 ≤ m ≤ M where p(t) is a pulse of duration T and Am’s denote the M possible amplitudes.

  • Example M = 2, p(t) is a real pulse

A1 = −A, A2 = A for a real number A

A

  • A

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SLIDE 16

Pulse Amplitude Modulation

  • Example M = 4, p(t) is a real pulse

A1 = −3A, A2 = −A, A3 = A, A4 = 3A

  • 3A
  • A

A 3A 16 / 21

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Phase Modulation

  • Complex envelope of phase modulated signals

sm(t) = p(t)e j π(2m−1)

M

, 1 ≤ m ≤ M where p(t) is a real baseband pulse of duration T

  • Corresponding passband signals

sp

m(t)

= Re √ 2sm(t)e j2πfct = √ 2p(t) cos π(2m − 1) M

  • cos 2πfct

− √ 2p(t) sin π(2m − 1) M

  • sin 2πfct

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SLIDE 18

Constellation for PSK

11 10 01 00 QPSK, M = 4 000 001 011 010 110 111 101 100 Octal PSK, M = 8

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Quadrature Amplitude Modulation

  • Complex envelope of QAM signals

sm(t) = (Am,i + jAm,q)p(t), 1 ≤ m ≤ M where p(t) is a real baseband pulse of duration T

  • Corresponding passband signals

sp

m(t)

= Re √ 2sm(t)e j2πfct = √ 2Am,ip(t) cos 2πfct − √ 2Am,qp(t) sin 2πfct

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SLIDE 20

Constellation for QAM

16-QAM

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Thanks for your attention

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