Lecture 02 Digital Modulation I-Hsiang Wang ihwang@ntu.edu.tw - - PowerPoint PPT Presentation

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Principle of Communications, Fall 2017 Lecture 02 Digital Modulation I-Hsiang Wang ihwang@ntu.edu.tw National Taiwan University 2017/9/21 Outline Digital-to-analog and analog-to-digital: a signal space perspective Pulse amplitude

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SLIDE 1

Principle of Communications, Fall 2017

Lecture 02 Digital Modulation

I-Hsiang Wang

ihwang@ntu.edu.tw National Taiwan University 2017/9/21

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SLIDE 2

Outline

  • Digital-to-analog and analog-to-digital: a signal space perspective
  • Pulse amplitude modulation (PAM), pulse shaping, and the Nyquist criterion
  • Quadrature amplitude modulation (QAM), and the equivalent complex baseband

representation

  • Symbol mapping and constellation set

2

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SLIDE 3

System architecture of digital modulation

  • Three major components (for the Tx):
  • Symbol mapping: bit sequence → symbol sequence
  • Pulse shaping: symbol sequence → (baseband) waveform
  • Up conversion: baseband waveform → passband waveform

3

passband waveform

Symbol Mapper Pulse Shaper Sampler + Filter Symbol Demapper

discrete sequence

Up Converter Down Converter

baseband waveform

Noisy Channel coded bits

{ci} {ˆ ci}

{um} {ˆ um} xb(t) yb(t)

y(t) x(t)

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SLIDE 4

Symbol mapping

  • To be designed: the constellation set and how to map bits to symbols
  • Constellation sets to be covered: standard PSK, standard PAM, standard QAM
  • Mapping: Gray mapping

4

passband waveform

Symbol Mapper Pulse Shaper Sampler + Filter Symbol Demapper

discrete sequence

Up Converter Down Converter

baseband waveform

Noisy Channel coded bits

{ci} {ˆ ci}

{um} {ˆ um} xb(t) yb(t)

y(t) x(t)

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SLIDE 5

Conversion between sequence and waveform

  • A pragmatic approach: Pulse Amplitude Modulation (PAM)
  • To be designed: the modulating pulse
  • System parameter: bandwidth
  • Nyquist criterion: a sufficient condition for the pulse to satisfy in order to avoid

aliasing effect

5

passband waveform

Symbol Mapper Pulse Shaper Sampler + Filter Symbol Demapper

discrete sequence

Up Converter Down Converter

baseband waveform

Noisy Channel coded bits

{ci} {ˆ ci}

{um} {ˆ um} xb(t) yb(t)

y(t) x(t)

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SLIDE 6

Conversion between basedband and passband

  • A pragmatic approach: Quadrature Amplitude Modulation (QAM)
  • Essentially speaking, PAM with two branches:
  • ne mixed with cosine, the other with sine
  • system parameter: carrier frequency
  • Equivalent complex baseband representation

6

passband waveform

Symbol Mapper Pulse Shaper Sampler + Filter Symbol Demapper

discrete sequence

Up Converter Down Converter

baseband waveform

Noisy Channel coded bits

{ci} {ˆ ci}

{um} {ˆ um} xb(t) yb(t)

y(t) x(t)

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SLIDE 7

7

Part I. Signal Space

A Linear Algebraic Point to View for the Conversion between Sequences and Waveforms

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SLIDE 8

8

Fourier series for time-limited signals

x(t) → x[m] Analysis (waveform → sequence) Synthesis (sequence → waveform) x[m] → x(t) x[m] = 1 T

  • T

x(t)e− j2πm

T

t dt

x(t) =

∞

  • m=−∞

x[m]ej 2πm

T

t

x[m] = ∞

−∞

x(t) 1 √ T e− j2πm

T

t dt

x(t) =

∞

  • m=−∞

x[m] 1 √ T ej 2πm

T

t

φm ≡ φm(t) 1 √ T exp(j2π T mt), m ∈ Z, Fourier Basis:

φ∗

m(t)

φm(t)

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SLIDE 9

9

Sampling theorem for band-limited signals

x(t) → x[m] Analysis (waveform → sequence) Synthesis (sequence → waveform) x[m] → x(t) Sinc Basis:

φ∗

m(t)

φm(t)

x[m] = x(t)|t= m

2W = x

m 2W

  • x(t) =

∞

  • m=−∞

x[m] sinc(2Wt − m) x[m] = 1 √ 2W x(m/2W) x(t) =

∞

  • m=−∞

x[m] √ 2Wsinc(2Wt − m) = ∞

−∞

x(t) √ 2Wsinc(2Wt − m) dt

check!

φm ≡ φm(t) √ 2Wsinc(2Wt − m), m ∈ Z

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SLIDE 10

Signal space intrepretation

10

x(t) → x[m] waveform → sequence sequence → waveform x[m] → x(t) {x[m]} → {φm(t)} → x(t) =

∞

  • m=−∞

x[m]φm(t) x(t) → φm(t) → x[m] = ∞

−∞

x(t)φ∗

m(t) dt

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SLIDE 11

Signal space intrepretation

11

x(t) → x[m] waveform → sequence sequence → waveform x[m] → x(t) {x[m]} → {φm} → x =

∞

  • m=−∞

x[m]φm

expansion over an

  • rthonormal basis

x φm x[m] = x, φm

projection onto an

  • rthonormal basis

waveform ⟷ vectors integration ⟷ inner product

u, v ∞

−∞

u(t)v∗(t) dt x(t) x

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SLIDE 12

12

Part II. Pulse Amplitude Modulation

A pragmatic approach to convert symbols to baseband waveforms and back

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SLIDE 13

Modulation basis as time-shifted pulses

  • T = 1/2W : transmission interval
  • W : operational bandwidth
  • Desired properties of the pulse function p(t):
  • Time-limited (approximately)
  • Band-limited

13

Pulse Shaper {um} xb(t) xb(t) =

∞

  • m=−∞

um p(t − mT). φm(t) = p(t − mT), T = 1 2W {um} → {φm(t)} → xb(t)

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SLIDE 14

PAM modulation and demodulation

  • Key question: how to design the pulse p(t) and the filter q(t)?

14

yb(t) Filter q(t) T =

1 2W

ˆ um = ∞

−∞

yb(τ)q(mT − τ) dτ Pulse Shaper {um} xb(t)

xb(t) =

∞

  • m=−∞

um p(t − mT).

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SLIDE 15

ISI-free condition when the channel is perfect

15

yb(t) Filter q(t) T =

1 2W

ˆ um = ∞

−∞

yb(τ)q(mT − τ) dτ Pulse Shaper {um} xb(t)

xb(t) =

∞

  • m=−∞

um p(t − mT). xb(t) xb(τ)

Want: ˆ um = um, ∀ m.

g(t) (p ∗ q)(t)

A sufficient condition: g(ˆ kT) =

  • if ˆ

k = 0 1 if ˆ k = 0

ˆ um = (xb ∗ q)(mT) =

∞

  • k=−∞

uk g(mT − kT) =

∞

  • k=−∞

uk g((m − k)T))

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SLIDE 16

Ideal Nyquist and the Nyquist criterion

16

A sufficient condition (in time domain) g(ˆ kT) =

  • if ˆ

k = 0 1 if ˆ k = 0 An equivalent condition (in frequency domain) T rect(Tf) =

  • m

ˆ g

  • f − m

T

  • rect(Tf)

f 1

rect(f)

1 2

− 1

2

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SLIDE 17

Nyquist criterion

17

ˆ g(f) f

1 2T ≡ W

− 1

2T ≡ −W

T ≡

1 2W

ˆ g(f − m/T) ˆ g(f + m/T)

· · · · · · · · · · · ·

1

  • T rect(Tf) =
  • m

ˆ g

  • f − m

T

  • rect(Tf)

Excessive bandwidth: Bb − W ← this should not be too large Typical choice: W ≤ Bb ≤ 2W

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SLIDE 18

Band-edge symmetry

18

When taking the typical choice: W ≤ Bb ≤ 2W the Nyquist criterion can be simplified to the following band-edge symmetry:

ˆ g(f) T

✟ ✟ ✯

T − ˆ g(Wb−∆) f ˆ g(Wb+∆)

✟ ✟ ✙

Wb Bb

T − Re{ˆ g(W − ∆)} Re{ˆ g(W + ∆)} Re{ˆ g(f)} W

ˆ g∗(W − ∆) + ˆ g(W + ∆) = T, ∀ ∆ ∈ [0, W]

⇐ ⇒

  • Re {ˆ

g(W − ∆)} + Re {ˆ g(W + ∆)} = T Im {ˆ g(W − ∆)} = Im {ˆ g(W + ∆)} , ∀ ∆ ∈ [0, W]

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SLIDE 19

Excessive bandwidth and rolloff factor

19

Excessive bandwidth: Bb − W

ˆ g(f) T

✟ ✟ ✯

T − ˆ g(Wb−∆) f ˆ g(Wb+∆)

✟ ✟ ✙

Wb Bb

T − Re{ˆ g(W − ∆)} Re{ˆ g(W + ∆)} Re{ˆ g(f)} W

Rolloff factor: Bb W − 1

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SLIDE 20

Raised cosine pulse

20

time domain

gβ(t) =

  • π

4 sinc

  • 1

2β

  • ,

if |f| = T

2β

sinc t

T

cos( πβt

T )

1−4 β2t2

T 2

,

  • therwise

rolloff factor = β

β = 0 β = 0.3 β = 0.5

Decay to zero with speed ∼ 1 t3 as t → ∞ when β > 0

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SLIDE 21

ˆ gβ(f) =      T if |f| ≤ 1−β

2T

if |f| > 1+β

2T

T cos2( πT

2β (|f| − 1−β 2T ))

if 1−β

2T < |f| ≤ 1+β 2T

Raised cosine pulse

21

frequency domain rolloff factor = β

β = 0 β = 0.3 β = 0.5

The larger it is, the smoother it transits from T to 0 in the frequency domain, and hence converges to zero faster in the time domain.

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SLIDE 22
  • A theorem:
  • The principle of designing p(t) and q(t)
  • Choosing the shifted pulses as an orthonormal set

22

{p(t − mT) : m ∈ Z} form an orthonormal set ⇐ ⇒ |ˆ p(f)|2 satisfies the Nyquist Criterion Choose ˆ q(f) = ˆ p∗(f) Choose ˆ p(f) such that |ˆ p(f)|2 satisfies the Nyquist Criterion If p(t) ∈ R (which is normally the case), then ˆ q(f) = ˆ p∗(f) = ˆ p(−f) and hence q(t) = p(−t). For faster decay in the time-domain (less approximation error) in t = ⇒ need "larger room" for smoother transition from T to 0 in the frequency domain.

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SLIDE 23

23

Part III. Quadrature Amplitude Modulation

A pragmatic approach to convert baseband to passband waveforms and back

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SLIDE 24

24

How to shift the frequency response?

  • We want to shift baseband signals to passband with center frequency fc:
  • Recall the frequency-shift property of Fourier Transform:
  • So, a naive way is to multiply the signal by a complex sinusoid
  • But, at this point we don’t know how to implement a complex signal in real world
  • We can take the real part after multiplying with the complex sinusoid:
  • But this is a waste of spectrum.

exp(j2πf0t)s(t)

F

← → ˆ s(f − f0) Re {exp(j2πf0t)s(t)} = s(t) cos(2πfct) s(t) ∈ R

real part imaginary part

         Re{ˆ s(f)} = Re{ˆ s(−f)} Im{ˆ s(f)} = −Im{ˆ s(−f)} |ˆ s(f)| = |ˆ s(−f)| ∠ˆ s(f) = −∠ˆ s(−f) mod 2π

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SLIDE 25

Two degrees of freedom for complex signal

  • Why not multiplex two individual baseband waveforms?
  • Quadrature amplitude modulation (QAM):

25

x(t) = x(I)

b (t)

√ 2 cos(2πfct) − x(Q)

b

(t) √ 2 sin(2πfct)

PAM p(t) √ 2 cos(2πfct) PAM p(t) − √ 2 sin(2πfct) {u(Q)

m }

{u(I)

m }

x(I)

b (t)

x(Q)

b

(t)

x(t)

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SLIDE 26

QAM modulation: real-domain implementation

26

x(t) = x(I)

b (t)

√ 2 cos(2πfct) − x(Q)

b

(t) √ 2 sin(2πfct)

PAM p(t) √ 2 cos(2πfct) PAM p(t) − √ 2 sin(2πfct) {u(Q)

m }

{u(I)

m }

x(I)

b (t)

x(Q)

b

(t)

x(t)

in-phase component quadrature component

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SLIDE 27

QAM modulation: equivalent complex

27

xb(t) x(I)

b (t) + jx(Q) b

(t) um u(I)

m + ju(Q) m

PAM p(t)

x(t)

{um} xb(t) √ 2 exp(j2πfct) Re{·}

x(t) = x(I)

b (t)

√ 2 cos(2πfct) − x(Q)

b

(t) √ 2 sin(2πfct) = √ 2Re {xb(t) exp(j2πfct)}

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SLIDE 28

Up conversion

28

PAM p(t) √ 2 cos(2πfct) PAM p(t) − √ 2 sin(2πfct) {u(Q)

m }

{u(I)

m }

x(I)

b (t)

x(Q)

b

(t)

x(t)

PAM p(t)

x(t)

{um} xb(t) √ 2 exp(j2πfct) Re{·}

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SLIDE 29

29

PAM p(t) √ 2 cos(2πfct) PAM p(t) − √ 2 sin(2πfct) {u(Q)

m }

{u(I)

m }

x(I)

b (t)

x(Q)

b

(t)

x(t)

PAM p(t)

x(t)

{um} xb(t) √ 2 exp(j2πfct) Re{·}

Re{ˆ xb(f)}

  • 1
  • Re{ˆ

x(I)

b (f)}

  • Im{ˆ

x(Q)

b

(f)}

1 1

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SLIDE 30
  • ≈

≈

  • √

2

  • ≈

≈ ≈ ≈

  • 1

√ 2 1 √ 2

Real part Real part

30

PAM p(t) √ 2 cos(2πfct) PAM p(t) − √ 2 sin(2πfct) {u(Q)

m }

{u(I)

m }

x(I)

b (t)

x(Q)

b

(t)

x(t)

PAM p(t)

x(t)

{um} xb(t) √ 2 exp(j2πfct) Re{·}

√ 2 cos(2πfct) = 1 √ 2

  • ej2πfct + ej2πfct

− √ 2 sin(2πfct) = j √ 2

  • ej2πfct − ej2πfct

Real part

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SLIDE 31
  • ≈

≈ ≈ ≈

  • 1

√ 2 1 √ 2

Real part Real part

31

PAM p(t) √ 2 cos(2πfct) PAM p(t) − √ 2 sin(2πfct) {u(Q)

m }

{u(I)

m }

x(I)

b (t)

x(Q)

b

(t)

x(t)

PAM p(t)

x(t)

{um} xb(t) √ 2 exp(j2πfct) Re{·}

  • ≈

≈

  • 1

√ 2

Re{s(t)}

F

← → 1 2(ˆ s(f) + s∗(−f))

Real part

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SLIDE 32

32

Down conversion

y(t)

√ 2 cos(2πfct) − √ 2 sin(2πfct) LPF 1 {|f| ≤ Bb} LPF 1 {|f| ≤ Bb} y(I)

b (t)

y(Q)

b

(t)

y(t)

Step Filter 1 {f ≥ 0} yb(t) √ 2 exp(−j2πfct)

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SLIDE 33

33

y(t)

√ 2 cos(2πfct) − √ 2 sin(2πfct) LPF 1 {|f| ≤ Bb} LPF 1 {|f| ≤ Bb} y(I)

b (t)

y(Q)

b

(t)

y(t)

Step Filter 1 {f ≥ 0} yb(t) √ 2 exp(−j2πfct)

  • ≈

≈

  • 1

√ 2

  • ≈

≈

  • 1

√ 2

  • ≈

≈

  • 1

√ 2

Real part

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SLIDE 34
  • ≈

≈

1 Real part

34

y(t)

√ 2 cos(2πfct) − √ 2 sin(2πfct) LPF 1 {|f| ≤ Bb} LPF 1 {|f| ≤ Bb} y(I)

b (t)

y(Q)

b

(t)

y(t)

Step Filter 1 {f ≥ 0} yb(t) √ 2 exp(−j2πfct)

  • ≈

≈

  • 1

√ 2

√ 2 cos(2πfct) = 1 √ 2

  • ej2πfct + ej2πfct

− √ 2 sin(2πfct) = j √ 2

  • ej2πfct − ej2πfct

Imaginary part

  • ≈

≈

1 Real part

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SLIDE 35
  • ≈

≈

  • 1

Imaginary part

  • ≈

≈

  • 1

Real part

35

y(t)

√ 2 cos(2πfct) − √ 2 sin(2πfct) LPF 1 {|f| ≤ Bb} LPF 1 {|f| ≤ Bb} y(I)

b (t)

y(Q)

b

(t)

y(t)

Step Filter 1 {f ≥ 0} yb(t) √ 2 exp(−j2πfct)

  • ≈

≈

  • 1

√ 2

√ 2 cos(2πfct) = 1 √ 2

  • ej2πfct + ej2πfct

− √ 2 sin(2πfct) = j √ 2

  • ej2πfct − ej2πfct

Real part

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SLIDE 36
  • ≈

≈

  • 1

Re{ˆ y(I)

b (f)}

36

y(t)

√ 2 cos(2πfct) − √ 2 sin(2πfct) LPF 1 {|f| ≤ Bb} LPF 1 {|f| ≤ Bb} y(I)

b (t)

y(Q)

b

(t)

y(t)

Step Filter 1 {f ≥ 0} yb(t) √ 2 exp(−j2πfct)

  • ≈

≈

  • 1

√ 2

  • ≈

≈

  • 1

Im{ˆ y(Q)

b

(f)}

Real part

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SLIDE 37
  • ≈

≈

  • 1

√ 2

37

y(t)

√ 2 cos(2πfct) − √ 2 sin(2πfct) LPF 1 {|f| ≤ Bb} LPF 1 {|f| ≤ Bb} y(I)

b (t)

y(Q)

b

(t)

y(t)

Step Filter 1 {f ≥ 0} yb(t) √ 2 exp(−j2πfct)

  • ≈

≈

  • 1

Re{ˆ y(I)

b (f)}

  • ≈

≈

  • 1

Im{ˆ y(Q)

b

(f)}

Real part

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SLIDE 38
  • 1

38

y(t)

√ 2 cos(2πfct) − √ 2 sin(2πfct) LPF 1 {|f| ≤ Bb} LPF 1 {|f| ≤ Bb} y(I)

b (t)

y(Q)

b

(t)

y(t)

Step Filter 1 {f ≥ 0} yb(t) √ 2 exp(−j2πfct)

  • ≈

≈

  • 1

Re{ˆ y(I)

b (f)}

  • ≈

≈

  • 1

Im{ˆ y(Q)

b

(f)} Re{ˆ yb(f)}

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SLIDE 39

39

QAM demodulation

y(t)

√ 2 cos(2πfct) − √ 2 sin(2πfct) Filter q(t) Filter q(t) T =

1 2W

T =

1 2W

{ˆ u(Q)

m }

{ˆ u(I)

m }

LPF 1 {|f| ≤ Bb} LPF 1 {|f| ≤ Bb} y(I)

b (t)

y(Q)

b

(t)

y(t)

Step Filter 1 {f ≥ 0} yb(t) √ 2 exp(−j2πfct) Filter q(t) T =

1 2W

{ˆ um}

merge

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SLIDE 40

40

QAM demodulation

y(t)

√ 2 cos(2πfct) − √ 2 sin(2πfct) Filter q(t) Filter q(t) T =

1 2W

T =

1 2W

{ˆ u(Q)

m }

{ˆ u(I)

m }