tsks01&digital&communication recall:&pulsemamplitude ......

TSKS01&Digital&CommunicationLecture&6

Dispersive&channels,&ML&sequence&detection

Emil&Björnson

Department&of&Electrical&Engineering&(ISY)Division&of&Communication&Systems

Recall:&Nyquist&Criterion

Recall:&PulseMAmplitude&Modulation&(PAM):

! " =$% & '(" − &*)�

-

Receiver&filtering&. " , Γ(1):

. ∗ ! " =$% & (. ∗ ')(" − &*)�

-

Nyquist&criterion:

$ Γ 1 −3*

4 1 −3*

5

6785

= Constant

2016M10M07 TSKS01&Digital&Communication&M Lecture&6 2

Called:&'(") and&.(") are&Nyquist(together

Bandwidth more than 1/(2*):Many Nyquist&pulses


' " BandB. " BareBFGHIJKLBLMNOLPOQBif $ Γ 1 −3* 4 1 −

3*

5

6785

isBconstant

$ Γ 1 −3*

4 1 −3*

5

6785

Γ 1 4 1


OneMway&Digital&Communication&System

Analog(channel

Source

Destination

Channel(encoder Modulator

Channel(decoder

De8modulator

Channel&coding

Error&control

Error&correction

Digital to&analog

Analog&to&digital

Medium

Digital&modulation

Channel(propertiesImpulse response

ℎ(")Additive&noise&V(")

Channel&Filtering

! Transmitted&PAM&signal

W " =$% & '(" − &*)�

-

! Filtered&by&channel,&impulse&response&ℎ("):

! Noise&is&added

X " = W ∗ ℎ " +V(")


ℎ(")W(") W ∗ ℎ (")

How will this affect the&Nyquist&criterion?

Example

Nyquist&Criterion&with&Channel&Filtering&(1/2)

Signal&after&Receiver&filtering&. " , Γ(1):

. ∗ ! " =$% & (. ∗ ' ∗ ℎ)(" − &*)�

-

+ . ∗ V (")

Effective&pulse:. ∗ ' ∗ ℎ (")

Nyquist&criterion:

$ Γ 1 −3*

Z 1 −3*

4 1 −3*

5

6785

= Constant


Can(this be(achieved in(practice?

Nyquist&Criterion&with&Channel&Filtering&(2/2)

! If&Z 1 ≠ 0 everywhere&with&P 1 ≠ 0 and&P 1 is&Nyquist! Pick&Γ 1 = 1/Z(1) Impractical&to&depend&on&Z(1)!

! If&Z 1 ≈ constant where&P 1 ≠ 0

! Pick&Γ 1 and&4(1) as&Nyquist&together


FGHIJKLB^QJLOQJM_ $ Γ 1 −3*

Z 1 −3*

4 1 −3*

5

6785

BisBconstant

Example:Small(bandwidth:

Approximately constantOne reason to&use OFDM

|Z 1 |

What&if&Nyquist&Criterion&is&not&satisfied?

! Sampled&output

a b =$% & (. ∗ ' ∗ ℎ)(b* − &*)�

-

+ . ∗ V (b*)

! Define&c d = (. ∗ ' ∗ ℎ)(d*)

! InterMsymbol&interference&if&c d ≠ 0 for&some&d ≠ 0

! Causal&and&timeMlimited&pulses&and&impulse&response! c d ≠ 0 for&d = 0, … , f − 1

a b = $% b − d c[d]i8j

k7l

+Vm[b]


Vm[b]

fMtap&Channel&with&Gaussian&Noise

! Sampled&output

a b = $% b − d c[d]i8j

k7l

+Vm[b]

where&Vm[b] is! Gaussian&with&zero&mean&and&variance&nl/2

! Independent&for&different&values&of&b


This is&called a&dispersive channel

How to(perform detection? This is&the&topic of this and&next lecture


Detecting&Sequence&of&o symbols

ChannelSender ReceiverSource Destination

Noise

Message&set: pq q7lrs8j Size:&tu

Mapping: pq → %q 0 , %q 1 , … %q[o − 1]

for&w ∈ 0,1, … ,tu − 1

A&message consists of oBsymbols&from&an&tMsize constellation (n = 1)

% 0 , % 1 , … %[o − 1]

a 0 , a 1 , …a[o − 1]

Assume: % −f + 1 ,… , %[−1] are known

ML&Sequence&Detection&(1/3)

! Recall:

! In&this&case:

! Note

a b = $% b − d c[d]i8j

k7l

+Vm[b]

is&Gaussian&with&variance&nl/2 and mean&∑ % b − d c[d]i8jk7l


ML(decision(rule: Set&pz = pq if 1{̅|}(~̅|p-) is&maximized for&& = w.


! Define:

� %- b , … , %-[b − f + 1] = $%- b − d c[d]i8j

k7l! We&obtain:

! Maximized&by&minimizing



! Define

! We&have

! Euclidean&distance&between&~̅ and&received&signal&without&noise


ML(decision(rule:&Set&pz = pq&if&Ä ~̅, �Å- &is&minimized&for&& = w.


1 3

3

1

–1

Geometric&Illustration

?

Received&vector: z

z

Goal:&Minimize&the&error&probability

Pr É ≠ ÉÑ|a̅ = ~̅

Estimated&message

Sent&message

Received&vector

Observed&realization

Same(setup(as(beforeAre&we&done&now?

�Åj

�ÅÖ

�Ål

Complexity&Considerations

! ML&sequence&detection! Compute&Ä ~̅, �Å- for&& = 0,… ,tu − 1

! Select&the&smallest&distance

! Ways&to&reduce&complexity1. Send&one&symbol&and&be&silent&for&f − 1 symbols2. Design&some&approximate&detection&algorithm3. Find&a&way&to&implement&ML&detection&more&efficiently


Number(of(messages(grows(very(fast!

This(is(what(we(will(do!www.liu.se

tsks01&digital&communication recall:&pulsemamplitude ......

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