Fractionally Spaced Equalization and Frequency Diversity...

Fractionally Spaced Equalization and Frequency Diversity Methods

for Block Transmission with Cyclic Prefix

Yuki Yoshida, Kazunori Hayashi, Hideaki SakaiDepartment of System Science,

Graduate School of Informatics, Kyoto University

Outline

Background & Motivation

The System Configuration of the Proposed Fractionally Spaced Equalizer (FSE)

Derivation of the Equalizer Weights

Frequency Diversity Method using the FSE

Simulation Results

Summary

Background & Motivation

・Block Transmission with Cyclic Prefix (CP)OFDM (Orthogonal Frequency Division Multiplexing)

DTV, DAB, Wireless LAN

SC-CP (Single Carrier block transmission with CP)

DMT (Discrete Multitone) ADSL

・FDE (Frequency Domain Equalization)Simple Implementation using FFTRobustness to Frequency Selective Fading Channels

Existing FDEers work at symbol rate (SSE : Symbol Spaced Equalizer )

Background & Motivation (cont’d)

Fractionally Spaced Equalizer (FSE)based on oversampling the received signal

ex) 4 times oversampling → T/4 - FSE

Zero- Forcing (ZF) based T/2-FSE has been proposed by Vaidyanathan and Vrcelj, 2002

How can we adopt the FSE to FDE systems?

Extension of Vaidyanathan’s method to general oversampling factor cases (T/K-FSE) Derivation of ZF and minimum mean-square-error (MMSE) weights of the T/K-FSEA simple frequency diversity method using the T/K-FSE

Fractionally Spaced EqualizationSSE

FSE (Oversampling Factor K)

Channel SSENoise

Equalizer Output

Transmitted Signal

FSE

(Uniformly Sampled Version of Continuous quantity )

DecimatorExpander

(If m is multiple of K)(else)

SC-CP SchemeSystem Configuration of SC-CP Scheme (Single Carrier with CP)

S/P P/S

Txfilter

Rx filter S

/P

P/S

DFT

IDFTFD

E

Add CP

Remove CP

Fading

Additive Noise&

Frequency Domain Equalization

DFT IDFTChannel Matrix Equalizer Weights

Noise Block

Received Signal Block

Block Diagram of SC-CP Scheme

n-th Input Signal Block

(Circulant) Matrix Matrix Matrix

n-th Equalizer Output Block

SC-CP SchemeS

/P P/S

Txfilter

Rx filter S

/P

P/S

DFT

IDFTFD

E

Add CP

Remove CP

Frequency Domain Equalization

System Configuration of SC-CP Scheme (Single Carrier with CP)

DFT IDFTChannel Matrix Equalizer Weights

Noise Block

Received Signal Block

Block Diagram of SC-CP Scheme

Fading

Additive Noise&

(Circulant) Matrix Matrix Matrix

n-th Equalizer Output Blockn-th Input Signal Block

CP Length: （Channel Order）

M×M Circulant Matrix

:Channel Impulse Response, （ ：Symbol Spacing ）

SC-CP SchemeSystem Configuration of SC-CP Scheme (Single Carrier with CP)

S/P P/S

Txfilter

Rx filter S

/P

P/S

DFT

IDFTFD

E

Add CP Frequency Domain Equalization

Remove CP

DFT IDFTChannel Matrix Equalizer Weights

Noise Block

Received Signal Block

Block Diagram of SC-CP Scheme

Fading

Additive Noise&

n-th Input Signal Block

(Circulant) Matrix Matrix Matrix

n-th Equalizer Output Block

M×M DFT Matrix

（Unitary）

SC-CP SchemeSystem Configuration of SC-CP Scheme (Single Carrier with CP)

S/P P/S

Txfilter

Rx filter S

/P

P/S

DFT

IDFTFD

E

Add CP

Remove CP

Frequency Domain Equalization

DFT IDFTChannel Matrix Equalizer Weights

Noise Block

Received Signal Block

Block Diagram of SC-CP Scheme

Fading

Additive Noise&

n-th Input Signal Block

(Circulant) Matrix Matrix Matrix

n-th Equalizer Output Block

Diagonalized by DFT

DFT IDFTDiagonal

Efficiently Equalized

OFDM SchemeSystem Configuration of OFDM Scheme

(Orthogonal Frequency Division Multiplexing)

S/P P/S

Txfilter

Rx filter S

/P

P/S

DFT

FDE

Add CP

Remove CPID

FT

Fading

Additive Noise&

Frequency Domain Equalization

Block Diagram of OFDM Scheme

n-th Input Signal Block

Channel Matrix(Circulant)

Noise BlockReceived Signal Block

n-th Equalizer Output Block

DFTMatrix

Equalizer WeightsMatrixIDFT

Matrix

Fractionally Spaced EqualizationSSE

Equalizer Output

Transmitted Signal

SSEChannel Noise

FSE (Oversampling Factor K)FSE

DecimatorExpander

(If m is multiple of K)(else)

Configuration of the SC-CP Scheme with Proposed T/K-FSE

S/P P/S Tx

filterTransmitter:

Frequency Selective Fading

Additive Noise&

Add CP

Rx filter

S/P

S/P

KM

-poi

nt D

FT

Remove CP

One-tap FDE using KM-point DFT

Decimator

K times Oversampling

KM

-poi

nt ID

FT

Receiver:

Signal Modeling

Received Signal：

Equalizer Output：

DFT IDFTExpander DecimatorEqualizer Weights

Channel(Circulant)

(Diagonal)

Noise Received SignalEqualizer OutputInput Signal

Signal Modeling

Received Signal：

Equalizer Output：

DFT IDFTExpander DecimatorEqualizer Weights

Channel(Circulant)

(Diagonal)

Noise Received SignalEqualizer OutputInput Signal

KM×M Expander Matrix

denotes the M x KM Decimator Matrix

Signal Modeling

Received Signal：

Equalizer Output：

DFT IDFTExpander DecimatorEqualizer Weights

Channel(Circulant)

(Diagonal)

Noise Received SignalEqualizer OutputInput Signal

KM×KM Channel Matrix (Circulant)

: Channel Response including Tx and Rx filter:Symbol Period :Channel Order

Signal Modeling

Received Signal：

Equalizer Output：

DFT IDFTExpander DecimatorEqualizer Weights

Channel(Circulant)

(Diagonal)

Noise Received SignalEqualizer OutputInput Signal

Simplification of T/K-FSE

Equalizer Output：

Same as the (m, n) element of M-point DFT matrix

: M x M Identity Matrix）（

Simplification of T/K-FSE (cont’d)

Equalizer Output:

is M×M diagonal submatrix of (k=1, 2, ・・・, K)

Simplification of T/K-FSE (cont’d)Received Signal Equalizer Output

Block Diagram of the proposed T/K-FSE

M-point IDFT

KM-point DFT

DFT IDFT DecimatorEqualizer Weights

Input and Output relation

is M×M submatrix of (k=1, 2, ・・・, K)

ZF Weights of Proposed T/K-FSE

ZF Condition： s.t.

SSE (K=1) Uniquely determined

Channel nulls result in Noise enhancement

T/K-FSE (K>1)

Certain Degree of Freedom in the choice of Equalizer Weights

ZF Weights of Proposed T/K-FSE (cont’d)

Minimization of the noise power at the equalizer output

Noise component at the equalizer output：

Minimization Problem

min

s.t.

ZF Weights of Proposed T/K-FSE (cont’d)

Minimization Problem

min

s.t.

ZF Weights of Proposed T/K-FSE:

MMSE Weights of Proposed T/K-FSE Cost function ( Mean-Square-Error ):

MMSE Weights of Proposed T/K-FSE:

Optimum Linear MMSE T/K-FSE

Optimum Linear MMSE T/K-FSE:

Expander ChannelEqualizer(M X KM)

Optimum Linear MMSE Equalizer can’t be realized by the one-tap FDE because of the colored noise

Simulation Settings

Mod/Demod. Scheme QPSK / Coherent DetectionBlock Size M = 256

Length of CP 32Channel Model 10-path rayleigh fading channels with

an exponentially decaying power profile

Tx & Rx Filter square-root raised-cosine filter (roll-off factor α = 0.5)

Channel Estimation IdealChannel Noise AWGN

# of Iteration 10,000

Oversampling Rate K = 1, 2, 4 (i.e. SSE, T/2-FSE, T/4-FSE)

BER Performance of Proposed T/K-FSE

OFDM Scheme SC-CP Scheme

Performance of T/K-FSEPassband width and Performance Improvement via FSE

No energy on this band

Exploiting Band

Not exploited in SSE case

We can’t expect any performance Improvement in K >2

Frequency Diversity Method using Proposed T/K-FSE

S/P P/S Tx

filter

CP InsertionOriginal Spectrum

(P-1) CopiesP times Symbol Rate

Transmit Signal Spectrum

Band Width:

Rx filter S

/P

S/P

IDFT

CP RemovalM

M

KM

-poi

nt D

FT

M-pointK times Oversampling

Equalization and diversity combining are achieved simultaneously

BER Performance of the SC-CP via Proposed FDE/DC

0 5 10 15 20 25

10

10

10

10

10

-1

-2

-3

-4

-5

1

[dB]

BER

K=P=1

K=P=2K=P=4

ZFMMSE

Mod./ Demod. QPSK Block Size M = 256CP Length N=32

Channel:10-path rayleigh fading channels

Order of Channel： L=30

Tx/Rx Filter:square-root raised-cosine filter ( α=0.5)

Channel Estimation: Ideal

（Energy per Bit over Noise Power Density）

BER Performance of the OFDM via Proposed FDE/DC

0 5 10 15 20 25

10

10

10

10

10

-1

-2

-3

-4

-5

1

[dB]

BER

Mod./ Demod. QPSK Block Size M = 256CP Length N=32

Channel:10-path rayleigh fading channels

Order of Channel： L=30

Tx/Rx Filter:square-root raised-cosine filter ( α=0.5)

Channel Estimation: Ideal

ZF & MMSE

K=P=1

K=P=2

K=P=4

（Energy per Bit over Noise Power Density）

Applications of FDE/DC

Robust Communication with Poor Receiversex) Sensor Network, Traffic lights Network, Inter Vehicle Communication, etc

One Alternative Rate Reduction Technique for Adaptive Modulation Systems

ex) Wi-Fi

QPSK

Im

Re

Im

Re

BPSK

1 2 3 4 5 6 7 8

1 0 2 0 3 0 4 0 Expanded Version

Original Block

FDE/DC

Rate Reduction Technique via FDE/DCOFDM

DFTIDFTExpander DecimatorEqualizer Weights

Channel

DFTIDFT Equalizer Weights

Channel

Change the WeightsAdd Decimator

Reduce the RateAdd Expander

BER Performance of the SC-CP via FDE/DC for a given transmission rate

10

10

10

10

10

-1

-2

-3

-4

-5

1

BER

0 5 10 15 20 25[dB]

K=P=1 M=512 BPSK ZF

K=P=2 M=256 QPSK ZF

K=P=4 M=128 16QAM ZF

Mod./ Demod.BPSK, QPSK, 16QAM Block SizeM =128, 256, 512(FFT size = 512)CP Length N=32

Channel:10-path rayleigh fading channels

Order of Channel： L=30

Tx/Rx Filter:square-root raised-cosine filter ( α=0.5)

Channel Estimation: Ideal

Pre- and Post- Equalization

DFT IDFTExpander DecimatorFDEChannel

Post-FDE/DC

Expander DecimatorChannel

(Frequency Domain Equalizer and Diversity Combiner)

FDEDFT IDFT

Pre-FDE/DC

Pre- and Post- Equalization

DFT IDFTExpander DecimatorFDEChannel

Post-FDE/DC

Expander DecimatorChannel

(Frequency Domain Equalizer and Diversity Combiner)

FDEDFT IDFT

Pre-FDE/DC

Configuration of the SC-CP Scheme with Proposed Pre-FDE/DC

S/P

P/S

CP Insertion

Transmitted Signal

DFT

IDFT

Transmitter:

Rx filter

P times Symbol RateBand Width:

Receiver:

S/P

CP Removal

P/S

Output

Rx filter

Sampling at Symbol Rate

Weights of Proposed Pre-FDE/DC

ZF based:

MMSE based:

BER Performance of the SC-CP via Proposed Pre-FDE/DC

0 5 10 15 20 25

10

10

10

10

10

-1

-2

-3

-4

-5

1

[dB]

BER

ZFMMSE

K=P=1

K=P=2K=P=4

Mod./ Demod. QPSK Block Size M = 256CP Length N=32

Channel:10-path rayleigh fading channels

Order of Channel： L=30

Tx/Rx Filter:square-root raised-cosine filter ( α=0.5)

Channel Estimation: Ideal

AWGN

BER Performance of the OFDM via Proposed Pre-FDE/DC

0 5 10 15 20 25

10

10

10

10

10

-1

-2

-3

-4

-5

1

[dB]

BER

ZFMMSE

K=P=1

K=P=2K=P=4

Mod./ Demod. QPSK Block Size M = 256CP Length N=32

Channel:10-path rayleigh fading channels

Order of Channel： L=30

Tx/Rx Filter:square-root raised-cosine filter ( α=0.5)

Channel Estimation: Ideal

AWGN

Summary

T/K-FSE methods for block transmission with cyclic prefix are proposed

ZF and MMSE weights of the proposed T/K-FSE are derived

Based on the idea of T/K-FSE, two simple frequency diversity method is also proposed

Computer simulations reveal the performance improvements by proposed methods

PAPR of the SC-CP via Proposed Pre-FDE/DC

CCDF=complementary cumulative distribution function

0 2 4 6

10

10

10

10

-1

-2

-3

-4

1

8 10 12 14 16

CC

DF

PAPR [dB]

OFDM(M=256)

ZFMMSE

SC-CP(M=256)

K=P=２ Mod./ Demod. QPSK Block Size M = 256CP Length N=32

Order of Channel： L=30

Tx/Rx Filter:square-root raised-cosine filter ( α=0.5)

CCDF=complementary cumulative distribution function

PAPR of the OFDM via Proposed Pre-FDE/DC

0 2 4 6

10

10

10

10

-1

-2

-3

-4

1

8 10 12 14 16

CC

DF

PAPR [dB]

K=P=2

ZFMMSE

OFDM(M = 256)

Mod./ Demod. QPSK Block Size M = 256CP Length N=32

Order of Channel： L=30

Tx/Rx Filter:square-root raised-cosine filter ( α=0.5)