Sphere Decoding for Noncoherent Channels
Lutz Lampe
Deptartment of Electrical & Computer Engineering
The University of British Columbia, Canada
joint work with Volker Pauli, Robert Schober, and Christoph Windpassinger
Motivation 2
in various propagation environmentshigh frequency bandsOffer high fidelityOperate in
Wireless communication systems
Use cheap consumer devices
L. Lampe: Sphere Decoding for Noncoherent Channels
Motivation 3
⇒ E.g. low cost local oscillators
⇒ no perfect carrier phase synchronization feasible
⇒ noncoherent communication is a favorable choice
L. Lampe: Sphere Decoding for Noncoherent Channels
Motivation 3
⇒ E.g. low cost local oscillators
⇒ no perfect carrier phase synchronization feasible
⇒ noncoherent communication is a favorable choice
L. Lampe: Sphere Decoding for Noncoherent Channels
Motivation 3
⇒ E.g. low cost local oscillators
⇒ no perfect carrier phase synchronization feasible
⇒ noncoherent communication is a favorable choice
L. Lampe: Sphere Decoding for Noncoherent Channels
Motivation 4
Noncoherent channel:
n[k]h[k]ejθ[k]
r[k]s[k]
Noncoherent multipath channel:
“noncoherent” ≡ “noncoherent reception without channel state information (CSI)”
L. Lampe: Sphere Decoding for Noncoherent Channels
Motivation 4
Noncoherent channel:
n[k]h[k]ejθ[k]
r[k]s[k]
Noncoherent multipath channel:
n[k]
h[k]︷ ︸︸ ︷
a[k]ejθ[k]
r[k]s[k]
“noncoherent” ≡ “noncoherent reception without channel state information (CSI)”
L. Lampe: Sphere Decoding for Noncoherent Channels
Motivation 4
Noncoherent channel:
n[k]h[k]ejθ[k]
r[k]s[k]
Noncoherent multipath channel:
n[k]
h[k]︷ ︸︸ ︷
a[k]ejθ[k]
r[k]s[k]
“noncoherent” ≡ “noncoherent reception without channel state information (CSI)”
L. Lampe: Sphere Decoding for Noncoherent Channels
Outline 5
� Transmitter: Differential Encoding
� Receiver:
? Multiple-Symbol Differential Detection
? Multiple-Symbol Differential Sphere Decoding
� Performance Results
� Summary
L. Lampe: Sphere Decoding for Noncoherent Channels
Outline 5
� Transmitter: Differential Encoding
� Receiver:
? Multiple-Symbol Differential Detection
? Multiple-Symbol Differential Sphere Decoding
� Performance Results
� Summary
L. Lampe: Sphere Decoding for Noncoherent Channels
Differential Encoding 6
� Consider phase-shift keying (PSK) signal set: s[k] = ej2πM m[k]
L. Lampe: Sphere Decoding for Noncoherent Channels
Differential Encoding 6
� Consider phase-shift keying (PSK) signal set: s[k] = ej2πM m[k]
Im
Re
r[k] = ejθs[k] (h[k] + n[k])
s[k] ? (θ ?)
L. Lampe: Sphere Decoding for Noncoherent Channels
Differential Encoding 6
� Consider phase-shift keying (PSK) signal set: s[k] = ej2πM m[k]
Im
Re
r[k] = ejθs[k] (h[k] + n[k])
s[k] ? (θ ?)
L. Lampe: Sphere Decoding for Noncoherent Channels
Differential Encoding 6
� Consider phase-shift keying (PSK) signal set: s[k] = ej2πM m[k]
Im
Re
r[k] = ejθs[k] (h[k] + n[k])
s[k] ? (θ ?)
L. Lampe: Sphere Decoding for Noncoherent Channels
Differential Encoding 6
� Consider phase-shift keying (PSK) signal set: s[k] = ej2πM m[k]
Im
Re
r[k] = ejθs[k] (h[k] + n[k])
s[k] ? (θ ?)
� Solution: encode data in phase difference arg{s[k]} − arg{s[k − 1]}
L. Lampe: Sphere Decoding for Noncoherent Channels
Differential Encoding 7
� Differential phase-shift keying (DPSK): s[k] = v[k] · s[k − 1]
L. Lampe: Sphere Decoding for Noncoherent Channels
Differential Encoding 7
� Differential phase-shift keying (DPSK): s[k] = v[k] · s[k − 1]
� Structure: v[k]
s[k − 1]
s[k]
Delay
L. Lampe: Sphere Decoding for Noncoherent Channels
Outline 7
� Transmitter: Differential Encoding
� Receiver:
? Multiple-Symbol Differential Detection
? Multiple-Symbol Differential Sphere Decoding
� Performance Results
� Summary
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Detection 8
� Write: r[k] = s[k] · h[k] + n[k]
= v[k]s[k − 1] · h[k − 1] + n[k]
= v[k]r[k − 1] − v[k]n[k − 1] + n[k]
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Detection 8
� Write: r[k] = s[k] · h[k] + n[k]
= v[k]s[k − 1] · h[k − 1] + n[k]
= v[k]r[k − 1] − v[k]n[k − 1] + n[k]
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Detection 8
� Write: r[k] = s[k] · h[k] + n[k]
= v[k]s[k − 1] · h[k − 1] + n[k]
= v[k]r[k − 1] − v[k]n[k − 1] + n[k]
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Detection 8
� Write: r[k] = s[k] · h[k] + n[k]
= v[k]s[k − 1] · h[k − 1] + n[k]
= v[k]r[k − 1] − v[k]n[k − 1] + n[k]
� Conventional differential detection:
v̂[k] = argmaxv
{Re{r∗[k]v r[k − 1]}}
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Detection 8
� Write: r[k] = s[k] · h[k] + n[k]
= v[k]s[k − 1] · h[k − 1] + n[k]
= v[k]r[k − 1] − v[k]n[k − 1] + n[k]
� Conventional differential detection:
v̂[k] = argmaxv
{Re{r∗[k]v r[k − 1]}}
��
(·)∗r∗[k − 1]
v̂[k]r[k]
Delay
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Detection 9
� Idea: Use observation window of length N > 2 received samples
� Multiple-symbol differential detection (MSDD)
Joint decision on N−1 data symbols based on observation of N received samples
� Performance ⇔ Complexityexploit memory “tree” search
(“memory increases capacity”) (complexity exponential in N)
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Detection 9
� Idea: Use observation window of length N > 2 received samples
� Multiple-symbol differential detection (MSDD)
Joint decision on N−1 data symbols based on observation of N received samples
� Performance ⇔ Complexityexploit memory “tree” search
(“memory increases capacity”) (complexity exponential in N)
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Detection 9
� Idea: Use observation window of length N > 2 received samples
� Multiple-symbol differential detection (MSDD)
Joint decision on N−1 data symbols based on observation of N received samples
r[k − N + 1] . . . r[k]r[k − 1] . . .. . .
observation window
� Performance ⇔ Complexityexploit memory “tree” search
(“memory increases capacity”) (complexity exponential in N)
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Detection 9
� Idea: Use observation window of length N > 2 received samples
� Multiple-symbol differential detection (MSDD)
Joint decision on N−1 data symbols based on observation of N received samples
r[k − N + 1] . . . r[k]r[k − 1] . . .. . .
observation window
� Performance ⇔ Complexityexploit memory “tree” search
(“memory increases capacity”) (complexity exponential in N)
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Detection 10
� Maximum-likelihood (ML) MSDD
? Wilson et al., 1989
? Divsalar & Simon, 1990
? Leib & Pasupathy, 1991 . . .
� ML MSDD for static channels
? Mackenthun, 1994
� Suboptimum MSDD
– Linear-prediction based decision-feedback differential detection (DF-DD)
? Kam & Teh, 1983
? Svensson, 1994
? Schober et al., 1999 . . .
– Two-step algorithms
? Xiaofu & Songgeng, 1998 . . .
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Detection 10
� Maximum-likelihood (ML) MSDD
? Wilson et al., 1989
? Divsalar & Simon, 1990
? Leib & Pasupathy, 1991 . . .
� ML MSDD for static channels
? Mackenthun, 1994
� Suboptimum MSDD
– Linear-prediction based decision-feedback differential detection (DF-DD)
? Kam & Teh, 1983
? Svensson, 1994
? Schober et al., 1999 . . .
– Two-step algorithms
? Xiaofu & Songgeng, 1998 . . .
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Detection 10
� Maximum-likelihood (ML) MSDD
? Wilson et al., 1989
? Divsalar & Simon, 1990
? Leib & Pasupathy, 1991 . . .
� ML MSDD for static channels
? Mackenthun, 1994
� Suboptimum MSDD
– Linear-prediction based decision-feedback differential detection (DF-DD)
? Kam & Teh, 1983
? Svensson, 1994
? Schober et al., 1999 . . .
– Two-step algorithms
? Xiaofu & Songgeng, 1998 . . .
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Detection 10
� Maximum-likelihood (ML) MSDD
? Wilson et al., 1989
? Divsalar & Simon, 1990
? Leib & Pasupathy, 1991 . . .
� ML MSDD for static channels
? Mackenthun, 1994
� Suboptimum MSDD
– Linear-prediction based decision-feedback differential detection (DF-DD)
? Kam & Teh, 1983
? Svensson, 1994
? Schober et al., 1999 . . .
– Two-step algorithms
? Xiaofu & Songgeng, 1998 . . .
L. Lampe: Sphere Decoding for Noncoherent Channels
Outline 10
� Transmitter: Differential Encoding
� Receiver:
? Multiple-Symbol Differential Detection
? Multiple-Symbol Differential Sphere Decoding
� Performance Results
� Summary
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 11
� Block diagram
v̂[k]v[k]
T
ŝ[k]
()∗
MSDSD
T
s[k]
h[k]
r[k]
n[k]
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 11
� Block diagram
v̂[k]v[k]
T
ŝ[k]
()∗
MSDSD
T
s[k]
h[k]
r[k]
n[k]
� Rayleigh fading channel h[k]
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 11
� Block diagram
v̂[k]v[k]
T
ŝ[k]
()∗
MSDSD
T
s[k]
h[k]
r[k]
n[k]
� Rayleigh fading channel h[k]
� ML MSDD
r , [r[k − (N − 1)], r[k − (N − 2)], . . . , r[k]]T = [r1, r2, . . . , rN ]]T
s , [s[k − (N − 1)], s[k − (N − 2)], . . . , s[k]]T = [s1, s2, . . . , sN ]]T
h , [h[k − (N − 1)], h[k − (N − 2)], . . . , h[k]]T = [h1, h2, . . . , hN ]]T
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 12
� Vector channelr = diag{s}h + n
� ML decision rule [Ho&Fung, 1992]
ŝ = argmins
{rHR−1rr r}
where
Rrr , E{rrH|s}
= diag{s}(
E{hhH} + σ2nIN︸ ︷︷ ︸C
)diag{s∗}
and
(diag{s})−1 = diag{s∗} ,
diag{s∗}r = diag{r}s∗
ŝ = argmins
{sT diag{r}HC−1 diag{r}s∗}
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 12
� Vector channelr = diag{s}h + n
� ML decision rule [Ho&Fung, 1992]
ŝ = argmins
{rHR−1rr r}
where
Rrr , E{rrH|s}
= diag{s}(
E{hhH} + σ2nIN︸ ︷︷ ︸C
)diag{s∗}
and
(diag{s})−1 = diag{s∗} ,
diag{s∗}r = diag{r}s∗
ŝ = argmins
{sT diag{r}HC−1 diag{r}s∗}
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 12
� Vector channelr = diag{s}h + n
� ML decision rule [Ho&Fung, 1992]
ŝ = argmins
{rHR−1rr r}
where
Rrr , E{rrH|s}
= diag{s}(
E{hhH} + σ2nIN︸ ︷︷ ︸C
)diag{s∗}
and
(diag{s})−1 = diag{s∗} ,
diag{s∗}r = diag{r}s∗
ŝ = argmins
{sT diag{r}HC−1 diag{r}s∗}
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 12
� Vector channelr = diag{s}h + n
� ML decision rule [Ho&Fung, 1992]
ŝ = argmins
{rHR−1rr r}
where
Rrr , E{rrH|s}
= diag{s}(
E{hhH} + σ2nIN︸ ︷︷ ︸C
)diag{s∗}
and
(diag{s})−1 = diag{s∗} ,
diag{s∗}r = diag{r}s∗
ŝ = argmins
{sT diag{r}HC−1 diag{r}s∗}
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 12
� Vector channelr = diag{s}h + n
� ML decision rule [Ho&Fung, 1992]
ŝ = argmins
{rHR−1rr r}
where
Rrr , E{rrH|s}
= diag{s}(
E{hhH} + σ2nIN︸ ︷︷ ︸C
)diag{s∗}
and
(diag{s})−1 = diag{s∗} ,
diag{s∗}r = diag{r}s∗
ŝ = argmins
{sT diag{r}HC−1 diag{r}s∗}
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 13
� Cholesky factorization
C−1 = LLH
U , (LH diag{r})∗
⇒
ŝ = argmins
{||Us||2}
� M = 8, N = 10 → 227 hypothetical vectors s
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 13
� Cholesky factorization
C−1 = LLH
U , (LH diag{r})∗
⇒
ŝ = argmins
{||Us||2}
� M = 8, N = 10 → 227 hypothetical vectors s
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 13
� Cholesky factorization
C−1 = LLH
U , (LH diag{r})∗
⇒
ŝ = argmins
{||Us||2}
� M = 8, N = 10 → 227 hypothetical vectors s
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 14
� ML decoding in multiple-input multiple-output (MIMO) communications with CSI
ŝ = argmins
||r − Hs||2= argmins
||U (s − sLS)||2
with Cholesky factorization HHH = UHU
and unconstrained least-squares solution sLS
� Efficiently be solved by the application of sphere decoding (SD)
? Viterbo & Boutros, 1999
? Damen et al., 2000
? Agrell et al., 2002
. . .
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 14
� ML decoding in multiple-input multiple-output (MIMO) communications with CSI
ŝ = argmins
||r − Hs||2= argmins
||U (s − sLS)||2
with Cholesky factorization HHH = UHU
and unconstrained least-squares solution sLS
� Efficiently be solved by the application of sphere decoding (SD)
? Viterbo & Boutros, 1999
? Damen et al., 2000
? Agrell et al., 2002
. . .
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 14
� ML decoding in multiple-input multiple-output (MIMO) communications with CSI
ŝ = argmins
||r − Hs||2= argmins
||U (s − sLS)||2
with Cholesky factorization HHH = UHU
and unconstrained least-squares solution sLS
� Efficiently be solved by the application of sphere decoding (SD)
? Viterbo & Boutros, 1999
? Damen et al., 2000
? Agrell et al., 2002
. . .
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 13
� Cholesky factorization
C−1 = LLH
U , (LH diag{r})∗
⇒
ŝ = argmins
{||Us||2}
⇒ Sphere decoding for noncoherent channels
Multiple-Symbol Differential Sphere Decoding (MSDSD)
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 13
� Cholesky factorization
C−1 = LLH
U , (LH diag{r})∗
⇒
ŝ = argmins
{||Us||2}
⇒ Sphere decoding for noncoherent channels
Multiple-Symbol Differential Sphere Decoding (MSDSD)
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 15
� SD concept
||Us||2 ≤ R2s1s2...
sN−1sN
? i = N
|uNNsN |2 , d2N
?≤ R2 → ŝN
? i = N − 1
|uN−1N−1sN−1 + uN−1N ŝN |2 + d2N , d
2N−1
?≤ R2 → ŝN−1
...
? i = 1ŝ : update R := ||Uŝ||
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 15
� SD concept
||Us||2 ≤ R2s1s2...
sN−1sN
? i = N
|uNNsN |2 , d2N
?≤ R2 → ŝN
? i = N − 1
|uN−1N−1sN−1 + uN−1N ŝN |2 + d2N , d
2N−1
?≤ R2 → ŝN−1
...
? i = 1ŝ : update R := ||Uŝ||
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 15
� SD concept
||Us||2 ≤ R2s1s2...
sN−1sN
? i = N
|uNNsN |2 , d2N
?≤ R2 → ŝN
? i = N − 1
|uN−1N−1sN−1 + uN−1N ŝN |2 + d2N , d
2N−1
?≤ R2 → ŝN−1
...
? i = 1ŝ : update R := ||Uŝ||
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 15
� SD concept
||Us||2 ≤ R2s1s2...
sN−1sN
? i = N
|uNNsN |2 , d2N
?≤ R2 → ŝN
? i = N − 1
|uN−1N−1sN−1 + uN−1N ŝN |2 + d2N , d
2N−1
?≤ R2 → ŝN−1
...
? i = 1ŝ : update R := ||Uŝ||
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 16
� Phase ambiguities
? Fix sN = 1 and start sphere decoding with i = N − 1
� Schnorr-Euchner search strategy
? Ordering of hypothetical symbols si according to di
a) Find the phase index mi (si = ej2πmi/M) of the best candidate point
b) Zigzag
? Account for finite constellation size
� Initial radius
? Works without initial radius
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 16
� Phase ambiguities
? Fix sN = 1 and start sphere decoding with i = N − 1
� Schnorr-Euchner search strategy
? Ordering of hypothetical symbols si according to di
a) Find the phase index mi (si = ej2πmi/M) of the best candidate point
b) Zigzag
? Account for finite constellation size
� Initial radius
? Works without initial radius
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 16
� Phase ambiguities
? Fix sN = 1 and start sphere decoding with i = N − 1
� Schnorr-Euchner search strategy
? Ordering of hypothetical symbols si according to dia) Find the phase index mi (si = e
j2πmi/M) of the best candidate point
b) Zigzag
1 2 3 4 5 6 70 0
? Account for finite constellation size
� Initial radius
? Works without initial radius
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 16
� Phase ambiguities
? Fix sN = 1 and start sphere decoding with i = N − 1
� Schnorr-Euchner search strategy
? Ordering of hypothetical symbols si according to dia) Find the phase index mi (si = e
j2πmi/M) of the best candidate point
b) Zigzag
1 2 3 4 5 6 70 0
? Account for finite constellation size
� Initial radius
? Works without initial radius
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 16
� Phase ambiguities
? Fix sN = 1 and start sphere decoding with i = N − 1
� Schnorr-Euchner search strategy
? Ordering of hypothetical symbols si according to dia) Find the phase index mi (si = e
j2πmi/M) of the best candidate point
b) Zigzag
1 2 3 4 5 6 70 0
? Account for finite constellation size
� Initial radius
? Works without initial radius
L. Lampe: Sphere Decoding for Noncoherent Channels
function MSDSD(U , M, N, R)input: N × N upper triangular matrix U , constellation size M , dimension N , initial radius Routput: Maximum-likelihood decision ŝ
1 dN := |uNN | // initialize length2 sN := 1 // fix last component of s3 i := N − 1 // start with component i = N − 14 ki := u(N−1)N // sum of last N − i components5 [mi, stepi, ni] = findBest (ki, uii, M) // find best candidate point
6 〈loop〉
7 d2i
:=∣∣∣ki + uii · e
j 2πM
mi
∣∣∣
2
+ d2i+1 // update squared length
8 if di < R and ni ≤ M { // check radius and constellation size
9 si := ej 2πM
mi // store candidate component10 if i 6= 1 { // component 1 not reached yet11 i := i − 1 // move down
12 ki :=∑
N
ι=i+1 uiιsι // add last N − i components13 [mi, stepi, ni] := findBest (ki, uii, M) // find best candidate point14 }15 else { // first component reached16 ŝ := s // best point so far17 R := di // update sphere radius18 i := i + 1 // move up19 [mi, stepi, ni] := findNext (mi, stepi, ni) // next point examined for component i20 }21 }
22 else {23 do {24 if i == N − 1 return ŝ and exit // outside sphere and no component left25 i := i + 1 // move up26 } while ni == M // while all constellation points examined27 [mi, stepi, ni] := findNext (mi, stepi, ni) // next point examined for component i28 }
29 goto 〈loop〉
subfunction [mi, stepi, ni] = findBest (ki, uii, M) // Finds best MPSK signal point
F1-1 p := M2π
(
angle(
− kiuii
))
// unconstrained phase index (p ∈ IR)
F1-2 mi := bpe // constrained phase index (mi ∈ ZZ)F1-3 ni := 1 // counter of examined candidatesF1-4 stepi := sign(p − mi) // step size for phase index
subfunction [mi, stepi, ni] = findNext (mi, stepi, ni) // Selects next MPSK signal point// according to Schnorr-Euchner strategy
F2-1 mi := mi + stepi // zig-zag through MPSK constellationF2-2 stepi = −stepi − sign(stepi) // update step sizeF2-3 ni := ni + 1 // count examined candidates
Multiple-Symbol Differential Sphere Decoding 18
� Example
L. Lampe: Sphere Decoding for Noncoherent Channels
Multiple-Symbol Differential Sphere Decoding 19
� MSDSD vs. DF-DD
uiι = r∗ι · p
N−iι−i /σe,N−i
piι : ιth coefficient , pi0 , −1
σ2e,i : error variance
of an ith order linear backward minimum mean-squared error (MMSE) predictorfor the discrete-time random process h[k] + n[k]
s1s2...
sN−1sN
⇒ Estimation of si based on (tentative) decisions ŝl, i + 1 ≤ l ≤ N , can beinterpreted as linear-prediction based DF-DD with window size N − i + 1
L. Lampe: Sphere Decoding for Noncoherent Channels
Outline 19
� Transmitter: Differential Encoding
� Receiver:
? Multiple-Symbol Differential Detection
? Multiple-Symbol Differential Sphere Decoding
� Performance Results
� Summary
L. Lampe: Sphere Decoding for Noncoherent Channels
Performance Results 20
� Figures of merit
? Power efficiency
? Computational complexity
� System parameters
? Fading according to Clarke’s model with normalized bandwidth BfT = 0.03
? 4 and 8DPSK
� Comparison with
? Mackenthun’s algorithm (MA)
? Prediction-based DF-DD
? Xiaofu and Songgeng’s algorithm (X&S-A) (only 4PSK)
L. Lampe: Sphere Decoding for Noncoherent Channels
Performance Results 20
� Figures of merit
? Power efficiency
? Computational complexity
� System parameters
? Fading according to Clarke’s model with normalized bandwidth BfT = 0.03
? 4 and 8DPSK
� Comparison with
? Mackenthun’s algorithm (MA)
? Prediction-based DF-DD
? Xiaofu and Songgeng’s algorithm (X&S-A) (only 4PSK)
L. Lampe: Sphere Decoding for Noncoherent Channels
Performance Results 20
� Figures of merit
? Power efficiency
? Computational complexity
� System parameters
? Fading according to Clarke’s model with normalized bandwidth BfT = 0.03
? 4 and 8DPSK
� Comparison with
? Mackenthun’s algorithm (MA)
? Prediction-based DF-DD
? Xiaofu and Songgeng’s algorithm (X&S-A) (only 4PSK)
L. Lampe: Sphere Decoding for Noncoherent Channels
Performance Results 21
10 15 20 25 30 35 4010
−4
10−3
10−2
10−1
10log10
( Eb/N
0) [dB] −−−>
BE
R −
−−
>
N=2 N=6 N=10coherent
N=2
coherent
MSDSD
DF-DD
MA
4DPSK
L. Lampe: Sphere Decoding for Noncoherent Channels
Performance Results 22
0 2 4 6 8 10 12 14 16 18 20
37
38
39
40
41
42
43
requ
ired
10lo
g 10(
Eb/
N0)
[dB
] −
−−
>
position i −−−>
coherent
N = 20
N = 16
N = 10
N = 8N = 6
numerical resultssimulation results
4DPSK
(see paper for details of subset MSDSD and error-rate analysis)
L. Lampe: Sphere Decoding for Noncoherent Channels
Performance Results 23
10 15 20 25 30 35 4010
2
103
104
10log10
( Eb/N
0) [dB] −−−>
aver
age
# of
mul
tiplic
atio
ns
−−
−>
lower bound
MSDSD
DF-DD
MA
for MSDSD
8DPSK4DPSK
L. Lampe: Sphere Decoding for Noncoherent Channels
Performance Results 24
5 10 15 20 25 302
2.5
3
3.5
4
4.5
5
Observation window length N
log
N−
1(av
erag
e #
of m
ultip
licat
ions
)
10log10
( Eb/N
0) = 10 dB
10log10
( Eb/N
0) = 20 dB
10log10
( Eb/N
0) = 30 dB
10log10
( Eb/N
0) = 40 dB
for MSDSDlower bound
X&S-A
MSDSD
DF-DD
MA
4DPSK
L. Lampe: Sphere Decoding for Noncoherent Channels
Performance Results 25
103
104
105
106
10−4
10−3
10−2
maximum allowed # of multiplications −−−>
BE
R −
−−
>
10log10
(Eb/N
0)=20dB
10log10
(Eb/N
0)=30dB
10log10
(Eb/N
0)=40dB
average #
BER and # of multiplicationsfor MSDSD without limitation
maximum #
4DPSK
L. Lampe: Sphere Decoding for Noncoherent Channels
Outline 26
� Transmitter: Differential Encoding
� Receiver:
? Multiple-Symbol Differential Detection
? Multiple-Symbol Differential Sphere Decoding
� Performance Results
� Summary
L. Lampe: Sphere Decoding for Noncoherent Channels
Summary 27
� Low-complexity ML MSDD has been a long-standing problem
→ Solution via application of sphere decoding
→ Multiple-symbol differential sphere decoding (MSDSD)
� Expected complexity is orders of magnitudes below that of brute-force search
� Excellent performance versus complexity trade-off
Gains in power efficiency almost for free
� Variations of MSDSD
? Subset MSDSD
? MSDSD with limited maximum complexity
? MSDSD with different initialization
L. Lampe: Sphere Decoding for Noncoherent Channels
Summary 27
� Low-complexity ML MSDD has been a long-standing problem
→ Solution via application of sphere decoding
→ Multiple-symbol differential sphere decoding (MSDSD)
� Expected complexity is orders of magnitudes below that of brute-force search
� Excellent performance versus complexity trade-off
Gains in power efficiency almost for free
� Variations of MSDSD
? Subset MSDSD
? MSDSD with limited maximum complexity
? MSDSD with different initialization
L. Lampe: Sphere Decoding for Noncoherent Channels
Summary 27
� Low-complexity ML MSDD has been a long-standing problem
→ Solution via application of sphere decoding
→ Multiple-symbol differential sphere decoding (MSDSD)
� Expected complexity is orders of magnitudes below that of brute-force search
� Excellent performance versus complexity trade-off
Gains in power efficiency almost for free
� Variations of MSDSD
? Subset MSDSD
? MSDSD with limited maximum complexity
? MSDSD with different initialization
L. Lampe: Sphere Decoding for Noncoherent Channels
Summary 27
� Low-complexity ML MSDD has been a long-standing problem
→ Solution via application of sphere decoding
→ Multiple-symbol differential sphere decoding (MSDSD)
� Expected complexity is orders of magnitudes below that of brute-force search
� Excellent performance versus complexity trade-off
Gains in power efficiency almost for free
� Variations of MSDSD
? Subset MSDSD
? MSDSD with limited maximum complexity
? MSDSD with different initialization
L. Lampe: Sphere Decoding for Noncoherent Channels
