A Perfect Reconstruction Paradigm for Digital Communication* Petros. G. Voulgaris,** Christoforos. N. Hadjicostis and Rouzbeh Touri Coordinated Science Laboratory University of Illinois at Urbana-Champaign Abstract— In this paper we present a deterministic worst-case framework for reconstruction of discrete data transmissions through dispersive communication channels. This framework can be explored based on robust control ideas and formulations and serves as a complement to existing approaches that reconstruct data by optimizing probabilistic criteria. The particular problems touched upon are: (i) necessary and sufficient conditions for causal and non-causal perfect reconstruction under deterministic magnitude bounded noise for single-input singleoutput (SISO) and multi-input multi-output (MIMO) channels, (ii) perfect reconstruction based on decision feedback equalizer (DFE) structures, and (iii) necessary and sufficient conditions for perfect reconstruction with DFEs in the presence of channel fading. The `1 control theory emerges as the natural key player for analysis and synthesis of perfectly reconstructing receivers in this framework.
I. I NTRODUCTION The study of data transmission and reconstruction has been based almost entirely on stochastic formulations of the various problems involved (e.g., [1], [2]). In these formulations, the measure of performance for a communication system is characterized primarily in terms of the probability of error under various stochastic assumptions on the noise and channel behavior. Designing a system that minimizes this probability is a hard problem and the proposed algorithms are characterized by high complexity (e.g., Viterbi’s algorithm [1]). In this paper we consider the problem of accurate reconstruction of discrete (source) data and present an alternative, deterministic worst-case framework that can be explored based on robust control ideas and formulations. The particular problems touched upon include: (i) necessary and sufficient conditions for causal (without delay) and noncausal (with delay) perfect reconstruction under deterministic, magnitude-bounded noise for single-input single-output (SISO) and multi-input multi-output (MIMO) channels, (ii) perfect reconstruction based on decision feedback equalizer (DFE) and linear structures, (iii) perfect reconstruction under channel uncertainty. As shown in the paper, our approach leads to novel and attractive designs of DFE or linear equalizers. Although DFE *This material is based upon work supported in part by the National Science Foundation under NSF Career Award No 0092696 and NSF ITR Award No 0085917, and in part by the Air Force Office of Scientific Research under Award No AFOSR DoD F49620-01-1-0365URI. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of NSF or AFOSR. **Address for correspondence: 161 CSL, 1308 West Main Street, Urbana, IL 61801-2307, USA. E-mail:
[email protected].
analysis and design has received considerable attention over the last forty years, there are many issues of current interest (e.g., [3] and references therein). A common assumption in the DFE design literature is that of correct past decisions, something that is arguably a strong assumption to coexist with the notion of optimality of the design procedure [4]. In our approach, since we provide the exact conditions for the existence of perfectly reconstructing DFEs (as well as an explicit construction of such DFEs if at all possible), this assumption is not present. Thus, our analysis can be used to study the tradeoffs between the ability for perfect reconstruction, the resulting delay in making the decision, the number of filter taps, and the size of noise and channel uncertainty. Of course, if the noise level is higher than the maximum allowed for perfect reconstruction under the chosen receiver structure, errors will occur and one has to analyze how these errors will propagate in the system. This is not explicitly done in this paper, although our work has touched some relevant issues in the case of first-order FIR channels. Naturally, our simulations include the effects of error propagation. For a detailed discussion on these matters we refer to works in the literature [5], [6], [7]. The notation in the paper is as follows: kxk := supk |x(k)| is the `∞ norm of a sequence x = {x(k)}∞ k=0 ; kT k1 := P∞ 1 |t(k)| is the ` norm of the linear time-invariant (LTI) k=0 ∞ ˆ(λ) := system T having unit pulse response {t(k)} ; T k=0 P∞ k k=0 t(k)λ is the λ-transform of T . For a vector valued signal x = (x1 , x2 , . . . , xn )0 , kxk := maxi kxP i k and for MIMO systems T = {Tij }, kT k1 := maxi j kTij k1 ; q kT kH∞ := sup|λ|≤1 ρ[Tˆ∗ (λ)Tˆ(λ)] where ρ[•] stands for the maximum modulus eigenvalue of the matrix argument and ∗ stands for complex conjugate transpose; kSk∞−∞ := ∞ supx6=0 kSxk kxk is the ` -induced norm of a possibly timevarying and/or nonlinear system S (note that kSk∞−∞ = kSk1 if S is LTI); S is stable if kSk∞−∞ < ∞. II. P ROBLEM D EFINITION The basic problem we are concerned with is depicted in Figure 1 where s is a binary signal to be transmitted with s(k) ∈ {−1, 1} for all k = 0, 1, . . .; n is the noise with |n(k)| ≤ b, where b is a known constant; H = {h0 , h1 , . . .} is an LTI system that represents the channel dynamics which are assumed known a priori for now. We want to accurately and causally reconstruct s via the receiver structure R, i.e., we would like to find the necessary and sufficient conditions for sˆ(k) = s(k) for all times k = 0, 1, . . .,
and the corresponding receiver structure R. Our analysis in n
The above test for perfect reconstructability requires solving a mixed integer linear program [9]. For certain instances, such as one-step delay, analytical results can be obtained.
PSfrag replacements
s
+
H
Fig. 1.
sˆ
r R
Basic set-up
[8] showed that perfect reconstruction requires |h0 | > b. Furthermore, the construction of a receiver R (refer to Figure 1) can be obtained as the decision feedback equalizer (DFE) shown in Figure 2, where F := H − h0 and Λ is the unit step delay operator. The above setup and analysis can be r
Sfrag replacements
˜ H
r˜
+
sˆ 1 h0
sgn
-
B. MIMO channels Generalizations are also possible for MIMO channels. In the case of m transmitters and p receivers the (equivalent) channel dynamics can be represented by a (stable) p × m transfer function H with pulse response H = {H0 , H1 , . . .}, where each Hi is a p×m matrix. The motivation for problems of this sort comes from multiple antenna systems designed to combat fading channels and the detection of multiuser code division multiple access (CDMA) signals. The set-up is as before, i.e., r = Hs + n, where s and n are vector valued sequences such that si (k) ∈ {−1, 1} for all components si of s, and |ni (k)| ≤ b for all components ni of n. Under this setup, one can prove the following: if vi ∈ {−1, 0, 1}, i = 1, . . . , m, then the necessary and sufficient condition for perfect reconstruction causally in time is min
vi ∈{−1,0,1} not all equal 0
where
ΛF
Fig. 2.
DFE structure
A. Non-causal reconstruction The case of non-causal reconstruction (smoothing) can also be considered in the same framework. In this case we are allowed to estimate s(k) by incorporating K future receptions r(k + 1), . . . , r(k + K) as well as r(0), r(1), . . . , r(k). In other words, we allow a delay of K steps in reconstructing s. The necessary and sufficient condition for perfect reconstruction is that there are no sequences s1 and s2 such that if they are indistinguishable at any time t they remain so for the next K time steps. Following the same line of argument as in [8], we obtain that the necessary and sufficient condition is min
max{|a(0)|, |a(1)|, . . . , |a(K)|} > b ,
where v(0) 6= 0 and a(0) h0 a(1) h1 .. := .. . . a(K) hK
a0 a1 . = H0 .. ap
h0 ..
...
. ...
h0
v(K)
vm
III. R ECONSTRUCTION BASED ON D ECISION F EEDBACK E QUALIZATION In the previous section, causal perfect reconstruction in SISO channels led naturally to a DFE structure. Herein we elaborate more on the optimality of such a structure for noncausal and MIMO problems. The general DFE structure n
(1) +
Q
+
Θ
-
.
sˆ
s˜
H
PSfrag replacements
v(0) v(1) .. .
.
Checking the above condition is a mixed integer linear program (MILP) and a closed-form solution is, in general, hard to obtain. As in the SISO case, one can look at noncausal reconstruction for MIMO channels; the problem to solve is again an MILP. In general, the construction of a perfectly reconstructing R (causal or noncausal) can be quite complex. This is a motivation for the specific (DFE and linear) structures of reconstructors that we consider in the next section.
s
v0 v1 .. .
generalized to cases where s(k) belongs to a set of equally spaced numbers in [−1, 1]. For instance, if s(k) ∈ {j/N, j = −N, −N +1, . . . , 0, . . . , N }, i.e., if there are 2N +1 numbers equally spaced by intervals of size 1/N , the condition for perfect reconstruction becomes |h0 | > 2N b, and the decision structure is an obvious extension of the structure in Figure 2.
v(i)∈{−1,0,1}
max{|a0 |, |a1 |, . . . , |ap |} > b, (2)
ΛF
Fig. 3.
General DFE structure for causal reconstruction
n
sˆ
s˜
s
H
Q
+
+
Θ
-
PSfrag replacements
(b) k(I − QH + ΛF Qb)k1 < 1 for some Q and F . (c) There are Q and F so that, for all k, |s(k) − s˜(k)| < 1 − ²0 for some ²0 > 0 and all s and n with knk ≤ b. Based on the above proposition we can define the relevant `1 -optimization as µ
ΛF
:=
inf Jb ,
Q,F
where Jb
Fig. 4. Equivalent DFE structure under the perfect reconstruction hypothesis
is depicted in Figure 3 where Q is a (stable) linear forward filter Q = {q0 , q1 , . . .}; ΛF is a feedback filter with Λ ˆ being the one-step delay operator (i.e., Λ(λ) = λ) and F = {f0 , f1 , . . .}; Θ is a thresholding operator that produces −1 or 1 depending on which one has the closest distance to s˜. In this particular case, (Θ˜ s)(k) = sgn [˜ s(k)]. It should be clear that for this structure to perfectly reconstruct s(k) causally in time for each time k, it is necessary and sufficient that s˜(k) > 0 if s(k) > 0 (and s˜(k) < 0 if s(k) < 0) for all possible signal sequences s and all possible noise sequences n. The perfect reconstruction requirement/assumption, allows us to interpret the situation in Figure 3 as in Figure 4: there are (stable) Q and F and an (arbitrarily small) ² > 0, such that for all time-steps k with s(k) = 1, s˜(k) = ((QH − ΛF )s + Qn)(k) > ² > 0 for all sequences s and knk ≤ b. Denoting by X := QH − ΛF = {x0 , x1 , . . .}, we have that the above condition equivalently means s˜(k)
=
k X
xk−i s(i) +
i=0
k X
qk−i n(i) > ² > 0
i=0
for all times k and for all sequences s with s(k) = 1 and knk ≤ b (for s(k) = −1 same condition can be easily obtained). Equivalently, we have that x0
>
k X
|xi | + b
i=1
k X
|qi | + ²
∀k = 1, 2, . . . .
(3)
Note that the above is also equivalent to x0
>
∞ X i=1
|xi | + b
Then, the problem of perfect reconstruction has a DFE solution if and only if µ < 1. If we denote G := QH = {g0 , g1 , . . .}, it follows from the structure of the `1 norm that, in order to minimize Jb , the optimal F for any Q should be selected as F = {g1 , g2 , . . .} so that ΛF cancels all terms in G except the feed-through term g0 = q0 h0 (any other choice for F will increase Jb ). Given that, it also follows that the minimizing Q has to be a constant Q = q0 , where q0 minimizes µ = min |1 − q0 h0 | + b|q0 | . q0
This gives us µ = 1 if |h0 | ≤ b and µ = b/|h0 | if |h0 | > b. The minimizing Q is Q = 0 and Q = h10 respectively. In other words, the DFE structure leads to, as it was supposed to, the same conclusion for perfect reconstructability as in the previous section. A point to be made here is that having Jb > 1 does not always imply that the DFE structure with the particular Q and F does not perfectly reconstruct s, unless 0 < g0 = q0 h0 ≤ 1. If g0 > 1, Condition (4) needs to be checked to determine whether perfect reconstruction is possible at the given noise level b. A. Noncausal reconstruction To capture K-step noncausal reconstruction we allow the forward filter to be of the form Λ−K Q (i.e., noncausal). It then follows that the problem of interest is to find µ
∞ X
=
inf Jb ,
Q,F
where now Jb =
i=0
= k(I 0) − (QH − ΛF Qb)k1 .
° ¶° µ ° s ° ° ° [(I 0) − (Λ−K QH − ΛF Λ−K Q)] ° n ° ksk≤1,knk≤b sup
or, equivalently, |qi |.
(4)
i=0
Since the xi ’s are (linear) functions of the qi ’s and the fi ’s, the problem of checking the above condition for a given b > 0 is a linear programming (LP) feasibility problem with infinite variables {qi , fi }∞ i=0 . We now reduce Condition (4) to equivalent conditions. The proof of the proposition below is omitted due to space limitations. Proposition 3.1: The following are equivalent: (a) Condition (3) is satisfied for some Q and F .
Jb
= k(ΛK 0) − (QH − ΛK+1 F
Qb)k1 .
Again, it is true that the best F , for any choice of Q in the above optimization, is to cancel the coefficients of G = QH of order K +1 and above, i.e., F = {gK+1 , gK+2 , . . .}. With this done, it also follows that the minimizing choice for Q is FIR of order K, i.e., Q = {q0 , q1 , . . . , qK , 0, . . .} with the parameters q0 , . . . , qK solving the linear program µ = min |1 − gK | + b|qK | + q0 ,...,qK
K−1 X i=0
(|gi | + b|qi |).
(5)
Hence, the DFE structure will perfectly reconstruct s with a delay of K steps if and only if the linear programming problem in (5) leads to a cost µ < 1. A natural question to answer is how closely the maximum allowable noise bound b for perfect reconstructability with a DFE structure relates to the absolute bound of Condition (1). For the one-step delay case (K = 1) it can be shown that the bounds are the same. B. MIMO channels In the case of MIMO channels with a DFE receiver structure one needs to check whether ksi − s˜i k < 1 for all of the i source data si transmitted. In particular, the relevant problem to solve is µ
=
inf k(I 0) − (QH − ΛF
Q,F
QB)k1 ,
where all systems are MIMO and B is a scaling noise matrix with B := diag(bi ), where bi is the noise bound on channel i. The DFE perfectly reconstructs s causally in time iff µ < 1. Using the same arguments as in the previous two subsections, a minimizing F is F = {G1 , G2 , . . .} where G = QH, and a minimizing Q is Q = Q0 , a constant matrix that solves the finite-dimensional LP µ
=
min k(I − Q0 H0 Q0
− Q0 B)k1 .
(6)
In general, assuming (for simplicity) that B = diag(b), the maximum bound on the noise b for perfect reconstructability obtained by requiring that µ < 1 can be different (smaller) than the absolute bound of Condition (2). Note that noncausal reconstruction problems lead to analogous results as in the scalar case. The relevant problem to consider for K-step delayed reconstruction is the (finitedimensional) LP µ
= ...
min
Q0 ,...,QK
k(I − GK
−G0 − QK B . . .
− GK−1 . . . − Q0 B)k1 < 1 .
(7)
C. Reconstruction Based on Linear Equalizers In this case we restrict the structure of R to be R = Q, where Q = {q0 , q1 , . . .} is a linear time-invariant filter. This is a special case of the DFE structure when F is set to zero, thus all of the previous discussion carries over. Therefore , for perfect causal reconstruction, it is necessary and sufficient that µ
=
inf k(I 0) − Q(H Q
bI)k1 < 1 ;
(8)
for K-step noncausal reconstruction it is necessary and sufficient that ° ° µ = inf °ΛK (I 0) − Q(H bI)°1 < 1 ; (9) Q
for MIMO channels the problem to solve is µ
=
inf k(I 0) − Q(H Q
BI)k1 < 1 ,
(10)
where all systems are MIMO and B is a scaling noise matrix with B := diag(bi ), where bi is the noise bound on channel i. D. Remarks We note that more complex alphabets can be handled in a similar manner. For example, when s(k) belongs to a set of equally spaced numbers in [−1, 1], such as s(k) ∈ {j/N, j = −N, −N + 1, . . . , 0, . . . , N − 1, N }, assuming that the thresholding now changes to produce the closest j/N to s˜(k), the previous approach for causal reconstruction leads 1 to the condition µ < 2N . When Q and F are restricted to be FIR filters of some desired order the associated LPs will be infinite-dimensional if (and only if) H is IIR. This poses no serious difficulty as standard `1 algorithms (as in the case of linear equalization) can be utilized. The noncausal and/or MIMO reconstruction problem with DFE also leads in this case to an `1 problem to be solved. Finally, in MIMO channels one can solve without additional difficulty decentralized reconstruction problems by imposing structural constraints on Q and F . For example, the ith receiver (decision maker) can be restricted to obtain information only from its neighboring sites in a multiple antenna system. IV. ROBUSTNESS TO CHANNEL UNCERTAINTY In the previous section we investigated DFE structures under the assumption that the channel H is known. We now consider the case of uncertain channel dynamics which we model as shown on the left of Figure 5. The uncertainty here is given in terms of an additive weighted block ∆W , where ∆ is assumed to be an unknown perturbation, possibly time-varying, that has a bounded `∞ -induced norm k∆k∞−∞ ≤ 1. The weight W is a known stable LTI dynamical system that may reflect magnitude normalizations and partial information on the magnitude of the uncertainty over different frequencies (i.e., it “shapes” the uncertainty block). As an example, consider the actual channel Ha as Ha = H + E where H = {h0 , h1 , . . .} is the nominal LTI channel and E represents possibly time-varying perturbations on the parameters of H leading to a response (Ha s)(k) =
k X
(hk−i + ²ki )s(i) ,
i=0
with the perturbation ²ki bounded as |²P ki | ≤ ²i for all ∞ k = 0, 1, 2, . . ., but otherwise arbitrary. If i=0 ²i = ², then this amounts to modeling E as E = ∆W with W = ² and k∆k∞−∞ ≤ 1. Similarly, if the first N channel coefficients are not changing (²0 = . . . = ²N −1 = 0) but there is ˆ (λ) = ²λN . uncertainty in the higher order terms, then W Since we deal with the problem of perfect reconstruction, we consider the situation in Figure 4 where H in that figure is understood to depict the “actual” channel ¡ H ¢ a . Let Ja represent the `∞ -induced norm of the map ns → s − s˜,
°¡ ¢ ° i.e., Ja = ° ns → s − s˜°∞−∞ . This is a function of the uncertainty ∆. We assume that when no uncertainty is present, i.e., when ∆ = 0, Ja < 1 and hence perfect reconstruction is achieved for some DFE parameters Q and 1 F . In this case, Ja ¡= ¢J where J represents °¡the ¢ ` norm°of s s ° the (nominal) map n → s − s˜, i.e., J = n → s − s˜°1 . What we would like to ensure is robust performance (RP) in the presence of all possible k∆k∞−∞ ≤ 1, i.e., we want to find conditions such that Ja < 1 for all k∆k∞−∞ ≤ 1. If we denote Φ1 := I − QH + ΛF and Φ2 := −Qb, from the definition of the 1-norm we have J
=
:= =
µ
0 k(W 0)k1 k − Qk1 k(Φ1 Φ2 )k1 µ ¶ 0 kW k1 . kQk1 J
H11 = (0 − D −1 WD ),
H12 = D−1 ,
H21 = (WN
H22 = H, ∆ =
− HWD ),
∆N ∆D
¶
.
¶
J¯
° ° ° ° ° (I 0 0) − (QH − ΛF Qb kH12 k1 QH21 )° . (12) ° ° 1 − kH11 k1 1
=
Using the uncertainty description and redrawing Figure 4 as For the case of noncausal reconstruction with K-step delay, on the right of Figure 5, it can be shown [10] that PSfrag RP replacements is obtained if and only if ∆ ∆ p 2 n ¯ ) < 1 ⇔ J + J + 4kW k1 kQk1 < 1 . ρ(M 2 w z à ! à ! Equivalently, kW k1 kQk1 + J < 1. Note that the left side is s s r H H H H 11
12
H21 H22
PSfrag replacements
w s
Fig. 5. (right)
+
+
s
12
H21 H22
s − s˜
n
z Ã
0 (W 0) −Q (Φ1 Φ2 )
!
s − s˜
A fading channel model (left) and its robustness analysis loop
equal to J¯ := =
n
Fig. 6. A more general fading model (left), and its robustness analysis loop (right)
n
H
11
+
∆
∆
W
r
µ
Redrawing the system to analyze robustness as in Figure 6 we can use the same spectral radius condition as before [10], to conclude that for perfect reconstruction for all possible perturbations k∆k∞−∞ ≤ 1 (i.e., k∆N k∞−∞ ≤ 1, k∆D k∞−∞ ≤ 1), it is necessary and sufficient that there ¯ ) < 1 , which equivalently leads exist Q , F such that ρ(M ¯ to kH11 k1 , J < 1 whith J¯ defined by
kΦ1 k1 + kΦ2 k1 = k(Φ1 Φ2 )k1 .
¯ be Let M ¯ M
with the sources of dynamical uncertainty lumped in ∆. −1 For example, consider a channel given by Ha = NH DH , where NH = N + WN ∆N and DH = D + WD ∆D with D, N being the nominal (stable) “numerator” and “denominator” respectively (i.e., H = N D −1 ). Assume that D and N are coprime [10], and that ∆N and ∆D are normalized perturbations with known shaping weights WN , WD . Putting this in the mold of Figure 6 (left) the following correspondence is obtained:
k(I 0 0) − (QH −ΛF Qb QkW k1 )k1 k(I 0) − (QH −ΛF Q(b + kW k1 ))k1 .
Hence, is equivalent to minimizing ° ° ¡ ¢ RP optimization ° s → s − s˜° , where n ¯ is bounded as k¯ nk ≤ b + kW k1 . n ¯ 1 That is, one has to consider the nominal channel with the “equivalent” noise bound given as b + kW k1 . Based on the results of the previous section, perfect causal reconstruction is possible with a DFE iff |h0 | > b + kW k1 .
(11)
A more general situation is depicted on the left of Figure 6, where H22 = H and the Hij can be general (stable) dynamical LTI systems that connect the nominal channel
the same arguments hold and lead us to conclude that the necessary and sufficient condition for RP is that kH11 k1 , J¯ < 1 where ° ° K K+1 F J¯ = ° ° (Λ 0 0) − (QH − Λ
Qb
° ° kH12 k1 QH21 )° ° . 1 − kH11 k1 1
To check whether there exist Q, F so that the DFE perfectly reconstructs s in the presence of unmodeled dynamics, the above conditions lead to the following `1 -optimization problem µ = inf J¯ < 1 , Q,F
assuming kH11 k1 < 1 holds (which can be easily checked since H11 does not depend on the DFE). For this problem it is clear that, for any choice of Q, filter F should cancel the coefficients of G = QH of order K+1 and above, i.e., F = {gK+1 , gK+2 , . . .}. In general, the optimal Q may not be FIR as the LP is infinite-dimensional. Nonetheless, arbitrarily close to optimal solutions can be obtained using standard `1 methods. In the case of more complex alphabets as in the case when s(k) belongs to a set of equally spaced numbers in [−1, 1],
i.e., when s(k) ∈ {j/N, j = −N, −N + 1, . . . , 0, . . . , N }, assuming that the thresholding device now changes to produce the closest j/N to s˜(k), the previous approach leads to the conditions kH11 k1 , J¯
