612
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 4, APRIL 2001
On Reverse Concatenation and Soft Decoding Algorithms for PRML Magnetic Recording Channels Luca Reggiani, Student Member, IEEE, and Guido Tartara, Senior Member, IEEE
Abstract—High-density magnetic recording systems require increasingly sophisticated signal processing techniques. In magnetic recording channels, the information bits are encoded by the concatenation of an outer nonbinary error correcting code (ECC) and an inner line code. Furthermore, the high intersymbol interference that characterizes the channel is controlled by “partial response” equalization. This paper presents a study on a better-integrated decoding procedure between the inner and outer codes. Inversion of the concatenated codes (reverse concatenation) allows the direct mapping of soft information from the partial response channel to the outer ECC. A simplified soft decoding technique, based on the use of erasures, is applied to two typical magnetic recording systems; performance curves are obtained by an analysis of the distributions of the reliability measures associated with incorrect and correct symbols. Index Terms—Concatenated codes, magnetic recording systems, nonbinary linear block codes, partial response maximum-likelihood, soft decoding.
I. INTRODUCTION
T
HIS paper concerns the problem of simple soft decoding algorithms for the improvement of conventional partial response maximum-likelihood (PRML) detectors in magnetic recording systems. A magnetic recording system usually includes an error correcting code (ECC) (Reed–Solomon with 8-bit symbol), a block interleaver, and an inner code (typically, a nonlinear block code). In Fig. 1, the first three blocks of the encoding stage are followed by the channel that returns the stream of impulses read from the track. The receiver normally comprises an analog input filter, an equalization block, and corresponding decoding blocks; the block de-interleaver, which operates at a symbol level, decorrelates the error bursts from the channel. This channel is characterized by high intersymbol interference (ISI): this interference is controlled by a “partial response” equalization, followed by a Viterbi detector (VD) that selects the most likely sequence according to the desired target (partial response maximum-likelihood sequence detection—PR MLSD [20]). The transmitted sequence must respect some general line constraints that basically concern the maximum number of consecutive zeros or ones for reliable timing recovery. In addition to these constraints, the line code (LC) plays a relevant role in improving the performance of the overall scheme. It is especially important that the code rate in magnetic recording sysManuscript received March 1, 2000; revised June 1, 2000. This work was supported by STMicroelectronics, Milan, Italy. The authors are with the Dipartimento di Elettronica ed Informazione, Politecnico di Milano, 32-20133 Milano, Italy (e-mail:
[email protected];
[email protected]). Publisher Item Identifier S 0733-8716(01)01755-3.
tems be high as impulse energy depends on the rate of transmission. One of the most delicate problems associated with standard code concatenation is error propagation, which can be very marked for line codes with high rate and very large codewords. Section II gives a summary of a particular coding technique, which consists in inversion of the inner and outer codes (reverse concatenation); this technique limits error propagation and can favor the introduction of soft decoding procedures. Section III considers a suboptimal soft decoding algorithm for the cascade of the channel, the LC, and the outer ECC. Performance estimation of this system can be interesting for a comparison of the soft decoding techniques in magnetic recording channels. Section IV presents a method for analytical performance evaluation and some numerical results. II. REVERSE CONCATENATION As mentioned in the introduction, the inner1 code LC should have a rate as close as possible to 1; this achievement implies the use of long codewords and the possibility of serious error propagation between the VD and the systematic ECC (Fig. 1). In [5], Bliss proposed a solution that allows high-rate coding without any significant increase in error propagation: it consists essentially in the inversion of the inner and outer codes, a technique we will refer to as “reverse concatenation” (Fig. 2). It means that the ECC is applied to the line-encoded symbols rather than to the user symbols: the ECC sees the error events directly from the PR VD without the effect of error propagation due to the LC (Fig. 3). The disadvantages of this technique are as follows: • increase of the ECC codeword length, low in this case because of the high rate; • necessity of a second line code LC2 for the parity symbols at the output of the systematic ECC. The LC2 code should have small codewords for small error propagation: its worst rate does not affect the overall code rate significantly as parity checks are a very small portion of the codeword length. In [14], Immink proposed a method to reduce the expansion of the ECC codewords and the application of enumerative encoding techniques to the line constraints. In [8], Fan and Calderbank analyzed the advantages of this technique in magnetic storage and pointed out the possibility of performing soft decoding using reliability information directly from the channel 1We prefer to use the notation “inner code” LC instead of “line code” because this code has often correction capabilities in addition to the LC features. In the literature, the “line code” is also known as “modulation code” or “constrained code.”
0733–8716/01$10.00 © 2001 IEEE
REGGIANI AND TARTARA: SOFT DECODING ALGORITHMS FOR PRML CHANNELS
Fig. 1.
613
A typical magnetic recording system with standard concatenation.
Fig. 2. Reverse encoding.
Fig. 3.
Reverse decoding.
detector. The idea of using the reverse concatenation technique for soft and turbo decoding has also appeared in more recent papers [1], [9]. In the next sections, we will investigate the potential performance gain provided by a simple soft decoding technique that uses the reverse concatenation principle. In Fig. 2, the reverse encoding scheme includes the following: LC Inner code ( , ). interleaver depth) at ITL Block interleaver ( ECC symbol level. DE-ITL The corresponding block de-interleaver. ECC Systematic nonbinary ECC. (A Reed–Solomon symbol length code over correction capability.) LC2 Line code 2 ( , ) applied to the parity symbols generated by the ECC. In Fig. 3, there is the corresponding decoding stage: from the VD, the parity check symbols are separated from the systematic part of the ECC codeword symbols. The parity symbols pass through the LC2 decoder before going into the ECC decoder; the inner LC decoder is moved after the ECC block. In soft decoding, a reverse concatenation eliminates the effect of propagation of soft information associated with incorrect bits between the VD and the ECC: the least reliable bits would
propagate their influence to more bits at the output of an LC soft decoder. In systems with standard concatenation, this propagation effect decreases the performance of successive soft algorithms. In addition to this basic advantage, reverse concatenation can simplify the implementation because it allows the elimination of the soft decoding step corresponding to LC, necessary in standard concatenation. In reverse concatenation, there are two possibilities for the role of the inner code LC: stressing its rate (that limits it to pure line constraints) or introducing error-correction capabilities. In the former option, the soft reliability measures obtained by the channel are mapped directly to the ECC; in the latter option, the greater structure of the inner code provides a refinement of the soft information associated with the codeword bits. In Section IV, we will study two examples that exploit these two possibilities. In soft reverse decoding, the ECC parity symbols at the channel output must pass through a soft decoder for LC2. This block can be greatly simplified by mapping the input reliability measures directly to the output symbols; after intercepting the ECC parity symbols encoded by LC2, this block should assign them the same reliability measures provided by the channel. This possibility can be obtained if the LC2 code maps one ECC , for instance 8/9 if symbol to one LC2 codeword ( the ECC symbol is 8 bits long). If the received codeword does not belong to LC2, the LC2 soft decoder can simply change its reliability measure to the minimum value. Generally speaking, a simplified soft decoding of this concatenated scheme can be thought of as being in two parts: a block that assigns the reliability measures to the channel outputs (either with or without the contribution of LC) and a soft algorithm that addresses the ECC decoding (Fig. 4). The memory of the channel (PR) suggests not only the use of reliability information associated with the nonbinary ECC symbols, but also an
614
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 4, APRIL 2001
Fig. 4. A simplified soft decoding procedure: the principle.
adequate block interleaving. This compromise reflects exactly the solution adopted in hard decoding. Fig. 4 shows the simplified soft decoding scheme using soft information associated with each ECC symbol, allowing the selection of appropriate sets of erasures. The following section discusses the analysis of a simple procedure to achieve an estimation of the soft decoding potential in magnetic recording systems.
(a)
III. A SUBOPTIMAL SOFT DECODING PROCEDURE FOR PRML CHANNELS This section considers a formalization of a simplified decoding algorithm in order to achieve an understanding of the possibilities of soft decoding in magnetic storage. Following the basic scheme in Fig. 4, Section III-A will deal with the choice of soft measures from the channel and Section III-B the simplified ECC soft decoder. The ECC soft decoder will be based on errors erasures algebraic decoding. A. Soft Information from the Channel A Viterbi algorithm usually detects the most likely sequence from the partial response channel; in a soft decoding context, a measure of reliability is required to improve the integration and the performance of the inner and outer codes. The association of a reliability measure with a symbol (binary or not) is an issue widely discussed in the literature with an even greater attention after the introduction of turbo codes. The maximum a posteriori (MAP) algorithm computes the a posteriori probability of the bits, accomplishing a forward and a backward recursion of the trellis [2]. More recently, the soft output Viterbi algorithm (SOVA), [13] was proposed as an improvement of the Viterbi algorithm for soft decoding. The channel soft detector in Fig. 4 can be interpreted as a soft input soft output (SISO) block [4]: it receives the probabilities of each channel symbol and outputs the probabilities of all encoded symbols. In the partial response channels used in Section IV, a simple version of the SOVA algorithm provides the soft information associated with each ECC symbol; a reliability measure turns out to be the minimum difference between the ML sequence and the sequence that differs from it in at least a bit. Hence, referring , we define to the VD matched to the channel PR function for the th symbol (1) where received sequence; ML sequence;
(b) Fig. 5. Example of probability density of reliability measures of (a) correct and (b) incorrect symbols (the measures come from an EPR4 channel with a 24/25 parity bit—it is clear the possibility of discriminating incorrectly from correct symbols).
ML sequence that differs from in at least one bit of the th symbol. is indicative of For every symbol, its reliability measure ; this defthe likelihood ratio is indicated also by because it is a difference inition of between the metrics and is always positive (the minimum value is zero, corresponding to maximum uncertainty). Fig. 5 shows an example of the probability density functions of the reliability information associated with incorrect and correct symbols: note s associated with incorrect symbols have statistics that the that concentrate most of the measures close to zero, as can be expected. These densities are of fundamental importance in the estimation of the performance of the entire system (Section IV). B. Soft Information from the Channel Our simplified soft decoding scheme is based on the use of sets of erasures chosen from the least reliable symbols. The algorithm in Fig. 6 shows the most intuitive and simple way to
REGGIANI AND TARTARA: SOFT DECODING ALGORITHMS FOR PRML CHANNELS
615
the reliability measures, and error events or segments of error events will generally affect the wrong symbols in a few bits. On the other hand, bit interleaving would be inefficient with a powerful nonbinary outer code (Reed–Solomon). The first step of the algorithm is the usual decoding with no and ). If the first attempt fails erasures (in Fig. 6, , ), a set of combinations of erasures is ( generated: the erasures are chosen from the set of the least . These combinations are sorted according reliable symbols , where is to the simple weight function the reliability measure associated with the th erasure. The algorithm ends when the first acceptable codeword passes the decoding failure test. The following basic correction strategy “CS” is adopted here: at each step of the algorithm, the set is constituted by all the possible combinations of (equal to ) erasures from . The optimal choice of parameter is , where is the minimum distance of the ECC, because, in principle, it allows the exploitation of the best performance of the code; in a real project, the choice of is affected by considerations on computational time in the worst possible case. IV. PERFORMANCE ESTIMATION AND SOME EXAMPLES OF SOFT DECODING SCHEMES
=
=
Fig. 6. ECC simplified soft decoder [k number of erasures, s step of the algorithm, L maximum number of erasures, L first L least reliable symbols, and L set of combinations of k erasures chosen in L ].
=
=
=
choose these erasures. An important issue is the investigation of an algebraic efficient way to detect errors, given sets of erasures at the input of the ECC, especially when the ECC has long codewords: several algorithms have been proposed for nonbinary ECCs (e.g., recently, [15], [21], [23]) and concern efficient ways for root calculation in polynomials with sets of errors and erasures. In Fig. 6, this simple soft decoding algorithm is shown in a way that resembles Chase’s algorithm for binary block codes [6]. If the ECC decoding without erasures fails, a list of patterns of erasures is chosen from the least reliable symbols. The main difference from the classic Chase algorithm is that we do not use error patterns but erasures patterns: this choice is due to the presence of a channel with memory,2 combined with a nonbinary ECC. The principle of the algorithm turns out to be similar to Forney’s generalized minimum distance (GMD) decoding [10]. In this system, the inner code can have a role in the detection of the ECC decoding failure and, possibly, in the detection of erasures. More precisely, a check on the systematic part of the ECC codewords, which are encoded by LC, can reveal ECC symbols surely incorrect, or can refine the associated reliability measures at each loop. We emphasize that this possibility is peculiar to reverse concatenation. As mentioned above, the choice of erasures is due to the fact that the channel soft decoder has a memory and the output soft information are correlated. The block interleaver decorrelates 2Memory,
which produces error events and not single bit errors.
In this section, we will estimate the performance of the CS defined in Section III when it is applied to some typical magnetic recording schemes. As mentioned in Sections I and II, partial response equalization controls the ISI introduced by the channel. For current channel densities of 2.5–3.0, the EPR4 equalization , ) is commonly used class ( and accepted as one of the best compromises concerning performance loss due to the equalization. The analysis is performed in terms of the input SNR and off-track noise. Off-track noise is one of the greatest sources of noise in a magnetic recording channel: it is due to the interference of the adjacent tracks on the signal read by the right current track (a sort of “co-channel” interference). The channel model is assumed to be a Lorentzian channel with MMSE equalization and colored Gaussian noise. Some simulations with a media noise model confirm the validity of a soft decoding procedure also for different noise sources. In Section IV-A, we will study a general method of analysis for the CS, based on the use of the reliability measure distributions. Section IV-B presents the performance estimation of two typical systems. A. Performance Estimation For the estimation of performance, it is assumed that the reliability measures are independent of each other (sufficient interleaving depth) and that the quantization effects are negligible. Given: Probability of errors in an ECC codeword. Probability of decoding failure with hard decoding and without any correction strategy. Probability of decoding failure with soft CS. ECC codeword length.
616
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 4, APRIL 2001
Fig. 7. Distribution of symbols sorted by reliability measures: symbols, j within the L least reliable.
i
incorrect
Clearly (2) and, for the CS errors
(3)
with equal to the number of guessed erasures inside the list of the first least reliable symbols. The decoding miscorrection is neglected; the decoding check provided by the inner code LC [8], where has a failure probability of about is the capacity of the inner code, is the minimum distance of is the symbol length in bits ( ). Furthe ECC, and thermore, in this specific case, the Reed–Solomon has a strong capability of detecting nonvalid codewords. errors or Let us now consider the probability there are incorrect the probability that inside the list symbols, given that is the total number of incorrect symbols. If , there will be correct symbols there are errors and ) correct symbols inside the first in the ECC codeword and ( least reliable symbols (Fig. 7). In Fig. 7, the ECC symbols is the value are sorted from the least to the most likely one: of the reliability measures associated with the last symbol of or, equivalently, with the th symbol of the list . Similarly, is the measure of the least reliable symbol after the first or the th of the list . Assuming the independence among the ’s associated with each symbol, it is possible to calculate the probability density and , given that is the total number of function of errors and
and
Fig. 8. Performance evaluation (sector failure probability) in terms of input SNR [CS (L ), EPR4 RLL 16/17 ITL(depth ) RS (N )]. ; t
181 = 5
=6
+
+
=3 +
=
The density functions of and have different curves, and an example is reported in Fig. 5. Thus, these probability densities depend on the signal-to-noise ratio (SNR): the greater the SNR, the greater the discrimination between the two curves. Namely, the greater the SNR, the greater the possibility of distinguishing the incorrect symbols from the correct ones by observing their reliability measures. and , the probability Given the two densities errors ( is equal to the number of guessed erasures within the first measures) can be evaluated by errors (6) This completes the evaluation of (3)
errors errors is clearly equal to zero for where The final formula is
(7) .
(4) and
(5)
and indicate a reliability measure aswhere sociated with a correct and an incorrect symbol, respectively. is computed considering the probability The density density function of the minimum of a set of values composed reliability measures associated with correct by associated with incorrect symbols [simisymbols and ]. larly
(8)
Comparing (8) with (2), it is easy to see that the factor of gain is due to the term (lower than 1) that multiplies the probabilities to . of input errors from
REGGIANI AND TARTARA: SOFT DECODING ALGORITHMS FOR PRML CHANNELS
617
any case, this kind of solution, combined with reverse concatenation, can be applied to any ECC inserted after the channel and before the outer Reed–Solomon code. Another disadvantage could be the time requirement for a sector decoding when the usual hard decoding fails and the soft algorithm starts. V. CONCLUSION
Fig. 9. Performance evaluation in terms of offtrack percent [CS (L EPR4 RLL 16/17 ITL(depth ) RS (N ; t )].
+
+
=3 +
= 181 = 5
= 6),
Reverse concatenation gives the possibility of a simpler use of soft information in the cascade of a VD matched to a partial response channel, an LC, and an ECC. We have discussed reverse concatenation and analyzed a simplified soft solution that improves the performance of the standard scheme. The investigation reveals that there is an appreciable gain in such performance. The solution is based on the use of erasures in the least reliable symbols. The role of the probability densities of the reliability measures associated with the correct and incorrect symbols was also discussed and used in the performance estimation. ACKNOWLEDGMENT The authors would like to thank the reviewers for their valuable suggestions, and A. Rossi and G. Betti for the stimulating discussions. REFERENCES
Fig.
10.
=
Performance evaluation for serious offtrack percent [CS (L ), EPR4 RLL Parity bit code ; : 24/25 (time-varying trellis) ITL(depth ) RS (N ; t )]. Similar findings confirm a performance gain of about 1.5 dB in terms of SNR.
10 00 10 50 11 00 :
;
:
+
= 6
=3
+
+ = 181 = 5
B. Examples In Figs. 8–10, we report some results from two typical systems: an EPR4 with a RLL code with rate 16/17 and an EPR4 with an LC that uses an additional parity check bit. The interleaving depth is equal to 3 or 5, the ECC is a Reed–Solomon code with 8-bit symbols, and the number of correctable sym5 or 4. The performance estimations in terms of SNR bols is indicate a gain of about 1–1.5 dB with respect to the hard solution. The evaluation of performance after the Reed–Solomon in (2) and (3)] is done, taking into account the output [ statistics after the channel detector; the analysis generally confirms the assumption of independent symbol errors for an interleaver depth greater than 3. About the implementation of this system, the main disadvantage is probably the complexity of the errors erasures algebraic decoding of a Reed–Solomon with long codewords. In
[1] K. D. Anim-Appiah and S. W. McLaughlin, “Towards soft output APP decoding for nonsystematic nonlinear block codes,” in Proc. Allerton Conf, 1999. [2] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. IT-10, pp. 284–287, Mar. 1974. [3] J. Bajcsy, J. A. Hunziker, and H. Kobayashi, “Iterative decoding for digital recording systems,” in Proc. IEEE GLOBECOM, Sydney, Australia, 1998, pp. 2700–2705. [4] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “A soft-input soft-output APP module for iterative decoding of concatenated codes,” IEEE Commun. Lett., vol. 1, pp. 22–24, Jan. 1997. [5] W. G. Bliss, “Circuitry for performing error correction calculations on baseband encoded data to eliminate error propagation,” IBM Tech. Discl. Bull., vol. 23, pp. 4633–4634, 1981. [6] D. Chase, “A class of algorithms for decoding block codes with channel measurement information,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 170–182, Jan. 1972. [7] A. G. A. Daraiseh and C. W. Baum, “Decoder error and failure probabilities for Reed–Solomon codes: Decodable vectors method,” IEEE Trans. Commun., vol. 46, pp. 857–869, July 1998. [8] J. L. Fan and A. R. Calderbank, “A modified concatenated coding scheme, with applications to magnetic data storage,” IEEE Trans. Inform. Theory, vol. 44, pp. 1565–1574, July 1998. [9] J. L. Fan and J. M. Cioffi, “Constrained coding techniques for soft iterative decoders,” in Proc. IEEE GLOBECOM, Rio De Janeiro, Brazil, 1999, pp. 723–727. [10] G. D. Forney, “Generalized minimum distance decoding,” IEEE Trans. Inform. Theory, vol. IT-12, pp. 125–131, Apr. 1966. [11] M. P. C. Fossorier, F. Burkert, S. Lin, and J. Hagenauer, “On the equivalence between SOVA and Max-Log-MAP decodings,” IEEE Commun. Lett., vol. 2, pp. 137–139, May 1998. [12] M. P. C. Fossorier and S. Lin, “Computationally efficient soft-decision decoding of linear block codes based on ordered statistics,” IEEE Trans. Inform. Theory, vol. 42, pp. 738–750, May 1996. [13] J. Hagenauer and P. Hoeher, “A Viterbi algorithm with soft-decision outputs and its applications,” in Proc. IEEE Globecom 1989, Dallas, TX, pp. 1680–1686. [14] K. A. S. Immink, “A practical method for approaching the channel capacity of constrained channels,” IEEE Trans. Inform. Theory, vol. 43, pp. 1389–1399, Sept. 1997.
618
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 4, APRIL 2001
[15] N. Kamiya, “On acceptance criterion for efficient successive errors-anderasures decoding of Reed–Solomon and BCH codes,” IEEE Trans. Inform. Theory, vol. 43, pp. 1477–1488, Sept. 1997. [16] K. J. Knudson, J. K. Wolf, and L. B. Milstein, “Producing soft decision information at the output of a Class IV partial response Viterbi detector,” in Proc. ICC, Denver, CO, 1991, pp. 820–824. [17] N. Kobayashi, H. Sawaguchi, M. Kondo, and S. Mita, “Application of erasure error correction to concatenated partial response channel code,” in Proc. ICC, Vancouver, BC, Canada, 1999, pp. 1627–1631. [18] S. Lin and D. J. Costello, Error Control Coding. Englewood Cliffs, NJ: Prentice-Hall. [19] C. Nill and C. E. W. Sundberg, “List and soft symbol output Viterbi algorithms: Extensions and comparisons,” IEEE Trans. Commun., vol. 43, pp. 277–287, Feb. 1995. [20] J. G. Proakis, Digital Communications: McGraw-Hill. [21] D. J. Taipale and M. J. Seo, “An efficient soft-decision Reed–Solomon decoding algorithm,” IEEE Trans. Inform. Theory, vol. 40, pp. 1130–1139, July 1994. [22] T. K. Truong, J. H. Jeng, and K. C. Hung, “Inversionless decoding of both errors and erasures of Reed–Solomon code,” IEEE Trans. Commun., vol. 46, pp. 973–976, Aug. 1998. [23] T. K. Truong and J. H. Jeng, “On decoding of both errors and erasures of a Reed–Solomon code using an inverse free Berlekamp–Massey algorithm,” IEEE Trans. Commun., vol. 47, pp. 1488–1494, Oct. 1999.
Luca Reggiani (S’00) received the honors degree in electronic engineering from the Politecnico di Milano, Italy, 1998. He is currently pursuing the Ph.D. degree in electronics and communications at Politecnico di Milano. He was a Visiting Scholar at the CMRR (Center for Magnetic Recording Research), University of California, San Diego, from September 1999 to April 2000. His research interests include coding and detection for magnetic recording systems and information theory.
Guido Tartara (M’86–SM’92) was born in La Spezia, Italy, on March 27, 1938. He received the Dr.Ing. degree in electrical engineering and the “Libera Docenza” in electrical communications from the Politecnico di Milano, Italy, in 1961 and 1971, respectively. Following his military service, he joined the Department of Electrical Engineering, Politecnico di Milano, in 1964. During the period 1967–1968, he was on leave at the University of California, Los Angeles, and then joined the Faculty of the Politecnico di Milano, where he became Professor of Communication System Theory in 1975. His research interests cover various aspects of communication system theory, digital transmission, and radio communication systems. He is the Director of the Space Telecommunications Research Center of CNR (National Research Council) at the Politecnico di Milano. This Center is active in basic research for satellite communications and in application programs. Dr. Tartara has been Chairman of the Committee for the Master Degree in Telecommunication Engineering, a member of the “Senato Accademico” at the Politecnico di Milano, and Vice President of AEI (Electrical Engineering Association, Italy). He was the recipient of the best paper award at the 14th Convegno Internazionale delle Comunicazioni, Genova, 1966, and he won the AEI (Italian Electrical and Electronic Society) Prize for the papers published in Alta Frequenza, 1976. He then won the AEI Prize for research achievements, 1989, and the International “Colombo” Award in Communication Science, City of Genova, 1989.
