WO2008037369A1

WO2008037369A1 - Spatial sampling grid recovery

Info

Publication number: WO2008037369A1
Application number: PCT/EP2007/008069
Authority: WO
Inventors: Felix Balado; Kevin M. Whelan; Guenole C. M. Silvestre; Neil J. Hurley
Original assignee: University College Dublin, National University Of Ireland, Dublin
Priority date: 2006-09-29
Filing date: 2007-09-17
Publication date: 2008-04-03

Abstract

A method of synchronized transmission of spatial data between a transmitter and a receiver and a synchronisation unit for use in such a method are disclosed. The method comprises the following steps. At the transmitter, encoding a set of symbols A using a spatial sampling in a data set and transmitting the data over a channel to the receiver. At the receiver, receiving the output z(t) of the channel, and sampling it to derive a matrix Z from which an estimate A of the set of symbols A is obtained, the sampling being done using a sampling grid that is a modification of that used by the transmitter to sample the data, the modification being introduced to correct spatial errors introduced by the channel. The sampling grid is modified using an estimate of the spatial error at each of its points, computed as a linear combination of past received samples. The modifications to the sampling grid are determined by a function that is dependent upon Z and already decoded symbols and/or upon pilots symbols within the transmitted symbols.

Description

Spatial sampling grid recovery

This invention relates to sampling grid recovery in two-dimensional digital communications systems, with application to higher-dimensional systems.

Several emerging communication systems are inherently two-dimensional or spatial in nature. These include two-dimensional barcode systems, steganographic information embedding (data hiding) in images, and two-dimensional data storage applications.

As with many communications systems, as a result of random or systematic changes in the domain of definition of the signal introduced by the channel, these spatial communications systems are subject to performance degradation caused by a loss of synchronisation between the transmitter and receiver. Due to the two-dimensional nature of the signals and communications channels involved, sampling the signal at the receiver presents the problem of maintaining spatial synchronisation.

Known synchronisation methods from conventional digital communications are intended to maintain temporal synchronisation only, and these methods are not straightforwardly extensible to applications in which spatial synchronisation must be maintained.

An aim of this invention is to provide methods and systems that can be used to recover spatial synchronisation in certain spatial communications systems.

The particular class of spatial communications system to which the embodiments of the invention may be applied are two-dimensional, multi-level baseband pulse amplitude modulated (PAM) systems. The two-dimensional signal released by the transmitter is communicated through a spatial communication channel, which is assumed to introduce a set of unknown spatial displacements and additive noise. To understand this, it is important that the term "communication channel" is interpreted broadly. One example of such a channel is when some information is first encoded into a spatial pattern of intensities. These are then encoded in a two-dimensional pattern, which is then printed onto a physical medium such as a paper label. The pattern is subsequently scanned optically for decoding, which generates a two-dimensional numerical array of data within a decoder. At the decoder, the spatial displacement introduced by the channel must be reversed, as sampling the received signal at incorrect locations can lead to a severe degradation in performance of the communication. An example of such displacement occurs because of imperfect alignment of the printed pattern during optical scanning. This gives rise to uncertainty of the position of any piece of encoded data within the scanned image. The task performed by embodiments of the invention within the receiver is to compensate for this spatial displacement introduced by the channel.

It is well known that phase-locked loops (PLLs) can be used to effect symbol timing recovery on 1 -dimensional signals. The task that this invention seeks to perform can be thought of as a digital PLL that operates in two dimensions. The term "PLL" (and the two-dimensional spatial extension to 2D-PLL) is used in this description following from its widespread use in literature that describes applications of sampling error tracking; however the use of the term

PLL does not imply that the invention requires estimation of the phase parameter of any sinusoidal signal, as the system considered is baseband.

The notation to be used in the following description will now be described. Uppercase bold symbols denote matrices while lowercase bold symbols denote column vectors. Thus:

E[«]denotes statistical expectation, or average, diag(x) denotes a diagonal matrix with diagonal elements being the elements of the vector x. From a first aspect, this invention provides a method of synchronised transmission of spatial data between a transmitter and a receiver comprising: a. at the transmitter, encoding a set of symbols A using a spatial sampling in a data set and transmitting the data over a channel to the receiver; b. at the receiver, receiving the output z(t) of the channel, and sampling it to derive a matrix Z from which an estimate A of the set of transmitted symbols A is obtained, the sampling being done using a sampling grid that is a modification of that used by the transmitter to sample the data, the modification being introduced to correct spatial errors introduced by the channel; c. the sampling grid being modified using an estimate of the spatial error at each of its points, computed as a linear combination of past received samples.

In typical embodiments of the invention, the data is encoded using pulse amplitude modulation (PAM).

Typically, in an embodiment of the invention, at a stage k the output z(t) is sampled using an estimate of the spatial displacement τ_k and a new estimate at stage k + 1 is formed by adding to τ_k an attenuated estimate z_k of the error signal ε_k , where ε_k = τ_k -τ_k . More specifically, the method may progress in steps such that τ_k+] = τ_k + diag(α) -έ_t where diag(α) is a gain matrix. The value ε_k typically, on average evaluates to the value of a specified function of the spatial displacement error f(ε) whereby E[ε_k J = f (ε) « ε . In preferred embodiments, the function f(ε) = (/i(ε), ^(ε)), such that f(ε) ~ ε, is computed as a linear combination of samples of the two-dimensional impulse response /z(t) offset by ε.

The method may operate in a "decision directed" mode or in a "data aided" mode or in a combination of these two modes. In the former, the modifications to the sampling grid are determined by a function that is dependent upon already decoded symbols. In the latter, the modifications to the sampling grid are determined by a function that is dependent upon pilots symbols within the transmitted symbols.

From a second aspect, the invention provides a synchronisation unit for use in synchronized transmission of spatial data between a transmitter and a receiver comprising, the data comprising a set of symbols A encoded at the transmitter using a spatial sampling pattern agreed prior to transmission of the data over a channel to the receiver; the synchronisation unit being operative to receive the output z(t) of the channel, and to sample it to derive a matrix Z from which an estimate A of the set of symbols A is obtained, the sampling being done using a sampling grid that is a modification of that used by the transmitter to sample the data, the modification being introduced to correct spatial errors introduced by the channel; and the synchronisation unit modifying the sampling grid by estimating the spatial error at each of its points as a linear combination of past received samples.

From a third aspect, the invention provides a system for synchronized transmission of spatial data comprising a transmitter and a receiver comprising, the data comprising a set of symbols A encoded at the transmitter using a spatial sampling in a data set prior to transmission of the data over a channel to the receiver, in which the receiver includes a synchronisation unit embodying the second aspect of the invention.

From a fourth aspect, the invention provides a method for correcting a spatial error in a sampling grid used for decoding a multi-valued array of sampled data, the method comprising estimating the error in the position of a sample as a linear combination of past received samples and adjusting the position of the sample in accordance with the estimated error.

An embodiment of the invention will now be described in detail, by way of example, and with reference to the accompanying drawings, in which:

Figure 1 is a model for a two-dimensional communication system within which an embodiment of the invention may find application;

Figure 2 shows a scheme employing a two-dimensional PLL for spatial sampling grid recovery within an embodiment of the invention;

Figure 3 shows the structure of a spatial displacement error detector;

Figure 4 shows the function of the spatial displacement error to be estimated from the channel output; Figure 5 shows the group of past samples used in estimation of the spatial displacement error;

Figure 6 shows the function of the spatial displacement error to be estimated from the channel output along with experimental values produced by the preferred embodiment of the invention;

Figure 7 is a block diagram of the preferred embodiment.

In the system of Figure 1 , a matrix A contains spatially arranged pulse amplitude modulation (PAM) data symbols to be transmitted from a transmitter to a receiver. For example, for 4- ary PAM, the element α_g at the position (i,j) of the matrix A is one of the four values in the set {-3P/2, -PIl, P 12, 3/72}, for a given parameter P > 0 used to control the transmitter power.

In Figure 1, h(t) denotes the impulse response of a certain two-dimensional channel, evaluated at the spatial position t = (t_\, t₂ )^τ. This function models the whole channel as it is "seen" by the receiver; that is, it includes a transmission filter used by the transmitter. For instance, in a 2-D barcode application, h(t) may be defined as a raised cosine filter, or any other filter with good intersymbol interference cancellation properties. If a constant spatial displacement τ = (τi, r₂)^r is introduced, this function becomes /z(t - τ). In a typical embodiment, the channel also adds white Gaussian noise to the transmitted signal.

The spatially continuous output of the channel, denoted z(t), is sampled in the receiver at spatial locations determined by a spatial synchronisation unit in the receiver. This sampling process yields a matrix Z of samples of the received signal, which a symbol decision device uses to form an estimate, A, of the transmitted symbols. If the transmission channel were perfect, then A and A would be identical. However, the spatial (i.e., desynchronisation) errors and noise introduced by the channel mean that this identity is not met in practice. The aim of the invention is to enable the recovery of A from A and Z.

A synchronisation unit for use in a receiver of a system embodying the invention will now be described. The synchronisation unit implements a two-dimensional PLL (2-D PLL) which estimates τ from the samples Z of the received signal and the transmitted data symbols A. A scheme of operation of a receiver making use of an embodiment of the invention is shown in Figure 2.

The receiver samples a continuous two-dimensional channel output z(t) on a spatial sampling grid that has been agreed between transmitter and receiver. Nevertheless, the spatial displacement introduced by the channel, as described above, implies that this is not the correct sampling grid for the received output, but just an initial estimate. The job of the

2-D PLL is to adjust the spatial sampling grid used by the receiver to bring it into spatial synchronism with the original grid on which A was arranged. The 2-D PLL does this by calculating and updating an estimate of the spatial displacement τ for each point in the sampling grid.

Denote by τ_k the estimate of the spatial displacement at some stage k calculated by the 2-D PLL. The new estimate at stage k + 1 is formed by adding to τ_k an attenuated estimate ε_k of the error signal ε_k , where ε_k = τ_k -τ_k . Specifically,

τ*₊₁ = τ_A +diag(α) -έ_t (1)

where diag(α) is a gain matrix of the 2-D PLL. This is similar to a two-dimensional version of the usual one-dimensional digital PLL. The updated sampling grid is used to resample the channel output z{t) at the corrected positions.

Next, there will be described a general procedure for computing ε_k to allow this technique to be applied in practice.

In temporal communications applications using digital PLLs, a device that computes an estimate of this error is known as a timing error detector (TED). As this invention deals with a spatial communications application, the equivalent device for use with 2-dimensional data will be called a "spatial displacement error detector" (SDED).

To compute ε_k , the SDED uses samples Z of the received signal and data symbols A (or estimates A of already decoded data symbols, computed by a symbol decision device in the receiver). Prior to exploiting this information, it is necessary to establish a scanning strategy (path) of the two-dimensional signal. Different scanning strategies will lead to different SDEDs. This method builds the SDED and describes how to tackle any arbitrary spatial scanning strategy.

The receiver follows sequentially this pre-established scanning strategy. At stage k, it obtains a decoding estimate of the transmitted symbol in the matrix A corresponding to this stage in the path. To this end, the SDED is first used to compute έ_t , and then z(t) is sampled using the estimate of the spatial displacement updated using (1). The decoded symbol at stage k is decoded from this sample of z(t). The k - 1 information symbols already decoded - and the corresponding already synchronised samples in Z - constitute the "past" of the scanning path, and are used by the SDED.

This mode of operation of the SDED, based on already decoded symbols, is called "decision- directed" (DD) mode. Alternatively, the SDED could have access to the transmitted symbols (pilots), which is called "data-aided" (DA) mode. These two modes of operation are indicated in Figure 3, which also shows that the SDED is simply a function

that produces an estimate ε_k of the spatial displacement error z_k at stage k.

The set of past samples actually used is usually chosen to be spatially close to the element z*, both to reduce complexity and due to the possibility that the spatial distortion could differ between points. Embodiments of the invention yield a generic way of dealing with different choices of these past samples, as will now be explained.

The SDED computes the spatial displacement error estimate έ^ as a linear combination of the group of past samples z*. In this case, ε_k = φ(z_k,a_k) = G_kz_k . The elements of the matrix GA ⁼ (gi,* I g₂,_k Ϋ are functions of the data symbols a*. These elements are chosen such that the spatial displacement error estimate ε_k will, on average, evaluate to the value of a specified function of the spatial displacement error f(ε). That can be expressed as: E[έ_A ] = f (ε) w ε . An additional constraint is that G* must be chosen such that a low variance estimate of ε* is also obtained. The computation described above is performed under the assumption that the decisions on the transmitted symbols a* are correct. This is equivalent to assuming an ideal situation where the receiver knows a. Indeed, this will be the case for an SDED operating in DA mode. When operating in DD mode, decisions a on the transmitted symbols may be incorrect. The result of these incorrect decisions is that the receiver will estimate a function f(ε) different to that which is required for accurate decoding, which would yield an incorrect estimate of ε_k .

The function f(ε) = (/i(ε), fι(z)), such that f(ε) ~ ε, is computed as a linear combination of samples of the two-dimensional impulse response h(t) offset by ε. In this type of SDED, the timing functions may always be written as

/_/(ε) = u[h , / = 1,2, (2)

with h being a vector containing the arbitrarily arranged unique samples of h{-) used to compute E[z* | a*] - that is, the expected value of the received signal conditioned to the already decoded symbols - and u_/ a vector of scalars that defines the function.

The weights G_t needed for the SDED are the solution to the following channel-independent linear problem:

A_tg_/tt = u,d_/it . / = 1,2, (3)

with dι_t/c a zero-mean random vector chosen to meet the problem constraints, and with A* a matrix of past information symbols such that E[z* | a*] = A^h.

Specifically, for this embodiment, the components of the function f(ε) are defined as follows:

where T_s is the spatial sampling period in each dimension. For two-dimensional impulse responses approximating the ideal two-dimensional pulse h(t) = sinc(Yi / T₅) ^■ sinc(t₂ / T₅) (i.e., in the case where there is intersymbol interference cancelling) the above components evaluate approximately to the value of the error parameter that is to be estimated i.e., f_x (ε) « ε_x and /₂ (ε) « ε₂. To demonstrate this, the curves in Figure 4 show the approximation /ι(ε) in (4) compared to the ideal case/i(ε) = εj when h(t) is the aforementioned ideal pulse.

This embodiment uses a simple left-to-right, line-by-line scanning of the coordinates of the elements of Z. The 'past' samples chosen for estimation of ε at decoding stage k are the sample at the corresponding position in the path and the one directly to the left of it, and the two samples directly above on the previous line. This is more clearly stated by considering the position corresponding to the current stage k in the scan as the coordinate pair (/,/). Then, the 'past' associated with this point is given by the following vector of samples:

= (*ι-lj-l . Z,-lj » Z,j-l » O (5)

Figure 5 shows graphically the selection of this particular group of past samples when a left- to-right, line-by-line scanning strategy is adopted. Note that better estimates can be obtained by using more samples, but this hampers finding a solution to equation (3). The choice above allows an approximation to be calculated with enough fidelity to the ideal case, while facilitating an algebraic solution to (3).

To satisfy the requirement that E[S₁J = f(ε), with the components of f(ε) given in (4) and the 'past' given by (5), the solution to the algebraic problem is the following:

which yields the following elements of ε_k :

fi - (- ^^α-'.; - ^.7-i^a'-1.7-_{l r} ^{+ 2} _r1.7^α'.7 + *,-.,;-■",,;-■ ) ₍~ u^~ 4ER ⁽⁷⁾ and

where E[α²] is the power of the transmission constellation employed. For instance, for the 4-ary PAM example previously mentioned, E[α²] = 5P² /4 with equally likely symbols.

To demonstrate the validity of this selection of G*, the markers in Figure 6 show experimentally measured values of E[έ, _k J for a typical set of system parameters with the

SDED operating in data aided mode. The lines in Figure 6 show/i(ε) for different values of ε. This shows that E[έ_u] = /j(ε) as required. As well as satisfying the expectation requirement, this selection of GA yields a low variance on the estimates z_k .

The above description is summarised in Figure 7. The path through Z is given by left-to- right line-by-line scanning of the elements of Z. The elements of Z selected for estimation of Z_k and the weighting matrix G^ used are shown on the right. The embodiment is shown operating in decision-directed mode.

Claims

1. A method of synchronized transmission of spatial data between a transmitter and a receiver comprising:

a. at the transmitter, encoding a set of symbols A using a spatial pattern and transmitting the data over a channel to the receiver;

b. at the receiver, receiving the output z(t) of the channel, and sampling it to derive a matrix Z from which an estimate A of the set of symbols A is obtained, the sampling being done using a sampling grid that is a modification of that used by the transmitter to sample the data, the modification being introduced to correct spatial errors introduced by the channel

c. the sampling grid being modified using an estimate of the spatial error at each of its points, computed as a linear combination of past received samples.

2. A method of transmitting spatial data according to claim 1 in which the data is encoded using pulse amplitude modulation.

3. A method of transmitting spatial data according to claim 1 or claim 2 in which at a stage k the output z(t) is sampled using an estimate of the spatial displacement is τ_k , and a new estimate at stage k + 1 is formed by adding to τ_k an attenuated estimate ε_k of the error signal ε_k , where ε_k = τ_k -τ_k .

4. A method of transmitting spatial data according to claim 2 in which τ_A+1 = τ_k + diag(α) • ε_k where diag(α) is a gain matrix.

5. A method of transmitting spatial data according to claim 3 or claim 4 in which ε_k on average evaluates to the value of a specified function of the spatial displacement error f(ε) whereby E[ε_k ] = f (ε) » ε .

6. A method of transmitting spatial data according to claim 5 in which the function f(ε) = (/i (ε), fι(z)), such that f(ε) ~ ε, is computed as a linear combination of samples of the two-dimensional impulse response h{t) offset by ε.

7. A method of transmitting spatial data according to any preceding claim in which the modifications to the sampling grid are determined by a function that is dependent upon already decoded symbols.

8. A method of transmitting spatial data according to any preceding claim in which the modifications to the sampling grid are determined by a function that is dependent upon pilots symbols within the transmitted symbols.

9. A synchronisation unit for use in synchronized transmission of spatial data between a transmitter and a receiver, the data comprising a set of symbols A encoded at the transmitter using a spatial sampling pattern agreed prior to transmission of the data over a channel to the receiver, the synchronisation unit being operative to receive the output z(t) of the channel, and to sample it to derive a matrix Z from which an estimate A of the set of symbols A is obtained, the sampling being done using a sampling grid that is a modification of that used by the transmitter to sample the data, the modification being introduced to correct spatial errors introduced by the channel, and the synchronisation unit modifying the sampling grid using an estimate of the spatial error at each of its points, computed as a linear combination of past received samples.

10. A synchronisation unit according to claim 9 operative to decode data that has been encoded using pulse amplitude modulation.

11. A synchronisation unit according to claim 9 or claim 10 in which at a stage k the output z(t) is sampled using an the estimate of the spatial displacement is τ_k , and a new estimate at stage k + 1 is formed by adding to τ_k an attenuated estimate έ_t of the error signal ε_k , where ε_k = τ_k -τ_k .

12. A synchronisation unit according to claim 11 in which τ_M - τ_k +diag(α) -έ_i where diag(α) is a gain matrix.

13. A synchronisation unit according to claim 11 or claim 12 in which ε_k on average evaluates to the value of a specified function of the spatial displacement error f(ε) whereby E[ε_k ] = f (ε) « ε .

14. A synchronisation unit according to claim 13 which computes the function f(^ε) ⁼ (/i(ε), ./Kε)), ^sucn that f(ε) ~ ε, as a linear combination of samples of the two-dimensional impulse response h(t) offset by ε.

15. A system for synchronized transmission of spatial data comprising a transmitter and a receiver comprising, the data comprising a set of symbols A encoded at the transmitter using a spatial sampling in a data set prior to transmission of the data over a channel to the receiver, in which the receiver includes a synchronisation unit according to any one of claims 9 to 14.

16. A method for correcting a spatial error in a sampling grid used for decoding a multi-valued array of sampled data, the method comprising estimating the error in the position of a sample as a linear combination of past received samples and adjusting the position of the sample in accordance with the estimated error.

17. A method according to claim 16 in which the sampled data is symbols A encoded by a transmitter using a spatial sampling pattern and transmitting the data over a channel to the receiver; at the receiver, receiving the output z(t) of the channel, and sampling it to derive a matrix Z from which an estimate A of the set of symbols A is obtained, the sampling being done using a sampling grid that is a modification of that used by the transmitter to sample the data, the modification being introduced to correct spatial errors introduced by the channel, the sampling grid being modified using an estimate of the spatial error at each of its points computed as a linear combination of past received samples.