The contents of this thesis are organized as follows: Before getting into the design of detection schemes, our assumption about the black box VHM has to be clarified. In Chapter 2, two kinds of channel models that are commonly applied are introduced.
These channel models constitute the basis for the following detector design. In Chapter 3, two iterative detection schemes are presented and compared, and the conclusion is reached with the adoption of the soft detection scheme, 2D-MAP, because of its superior bit-error rate (BER) performance. However, the extraordinary performance is achieved at the cost of a huge amount of arithmetic operations. Therefore, a suite of schemes are proposed to reduce the computational complexity of 2D-MAP detection in Chapter 4. These schemes are targeted for different dimensions of the 2D-MAP algorithm, while combined together, they would be able to realize significant complexity reduction. Based on these schemes, the implementation of the basic processing element in a 2D-MAP detector is proposed in Chapter 5. Some other implementation issues are also discussed in this chapter. Finally, the conclusion and outlook are given in Chapter 6 to end this thesis.
Chapter 2
Introduction to Holographic Data Storage Systems
2.1 Introduction
Similar to a book comprising many pages, the holographic data storage superimposes many so-called holograms in the same position on storage medium through angular multiplexing. A hologram is a recording of the optical interference pattern that forms at the intersection of two coherent optical beams. To make the idea more vivid, the working principle of holographic data storage could be demonstrated with the operation of recording and retrieval of data on a holographic disc:
(a) Recording: As shown in Figure 2.1(a), the aforementioned two optical beams are referred to as the object beam and the reference beam. Data taking the form of two-dimensional binary patterns become the object beam after passing through spatial light modulator (SLM) and the Fourier lens. And the reference beam is usually designed to be simple to reproduce. A common reference beam is a plane wave: a light beam that propagates without converging or diverging. When the object beam and a corresponding reference beam overlap with each other on the
holographic medium, the interference pattern between the two beams is recorded.
When the next data page comes in, a second hologram could be stored in the same location only by tilting the incident angle of the reference beam by a small degree.
(b) Retrieval: By illuminating the storage medium with the reference beam of a proper incident angle, a certain object beam is reproduced and then captured by the detector arrays at the receiving end, as clear illustrated in Figure 2.1(b).
[7]
Figure 2.1 In the holographic data storage system: (a) the recording process (b) the retrieval process
By the recording and retrieving scheme, holographic data storages are able to utilize all three dimensions at high resolution, which justifies its property of being a 3-D volume optical storage.
2.2 Channel Model
Owing to the dense packing of data pixels, holographic data storages have been suffering from severe inter-pixel interference (IPI), which is also the major problem that we wish to overcome. To demonstrate the significant influence of IPI, first we will need to introduce and clarify the channel model to be applied.
The first channel model that we are going to present is the complete channel model [5][6], which is the most complex model, yet providing the most accurate approximation of a realistic environment. Thus, the complete channel model is usually treated as the simulation of an actual channel. On the contrary, the second model, the incoherent intensity channel model is a simplified version of the former one. It is a linear model and is not generally applicable to all kinds of holographic data storage channel with different parameter settings. Nonetheless, as will be discussed in Chapter 4, a receiver based on the assumption of an incoherent intensity channel model shall enable a simplified design. As a result, the intensity model is included here in our scope to increase completeness.
2.2.1 Complete Channel Model
A schematic diagram of a 4-fL (focal length) holographic data storage system is shown in Figure 2.2. Each of the two Fourier lenses in this 4-fL architecture performs a Fourier-transform operation, so that the light from Spatial Light Modulator (SLM) is imaged onto the Charged Coupled Detector (CCD). The storage of the Fourier holograms instead of image holograms serves a similar purpose as interleaving, in which the information is distributed in a different transformed domain so as to decrease the possibility of burst errors. And the placement of the aperture in the back of the first Fourier lens is to reduce the size of effective recording area on the holographic medium so that storage density could be increased. As the aperture windows the Fourier transform of the signal beam, it also acts as a low-pass filter whose bandwidth
[6]
Figure 2.2 Schematic diagram of a holographic data storage system in the 4-fL architecture
is determined by the aperture width, thus introducing the inter-pixel interference at the same time. One more thing to note about system is that, here we have assumed the system is pixel matched, which means that each of the SLM pixels is imaged onto one CCD pixel. Pixel-matched imaging is favorable in real implementation since it helps the realization of high data rates.
Based on the 4-fL architecture, a channel model could be developed, as is illustrated in Figure 2.3. The important elements in this model will be introduced respectively as follows.
Figure 2.3 Block diagram of a complete channel model
- dij: The input binary data sequence, dij, takes on values in the set {1, 1/ε}. While a pixel ONE takes the value 1, a ZERO pixel takes the value 1/ε, in which ε is the amplitude contrast ratio, i.e., the ratio of amplitudes of the bit 1 and the bit 0. This non-ideal effect is referred to as “the limited contrast ratio of the input SLM”.
- p(x,y): The SLM’s pixel shape function can be formulated as
Where α represents the SLM’s linear fill factor (namely the square root of the area fill factor for a square pixel). The fill factor is the ratio of the area of the active part of a pixel to the total area of a pixel. The symbol Δ represents the pixel width, which is assumed to be same for both SLM and CCD, and
∏ ( )
x is the unitrectangular function. Now, the output from the SLM could be expressed as:
( )
( , ) kl , (2.2)
k l
s x y
=∑∑ d p x k
− Δ − Δy l
Where
k and l refer to the pixel location along the x and y direction.
- hA(x,y): As mentioned earlier, the aperture leads to a low-pass behavior, whose frequency response is represented as HA(fx,fy) in Figure 2.3. The effect of the Fourier transform, HA(fx,fy) , and the inverse Fourier transform combined together is equal to convolving the SLM output with an impulse response, hA(x,y). The width of hA(x,y) is inversely proportional to the frequency plane aperture width.
For a square aperture of width D, the impulse response of the aperture is described
as
f represents the lens’s focal length. And in this way, the counterpart in frequency
Ldomain, HA(fx,fy), is an ideal low-pass filter with a cut-off frequency equal to
2 L
The impulse response hA(x,y), or commonly referred to as the point-spread function (PSF), may be the most critical ingredient in the channel model, since it compactly characterizes the extent of inter-pixel interference. Now, the signal through the PSF can be expressed as
( )
Where ⊗ represents a 2-D convolution, and h(x,y), which integrate the effects of the SLM pixel shape function and PSF, is referred to as the pixel-spread function (PxSF). From (2.1) and (2.5), it could be observed that the extent of IPI also depends on the SLM fill factor. A high SLM fill factor would broaden the PxSF, while a low one tends to increase the PxSF roll-off.
-
∫∫
| |i 2: The CCD is inherently a square-law device and tends to detect the intensity of incident light. It transforms the signal from the continuous domain to the discrete domain by integrating it spatially. Thus, the output from the CCDdetector combined with the noises can be described as below:
Where
β
represents the CCD linear fill factor. Now we know that the CCD fillfactor also has some connection with the extent of IPI. As a high CCD fill factor normally implies higher signal levels, it also contributes to channel nonlinearity and results in more IPI. And no
(i,j) and n
e(i,j) represents the term of optical noise
and electrical noise respectively. Optical noise results from optical scatter, laser speckle, etc., and is generally modeled as a circularly symmetric Gaussian random process with zero mean and variance No. Electrical noise arises from the electronics in the CCD array, and is normally modeled as a additive white Gaussian noise with zero mean and variance Ne.In Summary, the complete channel model is formulated as:
( )
There are still a few imperfections in a holographic data storage channel that are not considered, such as the inter-page interference, lens aberration, misalignment (including magnification, tilt, rotation, and translation differences) between SLM and CCD, and so on. However, these effects are either beyond the scope of our research or
could be conveniently integrated with the above architecture, so they are not specifically mentioned earlier. In addition, these effects are regarded as having minor influences when compared to the inter-pixel interference and noises.
For the simulations, pages of size 512×512 pixels are applied, with the parameters:
λ ,
f , and
LΔ
, set as 515 nm, 89 mm, and 18 μm, respectively. These parameters areselected as the same as in a paper on the IBM DEMON system [8].
To facilitate adjustment on operating conditions and make the simulation independent from λ ,
f ,
LΔ
, and D, the normalized aperture width, w, is defined asN
w D
=
D
, in which Nf
LD
=λ
Δ is the Nyquist aperture width. To make the description of the extent of IPI even more intuitive, the normalized half-blur width, W, of the point-spread function (PSF) is introduced and defined as
D
N 1W
=D
= . Thus, W = 1w
corresponds to a system with Nyquist aperture width, which also implies a system with critical sampling so that the first zero of PSF coincides with the center of the nearest pixel. While a larger W corresponds to severer IPI, a W larger than 1 suggests operation beyond the classical resolution limit.
For simplicity, the amplitude contrast ratio (ε), the SLM fill factor (α ), and CCD fill factor (
β
) are all set to 1 in the simulation. In fact, their influences can somehowbe depicted through the adjustment of W. For the setting of the noises, it is observed in [6] that the system under an electrical-noise-dominated condition and that under an
optical-noise-dominated condition yield extremely similar simulation results, so, again, for the reason of compactness, we choose to consider only the case when electrical noise dominates. We use signal-to-noise ratio (SNRe) to characterize the extent of electrical noise, and it is defined as SNRe=10 log10(0.52/N0).
Finally, we define two channel conditions that apply the complete channel model in our simulations hereafter: CH-1, with W = 1, represents the case of medium IPI, and CH-2, with W = 1.25, represents the case of severe IPI.
2.2.2 Incoherent Intensity Channel Model
The incoherent intensity channel model is a simplified and linearized version of the complete channel model. It is named “intensity model” because it assumes linear superposition in intensity during image formation, as is shown in Figure 2.4. Besides, the operation of this model on data pages is entirely in the discrete domain; that is, the minimum operating unit is the pixel, as opposed to finer sub-pixel grids that are applied in the complete model. Basically, we can say that the effects of the SLM pixel shape function, the PSF, and the CCD detector have been integrated within one single channel IPI matrix, as is similar to the case for traditional inter-symbol interference (ISI) channel commonly applied in communication systems.
Figure 2.4 Block diagram of the incoherent intensity channel model
Considering an N×N pixel detector array, the system is modeled as linear and shift-invariant, so that it is formulated as: (The index is omitted for simplicity.)
(2.8)
Z
= ⊗ +A H N
Where Z, A, and N are N×N pixel matrices that represent the page being received, the page being sent, and the additive white Gaussian noise (AWGN) with zero mean and variance N0, so that the SNR of the received page could be defined as SNR=10 log10(0.52/N0). H represents the IPI, and ⊗ is a 2-D convolution. To be more specific,
H is a 3×3 IPI matrix constructed from a continuous PSF, which is defined here as:
2 2
Where W is the normalized half-blur width as is defined earlier for the complete model.
Then H is derived from its continuous counterpart (2.9) as:
( 1/ 2) ( 1/ 2)
nine coefficients are re-normalized so that they sum up to be one.
In our simulations, as in the case of complete channel model, two kinds of IPI conditions are set: CH-1, with W = 1.5, represents the case of medium IPI, and CH-2, with W = 1.8, represents the case of severe IPI. The PSF and IPI matrix corresponding to each case are shown in Figure 2.5.
Figure 2.5 The PSF and IPI matrices under the two different IPI conditions
Though less accurate than the complete channel model, the incoherent intensity model is, however, the model usually assumed. [11]-[16] According to [5], the intensity model is an appropriate assumption when the holographic data storage system has high fill factors, so the application of this model can still be justified for some category of holographic data storages. On the other hand, if other page-oriented memories are taken into consideration, this model exactly characterizes the case for an
incoherent optical system, such as the two-photon memory. [9][10] As a result, we have also included the incoherent intensity channel model in our simulations. We have to say that, a final reason is we believe the adoption of intensity model could help inspire some innovations owing to its linearized hypothesis with respect to channel.
This belief will be proved in Chapter 4 that the intensity model does enable a simpler design for the receiver.
A final remark about the channel models is that the application of sinc function in PSF is actually justified by the assumption of a square aperture. However, a square aperture is difficult to acquire in practice, and a circular aperture may be adopted most of the time. Theorectically, the PSF shall then be defined with the polar coordinate, but this increases computational complexity since the convolution with input page is defined in Cartesian coordinate system. Fortunately, we found that, as the PSF is described in the sub-pixel domain, the more sub-pixel into which a pixel is segmented, the more a square aperture resembles the circular aperture. This is shown in Figure 2.6.
When a pixel is segmented into 11×11 sub-pixels, the PSF already looks like resulting from a circular aperture. Furthermore, we compare the channel coefficients derived
from these two sorts of aperture shape by defining D as 20 log10 | |
| |
Through calculation, we found D is about 56 dB, indicating that the difference between
applying these two assumptions of aperture shape is negligible.
(a) Square Aperture ( =1△ 1) (b) Circular Aperture
Figure 2.6 Comparison between square aperture and circular aperture
2.3 The Impact of Inter-Pixel Interference
Based on the system model described above, our aim is clear: to recover the original pages from the corrupted pages. Owing to IPI, the bit-error-rate (BER) performance achieved by a simple threshold detection, which compares a received pixel value to a fixed threshold and makes the decision, is simply not acceptable, as can be observed in Figure 2.7 and Figure 2.8, which assume the complete channel model and the incoherent intensity channel model respectively. Therefore, a more powerful detection scheme has to be devised.
On the other hand, the optimal detection scheme would be maximum likelihood page detection (MLPD). A conceptually simple method of accomplishing it involves the use of a look-up table (LUT) storing the expected received pages of all possible
data page patterns, with the data page pattern as index. Then an arbitrary received page is compared against every element in LUT to determine the most possible page that
Figure 2.7 BER performance of simple threshold detection under the complete channel model
Figure 2.8 BER performance of simple threshold detection under incoherent intensity channel model
was recorded. However, this is not even feasible, since a system with N×N pixel pages would require a LUT with 2N×N entries, as is illustrated in Figure 2.9.
While there is no practical implementation for MLPD at present, we are going to pursue the sub-optimal detection schemes.
Figure 2.9 The MLPD requires the examination of all possible combinations of data page pattern
Chapter 3
Iterative Detection Schemes
3.1 Introduction
Since there is no feasible implementation for MLPD at present, the sub-optimal detection schemes are being pursued. The iterative detection schemes aim to approximate the idea of MLPD by iteratively making improvements on their decisions.
Two representative schemes are parallel decision feedback equalization (PDFE) [12]
and two-dimensional maximum a posteriori (2D-MAP) detection [13], as PDFE being a hard detection scheme and 2D-MAP detection being a soft one, so that some insights shall be inspired from the demonstration and comparisons of these two extremes of approaches. In following discussions, we will assume a perfect knowledge of the channel information, in order to put our focus on the detection at receiving end.
3.2 Parallel Decision Feedback Equalization (PDFE)
As a hard detection scheme, in each iteration, PDFE makes a hard decision at each pixel, based on the knowledge of the decisions of the corresponding eight neighbors in last iteration.
To be specific, this detection scheme can be explained by two steps of operation:
(a) Initialization: The hard decisions are made on all pixels with respect to a fixed threshold, as is clearly illustrated in Figure 3.1. Thus, in the detected page pattern, every pixel has been assigned a binary value to be either “1” or “0”.
Figure 3.1 Initialization of PDFE scheme
(b) Hard decision feedback: For every pixel, given the decisions from its neighbors together with the channel information, the expected pixel value (namely, the noise-free channel output) given that it is sent as a “1” (Z1), and given that it is sent as a “0” (Z0), can be computed. Then, a pixel is decided to be “1”, if
|Z(i,j)-Z1|<|Z(i,j)-Z0| (where Z(i,j) represents the received pixel value); otherwise, it is decided to be “0”. The procedure is executed simultaneously at all pixels on the page, and then all updates of hard decisions also take place simultaneously in
the end of each iteration. This step is iterative and will repeat itself until no further change of hard decisions in the detected page, or a predetermined iteration number is reached. In practice, the BER normally converges in no more than 3 iterations.
This step is illustrated in Figure 3.2, where Nij represents the corresponding neighborhood pattern of the current pixel. The incoherent intensity model is assumed here, so the channel information is represented by a 3×3 IPI matrix.
Figure 3.2 Hard decision feedback of PDFE scheme
3.3 Two-Dimensional Maximum A Posteriori (MAP) Detection
Two-dimensional maximum a posteriori (2D-MAP) detection, which was proposed in [13] as the 2D4 (Two-Dimensional Distributed Data Detection) algorithm, is actually the well-known max-Log-MAP algorithm. As opposed to the hard decision
in PDFE, the information kept by each pixel is a log-likelihood ratio (LLR), or more intuitively, a reliability value that indicates its probability of being “1” and being “0”.
To be specific, an LLR with a more positive value indicates a greater probability of a pixel to be a ‘0”, and a more negative LLR indicates a greater probability for it to be a
“1”. Following the same reason, an LLR value which is around zero then indicates that the decision concerning current pixel is still somewhat ambiguous. In each iteration, the reliability value at each pixel will be re-calculated, based on the knowledge of LLRs of the nearest neighbors in last iteration.
Again, to delve into the details of this algorithm, 2D-MAP detection could also be explained by steps:
(a) Likelihood feedback: Under the current assumption of equal a priori statistics (i.e., P[A(i,j)=1]=P[A(i,j)=0]), the MAP rule actually reduces to the maximum likelihood
(a) Likelihood feedback: Under the current assumption of equal a priori statistics (i.e., P[A(i,j)=1]=P[A(i,j)=0]), the MAP rule actually reduces to the maximum likelihood