Quantization-Based Data-Hiding Robust to Linear Time ... - CiteSeerX

1

Quantization-Based Data-Hiding Robust to Linear Time-Invariant Filtering Fernando P´erez-Gonz´alez and Carlos Mosquera

Abstract Quantization-based methods, such as Dither Modulation (DM), have gained wide acceptance due to their host rejection capabilities which afford significant performance gains over spread-spectrum-based methods in additive white Gaussian channels. Unfortunately, existing quantization-based schemes are not robust against simple linear-time-invariant (LTI) filtering, which is a common operation with multimedia signals. We propose a new algorithm, named Discrete Fourier Transform Rational Dither Modulation (DFT-RDM) which is robust against LTI filtering and yet does not assume any prior knowledge of the filter at either the embedder or the detector. DFT-RDM basically combines a DFT operation with a quantization-based scheme robust to amplitude scaling. Two easily implementable improvements over the basic DFT-RDM are proposed: windowing and spreading. In particular, the latter leads to performance gains that are much larger than those achieved with spreading in regular DM. We also provide a thorough analysis of our scheme which leads to both accurate predictions and bounds on the per-DFT-channel bit error rate, for the basic DFT-RDM and its combination with spreading and windowing. These tools let the designer choose the main embedding parameters without actually requiring any simulation. The results of several simulations for practical filters validating our analysis are presented as well. The benefits of combining DFT-RDM with windowing, spreading and Reed-Solomon channel coding are illustrated with an example.

EDICS Category: WAT-BINM

Dept. Teor´ıa de la Se˜nal y Comunicaciones, ETSI Telecom., Universidad de Vigo, 36310 Vigo, Spain Corresponding author. E-mail: [email protected]. Tel:+34-986-812124. Fax:+34-986-812116 This work was partially funded by Xunta de Galicia under projects PGIDT04 TIC322013PR and PGIDT04 PXIC32202PM; MEC project DIPSTICK, reference TEC2004-02551/TCM, and European Comission through the IST Programme under Contract IST-2002507932 ECRYPT. ECRYPT disclaimer: The information in this paper is provided as is, and no guarantee or warranty is given or implied that the information is fit for any particular purpose. The user thereof uses the information at its sole risk and liability.

December 8, 2007

DRAFT

2

I. I NTRODUCTION Until very recently, one of the main drawbacks of quantization-based schemes, as opposed to spread-spectrum ones, was thought to be their fragility when subject to gain (also known as valumetric) attacks. Motivated by this problem, several researchers have proposed different ways of estimating this gain at the decoder, either by embedding an auxiliary pilot signal known to the decoder [1], or through blind estimation, [2], [3], that is, exploiting certain statistical properties of the received signal due to the periodic structure of the distribution of the watermarked signal amplitude. A different possibility consists in designing codes whose codewords are placed on a high-dimensional hypersphere as a way to achieve robustness to gain attacks [4], [5]. Another approach was taken in [6] to construct a simple, yet effective gain-invariant method which retains most desirable properties of Dither Modulation (DM), which in turn is the most popular algorithm in the family of side-informed ones. Such method is based on the use of a gain-invariant adaptive quantization step-size which is used by the embedder and which holds the advantageous feature of allowing its robust estimation at the decoder. This adaptive stepsize is constructed as a nonlinear function of past watermarked samples. The memory of such function, i.e., the number of past samples used at a certain instant to compute the step-size is a crucial design parameter: a small memory is useful when the target is to cope with varying gains; on the other hand, the larger this memory, the more robust the step-size estimation at the decoder, therefore improving performance. This new method, named Rational Dither Modulation (RDM), has been proven to asymptotically achieve the performance of DM. Full details on RDM and its performance can be found in [7]. Since its inception, several other works have shown the feasibility of combining RDM with channel coding [8] or source coding [9], applying RDM to image data hiding with Watson-like perceptual distances [10], and extending it to account for nonlinear distortions [11]. Under the premise that the gain attack can be considered somewhat solved, it is interesting to mention that there is a much less studied but almost as simple attack to quantization based schemes: linear time-invariant (LTI) filtering. LTI filtering is a recurrent operation with multimedia signals which in many instances can be considered as an unintentional attack. For instance, for audio signals equalization or certain audio effects, such as echo or phasers, well fit in this category. In image processing LTI filtering is also a common operation with applications including denoising, deblurring, interpolation or image enhancement. Unfortunately, existing side-informed methods have not been designed to survive such attacks, which therefore pose considerable concerns upon their alleged superiority over other alternatives such as spread spectrum. To see how devastating those attacks can be, consider the outcome of the following experiment on binary time-domain DM when the host signal is white with power 25 dB above that of the watermark, and the watermarked signal is attacked by a lowpass filter of variable cutoff frequency. When this frequency is as high as 0.99π rad, the bit error rate (BER) is 0.5; when the cutoff frequency is 0.999π rad, the BER is still 0.11. Note that given the tiny fraction of bandwidth that is thrown away in both cases, the attacked signals are very likely to go perceptually unnoticed. The reason for this failure lies in the distortion error due to filtering that moves the watermarked December 8, 2007

DRAFT

3

signal away from the correct quantization centroids. This distortion error increases with the host signal power; in other words, for a fixed watermark power the larger the host power, the worse the performance. This is clearly in contrast to the performance of quantization-based schemes subject to additive white Gaussian noise (AWGN) attacks, where the host signal power is virtually irrelevant for most cases of practical interest. To the best of our knowledge the only other work dealing with linear filtering attacks is [12], which considers a two-band filter in which the high frequency band is amplitude-scaled. The approach taken there is to estimate the high-pass gain using the maximum likelihood criterion, which unfortunately demands a large number of samples in order to achieve an accurate enough estimate. Furthermore, the decoder is assumed to know the filter coefficients (save for the gain to be estimated), which may be not realistic in most practical cases. In this paper, we propose an extension of RDM that allows us to construct a high-rate scheme that is robust to LTI filtering attacks. Our solution does not assume any prior knowledge of the filter neither at the embedder nor at the decoder. The idea is to perform RDM over parallel channels in the DFT domain, taking advantage of the orthogonality of this transform and the fact that it converts a filtering operation in the time domain into an approximate multiplication on each channel. For this reason, the new scheme is termed DFT-RDM. As we will see, the inevitable distortions incurred during the filtering process cause the solution not to be truly LTI-filtering-invariant; however, by adequately choosing the different design parameters, it is possible to achieve any desired degree of resilience against this attack, at the expense of increased complexity or reduced payload. The remainder of the paper is organized as follows: the used notation and a review of RDM is given in Section II; Section III is devoted to discussing the consequences of filtering on conventional quantization-based methods. We present our new method in Section IV, and analyze its performance in Section V. Windowing and spreading are two mechanisms that considerably improve the performance of DFT-RDM; these are discussed in Section VI. Finally, Sections VII and VIII are devoted to presenting some experimental results and our final remarks, respectively.

II. N OTATION

AND

R EVIEW

OF

RDM

For simplicity, we assume one-dimensional real-valued hosts arranged in a vector x; extension to the 2-D case is possible but it has not been pursued here. For convenience, and without loss of generality, we will regard the one-dimensional host as being given in the time-domain. Let xl denote the lth element of x. We will also use {xl } when referring to the sequence whose samples are xl . In data-hiding applications a message is embedded by modifying x to a vector y which we will call watermarked signal. The difference w , y − x is termed watermark. The lth elements of y and w are denoted by yl and wl , respectively. For analytical purposes, it will be convenient to model the different sequences involved in the embedding and

December 8, 2007

DRAFT

4

decoding operations as realizations of random processes. We will reserve uppercase letters to denote random variables and lowercase letters to denote specific values. Then, Xl is a random variable modeling the lth sample of the host sequence, and {Xl } is the random process modeling the whole sequence {xl }. In particular,

throughout this paper we will assume that {Xl } is a zero-mean white process with variance σx2 . The case of

colored hosts can be taken into account in our analysis by combining the coloring filter with the attack filter. Boldface letters will be used to denote vectors: uppercase for random vectors, and lowercase for realizations. The vector containing the available samples at the decoder, also called observations throughout this paper, will be denoted by z. Those observations zl are obtained after passing the watermarked signal {yl } through the attack filter. As our proposed embedding and decoding schemes operates in the Discrete Fourier Transform ˜ m,k will denote the (DFT) domain, we will use the tilde to express variables in such domain; for instance, X kth coefficient of the DFT of the mth block of {Xl }. The superscript ∗ will be used to denote conjugate.

We will also need to define the embedding distortion Dw as the average power of the watermark, i.e.,     1 1 Dw , M E ||Y − X||2 = M E ||W||2 , where E[·] denotes statistical expectation, || · || stands for Euclidean

(i.e., `2 ) norm, and M is the length of W. The Document to Watermark Ratio (DWR), often expressed in

decibels, is given by σx2 /Dw . We will assume that this DWR is large, as it occurs in the vast majority of practical applications, due to perceptual reasons. Values larger than 20 dB are customary. Although the procedures here described can be extended to the case of multilevel and lattice-block quantizers, the underlying principles of both RDM and DFT-RDM are fully illustrated by considering the simpler binary case (i.e., every embedding operation is meant to conceal one bit), to which our discussion will be confined. For further details on RDM the reader is referred to [7]. Let us start by recalling the basic principles of binary DM data-hiding. Define the shifted lattices Λb , 2∆Z − b∆/2,

b = −1, 1

(1)

which describe the centroids for the respective quantizers Q−1 (·) and Q1 (·), here assumed to be based on Euclidean distances. Given the kth information symbol bk ∈ {−1, 1}, embedding in the kth sample is performed

using the rule yk = Qbk (xk ). After observing the kth sample zk , the decoded binary symbol ˆbk is decided according to a minimum Euclidean distance rule ˆbk = arg min |zk − Qb (zk )| k −1,1

(2)

which can be also seen to be equivalent to quantizing zk with a quantizer with step-size ∆. On the other hand, RDM is based on the following embedding and decoding equations [7]   xk yk = g(y k−1 )Qbk , g(y k−1 )   zk ˆbk = arg min zk . − Qbk −1,1 g(zk−1 ) g(zk−1 ) December 8, 2007

(3) (4)

DRAFT

5

where yk−1 , (yk−1 , yk−2, · · · , yk−L )T , zk−1 , (zk−1 , zk−2 , · · · , zk−L )T , with L a designer parameter that determines the memory of the system. The function g : RL → R is designed in such a way that g(ρy) = ρg(y),

for any ρ > 0 and any y ∈ RL . One convenient choice for the g function is any of the H¨older or `p vector-norms: !1/p k−1 1 X g(y k−1 ) = , p ≥ 1. (5) |ym |p L m=k−L

Knowing that best performance is obtained when tuning the p parameter to the shape parameter of the host distribution (when X follows a Generalized Gaussian) [7], we will assume from now on p = 2, which suits a Gaussian host distribution and leads to simplified expressions. To understand why RDM is gain-invariant, let us consider a fixed amplitude scaling attack with no noise; hence, the observed vector z becomes z = ρy,

with ρ the channel gain unknown to both embedder and detector. Then, substituting zk = ρyk into (4) and taking into account the fact that g(zk−1 ) = ρg(y k−1), it is immediate to see that the ρ in the numerator and the denominator cancel out and the decision ˆbk is identical, irrespective of the value of ρ. III. LTI-F ILTERING

ATTACKS

In the remainder of this paper we will consider attacks of the type zl = yl ∗ hl , where ∗ denotes convolution and hl is the impulse response of a real-valued LTI filter. We will assume that the cursor of the filter (i.e., the largest magnitude coefficient) is located at the origin to facilitate the analysis. Note that in practice this requires that some synchronization measures are effected to align the watermarked signal and the observations. Also note that in the considered model there is no explicit noise component; this is because the filtered-host interference (FHI) will be the dominant impairment. However, in the event such noise were present, it could be easily incorporated in the subsequent developments. To quantify this FHI, suppose that DM is used to hide one bit per sample applying the rule given above. The observed sequence can be written in a way that makes explicit the influence over the desired informationconveying sample yl zl = yl ∗ hl = h0 yl +

X i6=0

hi yl−i ≈ h0 yl +

X

hi xl−i

(6)

i6=0

where the given approximation follows from the large DWR assumption. The second term in the right hand side of (6) corresponds to the FHI. For i.i.d. zero-mean Gaussian hosts, this FHI will be also zero-mean Gaussian, independent of xl and with variance σf2 given by σf2

=

σx2

X l6=0

σ2 |hl | = x 2π 2

Z

π

−π

|h0 − H(ejω )|2 dω,

(7)

where H(ejω ) denotes the Fourier Transform of hl . Now, assuming that h0 = 1, it is possible to directly apply the results in [13] to derive the bit error probability for DM under LTI attacks:      ∞   X (4k + 3)∆ (4k + 1)∆ ∆ =2 −Q Q Pe,DM σf 2σf 2σf

(8)

k=0

December 8, 2007

DRAFT

6

with Q(x) ,

1 2π

R∞ x

e−τ

2

/2 dτ .

(See also [13] for several useful approximations to the above equation.) In any

case, it is worth noting that while for large DWRs the performance of DM in an AWGN channel is solely governed by the so-called Watermark to Noise Ratio (WNR), here the role of the noise is played by the FHI, so performance will depend in exactly the same way on the Watermark to Interference Ratio (WIR), defined as WIR , Dw /σf2 . The problem is that the WIR is inversely proportional to the DWR, so if the DWR is increased in, say, 10 dB, so as to make the watermark less perceptible, the WIR will be correspondingly decreased in 10 dB. This reveals the strong dependence of the performance of DM upon the operating DWR, which typically imposes hard-to-meet constraints, as illustrated by the following example. Example 1: From (8) it is immediate to calculate that a WIR of 10 dB will guarantee a bit error probability of 6 · 10−3 , which may be reasonable for many applications. Suppose that hl is an ideal low-pass filter with cutoff frequency ω = ωc and such that h0 = 1. Then, from (7) we have Z  σx2 π ωc  2 σf = dω = σx2 1 − π ωc π

(9)

From here, it is possible to calculate, for a given DWR, the cutoff frequency that yields a WIR of 10 dB. For instance, when DWR=25 dB, we have that ωc = π(1 − 3.16 · 10−4 ) rad, while for DWR=15 dB, ωc =

π(1 − 3.16 · 10−3 ) rad. Observe how narrow the stop-band is for both cases.

It is also important to remark that if h0 6= 1 then, besides the FHI problem, DM would be facing a scaling attack which, as we have noted above, is not able to cope with in an effective manner. Even though this problem can be overcome by using RDM, the existing FHI would still render quite a poor performance. A possible countermeasure, proposed in [12] is to “invert” (equalize) the filtering operation; this of course requires accurately estimating the filter response which in turn needs either transmitting pilot signals or dealing with a very large number of samples at the decoder. Hence, this approach seems only feasible for relatively simple filters and for a decoder with prior knowledge of the attack subbands, as is the case for the two-band filter addressed in [12]. Another potential solution would be to spectrally shape (predistort) the watermark in such a way that the filtering takes no effect on it, but this clearly presumes too much knowledge about the attack at the embedder, and so it is feasible only for a very limited set of unintentional attacks. In the next section we discuss a third possibility that consists in applying RDM in the Discrete Fourier Transform (DFT) domain.

IV. D ISCRETE F OURIER T RANSFORM RDM The key idea for constructing a quantization-based data-hiding scheme robust to LTI filtering is the gaininvariance property of standard RDM and the convolution theorem [14] that allows to write the output of the filter as a multiplication in the Fourier domain, i.e., Z(ejω ) = Y (ejω )H(ejω ). Hence, in principle, linear filtering can be addressed by applying RDM in the Fourier domain. The problem with this approach is that December 8, 2007

DRAFT

7

Fig. 1.

exp( jg )

g

RDM embed

… …

… …

S/P

…

xl

R

y% m ,k

bm , k

y%mN , /2

Duplicate coeffs + N-point IDFT

…

y% m ,0

x%m,0

N-point DFT x% m ,k + select coeffs x%m, N / 2

P/S

yl

Embedding in DFT-RDM, detailing the operation in the kth channel.

it requires to work with the full sequences, which makes it impractical due to both computational complexity reasons and the fact that RDM requires a vector of past samples to build up the argument of its g function. This suggests operating in a block-by-block basis and approximating the Fourier transform by means of the DFT. Unfortunately, it is not possible to write the effect of the filter on each DFT coefficient as purely multiplicative, as RDM would require. This is a consequence of the fact that exact multiplications in the DFT domain would only be achieved with circular convolutions, while the filter performs ordinary convolutions. We remark that there exist two techniques that can be used to convert ordinary convolutions into circular ones: zero padding and cyclic prefixing [15]. In the first case, blocks are zero-padded so that after filtering block overlaps are avoided; in the second case, blocks are prefixed with a copy of their tail samples so that the result of the ordinary convolution coincides with that of the periodic convolution for the desired samples. However, none of these techniques can be used for the problem at hand because they would entail a significant distortion of the host. Accepting that exact multiplications cannot be achieved, we will proceed by analyzing the errors due to the DFT implementation. As we will see in Section V, by increasing the DFT size (thus approaching the Fourier transform) the errors can be driven to any desired level. This can be further improved by using windows to smooth block boundaries and by spreading, both to be discussed in Section VI. Some additional notation is required to describe the operation in the DFT domain. We will consider a nonoverlapping block-wise operation, with block-size N . Let xm denote the mth block of N samples of the host signal, then x ˜m,k will denote the kth coefficient of the length-N DFT of such block, i.e., x ˜m,k ,

N −1 X l=0

xm,l e−j(2π/N )lk =

mNX +N −1

xl e−j(2π/N )lk

(10)

l=mN

Even though in principle it would be possible to independently watermark the real and the imaginary parts of every DFT coefficient, a problem arises when multiplying such coefficient by a complex factor (as it turns out to be the case for most filters) because both parts mix in a way that depends on the phase of that complex

December 8, 2007

DRAFT

8

Fig. 2.

zl

S/P

g

RDM decod

g

RDM decod

g

RDM decod

bˆm ,0

… …

H (e jw )

… …

yl

…

z%m ,0

N-point DFT z% m ,k + select coeffs z%m, N / 2

bˆm , k

bˆm,N/2

Decoding in DFT-RDM

factor. For this reason, it is preferable to watermark only the magnitude of each DFT coefficient.1 Then, for the DFT-domain implementation of RDM (DFT-RDM in the sequel), the modulus of the kth coefficient of the mth block of the watermarked signal, y˜m,k , is obtained as  |˜ ym,k | = g(˜ ym−1,k )Qbm,k

|˜ xm,k | g(˜ y m−1,k )



,

(11)

where bm,k denotes the mth (m > 0) symbol sent through the kth channel (0 ≤ k ≤ N/2), see Figure 1, and the

˜ m−1,k , (˜ function g(·), defined in (5), operates on vector y ym−1,k , y˜m−2,k , · · · , y˜m−L,k )T . Thus, the proposed

scheme essentially constructs an RDM-like scheme for each DFT channel. The phase of y˜m,k is simply set to that of x ˜m,k to minimize the embedding distortion. This in turn implies that the watermark w ˜m,k is collinear with x ˜m,k . The remaining watermarked samples y˜m,k , N/2 + 1 < k < N − 1, are chosen to preserve the symmetry in such a way that the resulting sequence in the time domain {yk } is real. This is accomplished by ∗ making y˜m,k = y˜m,N −k .

The time-domain watermarked sequence {yl } is obtained as the inverse DFT (IDFT) of the watermarked DFT coefficients in a non-overlapping block-by-block basis: N −1 1 X y˜m,k ej(2π/N )lk , mN ≤ l ≤ mN + N − 1 yl = N

(12)

k=0

On the other hand, given the kth coefficient of the mth block of the observed signal z˜m,k , decoding uses the same rule as RDM in (4) with |˜ zm,k | being the input to the decoder, see Figure 2. The proposed scheme allows to transmit N/2 + 1 bits per DFT-block of size N , so without further channel coding it achieves an approximate rate of 1/2 bits per host sample, that is, half of that attainable with time-domain binary RDM. This is the price to be paid for watermarking only the magnitude in the DFT domain. It is interesting to point out one important implementation issue due to embedding in just the magnitude of the DFT coefficients. For purposes of discussion only, assume that symbol bm,k is embedded in a DFT 1

An extended version of RDM working with complex samples can also be devised, using both magnitude and phase for information

embedding and achieving tolerance to complex channel gains. However, the use of the phase for embedding purposes is not advisable (not even in differential form) because for small magnitude values of the host there is a large concentration of codewords near the origin of the complex plane which would be very sensitive to noise. December 8, 2007

DRAFT

9

coefficient x ˜m,k such that x ˜m,k = ∆/2, and that g(˜ y m−1,k ) = 1. Then, if the centroids in each DFT channel are given by the shifted lattices in (1), it is easy to verify that no matter which value bm,k takes, y˜m,k will always become ∆/2, and, of course, a bit error will occur with probability 1/2. This defect can be attributed to the phase-preserving operation, since it does not allow for sign changes, as standard dither modulation requires. One simple solution is to admit changes of π radians on the phase of x ˜m,k ; however, for certain signals, such as audio, these phase changes may produce annoying clicking artifacts. For this reason, we have chosen a different alternative that for large DWRs yields exactly the same performance and which consists in employing the following centroids Λb , 2∆Z + (1 − b)∆/2,

b = −1, 1, that is, a ∆/2 shift of the sets described by (1).

It is straightforward to check that with this definition, sign changes never occur. To get some intuition on the robustness of DFT-RDM to LTI filtering attacks, given by their frequency response H(ejω ), note that if N is sufficiently large (so that the the filter response can be considered flat where ˜ k y˜m,k , with h ˜ k = H(ej2πk/N ). On the other the DFT window has significant spectral contents), then z˜m,k ≈ h

˜k y ˜ k |g(˜ ˜ m−1,k ) = |h hand, g(˜ zm−1,k ) = g(h y m−1,k ), where the last equality follows from the properties of the

function g. From here, one can easily conclude the robustness of the proposed method against LTI filtering attacks. Note that this approximation is better for larger values of N and for smoother frequency responses of the LTI filter. In [7] it is shown that if a large DWR is assumed, leading to the host having an almost flat pdf within each quantization bin, for p = 2 in the g function, the DWR of time-domain RDM can be well approximated by 3/∆2RDM , with ∆RDM the quantization step. Thanks to the orthogonality of the DFT it is easy to conclude that

an identical relation holds in the DFT-domain. This means that by using the same quantization step in each DFT channel as for the standard time-domain RDM an identical DWR is achieved. Hence, given a desired DWR, we can find the corresponding quantization step ∆DFT-RDM to be used in every DFT channel as r 3 ∆DFT-RDM = DWRDFT-RDM

(13)

Finally, for the performance analysis of DFT-RDM, we will later need to use the following expression for the variance of the host DFT coefficients which easily follows from (10) ˜ m,k ] = N Var[Xm,l ] = N σx2 . Var[X

V. P ERFORMANCE

(14)

ANALYSIS

The performance of DFT-RDM will be measured by the bit error probability. In general, this probability will be different for each DFT channel, being of prime importance to derive analytical approximations as well as manageable bounds to such probabilities. As one would expect, performance will strongly depend on the filter hl ; our next task is to determine in which way.

December 8, 2007

DRAFT

10

A. Problem Formulation Recalling that when the DFT size is large enough we can approximate the kth DFT channel at the output of ˜ k , we will start by writing the random the attack filter as a multiplication by the kth DFT coefficient of the filter, h ˜ k (X ˜ m,k + W ˜ m,k + N ˜m,k ), where variable representing the observed signal in the DFT-domain Z˜m,k as Z˜m,k = h ˜m,k is a zero-mean complex Gaussian random variable independent of both X ˜ m,k and W ˜ m,k . The rationale for N

this model comes from the proposition proven in Appendix I, which states that when H(z) is an FIR filter with no zeros on the unit circle, independence is achieved for all k ∈ {0, 1, · · · , N − 1} asymptotically as N → ∞.

˜m,k , which will be referred to as per-channel multiplication error, is due to the unavoidable use of The term N

ordinary convolutions with the filter impulse response instead of circular convolutions, and thus it is null for unit sample filters, i.e., hl = Kδl , with K ∈ R, and δl the Kronecker delta function. Let φl,k denote the impulse response of the kth DFT basis function, i.e., φl,k , exp(−j(2π/N )lk), for l, k = 0, · · · N − 1, and is zero otherwise. Recalling the filter-bank interpretation of the DFT [14], it is possible

to write Z˜m,k = Zl ∗ φ∗l,k l=mN +N −1 = (Xl + Wl ) ∗ hl ∗ φ∗l,k l=mN +N −1

(15)

˜ k (X ˜ k (Xl + Wl ) ∗ φ∗ ˜ ˜ ˜ m,k + W ˜ m,k + N ˜m,k ) = h Z˜m,k = h l,k l=mN +N −1 + hk Nm,k

(16)

˜m,k = (Xl + Wl ) ∗ fl,k ≈ X ∗ f N l l,k l=mN +N −1 l=mN +N −1

(17)

˜ k − δl ) ∗ φ∗ , k = 0, · · · , N − 1. fl,k , (hl /h l,k

(18)

On the other hand, we have

Combining (15) and (16) we find that the interference term can be expressed as

where fl,k is given by

The approximation on the right hand side of (17) is a direct consequence of our assumption of a large DWR. Now, as the process {Xl ∗ fl,k } is stationary,2 it is immediate to conclude that the per-channel multiplication ˜m,k will approximately have zero-mean and variance σ 2 P |fl,k |2 . Now it becomes necessary to analyze error N x l

˜m,k at each frequency k, keeping in mind that the phase is not used in the embedding scheme the effect of N

(11). This is addressed next.

B. Bit error probability As the decisions at the decoder are based on the magnitude of Z˜m,k , the performance analysis can be based on real quantities. This is formally proved in Appendix II, where the random variables included in the equivalent 2

Note that, in strict sense, the process {(Xl + Wl ) ∗ fl,k } is not stationary but only asymptotically stationary as l → ∞ [7]. However,

the large DWR assumption implies that such process is almost stationary because its statistics are dominated by those of {Xl ∗ fl,k }.

December 8, 2007

DRAFT

11

Fig. 3.

R ˜m,k Equivalent model for communication over the kth channel in DFT-RDM. {N } corresponds to real white noise.

˜ m,k such that it becomes real-valued; ˜ 0 is the rotation of X model depicted in Figure 3 are characterized: X m,k ˜ 0 , and Z˜ 0 is a process producing the same decoder ˜ R } is a white noise real process uncorrelated with X {N m,k m,k m,k

˜ R is shown in Appendix II to be output as |Z˜m,k |. The pdf of N m,k fN˜ R

m,k |Θ

2 (˜ nR ˜ ,k (θ)) m,k |θ) = N (0, σN

(19)

where σN˜ ,k (θ) ,

q

2 cos2 θ + σ 2 sin2 θ σ1,k 2,k

(20)

2 and σ 2 are given in Appendix II as functions of σ 2 with the random variable Θ ∼ U[0, 2π), and where σ1,k x 2,k

nR and fl,k in (18). Unfortunately, the marginal pdf fN˜ R (˜ m,k ) cannot be obtained in closed-form, so one has to m,k

keep the auxiliary random variable Θ. In essence, for each Θ = θ we have a classical RDM problem, in which ˜ k | in Figure 3 is compensated for by RDM operation, and the noise for each Θ = θ is zero-mean the gain |h

Gaussian. Then, for each Θ = θ , we can resort to the results in [7] to predict the bit error probability. Here we focus in the case were the memory L of the g function is large, but the methodology can be extended to any other values of L by rescuing the appropriate results from [7]. Thus, if Pe,k (θ) denotes the bit error probability in the kth channel of DFT-RDM for Θ = θ , we have ! ! √ √ Z ∞ ∆DFT-RDM N σx s ∆DFT-RDM N σx fg(Zk ) (s)Pe,DM Pe,k (θ) = ds , Pe,RDM L, σN˜ ,k (θ) σN˜ ,k (θ) 0

(21)

where [7]   2s (s2 − 1)2 fg(Zk ) (s) ≈ √ , s ≥ 0. exp − 2σr2 2πσr

(22)

p with σr , 2/L. Interestingly, when L → ∞, we have that fg(Zk ) (s) → δ(s − 1), and Pe,k (θ) →   √ Pe,DM ∆DFT-RDM N σx /σN˜ ,k (θ) . In arriving at (21) we have used (14) to substitute the host variance in

the DFT domain.

Finally, the bit error probability in the kth channel of DFT-RDM can be obtained by averaging Pe,k (θ) with respect to θ , i.e., Pe,k

December 8, 2007

1 = 2π

Z

π

Pe,k (θ)dθ.

(23)

−π

DRAFT

12

As we will confirm in the experimental section, the procedure just given predicts very accurately the actual probability of error; however, from the designer point of view it is desirable to have simpler expressions available. We derive next upper and lower bounds on such probability, which being easier to compute, are only 3 dB apart (as measured by the effective signal to noise ratio) when the memory of the g function is such that L → ∞.

C. Upper bound on the bit error probability For obtaining an upper bound, we will make use of the following Theorem. Theorem 1: The function ϕ : R+ 7→ [0, 1/2) defined as    ∞   X 4k + 1 4k + 3 √ √ ϕ(x) = 2 Q −Q 2 x 2 x

(24)

k=0

increases monotonically. Moreover, it is convex for x . 1/12. The proof is given in Appendix III.

Note that Pe,DM (1/x) = ϕ(x2 ), so the above theorem is useful in handling expressions involving Pe,DM for a wide range of noise variances. As a consequence of Theorem 1, the error probability of DM monotonically increases with the noise variance.3 Simple and meaningful bounds result when considering the total variance of the per-channel multiplication error. This variance, which we will denote by σk2 , can be obtained as 2 2 + σ2,k ≈ σx2 σk2 = σ1,k

X l

|fl,k |2

(25)

˜ R conditioned on Θ = θ can be by using the corresponding expressions in Appendix II. The variance of N m,k 2 (θ) ≤ σ 2 for all θ ∈ [0, 2π). Then, from Theorem 1 and Eqs. (21) and (23) straightforwardly bounded as σN ˜ ,k k

we can write the following upper bound ! ! √ √ Z ∞ ∆DFT-RDM N σx s ∆DFT-RDM N σx fg(Zk ) (s)Pe,DM Pe,k ≤ ds = Pe,RDM L, σk σk 0

(26)

D. Lower bound on the bit error probability In order to get a lower bound, first note that Pe,RDM (x) ≥ Pe,DM (x), for all x ∈ R+ [7]. Hence, " " √ !# √ !# ∆DFT-RDM σx N ∆DFT-RDM σx N ≥ EΘ Pe,DM (27) Pe,k = EΘ Pe,RDM L, σN˜ ,k (Θ) σN˜ ,k (Θ) √ √ where expectation is taken with respect to Θ. Now, if ∆DFT-RDM σx N /σN˜ ,k (θ) & 12 for all θ ∈ [0, 2π) we 2 (θ). Recalling that σ can apply Theorem 1 to show that Pe,DM in (27) is a convex function of σN ˜ ,k (θ) < σk ˜ ,k N

3

Note that, even though intuitively plausible, to the best of our knowledge this important result is formally proven here for the first

time.

December 8, 2007

DRAFT

13

√ √ for all θ ∈ [0, 2π), a sufficient condition for convexity is that ∆DFT-RDM σx N /σk & 12. Now, using (20) we

can write 2 EΘ [σN ˜ ,k (Θ)]

1 = 2π

Z

π −π

2 σN ˜ ,k (θ)dθ =

2 + σ2 σ1,k 2,k

2

=

σk2 2

With this result, and the convexity of Pe,DM , we can apply Jensen’s inequality, and arrive at  √ √ ! √  √ √ ∆DFT-RDM σx N N 2∆ σ ∆ σ DFT-RDM x N DFT-RDM x   & 12, , = Pe,DM Pe,k ≥ Pe,DM q 2 (Θ)] σk σk EΘ [σN ˜ ,k

(28)

(29) P , which is always possible since we are considering a finite P . For a white process {Xl } with variance σx2 we have E [Xl Xn∗ ] = σx2 δl−n ,

so Eq. (50) becomes h i ˜k N ˜ ∗ = σx2 (φT H0 φ∗ − φT φ∗ uT h) ˜k · X E h kN N kN kN kN N kN kN

(51)

where H0 is an N × N matrix obtained by deleting the first P columns of H. The latter expression can be simplified by rewriting the product H0 φ∗kN as Φ∗kN h, with Φk the lower triangular Toeplitz matrix whose first column is φk . This allows us to write h i ˜k N ˜ ∗ = σx2 φT Γ∗ Sh ˜k · X E h kN N N kN kN

where h = (h1 , · · · , hP )T , S is the N × P matrix that selects the  2πk 2π2k −v0 ej N −v0 ej N   2πk  (v0 − v1 ) −v1 ej N   2πk  (v1 − v2 )e−j N (v0 − v2 )   . .. ..  .  Γk ,  2π(N −2)k 2π(N −3)k j  (v N (vN −3 − vN −1 )e−j N  N −2 − vN −1 )e  2π(N −2)k 2π(N −1)k  vN −2 e−j N vN −1 e−j N   .. ..  . .   0 0

(52)

first N rows of Γ∗kN , and ···

−v0 e

j 2π(PN−1)k

···

−v1 e

−v2 ej

··· .. . ···



k j 2πP N

2π(P −2)k N

.. .

(vN −1−P − vN −1 )e−j

2π(N −1−P )k N

··· .. .

vN −P e−j .. .

2π(N −P )k N

···

vN −1 e−j

2π(N −1)k N

                   

(53)

The key observation is that due to the continuity of v(t) the differences (vl−p − vl ) for every p ≤ P , go to zero as N → ∞. With this in mind, and after some straightforward algebra, we can write P l−1 P l−1 h X i X X X ˜ ˜ ∗ 2 2 −jω0 l 2 ˜ lim E hkN NkN · XkN = σx vn e hl vn2 < ∞ |hl | ≤ σx N →∞ l=1

n=0

l=1

(54)

n=0

i h 2 ˜ ˜k N ˜k |2 = σ 2 P |h Let us consider now the term E |h x N N l kN fl,kN | . We are then interested in computing

˜ k fl,k when N goes to infinity. This can be written in matrix form as follows: let f 0 , the limit of h N N

˜ k (f0,k , f1,k , · · · fN +P −1,k )T (recall from (18) that fl,k has at most N + P non-zero coefficients), and h N N N N N (e)

let φkN the result of appending P zeros to φkN . Then (e)

f 0 = HφkN − φkN uTkN h = ΓkN h

December 8, 2007

(55)

DRAFT

26

The same arguments as above allow us to establish that  P    −vl Pn=l+1 hn e−j2πω0 (n−l) ,    ˜ k fl,k = 0, lim h N N  N →∞    P  −j2πω0 (n−l) ,  P n=l−N +1 hn vl−n e

0≤l ≤P −1

(56)

P ≤l ≤N −1 N ≤l ≤N +P −1

Then, it is immediate to write the following lower bound lim

N →∞

X l

˜ k fl,k |2 ≥ v 2 |hP |2 + v 2 |hP |2 > 0 |h N P −1 N −1 N

(57)

  ˜ k |2 σ 2 . Combining this with (54) and (57) we achieve Finally, the term E |XkN |2 is easily seen to be N |h x N

the desired proof.

A similar reasoning can be used to show that the complementary correlation coefficient i h ˜k X ˜k E N N N ρckN ,  h i1/2 i h ˜ k |2 ˜k |2 · E |X E |N N N

(58)

˜k in the limit follows from the fact that both ˜ k and N also goes to zero as N → ∞. Independence between X N N

are Gaussian distributed and both the correlation and the complementary correlation coefficients are zero.

A PPENDIX II N OISE

CHARACTERIZATION

˜m,k in (17) will be complex in general, so we will find it useful to The per-channel multiplication error N

consider the joint second-order statistics of its real and imaginary parts: 2 ˜m,k }] ≈ σ 2 σR,k , Var[Re{N x

X l

(Re{fl,k })2

;

˜m,k }Im{N ˜m,k }] ≈ CRI,k , E[Re{N

2 ˜m,k }] ≈ σ 2 σI,k , Var[Im{N x

σx2

X l

X l

(Im{fl,k })2

Re{fl,k }Im{fl,k }

(59)

Note that for the particular cases of k = 0 and k = N/2, the DFT coefficients of the host, the watermark 2 and C and the watermarked signal are real, so both σI,k RI,k become zero.

Let us introduce the following random variable Θm,k , tan−1

˜m,k } Im{X + Dm,k ˜m,k } Re{X

mod 2π

(60)

where the random process Dm,k takes both values 0, π with probability 1/2.5 As the decisions at the decoder are based on the magnitude of Z˜m,k , it is clear that the same results are obtained if the rotated variable Z˜m,k e−jΘm,k 5

The purpose of Dm,k is to extend the domain of Θm,k from [−π/2, π/2) to [−π, π), which leads to a simpler analysis.

December 8, 2007

DRAFT

27

˜0 ,W ˜ m,k e−jΘm,k take real ˜0 , X ˜ m,k e−jΘm,k and W is used instead. Notice that under this rotation, both X m,k m,k

values.6 If we decompose the rotated error into its real and imaginary parts: ˜I ˜ R + jN ˜m,k e−jΘm,k , N N m,k m,k

(61)

then, when |˜ ym,k |  |˜ nm,k |, which corresponds to our case of interest (i.e., a large Document to Interference Ratio), we can approximate the magnitude of Z˜m,k as follows

˜ k | · |X ˜0 + W ˜ 0 +N ˜R | |Z˜m,k | ≈ |h m,k m,k m,k

(62)

Furthermore, due to the symmetry about the origin of the used codebook, and the sign invariance of the g(·) function, it follows that the real random variable 0 ˜ k |(X ˜0 + W ˜ 0 +N ˜R ) Z˜m,k , |h m,k m,k m,k

(63)

produces the same output at the decoder as |Z˜m,k | and, hence, the same error probability. As a consequence of the previous transformations, we propose the equivalent model in Figure 3 to analyze ˜ 0 } is a white real the performance of communication over the kth channel of DFT-RDM. In this model {X m,k 0 } is the associated watermark, and {N ˜ R } is white noise with Gaussian process with variance N σx2 ; {Wm,k m,k

a distribution yet to be derived. Note that this model is identical to that of time-domain RDM in an AWGN channel, save for the fact that noise is not necessarily Gaussian. Taking into account that RDM operation ˜ k | in Figure 3 can be is invariant to scaling, now it is straightforward to see that the multiplication by |h

disregarded. ˜ R . For the special cases k = 0 and k = N/2 This leaves one with the problem of statistically characterizing N m,k ˜R = N ˜m,k its pdf does not represent any difficulty, because Z˜m,k is already real and Θm,k ∈ {0, π}, so either N m,k

˜m,k . This implies that N ˜ R is Gaussian with variance σ 2 given in (59). When k 6= 0, N/2, ˜ R = −N or N R,k m,k m,k

a complication arises because neither Zm,k nor Nm,k are real anymore. However, in this case, the density of ˜ m,k has circular symmetry in the complex plane [18] because the complementary correlation is null: X ˜ 2 ] = σx2 E[X m,k

N −1 X

ej

2π2kl N

= 0,

l=0

for k = 1, · · · , N/2 − 1, and k = N/2 + 1, · · · , N − 1

(64)

so Θm,k will be uniformly distributed in [0, 2π). ˜m,k exp(−jΘm,k ). This corresponds to a rotation with uniform probability of a Consider the random variable N

zero-mean complex Gaussian random variable with second-order statistics given by (59). The key observation to be made is that an identical distribution is obtained if one rotates with uniform probability the Gaussian ˜m,k , and the vectors s1 ˜ (2) are the principal components of N ˜ (1) and N ˜ (2) s2 , where N ˜ (1) s1 + N mixture N m,k m,k m,k m,k

and s2 are the corresponding directions. Due to the uniform rotation, we are not interested in determining the 2 and σ 2 . This is done by directions of those components, but just their variances, that we will denote by σ1,k 2,k

6

˜ m,k and W ˜ m,k are collinear. Recall that X

December 8, 2007

DRAFT

28

˜m,k }, Im{N ˜m,k })T whose entries are given determining the two eigenvalues of the covariance matrix of (Re{N

in (59). Therefore, the variances are q  1 2 2 2 2 2 − σ 2 )2 = σ1,k + (σR,k + 4CRI,k σI,k + σR,k I,k 2 q   1 2 2 2 2 2 − σ 2 )2 , k = 1, · · · , N/2 − 1 (65) + (σR,k − 4CRI,k = σ2,k σI,k + σR,k I,k 2 ˜m,k exp(−jΘm,k ) is identical to that of (N ˜ (1) + j N ˜ (2) ) exp(jΘ), As a consequence, the distribution of N m,k

with

˜ (1) N m,k

2 ), ∼ N (0, σ1,k

˜ (2) N m,k

m,k

2 ), and Θ ∼ U[0, 2π), where all random variables are mutually ∼ N (0, σ2,k

˜ R = Re{N ˜m,k exp(−jΘm,k )}, (cf. Eq. (62)), we can independent. Since we are interested in characterizing N m,k

state that it has the same distribution as the random variable ˜ (2) sin Θ ˜ eq , N ˜ (1) cos Θ − N N m,k m,k m,k

(66)

˜ (1) , N ˜ (2) and Θ as given above. This pdf cannot be obtained in closed-form, except in two special with N m,k m,k 2 , then N 2 ˜ R ∼ N (0, σ 2 ); 2) If σ 2 = 0, then it can be proven that f ˜ R (x) = = σ2,k cases: 1) If σ1,k 2,k 1,k m,k Nm,k √ 2 )), where K (·) is the modified Bessel function of the second kind. (A similar ( 2π/π 2 σ1,k )K 0 (x2 /(2σ1,k 0 2 = 0.) When Θ = θ , (66) becomes the sum of two zero-mean Gaussian random result follows when σ1,k

variables and (19) follows. A PPENDIX III M ONOTONICITY

AND CONVEXITY OF THE ERROR PROBABILITY IN

DM

For the monotonicity part, consider the function ϕ(x2 ), which by the chain rule is monotonic if and only ϕ(x) is so. We have that D1 (x) , =

∞   X (4k+3)2 (4k+1)2 dϕ(x2 ) 1 (4k + 1)e− 8x2 − (4k + 3)e− 8x2 =√ dx 2πx2 k=0   ∞ X 1 (2m + 1)2 m √ (−1) (2m + 1) exp − 8x2 2πx2 m=0

(67)

where we have used Leibniz’s rule for differentiating ϕ(x2 ) under the integral sign implicit in the Q(·) function. Now we want to prove that D1 (x) > 0 for 0 < x < ∞. As explicitly evaluating the sum in (67) is unfeasible, we will rather consider a sufficient condition: since (67) is an alternating sum, whose first term is positive, it is straightforward to see that if the absolute value of the summands decreases with m ≥ 0, then D1 (x) > 0.  Differentiation of (2m + 1) exp −(2m + 1)2 /8σ 2 with respect to m and a little algebra show that this is √ the case for x < 1/ log 3, where log denotes the natural logarithm. Unfortunately, the preceding approach √ does not lead to the desired proof for x ∈ (1/ log 3, ∞), thus making it necessary to work with an alternative expression for dϕ(x2 )/dx. To this end, we resort to the Fourier Transform. We have that ϕ(x2 ) in (24) can be

written as ϕ(x2 ) =

December 8, 2007

Z

∞ −∞

g(τ ) · Π(τ )dτ

(68)

DRAFT

29

where g(τ ) is N (0, x2 ) and Π(τ ) is a train of pulses of unit width: ∞ X

Π(τ ) ,

k=−∞

with

p (τ − (2k + 1))

  1, − 1 < τ < 1 2 2 p(τ ) ,  0, otherwise

By the properties of the Fourier Transform and the symmetry of the functions, we have that Z ∞ Z ∞ G(f ) · q (f )df g(τ ) · Π(τ )dτ =

(69)

−∞

−∞

 with G(f ) = exp −2π 2 σ 2 f 2 and

  ∞ 1 X (−1)m 2m + 1 1 δ f− q(f ) = δ(f ) − 2 π m=−∞ 2m + 1 2

If we replace in (69) the above expressions, we finally get an equivalent expression for ϕ(x2 ):  2 2  ∞ π x 1 2 X (−1)m 2 2 exp − (2m + 1) ϕ(x ) = − 2 π 2m + 1 2

(70)

m=0

which after differentiating with respect to x yields D2 (x) ,

  ∞ X dϕ(x2 ) π 2 x2 (−1)m (2m + 1) exp − = 2πx (2m + 1)2 dx 2

(71)

m=0

Obviously, we have that D1 (σ) = D2 (σ), although, term by term, both series differ. If we replicate the above analysis to investigate those values of x for which it is easy to verify that D2 (x) > 0, we find that now o n 2 2 √ 3 (2m + 1) exp − π 2x (2m + 1)2 decreases with m ≥ 0 whenever x > log 2π . Hence, we have shown that the √ √ derivative of ϕ(x2 ) is positive in the intervals (0, log 3) and ( log 3/(2π), ∞). Moreover, it can be readily

checked from (67) that limx→∞ dϕ(x2 )/dx = 0. Thus, we have established the monotonically increasing character of ϕ(x2 ) for all x ∈ R+ . For the convexity part, define ϕk (x) , Q

Then, noticing that ϕ(x) = 2

P∞

k=0 ϕk (x),



(4k + 1) √ 2 x



−Q



(4k + 3) √ 2 x



a sufficient condition is that ϕk (x) be convex for all k ≥ 0. Then,

ϕ(x) will be convex if d2 ϕk (x)/dx2 > 0 for all k ≥ 0. After taking the second derivative, the previous condition

can be seen to be equivalent to e

2k+1 x

>

(4k + 3)[(4k + 3)2 − 12x] (4k + 1)[(4k + 1)2 − 12x]

(72)

The right hand side of this inequality decreases with k ≥ 0 for all x, while the left hand side increases with k

for all x > 0. Hence, for any x ∈ R+ , the most restrictive condition in (72) is attained when k = 0. Numerically solving the inequality in this case, one finds that it is met whenever x < 0.83328 ≈ 1/12. This completes the proof.

December 8, 2007

DRAFT

30

R EFERENCES [1] J. J. Eggers, R. B¨auml, R. Tzschoppe, and B. Girod, “Scalar Costa scheme for information embedding,” IEEE Trans. on Signal Processing, vol. 4, pp. 1003–1019, April 2003. [2] F. Balado, K. Whelan, G. Silvestre, and N. Hurley, “Joint iterative decoding and estimation for side-informed data hiding,” IEEE Trans. on Signal Processing, vol. 53, pp. 4006–4019, October 2005. [3] I. Shterev and R. Lagendijk, “Amplitude scale estimation for quantization-based watermarking,” IEEE Trans. on Signal Processing, vol. 54, pp. 4146–4155, November 2006. [4] M. L. Miller, G. J. Doerr, and I. J. Cox, “Applying informed coding and embedding to design a robust, high capacity watermark,” IEEE Trans. on Image Processing, vol. 13, pp. 792–807, July 2004. [5] M. Barni and A. Abrardo, “Informed watermarking by means of orthogonal and quasi-orthogonal dirty paper coding,” IEEE Trans. on Signal Processing, vol. 53, pp. 824–833, February 2005. [6] F. P´erez-Gonz´alez, C. Mosquera, M. Barni, and A. Abrardo, “Ensuring gain-invariance in high-rate data-hiding,” in Security, Steganography, and Watermarking of Multimedia Contents VII, Proc. SPIE Vol. 5681 (P. W. Wong and E. J. Delp, eds.), (San Jose, CA, USA), January 2005. [7] F. P´erez-Gonz´alez, C. Mosquera, M. Barni, and A. Abrardo, “Rational dither modulation: a high-rate data-hiding method robust to gain attacks,” IEEE Trans. on Signal Processing, vol. 53, pp. 3960–3975, October 2005. [8] K. Whelan, G. Silvestre, and N. Hurley, “Iterative decoding of scale invariant image data-hiding,” in IEEE International Conference on Image Processing, (Genoa, Italy), September 2005. [9] A. Abrardo, M. Barni, F. P´erez-Gonz´alez, and C. Mosquera, “Improving the performance of RDM watermarking by means of trellis coded quantisation,” IEE Proceedings:Information Security, vol. 153, pp. 107–114, 2006. [10] Q. Li and I. Cox, “Rational dither modulation watermarking using a perceptual model,” in Proc. IEEE Workshop on Multimedia Signal Processing, MMSP’05, (Shanghai, China), October 2005. [11] P. Bas, “A quantization watermarking technique robust to linear and nonlinear valumetric distortions using a fractal set of floating quantizers,” in Proceedings of the 7th International Workshop on Information Hiding, (Barcelona, Spain), pp. 106–117, June 2005. [12] J. Wang, I. Shterev, and R. Lagendijk, “Scale estimation in two-band filter attacks on QIM watermarks,” in Security and Watermarking of Multimedia Contents V, Proceedings of SPIE, pp. 118–127, Jan 2006. [13] F. P´erez-Gonz´alez, F. Balado, and J. Hernandez, “Performance analysis of existing and new methods for data hiding with knownhost information in additive channels,” IEEE Trans. on Signal Processing, vol. 4, pp. 960–980, April 2003. [14] A. Oppenheim and R. Schafer, Discrete-time signal processing. Prentice-Hall, Inc. Upper Saddle River, NJ, USA, 1989. [15] B. Muquet, Z. Wang, G. Giannakis, M. de Courville, and P. Duhamel, “Cyclic prefixing or zero padding for wireless multicarrier transmissions?,” Communications, IEEE Transactions on, vol. 50, no. 12, pp. 2136–2148, 2002. [16] C. Baum and K. Conner, “A multicarrier transmission scheme for wireless local communications,” IEEE Journal on Selected Areas in Communications, vol. 14, no. 3, pp. 521–529, 1996. [17] S. Wilson, J. Bingham, M. Mallory, and J. Cioffi, “Erasure tagging in a slow Rayleigh-fading environment,” Global Telecommunications Conference, 1994. GLOBECOM’94.’Communications: The Global Bridge’., IEEE, vol. 2, 1994. [18] J. Barry, E. Lee, and D. Messerschmitt, Digital Communication. Kluwer Academic Publishers, 2003.

December 8, 2007

DRAFT