4.5.1 FUNDAMENTACIÓN TEÓRICA.

The watermarking methods presented in the two preceding sections use a matched filter technique based on the cross-correlation of the embedded PN sequence. The matched filter detection is optimal in the sense of SNR in the additive white Gaussian channel [2]. However, the host audio signal is generally far from the additive white Gaussian noise, which leads us to the optimal detection problem using a pre-processing of audio by the decorrelation of audio samples before detection. We proposed an audio decorrelation algorithm (Paper III) for a spread-spectrum watermarking that improves the robustness of the watermark detection and demonstrate a high resistance to attacks.

In a correlation detection scheme, used for watermark extraction process in spread- spectrum watermarking algorithms, it is often assumed that the host audio signal is white Gaussian process [134, 135, 136, 137]. However, real audio signals do not have white noise properties as adjacent audio samples are highly correlated. Therefore, the presump- tion for an optimal signal detection in the sense of signal to noise ratio is not satisfied, especially if extraction calculations is performed in short time windows of audio signal. Figure 4.12.a depicts a probability density function (pdf) of 5000 successive samples of a short clip of the watermarked audio signal. It is obvious that the pdf of watermarked audio is not smooth and has a large variance.

In order to decrease correlation between the samples of the audio signal, we use least squares Savitzky-Golay smoothing filters (with different polynomial order and window length), which are typically used to "smooth out" a noise signal whose frequency span is large [138]. Rather than having their properties defined in the Fourier domain, and then translated to the time domain, Savitzky-Golay filters derive directly from a particular for- mulation of the data-smoothing problem in the time domain. The Savitzky-Golay filters

Fig. 4.12. Probability density function of 5000 successive samples of a) watermarked audio signal b) watermarked signal after whitening process.

are optimal in the sense that they minimize the least square errors in fitting a polynomial to frames of noisy data. Equivalently, the idea is to approximate the underlying function within a moving window by a polynomial, typically quadratic. Figure 4.12.b shows the pdf of the 5000 consecutive samples of the residual signal after applying Savitzky-Golay filters, with the fourth order polynomial and 21 samples long time windowing. It can clearly be seen that the pdf of the residual signal has a more Gaussian-like distribution and a significantly smaller variance compared to the case of the pdf of the watermarked audio signal. We verified a Gaussian-like distribution of the residual signal using the Bera-Jarque parametric hypothesis test of composite normality [139] and a single sample Lilliefors hypothesis test [140]. Both tests have rejected hypothesis that watermarked au- dio has Gaussian distribution, with a significance level of 5%. On the other hand, both tests also showed that we cannot reject the hypothesis that the residual signal has a Gaus- sian distribution, using the same significance level.

4.5.1 Optimal watermark detection

Pre-processed audio sequence y may have an embedded watermark

y(i) =s(i) +w(i),0≤i≤N−1 (4.8) on the other hand, it may be an unwatermarked audio sequence

y(i) =s(i),0≤i≤N−1. (4.9)

The detection process verifies two hypotheses on the received content:

H0: watermarked audio content, so it is Gaussian white noise - residual signal of host

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H1: consists of decorrelated host audio and watermark

As decorrelation pre-processing was implemented, we can assume that the output of decorrelation filter y for a given w has the Gaussian distribution and the Likelihood Ratio Test can be performed.

In addition, the watermark part of the residual signal w is a sequence of samples w(i) with two equiprobable values, for examplew(i) ∈ {−²,+²} generated independently with respect to s. Parameter²is set based on temporal analysis within one block of host audio. As the same PN generation and perceptual shaping of the PN sequence can be done on the receiver side, the correlation detector performs the simple correlation calculation between the pre-processed audio and whitened watermark sequence:

C=yw= (s+w)·w=s·w+w·w=s·w+N ²2 (4.10)

whereN is the cardinality of involved vectors, and the correlation between two vectors a and b is defined asa·b=PN_i=0−1a(i)b(i). Since the host audio signal part of the residual audio clip s can be approximated as a Gaussian random vector s∼N(µx, σx), σx À²,

the normalized value of correlation can be written as:

Q= C N ²2 =ρ+ 1 ²N µ 0,√σx N ¶ (4.11) whereρ = 1if watermark is present andρ = 0if there is no watermark. The optimal detection rule is to declare that watermark is embedded in the host audio if the value ofQ exceeds a given threshold valueT. The selection of the thresholdTcontrols the trade-off between a false alarm probability and the probability of detection. Using derivations from the Central Limit Theorem, probability thatQ > T is equal to:

lim N→∞Pr(Q > T) = 1 2erfc à T√N σx √ 2 ! (4.12)

It is clear that the decorrelation of audio sequence leads to a decrease in variance value of signalσx(Figure 4.12), which again, according to the equations given above should lead

towards a better detection performance and smaller false alarm probability [141, 142]. The dominant factor of the detection algorithm is determined by the autocorrelation of the whitened watermark sequences[143, 144], while the "noise" associated with audio covert communications channel is additive white Gaussian [145, 146].

The experimental results (Paper III) showed a significantly improved detection per- formance of the described method, compared to the standard watermark detection, if a watermarked audio sequence is attacked with an mp3 compression and low pass filtering attacks. The reason is that the attacked audio sequences still keep their amplitude-pdf different from Gaussian pdf. Therefore, the correlation detection is not optimal in the sense of Signal to Noise Ratio, because the channel can not be modeled as an additive white Gaussian noise channel. The residual signal has in both cases properties consid- erably more similar to AWGN and detection is accordingly more precise and stable. In the case of the amplitude compression attack, no significant improvement (Paper III) in detection results is achieved using a decorrelation filter, because the attacked audio al- ready has a Gaussian-like pdf of amplitudes after an amplitude compression attack. In

general, the decorrelation algorithm improved the performance and stability of the water- mark detection, because similar test results were obtained in the presence of other standard watermarking attacks, such as resampling, equalization and noise addition.