Techniques for Improved Spread Spectrum Detection

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7.1 Matched Filter Approach

Some watermarking system employs a pseudo random sequence (PN sequence) for synchronization purpose. Matched filter is usually used in such cases to detect the existence of the PN sequence and to precisely locate the starting sample of the PN sequence.

In the watermarking system, the PN sequence is considered as noise added to the host signal. Since the PN sequence only lasts a very short period of time, it could be treated as transient noise pulses and detected by a filter whose impulse response is matched to the PN sequence. Such is the matched filter whose frequency response is defined as (Vaseghi, 2000):

(7.1)

Where K is a scaling factor, is the complex conjugate of the spectrum of PN sequence {u} and PSx(f) is the power spectrum of the host signal x.

In real world applications, the host signal is very close to a zero mean process with variance and is uncorrelated to the PN sequence. Then, Equation (7.1) becomes

(7.2) with impulse response

(7.3)

When the received signal contains the PN sequence, it is defined as

(7.4) where α is a strength control factor. Then the output of the matched filter is

(7.5)

Since the host signal and PN sequence are uncorrelated, d_μx is expected to be zero. The PN sequence itself is orthogonal to the host signal, and so d_μ will reach its maximum if the sequence is perfectly aligned, otherwise, it will be zero.

Therefore, the detected starting location of the PN sequence in a block will be

(7.6)

Figure 1 shows a typical output of such matched filter. Note that the peak location marks the beginning of the embedded PN sequence.

The described matched filter detection is optimal for additive white Gaussian noise (Cvejic, 2004). However, the host audio signal is usually far from being white Gaussian noise. Adjacent audio sample are usually highly correlated with large variance, which increases the detection error for spread spectrum. Several techniques have been developed to decrease such correlation.

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7.2 Savitzky-Golay Smoothing Filters

Cvejic (2004) applied least squares Savitzky-Golay smoothing filters in the time domain to smooth out a wideband noise audio signal. By doing so the variance of the host audio signal is greatly reduced.

Let’s recall the equal error extraction probability p as given by

(7.7)

It is clear that a reduced variance σ_x will result in a decreased detection error rate.