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## PATTERN COMPARISON TECHNIQUES

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**PATTERN COMPARISON TECHNIQUES**Test Pattern: Reference Pattern:**4.3 DISTORTION MEASURES-MATHEMATICAL CONSIDERATIONS**x and y: two feature vectors defined on a vector space X The properties of metric or distance function d: A distance function is called invariant if**PERCEPTUAL CONSIDERATIONS**Spectral changes that do not fundamentally change the perceived sound include:**PERCEPTUAL CONSIDERATIONS**Spectral changes that lead to phonetically different sounds include:**PERCEPTUAL CONSIDERATIONS**Just-discriminable change: known as JND (just-noticeable difference), DL (difference limen), or differential threshold**Spectral Distortion Measures**Spectral Density Fourier Coefficients of Spectral Density Autocorrelation Function**Spectral Distortion Measures**Short-term autocorrelation Then is an energy spectral density**Spectral Distortion Measures**Autocorrelation matrices**Spectral Distortion Measures**If σ/A(z) is the all-pole model for the speech spectrum, The residual energy resulting from “inverse filtering” the input signal with an all-zero filter A(z) is:**Spectral Distortion Measures**Important properties of all-pole modeling: The recursive minimization relationship:**CEPSTRAL DISTANCES**The complex cepstrum of a signal is defined as The Fourier transform of log of the signal spectrum.**CEPSTRAL DISTANCES**Truncated cepstral distance**Weighted Cepstral Distances and Liftering**• It can be shown that under certain regular conditions, the cepstral coefficients, except c0, have: • Zero means • Variances essentially inversed proportional to the square of the coefficient • index: If we normalize the cepstral distance by the variance inverse:**Weighted Cepstral Distances and Liftering**Differentiating both sides of the Fourier series equation of spectrum: This is an L2 distance based upon the differences between the spectral slopes**Cepstral Weighting or Liftering Procedure**h is usually chosen as L/2 and L is typically 10 to 16**A useful form of**weighted cepstral distance:**Likelihood Distortions**Previously defined: Itakura-Saito distortion measure Where and are one-step prediction errors of and as defined:**Likelihood Distortions**The residual energy can be easily evaluated by:**Likelihood Distortions**By replacing by its optimal p-th order LPC model spectrum: If we set σ2 to match the residual energy α : Which is often referred to as Itakura distortion measure**Likelihood Distortions**Another way to write the Itakura distortion measure is: Another gain-independent distortion measure is called the Likelihood Ratio distortion:**4.5.4 Likelihood Distortions**That is, when the distortion is small, the Itakura distortion measure is not very different from the LR distortion measure**4.5.4 Likelihood Distortions**Consider the Itakura-Saito distortion between the input and output of a linear system H(z)**4.5.5 Variations of Likelihood Distortions**Symmetric distortion measures:**4.5.5 Variations of Likelihood Distortions**COSH distortion**4.5.6 Spectral Distortion Using a Warped Frequency Scale**Psychophysical studies have shown that human perception of the frequency Content of sounds does not follow a linear scale. This research has led to the idea of defining subjective pitch of pure tones. For each tone with an actual frequency, f, measured in Hz, a subjective pitch is measured on a scale called the “mel” scale. As a reference point, the pitch of a 1 kHz tone, 40 dB above the perceptual hearing threshold, is defined as 1000 mels.**Examples of**Critical bandwidth**Warped cepstral distance**b is the frequency in Barks, S(θ(b)) is the spectrum on a Bark scale, and B is the Nyquist frequency in Barks.**4.5.6 Spectral Distortion Using a Warped Frequency Scale**Where the warping function is defined by**4.5.6 Spectral Distortion Using a Warped Frequency Scale**Mel-frequency cepstrum: is the output power of the triangular filters Mel-frequency cepstral distance**4.5.7 Alternative Spectral Representations and Distortion**Measures