Chirp Group Delay based Feature for Speech Applications (2024)

Abstract: Conventional Fast Fourier Transform (FFT),computed on the unit circle, gives an accurate representation of the spectrum if the signal under consideration is because of the sustained oscillations. However, practical signals are not sustained oscillations. For the signals that are either decaying/growing along time, the phase spectrum computed using conventional FFT is not accurate, and in turn, the magnitude spectrum too. Hence a feature, based on a variant of the group delay spectrum, namely the chirp group delay (CGD) spectrum, is proposed. The efficacy of the proposed feature is evaluated in Gaussian Mixture Model (GMM) and Convolutional Neural Network (CNN)-based speaker identification systems. Analysis reveals a significant increase in performance when using the CGD-based feature over the magnitude spectrum.

## introduction the characteristics and behaviour of speech signals are often inferred from their features. various well-known features include mel frequency cepstral coefficients (mfcc), linear predictive cepstral coefficients (lpcc), mel-frequency discrete wavelet coefficient @xcite , and modified group delay @xcite . however, recent neural approaches to speech processing predominantly involve the use of linear and mel spectrograms. these are derived from the fft magnitude spectrum, which has a multiplicative property and so closely spaced poles cannot be easily differentiated/resolved. to address this frequency resolution issue, the group delay spectrum, derived from the negative derivative of the phase spectrum, could be used instead to achieve better resolution. the properties of both the phase and group delay spectra are discussed in @xcite . since the group delay spectrum is additive in nature, poles and zeros are well-resolved as peaks and valleys @xcite . it has also been observed that a perfect minimum phase signal, derived from a one-sided inverse fourier magnitude spectrum, as described in @xcite , is energy bounded, and only when the causal region is considered, this function becomes uneven, resulting in a net phase change of zero (θ[π] -θ[0]). consequently, the group delay spectrum derived from this minimum phase signal behaves similar to the magnitude spectrum @xcite and can be used in spectrum estimation @xcite . in this minimum phase group delay function, poles and zeros can be distinguished easily, where peaks correspond to poles while valleys correspond to zeros @xcite . the phase spectrum, which is naturally wrapped between -π and π, can be used in various speech applications, including speech recognition systems and speech enhancement systems @xcite . however, the location of the poles on the unit circle @xcite leads to discontinuity in phase wrapping, which poses challenges in feature extraction. conventionally, the ft is computed on the unit circle. in such a case, if some singularities, especially zeros due to windowing, lie on the unit circle, the resultant group delay spectrum becomes spiky @xcite . to address these limitations, the chirp group delay spectrum may be used, which is computed at a radius that is not equal to 1, and this case, at a radius greater than 1 as shown in @xcite . this method is employed in @xcite , where the speech signal is assumed to be the ft spectrum of an arbitrary signal, and the group delay spectrum of the arbitrary signal measured at a radius greater than 1 is used to estimate the location of zeros lying outside the unit circle. previous studies by @xcite have also shown that chirp magnitude spectrum-based mfcc outperforms the conventional mfcc. in addition to retaining the error in phase spectrum (and magnitude spectrum) because of using chirp mfcc, if a feature derived from gd is also considered, the frequency resolution issue may also be reolved. therefore, building on this, the current work proposes a chirp group delay-based feature. the paper is organized as follows: section 2 describes the issues with group delay(gd) and the significance of chirp group delay spectrum(cgd). section 3 presents the experimental setup, the corpus used, and the performance analysis and comparison with conventional magnitude and group delay spectra, and section 4 summarizes the conclusions drawn. the group delay is defined as the negative derivative of the phase spectrum. as it is derived from the phase, it is additive in nature. as a result, it exhibits the higher resolution than magnitude spectrum, as discussed in the previous section. however, the group delay spectrum appears spiky when there are zeros on the unit circle. further, since speech signals are not sustained oscillations, the phase spectrum derived from the conventional ft, measured at a radius of 1, that is on the unit circle, may not be accurate, as described in @xcite . therefore, to address these issues, the chirp ft, measured at a radius not equal to 1, and hence the chirp group delay spectrum may be derived. ## performance analysis in order to assess the efficacy of the proposed feature, speaker identification systems are trained with the existing and proposed features and their performances are analyzed. the speech corpus used for training the speaker identification systems, the models and features used, and the results of the experiments are described below. ## conclusion in summary, these experiments confirm that the suggested chirp group delay spectrum is an extremely useful feature for speaker identification. the property of high frequency resolution and the effect of measuring the ft of a decaying signal outside the unit circle, make this feature highly effective compared to conventional features. the variation in chirp radius has a notable influence on the system's accuracy, with radii closer to the unit circle yielding the best results which is marginally better than the group delay spectrum, but significantly better than the conventional magnitude spectrum. thus, the chirp group delay spectrum offers a promising approach for future advancements in speech technology.