An Analysis/Synthesis Tool for Transient Signals

A flexible analysis/synthesis tool for transient signals, which is the frequency domain dual to sinusoidal modeling, is presented. This parametric model for transients allows the extension of current sinusoidal and sines+noise models for audio to sines+transients+ noise. The model not only permits the realistic analysis/synthesis of signals with sharp attacks, it also permits a wide range of transformations on (he analyzed signal. ~TRODUCTION Sinusoidal modeling has enjoyed a rich history in both speech and audio [1, 2, 3]. Providing a low order parametric model for a signal that allows meaningful transformations is one motivation for sinusoidal models. Because transient signals are not well suited by sinusoidal or sinusoidal+ noise models [2, 3], a flexible, low order model for transient signals was introduced in [4] and refined in [5]. The transient model is the frequency domain dual to sinusoidal modeling. It therefore compliments current models and allows a robust sines+transients+ noise model for audio. Here, the duality between the transient model and sinusoidal modeling is explored and an example demonstrating the complete sines+ transients+ noise model is presented. THE TRANSIENT MODEL The transient model is the frequency domain dual to sinusoidal modeling. Because of this duality, the parameters that characterize the sinusoidal components of a signal also characterize the transient components of a signal, although, as will be shown, in a different domain. There is a well known duality between well developed sinusoids and transients. This duality becomes apparent when observing the nature of these signals in the time and frequency domains. A slowly varying sinusoidal signal is impulsive in the frequency domain. This is why sinusoidal modeling is so effective at modeling slowly varying sinewaves. By performing a Short-Time Fourier Transform (Sm analysis on the time-domain signal and tracking spectral peaks (the tips of the impulsive signals) over time, we can easily model slowly varying sinewaves. Its contrast, transients, which are impulsive in the time-domain, cannot be easily tracked this way because its STFT analysis will not contain meaningful peaks. However due to the duality between time and frequency, if transients are impulsive in the time-domain, they must be osci~latory in the frequency domain. Therefore we can track transients by performing sinusoidal modeIing in a properly chosen frequency domain. The first step in the transient model is to map transient signals in the time domain to sinusoidal signals in some frequency domain. The Discrete Cosine Transform (DC~ provides such a mapping. Roughly speaking, an impulse that occurs toward the beginning of a frame results in a DCT domain signal that is a relatively low frequency cosine. If the impulse occurs toward the end of the frame, then the DCT of the signal is a relatively high frequency cosine. Transients encountered in real audio signals, however, are not generally ideal fionecker Delta functions. Figure 1(a) shows a more realistic transient which is a one sided exponentially decaying sinewave. Performing sinusoidal modeling on this signal would be difficult for many reasons including meaningful parameter estimation and the number of sinusoids required to represent such an impulsive signal. Figure 1(b) shows the DCT of the transient signal. In contrast to the time-domain signal, the DCT domain signal is exactly the type of signal that sinusoidal modeling performs best on; it is a slowly varying sinewave. Therefore by performing sinusoidal modeling in the DCT domain, we are actually modeling time-domain transients. The previous discussion leads to a simple algorithm for an effective analysis/synthesis transient modeling tool. During the analysis, take non-overlapping blocks of the input signal, On each block perform a DCT. Now perform sinusoidal modeling. This will result in model parameters that correspond to time-domain transients. The combination of the DCT then S~ analysis to find meaningful peaks takes the signal from the time-domain into the DCT frequency domain and then back into some type of time-like domain. Although it may seem redundant for the transient model to perform these transformations, theses operations rotate (unitary transforms simply rotate vector spaces) the signal in . —