## 1 Introduction
However, higher frequencies have more periods inside each frame with potentially more temporal variation lost by QSS. Additionally, the parameters of QSS are estimated using the center of the frame as the reference and the values are less accurate towards the edges because the DFT has a stationary basis [25]. This results in the loss of sharpness of attack known as pre-echo. The lack of transients and noise is perceptually noticeable in musical instrument sounds represented with QSS [17; 26]. [PERSON] and [PERSON] [1] proposed to decompose the musical instrument sound into a sinusoidal component represented with QSS and a residual component obtained by subtraction of the sinusoidal component from the original recording. This residual is assumed to be noise not captured by the sinusoids and commonly modeled by filtering white noise with a time-varying filter that emulates the spectral characteristics of the residual component [1; 17]. However, the residual contains both errors in parameter estimation and transients plus noise missed by the QSS [27]. The time-frequency resolution trade-off imposes severe limits on the detection of transients with the DFT. Transients are essentially localized in time and usually require shorter frames which blur the peaks in the spectrum. [PERSON] [28] reviews several techniques to detect and extract transients with sinusoidal models. Multi-resolution techniques [29; 30] use multiple frame sizes to circumvent the time-frequency uncertainty and to detect modulations at different time scales. Transient modeling synthesis (TMS) [26; 27; 31] decomposes sounds into sinusoids plus transients plus noise and models each separately. TMS performs sinusoidal plus residual decomposition with QSS and then extracts the transients from the residual. An alternative to multiresolution techniques is the use of high-resolution techniques based on total least squares [32] such as ESPRIT [33], MUSIC [34], and RELAX [35] to fit exponentially damped sinusoids (EDS). EDS are widely used to represent musical instrument sounds [11; 36; 37]. EDS are sinusoids with stationary (_i.e._, constant) frequencies modulated in amplitude by an exponential function. The exponentially decaying amplitude envelope from EDS is considered suitable to represent percussive sounds when the beginning of the frame is synchronized with the onsets [38]. However, EDS requires additional partials when there is no synchronization, which increases the complexity of the representation. ESPRIT decomposes the signal space into sinusoidal and residual, further ranking the sinusoids by decreasing magnitude of eigenvalue (_i.e._, spectral energy). Therefore, the first \(K\) sinusoids maximize the energy upon resynthesis regardless of their frequencies. Both the SM and EDS rely on sinusoids with stationary frequencies, which are not appropriate to represent nonstationary oscillations [21]. Time-frequency reassignment [39; 40; 41] was developed to estimate nonstationary sinusoids. Polynomial phase signals [20; 25] such as splines [21] are commonly used as an alternative to stationary sinusoids. [PERSON] and [PERSON] [20] were among the first to interpolate the phase values estimated at the center of the analysis window across frames with cubic polynomials to obtain nonstationary sinusoids inside each frame. [PERSON] _et al._[42] investigated the impact of the order of the polynomial used to represent the phase and concluded that order five does not improve the modeling performance sufficiently to justify the increased complexity. However, even nonstationary sinusoids leave a residual with perceptually important information that requires further modeling [25]. Sinusoidal models rely on spectral decomposition assuming that the lower end of the spectrum can be modeled with sinusoids while the higher end essentially contains noise. The estimation of the separation between the sinusoidal and residual components has proved difficult [27].