LTV system identification using the time-varying autocorrelation function and application to audio signal discrimination

In this paper, three algorithms for identifying and modeling linear, time-varying (LTV) systems are proposed. The first one extends the idea of the ACF of a stationary system to a time-varying autocorrelation function (TVACF) of an LTV system response; however, the second one uses the time-varying pseudo-autocorrelation function (TVPACF). In addition, we propose another algorithm based on the time-varying linear prediction (TVLP), which can be considered as an extended version of the well known linear prediction technique. Some LTV system identification and nonstationary signal modeling examples are given, using the above algorithms. The TVACF has also been applied for music and speech signal discrimination.

I. INTRODUCTION

The spectral representation of a stationary signal may be viewed as a sum of sinusoids with random amplitudes and phases. Thus, a stationary process can be expressed as [4] $$e(n)=\int_{-\pi}^{\pi}e^{jwn}dZ(w) \eqno{\hbox{(1)}}$$ where is a process with orthogonal increments. If the process is non-stationary this choice of family of sinusoids is no longer valid, since the sine and the cosine waves are themselves stationary and thus they form the basic elements in building up models of stationary processes. In this paper we propose new LTV system idintification procedures based on the time-varying autocorrelation function (TVACF). Given that the charecteristics of a non-stationary process change with time and that our model should reflect this change in the process, we need a more general form to represent it. According to the Wold-Cramer decomposition [4], we can represent a discrete-time non-stationary process as the output of a causal, linear, and time-variant (LTV) system with impulse response to a discrete-time stationary zero-mean, unit variance white noise process . Therefore, $$x(n)=\sum_{m=-\infty}^{n}h(n,m)e(n-m) \eqno{\hbox{(2)}}$$ and can be represented as in equation (1) with . From above, we get $$x(n)=\int_{-\pi}^{\pi}H(n, w)e^{jwn}dZ(w) \eqno{\hbox{(3)}}$$ where is the generalized transfer function of the LTV system. Hence a non-stationary process can be expressed as a sum of sinusoids with time-varying amplitudes and phases. In section II we discuss several algorithms for solving for the time-varying coefficients of an LTV system. The first algorithm is based on the TVACF of the LTV response. The second algorithm is based on the time-Varying linear prediction technique. However, the third one is by using the TVPACF. In section III we will show some simulations to show the effectness of our algorithms and will give an example of applying the TVACF algorithm for speech and music signals discrimination, however we conclude our work in section IV.