Atal B S, Chang J J, Mathews M V, Tukey J W
J Acoust Soc Am. 1978 May;63(5):1535-53. doi: 10.1121/1.381848.
We present numerical methods for studying the relationship between the shape of the vocal tract and its acoustic output. For a stationary vocal tract, the articulatory-acoustic relationship can be represented as a multidimensional function of a multidimensional argument: y=f(x), where x, y are vectors describing the vocal-tract shape and the resulting acoustic output, respectively. Assuming that y may be computed for any x, we develop a procedure for inverting f(x). Inversion by computer sorting consists of computing y for many values of x and sorting the resulting (y,x) pairs into a convenient order according to y; x for a given y is then obtained by looking up y in the sorted data. Application of this method for determining parameters of an articulatory model corresponding to a given set of formant frequencies is presented. A method is also described for finding articulatory regions (fibers) which map into a single point in the acoustic space. The local nature of f(x) is determined by linearization in a small neighborhood. Larger regions are explored by extending the linear neighborhoods in small steps. This method was applied for the study of compensatory articulation. Sounds produced by various articulations along a fiber were synthesized and were compared by informal listening tests. These tests show that, in many cases of interest, a given sound could be produced by many different vocal-tract shapes.
我们提出了用于研究声道形状与其声学输出之间关系的数值方法。对于静止的声道，发音 - 声学关系可以表示为多维自变量的多维函数：y = f(x)，其中x、y分别是描述声道形状和产生的声学输出的向量。假设对于任何x都可以计算出y，我们开发了一种求f(x)反函数的方法。通过计算机排序求反函数包括针对许多x值计算y，并根据y将所得的(y, x)对按方便的顺序排序；然后通过在排序后的数据中查找y来获得给定y对应的x。介绍了将此方法应用于确定与给定一组共振峰频率相对应的发音模型参数的情况。还描述了一种用于找到在声学空间中映射到单个点的发音区域（纤维）的方法。f(x)的局部性质通过在小邻域内进行线性化来确定。通过小步扩展线性邻域来探索更大的区域。该方法被应用于补偿性发音的研究。沿着纤维通过各种发音产生的声音被合成，并通过非正式听力测试进行比较。这些测试表明，在许多感兴趣的情况下，给定的声音可以由许多不同的声道形状产生。
