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在大多数打算使用沃尔什函数的应用中,二进制波形将代替比较一般的正弦波,因为快速沃尔什变换算法似乎使这些波形在许多种信号处理中具有很大的吸引力。本文首先简要论述沃尔什函数的特性及其应用。同时还介绍正弦波与沃尔什函数之间的一些老的和新的相互关系,但本文的主要目的是研究与傅里叶和沃尔什级数的使用有关的截断误差和舍入误差。 利用简化的近似式可以发现,对于平滑信号的长取样来说,用沃尔什表达式,项数要用得很多,而且对于一个给定的均方根总误差来说,要求这些项的系数有较高的精度。甚至对于不连续信号来说,沃尔什级数也可能要求很多项,这就抵消了快速沃尔什变换在计算上的优点。长波形沃尔什级数表达式的这种相对的不足可能就说明了为什么它在应用中不是特别有效的。
In most applications that intend to use Walsh functions, binary waveforms will replace the more general sine wave, as the fast Walsh transform algorithm seems to make these waveforms attractive for many kinds of signal processing. This article first briefly discusses the characteristics of Walsh functions and their applications. Some old and new relationships between sine waves and Walsh functions are also introduced, but the main purpose of this paper is to study the truncation and rounding errors associated with the use of Fourier and Walsh series. Using a simplified approximation, it can be found that for Walsh expressions, the number of terms is much more useful for long samples of the smoothed signal, and the coefficients of these terms are required for a given total root-mean-square error Have higher precision. Even for discontinuous signals, the Walsh series may require many terms, which negates the computational advantages of fast Walsh transformation. This relative lack of long-wave Walsh series expressions may explain why it is not particularly effective in applications.