Cascades of Polynomial-based and Fir Filters for Sampling Rate Conversion

where Fin = 1/Tin (Tin) and Fout = 1/Tout (Tout) are the original input sampling rate (period) and the sampling rate (period) after the conversion, respectively. The sampling rate conversion can be divided into two general cases. For R < 1, the original sampling rate is reduced and this process is known as decimation. For R > 1, the original sampling rate is increased and this process is known as interpolation [1]. When the decimation factor 1/R or the interpolation factor R is an integer or a ratio of two relatively small prime integers, then the sampling rate conversion can be performed conveniently with the aid of fixed digital filters [1]. If these factors are irrational, then fixed digital filters cannot be directly used. Furthermore, if the factor R is ratio of two relatively large prime integers, then, in the case of the conventional polyphase implementation, the required filter orders become very large [1]. In practice this means large number of coefficients that have to be stored in coefficient memory. One way to overcome this problem is to perform calculation of the coefficients while in operation for each output sample. However, sometimes the complexity of calculation of new coefficients may exceed the complexity of filtering operation. The complexity of calculation of new coefficient can be reduced by representing the impulse response of the underlying filter in relatively simple closed mathematical form. Probably the most convenient for implementation is impulse response represented as piecewise polynomial function of low order considered here. Sampling rate conversion by non-integer factor is a typical example where it is required to determine the values between existing samples. In this case it is very convenient to use interpolation filters. Among them, polynomial-based filters