Once we have obtained the transfer function of an FIR or IIR filter that approximates the desired specifications in the frequency domain or the time domain, our next step is to investigate as many filter structures as possible, before we decide on the optimal or suboptimal algorithm for actual implementation or application. A given transfer function can be realized by several structures or what we will call “circuits,” and they are all equivalent in the sense that they realize the same transfer function under the assumption that the coefficients of the transfer function have infinite precision. But in reality, the algorithms for implementing the transfer function in hardware depend on the filter structure chosen to realize the transfer function. We must also remember that the real hardware has a finite number of bits representing the coefficients of the filter as well as the values of the input signal at the input. The internal signals at the input of multipliers and the signals at the output of the multipliers and adders also are represented by a finite number of bits. The effect of rounding or truncation in the addition and multiplications of signal values depends on, for example, the type of representation of binary numbers, whether they are in fixed form or floating form, or whether they are in sign magnitude or two-complementary form. The effects of all these finite values for the number of bits used in hardware implementation ...

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