finiteprec - ECE 421 Sum 2010 Notes Set 9 Finite Precision...

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ECE 421 - Sum 2010 Notes Set 9: Finite Precision Effects 1 INTRODUCTION We previously have discussed the effects of using an Analog-to-Digital Converter to quan- tize the input signal to a digital filter. The original infinite precision input samples were therefore quantized to a finite number of bits in the digital representation of the signal. We saw that the result was equivalent to adding noise to the infinite precision input samples, and we called this effect signal quantization noise. Now we study a similar constraint imposed on the numbers within the digital filter structure. All coefficients, as well as the signal data, in digital multipliers, adders, storage locations, etc. have a numerical representation which allows only a finite number of bits. This also introduces additional “noise” in the digital filter, which propagates through to the output of the filter. Therefore, we must determine how much distortion exists between the ideal digital filter designed using infinite precision and the actual digital filter implemented using finite precision arithmetic. We will refer to the noise and distortion introduced by finite precision arithmetic as Finite Precision Effects. The goal of this Notes Set is to quantify the amount of difference or distortion between the output of the infinite precison filter (which we cannot implement) and the output of the actual finite precison filter (which we can implement). To quantify this distortion we will again need to use concepts from probability and statistics like we used in studying signal quantization noise. Additionally, we will limit the current work to a Fixed Point representation of signal, data, and coefficients. For introductory purposes, fixed point representation is somewhat easier to analyze that the alternative representation called Floating Point.
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ECE 421 - Sum 2010 Notes Set 9: Finite Precision Effects 2 MULTIPLIER OPERATION We have seen that analog signal quantization introduces a noise component in digital sig- nal processing. We will now show how using b -bit arithmetic in computations introduces another noise process into the algorithms. We will call this Finite Precision Effects. Once again b refers to the number of bits in the Magnitude of a Sign-Magnitude fixed-point representation. In this set of notes we will focus on multiplication. In general, multiplication of two b -bit fixed-point numbers gives a result which could need 2 b -bits for its errorless representation. Consider the table below for the multiplication of two b = 3-bit sign-magnitude numbers which span a dynamic range of D 2. The two numbers are expressed in binary and decimal representations: Sign-Magnitude Decimal 0 1 1 1 0 875 0 1 0 1 0 625 0 1 0 0 0 1 1 0 546875 The entry in the first column shows we need 6 bits to store the product with no error.
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