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and the first-order system function
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IMPLEMENTATION O F DISCRETE-TIME SYSTEMS
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Next, the coefficients q ( k ) are found using the recursion given in Eq. (8.9). Beginning with
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we then have
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This leads to the lattice filter implementation illustrated below.
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8.6 FINITE WORD-LENGTH EFFECTS
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In implementing a discrete-time system in hardware or software, it is important to consider the finite word-length o effects. For example, if a tilter is L be implemented on a tixed-point processor, the filter coefficients must be quantized lo a finite number of bits. This will change the frequency response characteristics of the filter. In this section, we look at the finite precision effects in digital tilter implementations.
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8.6.1 Binary Representation of Numbers
There are two basic systems for representing numbers in a digital system: fixed point and floating point. There is a trade-off in which type of representation to use. The dynamic range that is available in a floating-point representation IS much larger than with fixed-point numbers. However, fixed-point processors are typically faster and less expensive. Below, we briefly describe these number representations.
Fixed Point
In the binary representation of a real number, x, using B I bits, there are three commonly used formats: sign magnitude, one's complement. and two's complement, with two's complement being the most common. In these systems, the only difference is in the way that negative numbers are represented.
S g magnitude: With a sign-magni~ude in format, a number x is represented as
where X,, is an arbitrary scale factor and where each of the bits hi are either 0 or 1. Thus, ho is the sign bit, and the remaining bits represent the magnitude of the fractional number. Bit h l is called the most sign$canr bit (MSB). hB is called the leusr significant bit (LSB). For example, with X = 1, and ,
- ,y = -0.8125 = 1.1 1010
CHAP. 81
IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
One's complement: In one's complement form, a negative number is represented by complementing all of the bits in the binary representation of the positive number. For example, with X,, = I and x = 0.8125 = 0.1 1010, -X = -0.8125 = 0.11010 = 1.00101 Two's conzplement: With a two's complement format, a real number x is represented as
Thus, negatlve numbers are formed by complementing the bits of the positive number and adding I to the least significant bit. For example, with X , = I, the two's complement representation of .r = -0.8125 is x = -0.8125 =0.I1010+0.00001 = 1.00110 Note that with B
+ I bits, the smallest difference between two quantized numbers, the resolution, is
5 x < X,,
and all quantized numbers lie on the range -X,,
Floating Point
For a word length of B I bits in a fixed-point number system, the resolution is constant over the entire range of numbers, and the resolution decreases ( A increases) in direct proportion to the dynamic range, 2X,,. A floatingpoint number system covers a larger range of numbers at the expense of an overall decrease in resolution, with the resolution varying over the entire range of numbers. The representation used for floating-point numbers is typically of the form x = M ,2E where M, the mantissa, is a signed BM-bit fractional binary number with 5 I M I < I . and E , the exponent, is a BE-bit signed integer. Because M is a signed fraction, it may be represented using any of the representations described above for fixed-point numbers.
Quantization Errors in Fixed-Point Number Systems
In performing computations within a fixed- or floating-point digital processor, it is necessary to quantize numbers by either truncation or rounding from some level of precision to a lower level. For example, because multiplying two 16-bit fixed-point numbers will produce a product with up to 3 1 bits of precision, the product will generally need to be quantized back to 16 bits. Truncation and rounding introduce a quantization error
where x is the number to be quantized and Q[.Y]is the quantized number. The characteristics of the error depend upon the number representation that is used. Truncating numbers that are represented in sign-magnitude form result in a quantization error that is negative for positive numbers and positive for negative numbers. Thus, the quantization error is symmetric about zero and falls in the range
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