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The Process of Quantization and Error Characterizations 267
arithmetic including the sign bit. Then the resolution (∆) is given by
∆=2 −B {
absolute in the case of fixed-point arithmetic
relative in the case of floating-point arithmetic
(6.45)
6.7.1 FIXED-POINT ARITHMETIC
The quantizer block diagram in this case is given by
x −→ Quantizer
Infinite−precision
B, ∆
Q[·] −→
Q[x]
Finite−precision
where B, the number of fractional bits, and ∆, the resolution, are the parameters
of the quantizer. We will denote the finite word-length number,
after quantization, by Q[x] for an input number x. Let the quantization
error be given by
e = △ Q[x] − x (6.46)
We will analyze this error for both the truncation and the rounding
operations.
Truncation operation In this operation, the number x is truncated
beyond B significant bits (that is, the rest of the bits are eliminated) to
obtain Q T [x]. In MATLAB, to obtain a B-bit truncation, we have to first
scale the number x upward by 2 B , then use the fix function on the scaled
number, and finally scale the result down by 2 −B .Thus, the MATLAB
statement xhat = fix(x*2^B)/2^B; implements the desired operation.
We will now consider each of the 3 formats.
Sign-magnitude format If the number x is positive, then after truncation
Q T [x] ≤ x since some value in x is lost. Hence quantizer error for
truncation denoted by e T is less than or equal to 0 or e T ≤ 0. However,
since there are B bits in the quantizer, the maximum error in terms of
magnitude is
|e T | =0 00
} {{
···0
}
111 ···=2 −B (decimal) (6.47)
B bits
or
−2 −B ≤ e T ≤ 0, for x ≥ 0 (6.48)
Similarly, if the x