s89.1 Quantization Process and Error Representation of a general(b+1)-bit fixed-point fraction is shown below b s a-1a-2 b △ →· Smallest positive number that can be represented in this format will have a least significant bit (LSB)of l with remaining bits being all os
§9.1 Quantization Process and Error • Representation of a general (b+1)-bit fixed-point fraction is shown below −1 2 −2 2 −b 2 s −1 a a−2 • • • a−b • Smallest positive number that can be represented in this format will have a least significant bit (LSB) of 1 with remaining bits being all 0’s
s89.1 Quantization Process and Error Decimal equivalent of smallest positive number isδ=2b Numbers represented with(b+l) bits are thus quantized in steps of 2-b, called quantization step ° An original data x represented as a(β+1)-bit fraction is converted into a(b+1)-bit fraction Q()either by truncation or rounding
§9.1 Quantization Process and Error • Decimal equivalent of smallest positive number is =2-b • Numbers represented with (b+1) bits are thus quantized in steps of 2 -b , called quantization step • An original data x represented as a (b+1)-bit fraction is converted into a (b+1)-bit fraction Q(x) either by truncation or rounding
s89.1 Quantization Process and Error .. The quantization process for truncation or rounding can be modeled as shown below Q Q()
§9.1 Quantization Process and Error • The quantization process for truncation or rounding can be modeled as shown below x Q Q(x)
s89.1 Quantization Process and Error Since representation of a positive bina fraction is the same independent of y format being used to represent the negative binary fraction effect of quantization of a positive fraction remains unchanged The effect of quantization on negative fractions is different for the three dififerent representations
§9.1 Quantization Process and Error • Since representation of a positive binary fraction is the same independent of format being used to represent the negative binary fraction, effect of quantization of a positive fraction remains unchanged • The effect of quantization on negative fractions is different for the three different representations
89.1.1 Quantization of Fixed Point numbers · Truncation of a(β+1)- bit fixed- point number to(b+1) bits is achieved by simply discarding the least significant bits as shown below 2-12 2 2 b B s a 2 a-b To be discarded sala 2 b
§9.1.1 Quantization of FixedPoint Numbers • Truncation of a (b+1)-bit fixed-point number to (b+1) bits is achieved by simply discarding the least significant bits as shown below s a−1 a−2 a−b To be discarded s a−1 a−2 a−b • • • • • • −1 2 −2 2 −b 2 −b 2