89.1 Quantization Process and error Representation of a general (6+1)-bit fixed-point fraction is shown below l1-2 2-12 s a_lla Smallest positive number that can be represented in this format will have a least significant bit(LSB)of I with remaining bits being all 0s
§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 −2 a b a • • • − • 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
89.1 Quantization Process and error Decimal equivalent of smallest positive number is 8=2-b Numbers represented with(+1) bits are thus quantized in steps of 2-b, called quantization step An original data x represented as a(p+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 δ=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 (β+1)-bit fraction is converted into a (b+1)-bit fraction Q(x) either by truncation or rounding
89.1 Quantization Process and error The quantization process for truncation or rounding can be modeled as shown below Q
§9.1 Quantization Process and Error • The quantization process for truncation or rounding can be modeled as shown below x Q Q(x)
89.1 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 ne effect of quantization on negative fractions is different for the three different 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
°§9.11 Quantization of Fixed-point Numbers Truncation of a(B+l)- bit fixed-point number to(b+1)bits is achieved by simply discarding the least significant bits as shown below 2-12 sala To be discarded a ila
§9.1.1 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 ∆ s a−1 a−2 a−b To be discarded s a−1 a−2 a−b ∆ • • • • • • ↓ −1 2 ↓ −2 2 ↓ −b 2 ↓ −β 2