Lecture 12: Number representation and Quantization effects

1. Lecture 12: Number representation and Quantization effects Instructor: Dr. Gleb V. Tcheslavski Contact:gleb@ee.lamar.edu Office Hours: Room 2030 Class web site:http://ee.lamar.edu/gleb/dsp/index.htm

2. Representation of numbers Up to this point, we were considering implementations of discrete-time systems without any considerations of finite-word-length effects that are inherent in any digital realization, whether in hardware or software. Let us consider first two different representations of numbers. 1. Fixed-point representation. A real number X is represented as: (12.2.1) Where bi represents the digit, r is the radix or base, A is the number of integer digits, and B is the number of fractional digits. For example:

3. Representation of numbers We will focus our attention on the binary representation as most important for DSP. In this case r = 2 and the digits {bi} are called binary digits or bits. They take the values {0, 1}. The binary digit b-A is called the most significant bit (MSB) of the number, and the binary digit bB is called the least significant bit (LSB) of the number. The “binary point” between the digits b0 and b1 does not exist explicitly and the logics assumes location of this point. By using an n-bit integer format (A = n-1, B = 0), we can represent unsigned integer numbers from 0 to 2n-1. More frequently, the fractional format (A = 0, B = n-1) is used with a binary point between b0 and b1 that can represent numbers from 0 to 1-2-n. Any integer or mixed number can be represented in a fraction format by factoring out the term r A. There are three formats to represent negative numbers. The format for the positive numbers is the same: the MSB is set to zero (12.3.1)

4. Representation of numbers • The negative numbers can be represented by: • Sign-Magnitude format: MSB is set to 1 to represent “-” • One’s-Complement Format: • Where is the complement of bi (i.e., we replace ones by zeros and zeros by ones for all bits). • 3) Two’s-Complement Format: (12.4.1) (12.4.2) (12.4.3) (12.4.4) Where is modulo-2 addition. For example, -3/8 is obtained by complementing 0011 (3/8) to obtain 1100 and then adding 0001, which yields 1101 to represent -3/8 in the two’s-complement format.

5. Representation of numbers The basic operations of addition and multiplication depend on the format used. Most fixed-point digital signal processors use two’s-complement arithmetic, therefore, the range for (B + 1) bit number ranges from -1 to 1-2-B. In general, the multiplication of two fixed-point numbers each of b bits in length results in a product of 2b bits of length. The product is either truncated or rounded back to b bits resulting either in truncation or rounding errors. A fixed-point representation allows to cover a range of numbers, say, xmax – xmin with a fixed resolution: (12.5.1) where m = 2b is the number of levels and b is the number of bits.

6. Representation of numbers 2. Floating-point representation. Covers a larger dynamic range by representing the number X as (12.6.1) where M is a mantissa – the fractional part of the number: 0.5  M  1, E (exponent) is either negative or positive number. Both mantissa and exponent require additional sign bits for representing negative numbers. For example: Multiplication of two floating-point numbers is done by multiplying their mantissas and adding their exponents. Addition of two floating-point numbers requires that the exponents must be equal, which can be achieved by shifting the mantissa of the smaller number to the right and compensating by increasing the corresponding exponent. This, in general, may lead to loss of precision.

7. Representation of numbers Overflow occurs in the multiplication of two floating-point numbers when the sum of the exponents exceeds the dynamic range of the fixed-point representation of the exponent. The floating-point representation allows us to cover a larger dynamic range than the fixed-point representation by varying the resolution across the range. The distance between two successive floating-point numbers increases as the numbers increase in size. Also, the floating-point representation provides finer resolution for small numbers but coarser resolution for large numbers.

8. Quantization 1. Fixed-point: truncation To truncate a fixed-point number from (+1) bits to (b+1) bits, we just discard the least significant (-b) bits. The truncation error is denoted by (12.8.1) Here Q(X) is the truncated version of the number X. For a positive X, the error is equal to zero if all bits being discarded are zeros and is largest if all discarded bits are ones. (12.8.2)

9. Quantization • For a negative X, the truncation error will be different for three different formats: • Sign-Magnitude: • One’s-complement: • Two’s-complement: (12.9.1) (12.9.2) (12.9.3)

10. Quantization 2. Fixed-point: rounding In case of rounding, the number is quantized to the nearest quantization level. The rounding error does not depend on the format used to represent negative numbers: (12.10.1) In practice,  >> b, therefore, 2- 0 in all expressions considered.

11. Quantization 3. Floating-point (12.11.1) Considering a floating-point representation of a number (12.11.2) Quantization is carried out on the mantissa only in case of floating-point numbers. Therefore, it is more reasonable to consider the relative error. (12.11.3) In practice, a rounding quantizer can be modeled as follows: (12.11.4)