Lecture 25: Other Number Systems

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Transcription:

Lecture 25: Other Number Systems

Today s Goals Learn about Gray Codes Use scaled binary numbers Learn the IEEE floating point number format

Gray Codes Rotational Position Encoded with Standard Binary Gray Code 6 5 6 5 7 4 7 4 0 3 0 3 1 2 1 2 Gray Code, also known as the reflected binary code, is a binary number system in which two adjacent values are different by only one bit. From Digital System 1, Karnaugh map (K-map) uses Gray Code to simplify Boolean algebra expressions.

Generating a Gray Code Start t with a 1 bit code. 0 0 0 0 Double the number of values by mirroring the current code, appending 0 s to the beginning of the original copy and 1 s to the beginning of the new copy. Repeat this process until enough values are available. Note: 0 0 0 1 0 0 1 1 0 0 1 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 1 1 1 1 1 You can create a slightly smaller Gray Code by deleting the same number of values from the 1 1 1 0 top of the list as you do from the bottom. 1 0 1 0 A Gray Code must have an even number of 1 0 1 1 values. 1 0 0 1 1 0 0 0

Scaled Binary When a program requires very large numbers, very small numbers, or number with fractions, programmers often turn to floating point numbers. USAGDP DC.W 1 ; USA s gross domestic product ; in billions of dollars CANGDP DC.W 1 ; Canada s gross domestic product ; in billions of dollars WAVEON DC.W 1 ; Waveform on time in 0.001 s 001 s of a sec. WAVEONF DC.W 1 ; Waveform off time in 0.001 s of a sec. Note: The resolution is determined by the scaling factor, which is the drawback to this approach.

Using Binary Fractions Floating point arithmetic takes much more time to perform than integer arithmetic, and it should be minimally used. A binary number can contain a fraction component if the radix point is moved to the left. In decimal, a radix point is commonly called the decimal point. Moving the radix point has the same effect in binary that it does in decimal, as illustrated with the examples below. Examples: What decimal value does 000011.11 2 represent? 3.75 10 What decimal value does 11111.1111 2c represent? -0.0625 10 Notes: When performing arithmetic, all numbers must have the radix point in the same location. The radix point is tracked by the programmer, not the processor.

Converting to a Binary Fraction The integer portion and fraction portion are converted separately. The integer portion is converted as before. For the fraction, 1. Multiply the decimal fraction by two. 2. Record the digit to the left of the decimal point in the result as the next least significant digit. Repeat (note: ignore the integer portion) the above two steps until the fraction is zero (terminating fraction), or the fraction is the same as from a previous step (repeating fraction). Examples: Represent 6.375 10 in 8-bit unsigned binary. Place the radix point where desired. 00110.011 2 0110.0110 2 110.01100 2 Represent 0.4 10 in 8-bit unsigned binary. Place the radix point where desired..01100110 2

Floating Point Floating point numbers were designed d to be used when both very large and very small numbers needed to be used simultaneously. To do this, a floating point number includes the scaling factor that was assumed by the programmer when using scaled binary. A floating point number contains three values: Sign bit - most floating point systems used today are sign and magnitude) Exponent scaling factors are limited to powers of 2 Fraction

IEEE Floating Point IEEE established a standard d format, described d briefly below. S Exponent: 8 bits Fraction: 23 bits Except for 0, values are normalized. This means a number is scaled so that 1 fraction < 2. Since the normalized value is always 1.fraction, the leading 1 is assumed and not stored in the fraction. The value represented is (-1) S (2 (E-127) )(1.fraction) An exponent of 255 is reserved for infinity and not-anumber. An exponent of 0 either means 0 (if fraction is 0) or the An exponent of 0 either means 0 (if fraction is 0) or the minimum exponent (if fraction is not 0).

Examples S Exponent: 8 bits Fraction: 23 bits (-1) S (2 (E-127) )(1.fraction) More examples C11900000 16-9.5625 3CDC0000 16 0.02685546875 16 Calcuator: http://babbage.cs.qc.edu/ieee-754/32bit.html

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