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lab8 - Purdue University ECE438 Digital Signal Processing...

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Purdue University: ECE438 - Digital Signal Processing with Applications 1 ECE438 - Laboratory 8: Number Representation and Waveform Quantization October 6, 2010 1 Introduction This lab presents two important concepts for working with digital signals. The first section discusses how numbers are stored in memory. Numbers may be either in fixed point or floating point format. Integers are often represented with fixed point format. Decimals, and numbers that may take on a very large range of values would use floating point. The second issue of numeric storage is quantization. All analog signals that are processed on the computer must first be quantized. We will examine the errors that arise from this operation, and determine how different levels of quantization affect a signal’s quality. We will also look at two types of quantizers. The uniform quantizer is the simpler of the two. The Max quantizer is optimal, in that it minimizes the mean square error between the original and quantized signals. Questions or comments concerning this laboratory should be directed to Prof. Charles A. Bouman, School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907; (765) 494- 0340; [email protected]
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Purdue University: ECE438 - Digital Signal Processing with Applications 2 2 Review of number representations There are two types of numbers that a computer can represent: integers and decimals. These two numbers are stored quite differently in memory. Integers (e.g. 27, 0, -986) are usually stored in fixed point format, while decimals (e.g. 12.34, -0.98) most often use floating point format. Most integer representations use four bytes of memory; floating point values usually require eight. There are different conventions for encoding fixed point binary numbers because of the different ways of representing negative numbers. Three types of fixed point formats that ac- commodate negative integers are sign-magnitude, one’s-complement , and two’s-complement . In all three of these “signed” formats, the first bit denotes the sign of the number: 0 for positive, and 1 for negative. For positive numbers, the magnitude simply follows the first bit. Negative numbers are handled differently for each format. Of course, there is also an unsigned data type which can be used when the numbers are known to be non-negative. This allows a greater range of possible numbers since a bit isn’t wasted on the negative sign. 2.1 Sign-magnitude representation Sign-magnitude notation is the simplest way to represent negative numbers. The magnitude of the negative number follows the first bit. If an integer was stored as one byte, the range of possible numbers would be -127 to 127. The value +27 would be represented as 0 0 0 1 1 0 1 1 . The number -27 would represented as 1 0 0 1 1 0 1 1 .
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