 By John W. Leis

ISBN-10: 0470880910

ISBN-13: 9780470880913

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Extra info for Digital Signal Processing Using MATLAB for Students and Researchers

Sample text

Indd 47 2n 27 26 25 24 23 22 21 20 Decimal Example 128 0 64 0 32 1 16 0 8 1 4 0 2 0 1 1 (1 + 8 + 32 = 41) 4/13/2011 5:21:17 PM 48 CHAPTER 3 SAMPLED SIGNALS AND DIGITAL PROCESSING This is an 8-bit quantity, so we have N = 8 bits. The 20 bit position is usually called bit 0 or the least significant bit (LSB), and the 2N–1 bit position is called the most significant bit (MSB). Mathematically, the bits ai sum together to form a positive integer, A, according to A= N −1 ∑2 a i i for A ≥ 0 and ai = 0 or 1.

The key advantage of 2’s complement representation is that arithmetic operations perform as expected. For example, −2 + 5 = 3, −4 + 4 = 0, and so forth. 1, the range of unsigned numbers is 0 → (2N − 1) and that for signed numbers is −2N−1 → +(2N−1 − 1). For positive and negative numbers, the 2’s complement notation may be unified to become A = −2 N −1 aN −1 + N −2 ∑2 a . 1 Signed and Unsigned Interpretation of a 4-Bit Quantity Binary Unsigned Signed Binary Unsigned Signed 0000 0001 0010 0011 0100 0101 0110 0111 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 1000 1001 1010 1011 1100 1101 1110 1111 8 9 10 11 12 13 14 15 −8 −7 −6 −5 −4 −3 −2 −1 This is valid for both positive (sign bit aN−1 = 0) and negative (aN−1 = 1) numbers.

The 64- and 80-bit formats follow a similar pattern, with larger exponent and fraction fields, to give both a larger range of values able to be represented, as well as higher precision. to confusion in the terminology. On general-purpose CPUs, floating-point registers are usually 10 bytes, so there is no speed advantage in using lesser precision (and it may actually be slower due to the conversion required). This is summarized below for convenience. C language data type IEEE754 name Size (bits) Size (bytes) Float Double Long double Single precision Double precision Double extended 32 64 80 4 8 10 We now describe in more detail the 32-bit single-precision format of IEEE754.