ARITHMETIC 1
DIGITAL Milo D. Ercegovac and Toms Lang s a Morgan Kaufmann Publishers, an imprint of Elsevier Science, c 2004 COMMENTS AND ERRATA Updated: March 14, 2005
Chapter 1
page 36, line 14 and 16: replace w[j] with w[j + 1] page 39, lines 8, 10, and -4: Replace qn1j with qnj page 39, in Algorithm NRD, move endfor before Step 4. page 39, line 17: replace fractions with integers page 45: Exercise 1.22: replace right with left page 46, line -5: replace Hennessay with Hennessy
Chapter 2
page 64, line -2: replace TSRA with TSCRA page 64, line -1: after ... buers required insert for page 65, line 4: replace (2.21) with (2.22) page 70, line 1 in Section 2.5.2: (2.30) should be (2.31) Comment on delay calculation for Carry-lookahead adder (CLA): Because of the implementation of the CLG module shown in Figure 2.14, in delay expressions 2.43, 2.48, and 2.52 the term tclg corresponds to the delay between module input c0 and module output ci , i = 0. page 82, line -11: Change The number of cells is the same as for the basic scheme. to The number of cells is reduced by two compared to the basic scheme. page 89: include the delay of buers tbuf f in the expression (2.72). page 115: Exercise 2.2: replace Table 2.4 with Table 2.2
Digital Arithmetic - Ercegovac & Lang
Comments and errata (Updated March 14, 2005)
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Chapter 3
page 154, in Figure 3.13, add to the caption: (for p = 3) page 155, in Figure 3.14, line 13: replace d xxxx with d xxx (only 3 xs) page 156, line -11: replace T[4:2] < 2T[3:2] with t[4:2] < 2t[3:2]
Chapter 4
page 183, line 1: insert multiplying between for and magnitudes. page 186, Figure 4.3: eliminate Stage 3 in the top left corner. page 193, line -9: replace page 286 with page 198 page 196, Figure 4.9(b,c), column 4, row 4: replace xl with x1 page 216, expression (4.41): put parentheses around superscripts in the rightmost term. Comment on Figure 4.26 (page 220): The gure is generic: it does not imply that for a 8-bit result k = 6. The references cited on pages 236-237 should be consulted for details about determining truncation precision. page 227, line -7: replace with For the design exercises use the circuit data from Table 2.4 and Figure 5.4. Note also that the delays in Figure 5.4 are given in tN AN D2 units, whereas those in Table 2.4 are in nanoseconds. It might be best to report the results of the exercises in tN AN D2 units. page 229, in Exercise 4.8, replace Exercise 4.7 with Figure 4.7
Chapter 5
page 264, line 2: replace ps + pc > 0 with ps + pc < 0 page 264: in Radix-4 division algorithm with the residual in carry-save form there are some additional dierences with respect to the radix-2 algorithm in Figure 5.6 which should be considered. Namely, a) In the recurrence step, the number of iterations is N = n+2+1 (be2 cause of the initialization step and the guard bit) where n is the number of bits of the operands. b) In the termination step, instead of the radix-2 expression q = 2 (CON V ERT (Q [n + 1] , qn+2 1)) use the corresponding radix-4 expression q = 4 (CON V ERT (Q [N 1] , qN 1))
Digital Arithmetic - Ercegovac & Lang
Comments and errata (Updated March 14, 2005)
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and instead of the expression q = 2 (CON V ERT (Q [n + 1] , qn+2 )) use q = 4 (CON V ERT (Q [N 1] , qN )) page 266, Figure 5.6: The fact that y is represented by three integer bits and one fractional bit, does not imply that its range is [-4, 3.5]. The range of y in the selection function is obtained from expression (5.102) and determined by expression (5.107) for radix-2 division with carry-save adder. page 268, Figure 5.8: the output of the module SZ; Convert should be q. page 270, Figure 5.9, line 14: the least signicant bit of 4W C[2] should be 0, eliminate * page 270, Figure 5.9, line 16: the least signicant bit of w[3] should be 0 page 278: line -4, replace 0 d rn 1 with 0 < d rn 1 page 278: In Table 5.8 footnote replace Correction by Termination step page 278: In Section 5.4 there is ambiguity because of the use of r (radix) for two dierent purposes: On page 278 lines -5 and -4, the r refers to the radix in the representation of the operands. Usually, this radix will be 2. This also corresponds to the r in expression 5.46, 5.47, and 5.49. On page 279, line after expression 5.45, the r = 2k refers to the radix of the quotient-digit, as produced by the division algorithm. That, is for example in a radix-4 division algorithm, this radix would be 4. To avoid this ambiguity, the caption of Figure 5.15 should say n = 8 bits, instead of n = 4 (radix-4 digits), since the operands are in radix 2. page 279: Expression 5.44: replace the term ... = 2m x/d with ... = 2m (x/d ) page 279: replace expression 5.48 with N = (m + v)/k That is, the +1 is incorrect and the k is missing. The argument for eliminating the +1 is based on the fact that the quotient obtained by a fractional division algorithm is 1/2 q < 2, as indicated in item 2 of the same page.
Digital Arithmetic - Ercegovac & Lang
Comments and errata (Updated March 14, 2005)
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page 280: Example 5.1: replace the expression for N with N = (m + v)/2 = 4 page 282: expression 5.53 should be |w[j]| page d. 284: Figure 5.16(b): the label on X-axis should be d. page 288: line 14: replace {r[w]}c with {rw[j]}c page 297: Figure 5.25, number the mk constants beginning from m2 (0) to m2 (7). page 311: Exercise 5.8: line -11: Remove sentence Show all details.. page 312: Exercise 5.15: line -9: Replace sentence Draw the corresponding P-D diagrams (rst quadrant only). with Draw the corresponding P-D diagrams (give the portion for k = 6, rst quadrant only). page 313: Exercise 5.17: line 8: Expression qj+1 = integer (rw [j] + 0.5) is valid if w [j] is expressed in twos complement. When considering a sign and magnitude representation for the residuals, the expression has to be replaced by qj+1 = round (rw [j]). page 313: Exercise 5.17: line 14: Replace a fast radix-2 division algorithm. with other low radix division algorithms..
Chapter 6
page 352: line 15: replace sentence max (Lk (Ii )) j is positive with max (Lk (Ii )) the term depending on j is positive. page 358: Exercise 6.5, line 16: the expression for tcycle is tcycle = tSELSQRT + tbuf f + tmux + tHA + treg = 4 + 1 + 1 + 1 + 2 = 9tg page 359: Exercise 6.8, line 6: replace sentence Perform the integer division algorithm for radix 4 with residual in carry-save representation for x = 53 and d = 9. with Perform the integer square root algorithm for radix 4 with residual in carry-save representation for x = 53.
Chapter 7
page 371, expression (7.11): replace P [j] with R[j] page 382, line 11: replace R[j] with S[j] page 388: Exercise 7.6: a and b are dened in Exercise 7.5.
Digital Arithmetic - Ercegovac & Lang
Comments and errata (Updated March 14, 2005)
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Chapter 8
page 407, line 9: replace are not representable in the oating-point system with do not correspond to real numbers page 420, footnote No. 14: replace bised with biased page 427, line 9: replace put with plus. page 427, line 9, line -2: replace complemented with negated (two instances) page 428: In Figure 8.8 the Exponent Update module should have also an input to update when the signicand is shifted after the adder. page 434, Paragraph 3, line 4: replace signicants with signicands.
Chapter 9
pages 499 and 501, Figure 9.6 and 9.7: Replace Shift-Reg WC with Reg WC; replace Shift-Reg WS with Reg WS; replace 2w[j] with w[j] page 510, line 6: eliminate one the page 516, Table 9.4, row for r = 2: replace 6 with 3. Add to caption: The initial number of bits/operand is log2 r . page 535: In Exercise 9.3 the initial conditions are x[1] = y[1] = w[1] = 0. In the illustration of the input sequence, replace x[j] and y[j] with xj and yj , respectively.
Chapter 10
page 572: in the expression for the SEL digit selection function, replace the left-hand side with dkj+1 , and in the part with -1, replace wk[j] with vk[j]
Chapter 11
page 620: The sequence of scaling iterations is incorrect. A suitable sequence is (-1)(+2)(-5)(+8)(-10)(+15)(-17)(-19). Note that the scaling (-1) corresponds to a multiplication by 21 , so no scaling iteration is required. The sequence of scalings plus repetitions is correct. However, it assumes that the CORDIC iterations beg...