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طراحی مدارهای منطقی

طراحی مدارهای منطقی. دانشگاه آزاد اسلامی واحد پرند. نیمسال دوم 92-93. طراحی مدارهای منطقی. دانشگاه آزاد اسلامی واحد پرند. سیستم اعداد. Why Binary Numbers?. The switching devices used in digital systems are generally two-state devices:

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طراحی مدارهای منطقی

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  1. طراحی مدارهای منطقی دانشگاه آزاد اسلامی واحد پرند نیمسال دوم 92-93

  2. طراحی مدارهای منطقی دانشگاه آزاد اسلامی واحد پرند سیستم اعداد

  3. Why Binary Numbers? • The switching devices used in digital systems are generally two-state devices: • The output can assume only two different discrete values • Transistors, Diodes … • Because the outputs of most switching devices assume only two different values, it is natural to use binary numbers internally in digital systems • For this reason: • Binary numbers and number systems

  4. Number Systems • Base (مبنا) • مبناي r: ارقام محدود به [0, r-1] • دسيمال:(379)10 • باينري:(01011101)2 • اکتال:(372)8 • هگزادسيمال:(23D9F)16

  5. Number Systems • General N = (an-1…a2a1a0.a-1a-2…a-m)r = an-1 r n-1 + an-2r n-2 +… + a1r +a0 + a-1 r -1 + a-2r -2 +… + a-m r -m

  6. Number Systems • General N = (an-1…a2a1a0.a-1a-2…a-m)r = an-1 r n-1 + an-2r n-2 +… + a1r +a0 + a-1 r -1 + a-2r -2 +… + a-m r -m

  7. Number Systems • Binary Numbers • Computers: Strings of bits 0,1 • (101101.10)2 =125 + 024 + 123 + 122 + 021 + 120 + 12-1 + 02-2 32 16 8 4 2 1 .5 .25 .125 .0625 ( 1 1 0 1 0 1 . 1 0 1 1 ) = ( 53.6785 ) B D

  8. 2’s Powers

  9. Number Systems • Conversions • دسيمال  هر مبناي r • هر مبنای rدسيمال • دسيمال  باينري • اکتال  باينري و برعکس • هگزادسيمال  باينري و برعکس

  10. Number Systems • Conversions  هر مبنا به دسیمال

  11. Number Systems • Conversions  دسیمال به هر مبنا • تقسیمات متوالی  در بخش صحیح • خواندن باقیمانده ها از آخر به اول : معادل عدد از چپ به راست • ضرب متوالی  در بخش اعشاری • خواندن بخش صحیح از اول به آخر : معادل اعداد بعد از ممیز از چپ به راست

  12. Number Systems • Conversions  دسیمال به هر مبنا

  13. Number Systems • Conversions  دسیمال به باینری (روش متفاوت) • ورودی: عدد دسیمال N • بزرگترين توان 2 در N را پیدا کن (حفظ جدول توان 2 اینجا کاربرد دارد!) • عدد توان 2 را از N کم کن • يک عدد 1 در بیت معادل رقم توان قرار بده • مرحلة 1 را با عدد به دست آمده از مرحله 2 تکرار کن • توقف الگوریم: صفر شدن اختلاف

  14. Number Systems • Conversions  دسیمال به باینری (روش متفاوت) N = (717)10 717 – 512 = 205 = N1 512 = 29 205 –128 = 77 = N2 128 = 27 77 – 64 = 13 = N3 64 = 26 13 – 8 = 5 = N4 8 = 23 5 – 4 = 1 = N5 4 = 22 1 – 1 = 0 = N6 1 = 20 (717)10 = 29 + 27 + 26 + 23 + 22 + 20 = ( 1 0 1 1 0 0 1 1 0 1)2

  15. Number Systems • Conversions  اکتال به باینری و برعکس هر سه بیت باینری یک بیت اکتال (11010101000.1111010111)2 (011 010 101 000 . 111 101 011 100)2 ( 3 2 5 0 . 7 5 3 4 )8

  16. Number Systems • Conversions  هگزادسیمال به باینری و برعکس هر 4 بیت باینری یک بیت هگزادسیمال (110 1010 1000 . 1111 0101 11 )2 ( 0110 1010 1000 . 1111 0101 1100 )2 ( 6 A 8 . F 5 C )16

  17. Decimal, Binary, Octal, Hexadecimal

  18. Binary Arithmetic • Arithmetic operations in digital systems are usually done in binary: • Design of logic circuits is simpler in binary • Addition table for binary

  19. Binary Arithmetic • Arithmetic operations in digital systems are usually done in binary: • Design of logic circuits is simpler in binary • Subtraction table for binary

  20. Binary Arithmetic • Arithmetic operations in digital systems are usually done in binary: • Design of logic circuits is simpler in binary • Multiplication table for binary

  21. Representation of Negative Numbers • Sign and magnitude • First bit from left is sign bit • 2n-1 -1 : 2n-1 -1 • One’s Complement of N • 2n-1 -1 : 2n-1 -1 • Two’s Complement of N • 2n-1 : 2n-1 -1 3  0011 -3  1011 3  0011 -3  1100 3  0011 -3  1101

  22. Signed Binary numbers n=4

  23. مفهوم Overflow • When the word length is n bits: • We say that an overflow has occurred if the correct representation of the sum (including sign) requires more than n bits Overflow when  sign(A) = sign(B) ≠ sign (result)

  24. Addition of 2’s complement numbers

  25. Addition of 2’s complement numbers

  26. Binary Codes • Large computers work • Internally: binary numbers • The input output equipment: decimal numbers • Because most logic circuits only accept two-valued signals, the decimal numbers must be coded in terms of binary signals • Simplest Binary Coding  BCD (Binary Coded Decimal)

  27. Possible Binary Codes for Decimal Digits

  28. Another Useful Coding • ASCII  American Standard Code for Information Interchange • 7-bit code • Possible 27 = 128 characters can be coded

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