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UIT2201: Tutorial Set 9 (Fall 2016)
(Solution Sketch to Selected Problems)
** NOT TO BE GIVEN TO FUTURE UIT2201 STUDENTS **

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T9-D1: (Binary Representations) -- DIY (See solution sketch for T9-Q1.)

T9-D2: (ASCII and UNICODE encoding) -- SOLUTION SKETCH
Message: (Does not include the surrounding "" marks.)
    "What are you doing up so early in Cinnamon?" is 43 characters long.
(a) in ASCII code --- 43 bytes = 43x8 bits = 344 bits
(b) in UNICODE --- 86 bytes = 86x8 bits = 688 bits

• Problem 1(a): (133)₄ = 1x4² + 3x4¹ + 3x4⁰ = 16 + 12 + 3 = 31
• Problem 1(c): (1BA)₁₆ = 1x16² + 11x16¹ + 10x16⁰ = 256 + 176 + 10 = 442

• Problem 3(b): (11 0001)_2,unsigned = 1x2⁵ + 1x2⁴ + 1x2⁰ = 32 + 16 + 1 = 49

• Problem 3(d): (10 0000 0000)_2,unsigned = 1x2⁹ = 512

• Problem 4(b): (55)₁₀ = (0011 0111)_2,unsigned
  (Use the method: repeated-divide-by-2, take remainders [in reverse order])

• Problem 5(b): (01 1001 1000)_2,signed = + (2⁸ + 2⁷ + 2⁴ + 2³) = + (256+128+16+8) = + 408

• Problem 5(d): (10 0000 0000)_2,signed = – 0 (also equal to 0)
  (Note: Contrast this with the same bit-string in 3(d).)

• Problem 8(b):
  [Note: The textbook uses sign-magnitude representation for both mantissa and exponent.]
  – (20.25)₁₀ = – (10100.01)₂ = – (0.1010001) x 2⁵ = – (0.1010001) x 2^+(101)₂
      [Note: (0.25)₁₀ = (0.01)₂]

  Internal Representation:

  1 1 0100 0100 0 0 0101

• Problem 9(b),(c):

    Mike  = 0100 1101  0110 1001  0110 1011  0110 0101
                 M          i          k          e 
  
    $25.00 = 0010 0100  0011 0010  0011 0101  0010 1110  0011 0000  0011 0000
                  $          2          5          .          0          0       

(a) Uncompressed:
     25 Mega-Pixel image = (25 x 106) x 3 bytes = 75 x 106 bytes 
     A 32-GBytes = (32 x 109) bytes,
     can store (32 x 109) / (75 x 106) = 426.67,  (namely 426 images)

Alternatively, (using 1G = 2^30)
     a 32-GBytes = (32 x 230) = 34,359,738,368 bytes,
     can store (34,359,738,368) / (7.5 x 107) = 458.13,  (namely 458 images)

(b) Compressed 20:1 ratio: 426.66 x 20 = 8533.3,  (namely 8533 images)
            Alternatively, 458.13 x 20 = 9162.60, (namely, 9162 images)

(a) Uncompressed:   
     44100 samples/sec x 16 bits/sample x (60 x 4) sec = 169,344,000 bits
                                 = 169,344,000/8 bytes =  21,168,000 bytes per song
     A 32-GBytes = (32 x 109) bytes,
     can store (32 x 109) / (21,168,000) = 1511.72,  (namely 1511 songs)

Alternatively, (using 1G = 2^30)
     a 32-GBytes = (32 x 230) = 34,359,738,368 bytes,
     can store (34,359,738,368) / (21,168,000) = 1623.19,  (namely 1623 songs)
 
(b) Compressed 11:1 ratio: 1511.72 x 11 = 16628.87,  (namely 16628 songs)
            Alternatively, 1623.19 x 11 = 17855.12, (namely, 17855 song)

T9-Q4: (5 points) [Creative, Interesting Interpretations] --- DIY

T9-Q5: (10 points) [Boolean Logic Laws] ---- DIY
Using truth tables, prove the following logical laws.
        1. X*(X+Y) = X               [Absorption Law]
        2. ~(X*Y) = (~X) + (~Y)               [de Morgan's Law]
        3. X+(Y*Z) = (X+Y)*(X+Z)         [Distributive Law]