מערכות ספרתיות

✦ בנוי על ידי פהד גאנם ✦

N₁₀ = dₙ·10ⁿ + … + d₁·10¹ + d₀·10⁰   |   N₂ = bₙ·2ⁿ + … + b₁·2¹ + b₀·2⁰
N₁₆ = hₙ·16ⁿ + … + h₀·16⁰   |   C₁ = N̄   |   C₂ = N̄ + 1   |   A − B = A + C₂(B)
N₁₀ → N₂ : N2 = q , r   |   1 byte = 2 hex = 8 bit
45
18
45
2D
0010 1101
0100 0101
1101 0010
1101 0011
45 − 18
27
1

45 = 32 + 8 + 4 + 1 = 00101101 18 = 16 + 2 = 00010010
C₁(18) = 11101101 C₂(18) = 11101101 + 1 = 11101110
00101101 + 11101110 = 100011011
00011011 = 16 + 8 + 2 + 1 = 27

A + 0 = A   |   A · 1 = A   |   A + 1 = 1   |   A · 0 = 0
A + A = A   |   A · A = A   |   A + A' = 1   |   A · A' = 0
(A')' = A   |   A + A·B = A   |   A · (A+B) = A
A + A'·B = A + B   |   A·B + A·B' = A
A·B = A + B   |   A+B = A · B
F(A,B,C) = Σm(1,2,4,7) = A'·B'·C + A'·B·C' + A·B'·C' + A·B·C
1.2 s
0
0
0

1 : m1 m3 m7 m11 m15   |   X : m0 m2 m5   |   0 : m4 m6 m8 m9 m10 m12 m13 m14
m3 + m7 + m11 + m15 = C·D   |   416 = 14
m1 + m3 = A'·B'·D   ⇒   F = C·D + A'·B'·D
m1 + m3 + m5 + m7 = A'·D   ⇒   F = C·D + A'·D
m0 + m1 + m3 + m2 = A'·B'   ⇒   F = C·D + A'·B' C·D + A'·D  :  m0 = 0 , m2 = 0 , m5 = 1 C·D + A'·B'  :  m0 = 1 , m2 = 1 , m5 = 0
m9 = 1001 : D = 1  ,  F = 0   ⇒   F ≠ D

S = A⊕B   |   C = A·B   |   S = A⊕B⊕Cin   |   Cout = A·B + Cin·(A⊕B)   |   A>B = A·B̄   |   A=B = A⊙B   |   A<B = Ā·B   |   t̄bit = ttotaln
1.2 s
0
0
0

A=1 , B=0 , C0=0 → S0 = 1 , C1 = 0
A=1 , B=1 , C1=0 → S1 = 0 , C2 = 1
A=0 , B=1 , C2=1 → S2 = 0 , C3 = 1
A=1 , B=0 , C3=1 → S3 = 0 , C4 = 1
1011 + 0110 = 10001   |   11 + 6 = 17

Y = S1·S0·I0 + S1·S0·I1 + S1·S0·I2 + S1·S0·I3   |   2ⁿ = N   |   Yk = 1 ⟺ A = k   |   A1 = I3 + I2   |   A0 = I3 + I2·I1   |   V = I3 + I2 + I1 + I0
1.0 Hz
—
—
—
—
—

ABC = 000 : m0 = 0  |  001 : m1 = 1  |  010 : m2 = 0  |  011 : m3 = 1
ABC = 100 : m4 = 0  |  101 : m5 = 1  |  110 : m6 = 1  |  111 : m7 = 0
AB = 00 : F(C=0) = 0 , F(C=1) = 1
AB = 01 : F(C=0) = 0 , F(C=1) = 1
AB = 10 : F(C=0) = 0 , F(C=1) = 1
AB = 11 : F(C=0) = 1 , F(C=1) = 0
I0 = C  ,  I1 = C  ,  I2 = C  ,  I3 = C
AB = 11 , C = 0 : Y = I3 = C = 1 = m6 ✓
AB = 11 , C = 1 : Y = I3 = C = 0 = m7 ✓
F = C ⊕ (A·B)

Q⁺ = D   |   Q⁺ = T ⊕ Q   |   Q⁺ = J·Q + K·Q   |   fQ = fCLK2
0.80 Hz
0
1
0
0.80

Q⁺ = J·Q + K·Q = Q
Q(0) = 0   Q(1) = 1   Q(2) = 0   Q(3) = 1   Q(4) = 0   Q(5) = 1
fQ = fCLK2

f(Q0) = fclk2  |  f(Q1) = fclk4  |  f(Q2) = fclk8  |  f(Q3) = fclk16
f(Qn) = fclk2n+1   |   mod = 2n   |   tpd(Q3) = 4·tpd
1.00 Hz
10
1010 · 10
0000
0
0
0
0
0.06

f(Q0) = 160002 = 8000 Hz f(Q1) = 160004 = 4000 Hz f(Q2) = 160008 = 2000 Hz
f(Q3) = fclk24 = 1600016 = 1000 Hz = 1 kHz
mod = 24 = 16
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