2IC30Computer Systems

2IC30 · resource

Formula sheet

M2 Numbers

$Value in base b: ⟨a_{n-1}…a_0⟩_b = Σ a_i·b^i$
$2's-complement value: −a_{n-1}·2^{n-1} + Σ_{i<n-1} a_i·2^i$
$2's-complement range (n bits): −2^{n-1} … 2^{n-1}−1$
$Negate (2's-comp): invert bits, then +1$
$Horner: a_n y^n+…+a_0 = (((a_n)y + a_{n-1})y + …)+a_0 (n mults)$
$Half adder: s=a\oplusb, c=a\landb$
$Full adder: s=a\oplusb\oplusc_in, c_out=(a\landb)\lor(c_in\land(a\oplusb))$
$Overflow: V = carry_{n} ⊕ carry_{n-1} (signed)$

M1 Logic

$De Morgan: \neg(x\lory)=\negx\land\negy ; \neg(x\landy)=\negx\lor\negy$
$Distributivity: x\land(y\lorz)=(x\landy)\lor(x\landz) ; x\lor(y\landz)=(x\lory)\land(x\lorz)$
$Absorption: x\lor(x\landy)=x ; x\land(x\lory)=x$
$Complement-absorption: x\lor(\negx\landy)=x\lory$
$2-input MUX: f = a0\land\negs \lor a1\lands$
$n inputs → 2^n truth-table rows → 2^{(2^n)} possible functions$
$K-map group of 2^k cells removes k variables$

M3 Timing

$Clock period: T \geq t_pd,comb + t_pd,FF + t_su$
$Max frequency: f_max \leq 1/T$
$Always: 0 < t_su + t_dh ; 0 \leq t_oh \leq t_pd$
$Cascade FFs: t_dh \leq t_oh$

M5 Datapath

$Fetch: ΣA,IP \leftarrow IP(Bbus), IP(Abus)+1 (incA) ; ΣR \leftarrow RAM[ΣA] ; IR \leftarrow ΣR(Bbus) (loadB)$
$Cycle count \approx 3 (fetch) + 2\times(indirect levels) + 1 (execute)$
$Prefetch: ADD 4\to1, XOR[addr] 6\to3, STOR 5\to2, BEQ taken 4\to4 / not-taken 3\to1$

M6 Comparisons

$Unsigned: α β \Leftrightarrow C=0\landZ=0$
$Signed: α β \Leftrightarrow N=V\landZ=0$
$ARM shift: LSL #n = \times2^n ; LSR/ASR #n = \div2^n$
$Push = [--SP] ; Pull = [SP++]$