#$%&"'() #*+,-./0 10234./2.-0/ 5 #$%&6'() 10234./2.-0/ 7/8 9074:;./-08 5 <5=>$?< $@1<5A1>59A
B0*C8D E*42- FD %0*G; ;:7 A;4+
A*I/+J/0 6K &H""K "H)HH
Question 1 1.1 L"& M "D6 ,*4:.N
O4PP *-. /;23 0*Q *C .3/ C*PP*Q4:R .;JP/ Q4.3 .3/ 2*00/8,*:74:R 0/,0/8/:.;.4*: 4: *.3/0 J;8/8 *C .3/ :-+J/0 R4I/: 4: .3;. 0*QS 83*Q .3/ 7/.;4P8 *C T*-0 2;P2-P;.4*:8.
L;N LJN L2N L7N
7/24+;P
J4:;0T
3/M;7/24+;P
*2.;P
(a)
&6D6
1 1001.1
19.8
31.4
(b)
10.25
"H"HDH"
A.4
12.2
(c)
26
1 1010
"1
32
(d)
26
1 1010
1A
'&
25.510 U "( V W V " V X U 1 1001.12 U " "HH"D"HHHU19.816 U "" HH"D"HHU31.48 1010.012 U WV&V"YZ U 10.2510 U"H"HDH"HH U A.416 U " H"HDH"H U 12.28 1A16 U 1 10102 U "(V"H U 2610 = "" H"H U 32 Since the last 2 numbers in the last column is the same L'&N , , it means that the last 2 rows represent the same value, such that their elements are identical.
1.2 LZ ,*4:.8N
5: .3/ W[J4. 84R:/7 &\8 2*+,P/+/:. 0/,0/8/:.;.4*:K .3/ :-+J/0 *C 748.4:2. :-+J/08 48) L;N &6( C0*+ ]["&WK"&^_ LJN "&W L2N &66 L7N A*:/ *C .3/8/
1.3 L(V(VW U &H ,*4:.8N
^[J4. 0/R48./08 ;0/ -8/7 4: .348 `-/8.4*: .* 8.*0/ :-+J/08 /M,0/88/7 4: 2’s complement representation D L;N #*:I/0. .3/ C*PP*Q4:R .Q* 84R:/7 :-+J/08 .* J4:;0T *J8/0I4:R .3/ ;J*I/ ;88-+,.4*:) 1 U L[ '"N"H 1\8 +;R:4.-7/ 0/,0/8/:./7 *: ^ J4.8) '" U "( V W V Z V & V " U Ua 1 U ['" U &\8 2*+,P/+/:. LHH" """"N U 1"H HHH" b U LV ('N"H U 0"" """"
sgn
&6
&Z
&'
&&
&"
&H
0
H
"
"
"
"
"
LJN O4:7 .3/ &\8 2*+,P/+/:. *C .3/ 84R:/7 J4:;0T :-+J/08 1 ;:7 bK ;:7 R4I/ T*-0 0/8-P.8 4: 7/24+;PK .**) O4:74:R .3/ &\8 2*+,P/+/:. *C ; 84R:/7 J4:;0T :-+J/0 Q4PP :/R;./ .3/ :-+J/0) &\8 2*+,P/+/:. *C 1 U &\8 2*+,P/+/:. L""H HHH"N U 0H" """" &\8 2*+,P/+/:. *C 1 4: 7/24+;P) 7/2 4+;P) 0H" """"& U +'""H x D &\8 2*+,P/+/:. *C b U &\8 2*+,P/+/:. LH"" """"N U 1HH HHH"K Q3423 48 ; :/R;.4I/ :-+J/0K 8;T – x N of this number L-x N Q/ complement To find the magnitude L x Q/ complement it K since x U x U [ L[ xN xN U &\8 2*+,P/+/:. L x N) x N) x U &\8 2*+,P/+/:. L[ xN xN U &\8 2*+,P/+/:. L"HH HHH"N U H"" """" U V(' Ua &\8 2*+,P/+/:. *C b 4: 7/24+;P U - x U [ ('K Q3423 +;c/8 8/:8/K 84:2/ 2*+,P/+/:.4:R b U ('K Q/ :/R;./ 4.8 I;P-/ .* [('D L2N B/0C*0+ .3/ C*PP*Q4:R ;04.3+/.42 *,/0;.4*:8 4: 84R:/7[&\8 2*+,P/+/:. 0/,0/8/:.;.4*:K -84:R ; ^[ J4. 1deS 83*Q *,/0;.4*:8 ;:7 0/8-P.8 L4:2P-74:R 4:./0+/74;0T 8./,8NK J*.3 4: binary ;:7 4: decimal D 10/ .3/0/ ;:T *I/0CP*Q8f g*Q 2;: ; 2*+,-./0 7/./2. *I/0CP*Qf LV ('N"H V L[ '"N"H #;00T)
"
"
V(' [ '" V'&
"
"
"
"
"
0 " " " " " " 1 " H H H H " 0 " H H H H H
#*:I/0 I/0./7 .* 7/24+ 24+;P) ;P) V'&
L[ ('N"H [ L[ '"N"H #;00T)
H
H
"
"
"
"
"
[ (' 1 H H H H H " V'" 0 H " " " " " ['& 1 " H H H H H >3/ 0/8-P. 1"H HHHH 48 ; :/R;.4I/ :-+J/0K 8;T h x. h x. >* 2*:I/0. ""H HHHH L4D/DK [ xN xN .* 7/24+;PK C408. Q/ C4:7 x, C4:7 x, JT JT 2*+,P/+/:.4:R ""H HHHHK 4D/DK x U x U [ L[ x L[ xNN U &\8 2*+,P/+/:. L""H HHHHN U HH" """" V" U H"H HHHH U V'&K Ua .3/ 0/8-P. 4: 7/24+;P [ x [ x U ['&i Q3423 Q;8 /M,/2./7D 10/ .3/0/ ;:T *I/0CP*Q8f >3/0/ ;0/ :* *I/0CP*Q8 84:2/ 74CC/0/:.[84R: 7 4CC/0/:.[84R: :-+J/08 ;0/ ;77/7D g*Q 2;: ; 2*+,-./0 7/./2. *I/0CP*Qf "D 9I/0CP*Q U #;00T^ M*0 #;00T( *0 &D 9I/0CP*Q) j4R: 1 k j4R: b k Yj4R: 0/8-P. V Yj4R: 1 k Yj4R: b k j4R: 0/8-P.
Question 2 2.1 L'M& U ( ,*4:.8N l*- 3;I/ ; j?1< +/+*0T 234, Q4.3 ; 2;,;24.T *C Wc M Z
;N g*Q +;:T 4:,-.[*-.,-. 7;.; P4:/8 7*/8 4. 3;I/f Answer a) DD a) DDZDDDDDDDDDDDDDDDDDDDDDDDDDDD ZDDDDDDDDDDDDDDDDDDDDDDDDDDD JN g*Q +;:T ;770/88 P4:/8 7*/8 4. 3;I/f Answer b) DDDD"'DDDDDDDDDDDDDDDDDDDDDDDDD b) DDDD"'DDDDDDDDDDDDDDDDDDDDDDDDD 2N m3;. 48 4.8 2;,;24.T /M,0/88/7 4: nbit nbit 8nf 8nf
'
"H
&
"6
6
"H
Answer c) DDD'& c) DDD'& cJ4.8 U & & & U & U & & 2.2 L&M( U "& ,*4:.8N m3;. 48 Q0*:R Q4.3 .3/ C*PP*Q4:R .0;:8C/0 8.;./+/:.8 L?>dNf ;N M>) 1? 1?\K 1? H JN T>) B# 1?K B# B# V " 9:/ 2;::*. P*;7 .Q* 74CC/0/:. 7;.; 4: .3/ 8;+/ 0/R48./0o Question 3 (20 points)
=/84R: ; '[J4. +-P.4[C-:2.4*: 0/R48./0 A 0/R48./0 A L L A A& A" AHN Q3*8/ *,/0;.4*: 48 7/8204J/7 4: .3/ C*PP*Q4:R .;JP/K Q3/0/ #" ;:7 #H ;0/ .Q* 2*:.0*P J4.8D e8/ 4: T*-0 7/84R: pq CP4, CP*,8K P*R42 R;./8 ;:7 ;:T 74R4.;P 2*+,*:/:.8 L/:2*7/08K 7/2*7/08K +-P.4,P/M/08K /.2DNS 70;Q ; 7/.;4P/7 74;R0;+ *C .3/ P*R42 2402-4. *C .3/ +-P.4[C-:2.4*: 0/R48./0D O-:2.4*: .;JP/ *C '[J4. 0/R48./0 1D
#P*2c#" #H ↑ H H ↑ H " ↑ " H ↑ " "
I2
9,/0;.4*: A* 2 ;:R/ 5:20/+/:. JT ' =/20/+/:. JT ' d*;74:R /M./0:;P 4:,-.8K 8;T 5& 5" 5H
#3;0;2./048.42 >;JP/ J K QLnN QLnV"N H H H H H H " " H " H H H " " H " H H " " H " " " " H " " " " H
$M24.;.4*: >;JP/ QLnN QLnV"N J H H H H " " " H M " " M
A&V A"V AHV &
"
H
A& A" AH V ' A& A" AH V ' I & I " I H
#" #H
I1
Register
A2
A1
K H " H "
QLnV"N QLnN H " Q’ LnN
A0
?/+/+J/0 .3/ R/:/0;.4*: *C .3/ OO /M24.;.4*: /`-;.4*:8) K M M " H
#*,T .* J .* J .3/ .3/ :/M. 8.;./ QLnV"N 4C QLnNUH #*,T .* K .* K .3/ .3/ 2*+,P/+/:. *C .3/ :/M. 8.;./ QLnV"N 4C QLnNU" J ;:7 K ;:7 K 83*-P7 83*-P7 J/ rMs *.3/0Q48/
Answer: C1 C0 = 00 => no change
J H H " "
I0
tUa Ai Ln tUa A LnV"N U A U Ai Ln LnN S iU uHK"K&v Ua p& U HS q & U HS p" U HS q " U HS pH U HS q H U HD
C1 C0 = 01 =>Increment =>Increment by 3
B0/8/:. j.;./ A/M. j.;./ A& Ln LnN A" Ln LnN AH Ln LnN A&LnV"N A"LnV"N AHLnV"N H H H H " " " "
H H " " H H " "
H " H " H " H "
H " " " " H H H
J & A" AH HH H" "" "H A&
" H H " " H H "
J &
" H " H " H " H
H " " " M M M M
OO 5:,-.8 K & J " K " M M M M H " " "
" H M M " H M M
M M " H M M " H
J H " M " M " M " M
K & A" AH HH H" "" "H
J & U K U K & U A U AH V A V A"
A&
H
H
"
"
"
H
M
M
M
M
J " U K U K " U A U AH\
"
M
M
M
M
"
H
"
"
"
J H U K U K H U"
C1 C0 = 10 =>Decrement by 3 B0/8/:. j.;./
A& Ln LnN
A" Ln LnN
H H H H " " " "
H H " " H H " "
A/M. j.;./
AH Ln LnN A&LnV"N A"LnV"N AHLnV"N H " H " H " H "
" " " H H H H "
J & A" AH HH H" "" "H A&
K H M " M " M " M "
H " " H H " " H
J &
" H " H " H " H
" " " H M M M M
OO 5:,-.8 K & J " K " M M M M " " " H
H " M M H " M M
M M H " M M H "
J H " M " M " M " M
K H M " M " M " M "
K & A" AH HH H" "" "H
J & U K U K & U A U AH\ V A V A"\ U L A AH A"N\
A&
H
"
"
H
"
H
M
M
M
M
J " U K U K " U A U AH
"
M
M
M
M
"
"
"
H
"
J H U K U K H U"
C1 C0 = 11 =>Load =>Load I 2 I 1 I 0 to A2 A1 A0: tUa A tUa Ai Ln LnV"N U I U I i Ln LnN S iU uHK"K&v 0
AiLnN H H " "
I i AiLnV"N H H " " H H " "
J i K i H x " x x " x H
0 1 2
I2
J i U K U K i U I U I i M*0 A M*0 Ai *0 J i U I U I i S K i U I U I "\ *0 i Conclusion L> OO ;,,0*;23K 4D/DK p U q U >N)
0 1 2
I1
A2
U X
K Q R
M
J
0 1
1
I0
2
Q
A1
U X
K Q R
M
J
3
0
#H p& U q & p" U q " pH U q H H H H H " AH V A V A" AH\ " H L A AH A"N\ AH " " I & M*0 A M*0 A& I " M*0 A M*0 A" I H M*0 A M*0 AH
Q
J
M
3
0
#" H H " "
CLK
C1C0
Q
U X
A0
K Q R
3
R
D FF + adders approach:
0 1
C1C0
C1 0 1
0
!"#
%$Σ2
1
I2
0
3
0
Q
A2
X
K Q
1
M
J
0
!"#$(
Σ1
1
Q
A1
U 2
I1
3
X
K Q
M
J
0
!"#$)
Σ0
1
Q
A0
U 2
!&')
I0
J
C1
!&'(
1
M
U 2
!&'%
1
CLK
0 3
X
K Q
#*+,0/3/:84I/ 7/8204,.4*:K J-. P*:R/0 ;,,0*;23) #*:.0*P
OO 5:,-.8 #"#H A&LnN A"LnN AHLnN A&LnV"N A"LnV"N AHLnV"N J & K & J " K " J H K H HH H H H H H H H M H M H M H H " H H " H M H M M H H " H H " H H M M H H M H " " H " " H M M H M H M H H M H M " H H " H H M H H M M H " H " " H " M H M H H M " " H " " H M H M H M H " " " " " " H" H H H H " " H M " M " M H H " " H H " M H M M " H " H " H " " M M " " M H " " " " H " M M H M " " H H " " " M H " M " M " H " H H H M " H M M " " " H H H " M " M " " M " " " H " H M " M H M " "H H H H " H " " M H M " M
""
B0/8/:. j.;./
A/M. j.;./
H
H
"
"
"
H
" M " M M "
H H " " " "
" " H H " "
H " H " H "
" H H H H "
" H H " " H
" H " H " H
" H M M M M
H H H H " " " "
H H " " H H " "
H " H " H " H "
I & I & I & I & I & I & I & I &
I " I " I " I " I " I " I " I "
I H I H I H I H I H I H I H I H
M M " " " H
M M H " M M
H " M M H "
" M " M " M
M " M " M "
I & M I & M I & M I & M M I &\ M I &\ M I &\ M I &\
I " I " M M I " I " M M
M M I "\ I "\ M M I "\ I "\
I H x I H x I H x I H x
x I H\ x I H\ x I H\ x I H\
FF input equations
J & U K U K & U H J " U K " U H J H U K U K H U H
J & U K U K & U A U AH V A V A" J " U K U K " U A U AH\ J H U K U K H U"
J & U K U K & U A U AH\ V A V A"\ U L A AH A"N\ J " U K U K " U A U AH J H U K U K H U"
J & U I U I & K & U I U I &\ J " U I U I " K " U I U I "\ J H U I U I H K H U I U I H\
Question 4 (20 points)
=/84R: ; Z[J4. ;04.3+/.42 2402-4.K Q4.3 .Q* 8/P/2.4*: I;04;JP/8 x I;04;JP/8 x ;:7 ;:7 y ySS .3/ "[J4. I;04;JP/ z I;04;JP/ z 48 ;: 4:,-. *C C-PP ;77/0D >3/ 2402-4. R/:/0;./8 .3/ C*PP*Q4:R /4R3. ;04.3+/.42 *,/0;.4*:8 -:7/0 2*:.0*P *C x *C xKK y ;:7 y ;:7 z z ) LA*./) X LA*./) X \ 48 .3/ "\8 2*+,P/+/:. *C X *C X N x y y HH H" "H ""
z = 0 O U 1Vb L;77N O U 1 L.0;:8C/0N O U b\ L2*+,P/+/:.N O U 1Vb\
z = 1 O U 1VbV" O U 1V" L4:20/+/:.N O U b\V" L:/R;./N O U 1[b L8-J.0;2.N
=0;Q .3/ P*R42 74;R0;+ *C .3/ .Q* P/;8. 84R:4C42;:. J4.8 *C T*-0 ;04.3+/.42 2402-4.K onlyD onlyD e8/ C-PP ;77/08 ;:7 *.3/0 P*R42 2402-4.8 L+-P.4,P/M/0K R;./8N ;8 0/`-40/7D Answer: x y y HH H" "H ""
O' O& O" OH U O' O& O" OH U O' O& O" OH U O' O& O" OH U
Cy0 = z = 0 Cy0 = z = 1 1' 1& 1" 1H V b' b& b" bH V H 1' 1& 1" 1H V b' b& b" bH V " 1' 1& 1" 1H V H H H H V H 1' 1& 1" 1H V H H H H V " H H H H V b'\b&\b"\bH\V H H H H H V b'\b&\b"\bH\V " 1' 1& 1" 1H V b'\b&\b"\bH\V H 1' 1& 1" 1H V b'\b&\b"\bH\V "
O" U M\T\L1 M\T\L1" V b" V #T"N V M\T L1 M\T L1" V H V #T "N V MT\LH MT\LH V b"\ V #T"N V MT L1 MT L1" V b"\ V #T"N *0 O" U LM\T\1 LM\T\1" V M\T 1 M\T 1" V MT 1 MT 1"N V LM\T\ LM\T\b b" V M\T H M\T H V MT\b MT\b"\ V MT b MT b"\N V #T"U LMT\N\1 LMT\N\1" V LM\T\ LM\T\b b" V M\T H M\T H V MT\b MT\b"\ V MT b MT b"\N V #T" OH U M\T\L1 M\T\L1H V bH V #THN V M\T L1 M\T L1H V H V #T HN V MT\LH MT\LH V bH\ V #THN V MT L1 MT L1H V bH\ V #THNS Cy0 = z or: OH U LM\T\1 LM\T\1H V M\T 1H V MT 1HN V LM\T\b LM\T\bH V M\T H V MT\b MT\bH\ V MT bH\N V #THU LMT\N\1 LMT\N\1H V LM\T\b LM\T\bH V M\T H V MT\b MT\bH\ V MT bH\N V #TH >3/ .Q* ,*884JP/ 4+,P/+/:.;.4*:8 4+,P/+/:.;.4*:8 ;0/ ,0/8/:./7 J/P*Q)
x, y
A1
0 1
0
2 3
B1
0
0
!"#
%$!"
Σ1 !&'%
1 2 3
A0
0 1
0
2 3
B0
0
0
!"#$(
!#
Σ0 !&'(
1 2 3
z
