[CS61C FA20] Lecture 17.4 - Combinational Logic Blocks: Subtractor Design — Transcript

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1. 0:00this is it welcome back
2. 0:03we're here this is the last lecture of
3. 0:05this four lecture series
4. 0:08module of synchronous digital systems
5. 0:10this is it we're building
6. 0:12the climax come on band climax crescendo
7. 0:16to the final piece
8. 0:20where we're looking and we're trying to
9. 0:21build a subtractor the final design of
10. 0:24subtractor
11. 0:25and it is extremely clever and you'll
12. 0:28see why this is clever
13. 0:30i don't want to have to build just to
14. 0:31step back don't look at the screen don't
15. 0:32look over here look at me
16. 0:34i don't want to have to build 32 bits of
17. 0:37an adder which does its thing
18. 0:39and 32 bits of a subtractor and then
19. 0:41shoes in a mux between them i'd like to
20. 0:43be able to reuse that machinery
21. 0:45so can i possibly reuse the machinery
22. 0:47for my cascaded adder
23. 0:49for my subtractor here's the big idea i
24. 0:52love this idea
25. 0:54a minus b is the same as a plus
26. 0:58negative b like duh yeah i get that
27. 1:01but how do you how does that embody
28. 1:03itself in the circuit
29. 1:07i love this part see this little graph
30. 1:10over here this little truth table
31. 1:13is an xor do you remember that slide i
32. 1:16showed you before
33. 1:17i actually probably showed you this
34. 1:18truth table when x is zero
35. 1:22y the output is just y it's just y
36. 1:28when x is one it switches
37. 1:31it's not y we call this
38. 1:35a conditional inverter
39. 1:39so a plus negative b
40. 1:43sometimes i want to add a plus b
41. 1:46sometimes i want to add a plus
42. 1:47negative b how do you negate this is so
43. 1:50fun
44. 1:51how do you negate a two's complement
45. 1:53number
46. 1:56you invert the bits and add one
47. 2:01[Music]
48. 2:03subtract goes high when this line goes
49. 2:06hot i'm so pleased because i just love
50. 2:07this slide this is the crescendo of all
51. 2:09the fun
52. 2:10and it's really is very clever when this
53. 2:12subtract lone guy
54. 2:13line goes high i feed that into here
55. 2:17that is shared from each line of b i
56. 2:19need 32 of these guys
57. 2:21and end of these guys each of those bits
58. 2:25let's watch this each of those bits
59. 2:27independently is going to be flipped
60. 2:28remember
61. 2:29a two's complement number you flip the
62. 2:31bits this
63. 2:32does that each one of these gets their
64. 2:35bits
65. 2:36flipped when this goes high each of them
66. 2:38if it's a one becomes a zero
67. 2:39the output is zero if that if the input
68. 2:41for b of n minus one is a zero
69. 2:43the output is a one it flips the bits
70. 2:46and add one i had this port sitting over
71. 2:50here
72. 2:52i'm just too pleased with myself that
73. 2:54was like does it wiggling do i let it do
74. 2:56i let it go do i not plug anything into
75. 2:58it i said well let's put a zero there
76. 2:59and it's true two unsigned numbers i
77. 3:00better put a zero there but i want to be
78. 3:02able to flip the bits
79. 3:04and add one this
80. 3:07is your add one flip the bits
81. 3:11and add one and you are now calculating
82. 3:14a
83. 3:15minus b a plus negative b which is the
84. 3:18same as a minus b
85. 3:19so i can reuse my entire subtractor
86. 3:23my retire all the adders the one bit
87. 3:25adders from my adder
88. 3:27to be able to do a subtractor by just
89. 3:30flipping b's bits
90. 3:31and adding one and now i've flipped b
91. 3:33now it's negative b
92. 3:35now that representation is negative b
93. 3:40a plus negative b a minus b
94. 3:44this is my what this is my what this is
95. 3:46my signed overflow
96. 3:48okay my signed overflow by itself
97. 3:53is my unsigned overflow okay
98. 3:57so remember xor of both of these
99. 3:59channels and
100. 4:00c of n minus 1 and n we just talked
101. 4:02about that is my signed overflow
102. 4:04just take this line straight that's my
103. 4:06unsigned overflow unsigned i should
104. 4:08write this right
105. 4:08unsigned over
106. 4:12flow okay can't read my handwriting but
107. 4:13i can't either
108. 4:15when you have to be a phd you can write
109. 4:16scribble and people can't tell what's
110. 4:18happening
111. 4:19this is it i'm so excited so
112. 4:23in summary use muxes to be able to do a
113. 4:27flag
114. 4:28person on the line i got two signals to
115. 4:30come in but i only have one output
116. 4:31i use a mux to be able to drive who who
117. 4:33gets to drive the output who gets to
118. 4:34drive across the bridge okay
119. 4:36if i have s input bits i select from
120. 4:39among two to the s inputs we saw
121. 4:41one bit chooses among two two bits
122. 4:43chooses among four so
123. 4:45s bits chooses among s two to the s
124. 4:47inputs each input themselves
125. 4:49independently can be n bits wired that
126. 4:50could be
127. 4:50two that could be four one bit lanes
128. 4:53driving a one bit output
129. 4:54or or four n-bit lanes n-bit input lanes
130. 4:58all driving one n-bit
131. 5:00bridge okay that's pretty cool i can
132. 5:02implement those muxes hierarchically we
133. 5:04saw that thinking the ncaa
134. 5:05double-a tournament my aou can be
135. 5:07implemented using a mux i do all these
136. 5:09calculations
137. 5:10and then i just at the end use the muxes
138. 5:11to figure out who drives the output line
139. 5:14and i can use to build an n bit adder
140. 5:17subtractor
141. 5:18i can use n one bit adders cascade them
142. 5:21together
143. 5:22and use an xor to determine the signed
144. 5:25assigned overflow
145. 5:26that's pretty cool this is really good
146. 5:28stuff so we started we literally if i go
147. 5:30back four lectures ago
148. 5:33and several hours in one long recording
149. 5:35day for me
150. 5:37i started by talking about a transistor
151. 5:40and what
152. 5:40having voltage turning on and a gate to
153. 5:42be able to have what a bit means
154. 5:44how do you even go from current flowing
155. 5:46or not flowing to a bit
156. 5:48how to wire them to get some basic low
157. 5:50level structure how to make a nand gate
158. 5:52how to build up how to look at signals
159. 5:54and waveforms how to then
160. 5:55ignore those signals waveforms and think
161. 5:57about and so some timing in there
162. 5:58how to think about how would you
163. 6:02build combinational logic circuits
164. 6:03what's some boolean algebra behind that
165. 6:05how do you massage it how do you get
166. 6:06your hands all stuck with
167. 6:08lotion to massage a boolean expression
168. 6:10down to its simplest form
169. 6:11maybe fewest gates and then how to take
170. 6:14some of those basic gates
171. 6:15to build them to higher level blocks
172. 6:17we're going to use in our data path
173. 6:18to be able to build together a working
174. 6:20machine to process risk five machine
175. 6:23code and actually get this working
176. 6:24that's the next three lectures i cannot
177. 6:26wait to see you there awesome folks
178. 6:27thank you so much for coming to these
179. 6:28sts lectures we'll see the next time