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ECE 368 Sequential Multiplication. Shantanu Dutt ECE Dept. UIC. 1001 = 9 1011 = 11. 00000000 = P 0. + 1001. 00001001 = P 1. + 1001. 00011011 = P 2. + 0000. 00011011 = P 3. + 1001. 01100011 = P 4 = P = 99. 1001 = 9 1011 = 11. 00000000 = P 0. 1001. 10010000.

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Ece 368 sequential multiplication l.jpg

ECE 368Sequential Multiplication

Shantanu Dutt

ECE Dept.

UIC


Slide2 l.jpg

1001 = 9

1011 = 11

00000000 = P0

+ 1001

00001001 = P1

+ 1001

00011011 = P2

+ 0000

00011011 = P3

+ 1001

01100011 = P4 = P = 99


Slide4 l.jpg

1001 = 9

1011 = 11

00000000 = P0

1001

10010000

SHR: 01001000 = P1

1001

11011000

SHR: 01101100 = P2

0000

01101100

SHR: 00110110 = P3

1001

11000110

SHR: 01100011 = P4 = P = 99


Regular a s 2 s complement multiplication example l.jpg
Regular A&S 2’s Complement Multiplication Example

Multplicand Y = 11001101 = -51, Multiplier X = 01100111 = 103, -Y = 10110011= -77

Cout-AC-Q = 0| 00000000 | 01100111 M = 11001101

Iter 1: Q0 = 1, perform add AC+Y: 00000000

+ 11001101

0 11001101 = Cout-AC

No ovfl, ASR AC-Q: 11100110 | 10110011

Iter 2: Q0 = 1, perform add AC+Y: + 11001101

1 10110011 = Cout-AC

No ovfl, ASR AC-Q: 11011001 | 11011001

Iter 3: Q0 = 1, perform add AC+Y: + 11001101

1 10100110 = Cout-AC

No ovfl, ASR AC-Q: 11010011 | 01101100

Iter 4: Q0 = 0, only ASR AC-Q: 11101001 | 10110110

Iter 5: Q0 = 0, only ASR AC-Q: 11110100 | 11011011

Iter 6: Q0 = 1, perform add AC+Y: + 11001101

1 11000001 = Cout-AC

No ovfl, ASR AC-Q: 11100000 | 11101101

Iter 7: Q0 = 1, perform add AC+Y: + 11001101

110101101 = Cout-AC

No ovfl, ASR AC-Q: 11010110 | 11110110

Iter 8: Q0 = 0, only ASR AC-Q : 11101011 | 01111011

P = AC-Q = 1110101101111011 = -5253 = (-51) x 103. Needed to perform 5 +’s and 8 shifts


Regular a s 2 s complement multiplication example11 l.jpg
Regular A&S 2’s Complement Multiplication Example

Multplicand Y = 01001101 = 77, Multiplier X = 11100111 = -25, -Y = 10110011= -77

Cout-AC-Q = 0| 00000000 | 11100111 M = 01001101

Iter 1: Q0 = 1, perform add AC+Y: 00000000

+ 01001101

0 01001101 = Cout-AC

No ovfl, ASR AC-Q: 00100110 | 11110011

Iter 2: Q0 = 1, perform add AC+Y: + 01001101

0 01110011 = Cout-AC

No ovfl, ASR AC-Q: 00111001 | 11111001

Iter 3: Q0 = 1, perform add AC+Y: + 01001101

0 10000110 = Cout-AC

Ovfl, SR Cout-AC-Q: 01000011 | 01111100

Iter 4: Q0 = 0, only ASR AC-Q: 00100001 | 10111110

Iter 5: Q0 = 0, only ASR AC-Q: 00010000 | 11011111

Iter 6: Q0 = 1, perform add AC+Y: + 01001101

0 01011101 = Cout-AC

No ovfl, ASR AC-Q: 00101110 | 11101111

Iter 7: Q = 1, perform add AC+Y: + 01001101

0 01111011 = Cout-AC

No ovfl, ASR AC-Q: 00111101 | 11110111

Iter 8: Q = 1, perform sub AC-Y: +10110011

0 11110000 = Cout-AC

No ovfl, ASR AC-Q: 11111000 | 01111011

P = AC-Q = 1111100001111011 = -1925 = 77 x (-25). Needed to perform 6 +/- and 8 shifts


Slide14 l.jpg

inv_sign

(from CU)

Cout

reg.

Cin

sign of xi*Y “essentially” (considering Cin value) inverted when inv_sign=1


Slide16 l.jpg

001000

001000


Booth multiplication example l.jpg
Booth Multiplication Example

Multplicand Y = 01001101 = 77, Multiplier X = 11100111 = -25, -Y = 10110011= -77

Cout-AC-Q-q- = 0| 00000000 | 11100111 | 0 M = 01001101 (q- is the “prev” bit of Q)

Iter 1: q0-q- = 10, perform sub AC-Y: 00000000

+ 10110011

0 10110011 = Cout-AC

No ovfl, ASR AC-Q-q-: 11011001 | 11110011 | 1

Iter 2: q0-q- = 11, only ASR AC-Q-q-: 11101100 | 11111001 | 1

Iter 3: q0-q- = 11, only ASR AC-Q-q-: 11110110 | 01111100 | 1

Iter 4: q0-q- = 01, perform add AC+Y: +01001101

1 01000011 = Cout-AC

No ovfl, ASR AC-Q-q-: 00100001 | 10111110 | 0

Iter 5: q0-q- = 00, only ASR AC-Q-q-: 00010000 | 11011111 | 0

Iter 6: q0-q- = 10, perform sub AC-Y:+ 10110011

0 11000011 = Cout-AC

No ovfl, ASR AC-Q-q-: 11100001 | 11101111 | 1

Iter 7: q0-q- = 11, only ASR AC-Q-q-: 11110000 | 11110111 | 1

Iter 8: q0-q- = 11, only ASR AC-Q-q-: 11111000 | 01111011 | 1

P = AC-Q = 1111100001111011 = -1925 = 77 x (-25)

Needed only 3 +/-’s as opposed to 6 +/-’s needed in regular A&S mult.

# of shifts remain the same (8)


Slide26 l.jpg

Structural VHDL allows the designer to represent a system in terms of components and their interconnections. This module discusses the constructs available in VHDL to facilitate structural descriptions of designs.

Acknowledgement (this and next 3 slides): http://www.people.vcu.edu/~rhklenke/tutorials/vhdl/modules/m11_23/sld001.htm.

The next 3 slides appear in sld037.htm to sld039.htm of this module.


Slide27 l.jpg

: Structural Description terms of components and their interconnections. This module discusses the constructs available in VHDL to facilitate structural descriptions of designs.

Control signals

(from the CU)

The final example is of an RTL level datapath for an unsigned 8 bit multiplier. It illustrates the use of complex components in a structural description and the use of multiple levels of hierarchy in that the components are themselves structural descriptions of lower level components. This is the entity description for the datapath. The register controls (enable, mode) will go to the control unit when it is added.



Slide29 l.jpg

scan_out => OPEN, indications for the datapath.

This is the component instantiations for the datapath. The mapping of individual bits of the D and Q input and output of the shift registers is necessary to reverse the inputs and outputs. Recall that the shift register was a “shift up” type where the scan_in input goes to D(0) and D(0) to D(6) go to D(1) to D(7) when in shift mode. What is needed for the multiplier is a “shift down” type register where scan_in goes to D(7), etc.

It should have been possible (we believe) to use the syntax

d => multiplier(0 to 7)

in the shift_reg8_str PORT MAP to accomplish the same thing, but the QuickVHDL compiler gave a “warning” and the simulator crashed, so we don’t know if it really should work.


Slide30 l.jpg

Acknowledgement (this and next 7 slides): indications for the datapath.http://www.people.vcu.edu/~rhklenke/tutorials/vhdl/modules/m12_23/sld001.htm.

The next 7 slides appear in sld050.htm to sld056.htm of this module.


Slide31 l.jpg

: Behavioral Description indications for the datapath.

Now we are going to develop a behavioral description of the controller for the unsigned 8 bit multiplier. This is a flow chart of the algorithm that the controller uses.


Slide32 l.jpg

This is state diagram of the controller state machine. The outputs for each state aren’t shown for clarity. Notice that there is also a “count” variable that must be included in the state machine to count the number of iterations through the loop. The count variable is actually implemented as another state variable.


Slide33 l.jpg

The state machine actually has two state variables, the current state of the control state machine (e.g., initialize, shift, add), and the present loop count. The loop count is a state variable in that it has a present value and a next value, and it is updated in the clock process. However, the value of count only affects the next control state the machine goes to and doesn’t affect the outputs. The implementation is actually more like two state machines in the same architecture.


Slide34 l.jpg

Control signals current state of the control state machine (e.g., initialize, shift, add), and the present loop count. The loop count is a state variable in that it has a present value and a next value, and it is updated in the clock process. However, the value of count only affects the next control state the machine goes to and doesn’t affect the outputs. The implementation is actually more like two state machines in the same architecture.

(from the CU)

This the entity description for the unsigned 8 bit multiplier control unit. It hooks to the datapath via the control signals listed.


Slide35 l.jpg

This is the beginning of the architecture of the control unit. Note that the constrained subtype of integer is for synthesis - unconstrained integers are hard to synthesize! Also note the state variables are enumerated types. This allows the synthesis tools to encode the state variable using different schemes.

Also included here is the clock process. Note that it is edge triggered and that both present_state and present_count are updated on the clock edge. Also note the asynchronous reset signal.


Slide36 l.jpg

This is the state transition process for the state machine. Note the default assignment of next_state = present_state which is only really required for (some) synthesis tools.


Slide37 l.jpg

Reset active low Note the default assignment of next_state = present_state which is only really required for (some) synthesis tools.

x_enable = 1

allows reg. x

to be loaded

w/ its i/ps: inp.

bus or shift inp.

  • x_mode = 1

  • load x from

  • inp. bus

  • x_mode = 0

  • load x from

  • shift inp.

This is the output process. Note that the outputs are only dependent on the present_state variable (Moore machine).


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