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ariths-gen/ariths_gen/multi_bit_circuits/multipliers/wallace_multiplier.py

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from ariths_gen.multi_bit_circuits.adders.carry_lookahead_adder import UnsignedCarryLookaheadAdder
from ariths_gen.wire_components import (
Wire,
ConstantWireValue0,
ConstantWireValue1,
Bus
)
from ariths_gen.core.arithmetic_circuits import (
ArithmeticCircuit,
MultiplierCircuit
)
from ariths_gen.one_bit_circuits.one_bit_components import (
HalfAdder,
FullAdder,
FullAdderPG
)
from ariths_gen.one_bit_circuits.logic_gates import (
AndGate,
NandGate,
OrGate,
NorGate,
XorGate,
XnorGate,
NotGate
)
from ariths_gen.multi_bit_circuits.adders import (
CarrySaveAdderComponent,
UnsignedCarryLookaheadAdder
)
class UnsignedWallaceMultiplier(MultiplierCircuit):
"""Class representing unsigned wallace multiplier.
Unsigned wallace multiplier represents fast N-bit multiplier which utilizes
the functionality of wallace tree reduction algorithm proposed by Chris Wallace.
First partial products are calculated for each bit pair that form the partial product multiplication rows/columns.
At last the reduced pairs are inserted into the chosen multi bit unsigned adder to execute their summation and obtain the final output bits.
The multiplier can be build from carry save adders or fully connected half/full adders (greater delay).
The csa implementation uses carry save adder components to efficiently implement reduction of partial products utilizing the parallelism of the carry save adders. At last the reduced pairs are inserted into chosen multi bit unsigned adder to execute their summation and obtain the final output bits. It presents a faster version of multiplier opposed to the conventional architectures that are composed of interconnected half/full adders.
Wallace tree algorithm is described more in detail here:
https://en.wikipedia.org/wiki/Wallace_tree
It presents a smaller circuit in area opposed to an array multiplier but is slightly bigger then dadda because of less reduction stages.
```
CSA IMPLEMENTATION:
PP7 PP6 PP5 PP4 PP3 PP2 PP1 PP0
│ │ │ │ │ │ │ │
│ │ ┌▼──▼──▼┐ ┌▼──▼──▼┐
│ │ │ CSA │ │ CSA │
│ │ └─┬───┬─┘ └─┬───┬─┘
│ │ │c1 │s1 │c0 │s0
└┐ │ ┌──┘ └────┐ │ ┌┘
┌▼──▼──▼┐ ┌▼──▼──▼┐
│ CSA │ │ CSA │
└─┬───┬─┘ └─┬───┬─┘
│c4 │s4 ┌───────┘c3 │s3
│ └──┐ │ ┌────────┘
│ ┌▼──▼──▼┐
│ │ CSA │
│ └─┬───┬─┘
│ ┌────┘c5 │s5
│ │ ┌─────┘
┌▼──▼──▼┐
│ CSA │
└─┬───┬─┘
│c6 │s6
┌─▼───▼─┐
│ CPA │
└───┬───┘
o
```
Description of the __init__ method.
Args:
a (Bus): First input bus.
b (Bus): Second input bus.
prefix (str, optional): Prefix name of unsigned wallace multiplier. Defaults to "".
name (str, optional): Name of unsigned wallace multiplier. Defaults to "u_wallace_cla".
use_csa (bool, optional): Choose whether to use carry save adder architecture (True) or fully interconnected half/full adders (False). Defaults to True.
unsigned_adder_class_name (str, optional): Unsigned multi bit adder used to obtain final sums of products. Defaults to UnsignedCarryLookaheadAdder.
"""
def __init__(self, a: Bus, b: Bus, prefix: str = "", name: str = "u_wallace_cla", use_csa: bool = True, unsigned_adder_class_name: str = UnsignedCarryLookaheadAdder, **kwargs):
self.N = max(a.N, b.N)
super().__init__(a=a, b=b, prefix=prefix, name=name, out_N=self.N*2, **kwargs)
# Bus sign extension in case buses have different lengths
self.a.bus_extend(N=self.N, prefix=a.prefix)
self.b.bus_extend(N=self.N, prefix=b.prefix)
# CSA IMPLEMENTATION
if use_csa is True:
# Initialize all rows partial products forming AND gates matrix
self.rows = self.init_row_lengths()
# Zero extension of partial product rows
for i in range(0, len(self.rows)):
self.rows[i] = Bus(prefix=self.rows[i].prefix, wires_list=[ConstantWireValue0() for _ in range(0, i)] + self.rows[i].bus)
while len(self.rows) > 2:
# Gradual creation of unsigned csa adder components to reduce the pp rows to the total count of 2
pp_index = 0
while pp_index < len(self.rows) and (pp_index+2) < len(self.rows):
csa_reduction = CarrySaveAdderComponent(a=self.rows[pp_index], b=self.rows[pp_index+1], c=self.rows[pp_index+2], prefix=self.prefix+"_csa"+str(self.get_instance_num(cls=CarrySaveAdderComponent)), inner_component=True)
self.add_component(csa_reduction)
# 3 pp rows have been reduced to 2
[self.rows.pop(pp_index) for i in range(3)]
# Append rows of sum and carry results from csa calculation
csa_sums_N = self.out.N if csa_reduction.sum_bits.N > self.out.N-1 else csa_reduction.sum_bits.N
csa_sums = Bus(prefix=self.prefix+"_csa_s"+str(self.get_instance_num(cls=CarrySaveAdderComponent)), N=csa_sums_N)
csa_sums.connect_bus(connecting_bus=csa_reduction.out, end_connection_pos=csa_sums_N)
csa_carries_N = self.out.N if csa_reduction.carry_bits.N > self.out.N-1 else csa_reduction.carry_bits.N
csa_carries = Bus(prefix=self.prefix+"_csa_c"+str(self.get_instance_num(cls=CarrySaveAdderComponent)), N=csa_carries_N)
csa_carries.connect_bus(connecting_bus=csa_reduction.out, start_connection_pos=int(csa_reduction.out.N/2), end_connection_pos=int(csa_reduction.out.N/2)+csa_carries.N, offset=int(csa_reduction.out.N/2))
self.rows.insert(pp_index, csa_carries)
self.rows.insert(pp_index, csa_sums)
# Update of the number of pp rows
pp_index += 2
# Output generation
# 1 bit multiplier case
if self.N == 1:
self.out.connect(0, self.get_previous_component().out)
self.out.connect(1, ConstantWireValue0())
# 2 bit multiplier case
elif self.N == 2:
self.out.connect(0, self.components[0].out)
obj_ha = HalfAdder(a=self.components[1].out, b=self.components[2].out, prefix=self.prefix+"_ha"+str(self.get_instance_num(cls=HalfAdder)))
self.add_component(obj_ha)
self.out.connect(1, obj_ha.get_sum_wire())
obj_ha = HalfAdder(a=self.get_previous_component().get_carry_wire(), b=self.components[3].out, prefix=self.prefix+"_ha"+str(self.get_instance_num(cls=HalfAdder)))
self.add_component(obj_ha)
self.out.connect(2, obj_ha.get_sum_wire())
self.out.connect(3, obj_ha.get_carry_wire())
# Final addition of remaining bits using chosen unsigned multi bit adder
else:
# Obtain proper adder name with its bit width (columns bit pairs minus the first alone bit)
adder_name = unsigned_adder_class_name(a=a, b=b).prefix + str(self.rows[0].N)
adder_a = Bus(prefix="a", N=self.rows[0].N)
adder_b = Bus(prefix="b", N=self.rows[1].N)
[adder_a.connect(w, self.rows[0].get_wire(w)) for w in range(0, self.rows[0].N)]
[adder_b.connect(w, self.rows[1].get_wire(w)) for w in range(0, self.rows[1].N)]
final_adder = unsigned_adder_class_name(a=adder_a, b=adder_b, prefix=self.prefix, name=adder_name, inner_component=True)
self.add_component(final_adder)
[self.out.connect(o, final_adder.out.get_wire(o), inserted_wire_desired_index=o) for o in range(0, final_adder.out.N-1)]
# FULLY INTERCONNECTED HAs/FAs IMPLEMENTATION
else:
# Initialize all columns partial products forming AND gates matrix
self.columns = self.init_column_heights()
# Perform reduction until all columns have 2 or less bits in them
while not all(height <= 2 for (height, *_) in self.columns):
col = 0
while col < len(self.columns):
# If column has exactly 3 bits in height and all previous columns has maximum of 2 bits in height, combine them in a half adder
if self.get_column_height(col) == 3 and all(height <= 2 for (height, *_) in self.columns[0:col-1]):
# Add half adder and also AND gates if neccesarry (via add_column_wire invocation) into list of circuit components
obj_adder = HalfAdder(self.add_column_wire(column=col, bit=0), self.add_column_wire(column=col, bit=1), prefix=self.prefix+"_ha"+str(self.get_instance_num(cls=HalfAdder)))
self.add_component(obj_adder)
# Update the number of current and next column wires
self.update_column_heights(curr_column=col, curr_height_change=-1, next_column=col+1, next_height_change=1)
# Update current and next column wires arrangement
# add ha's generated sum to the bottom of current column
# add ha's generated cout to the top of next column
self.update_column_wires(curr_column=col, next_column=col+1, adder=self.get_previous_component(1))
# If column has more than 3 bits in height, combine them in a full adder
elif self.get_column_height(col) > 3:
# Add full adder and also AND gates if neccesarry (via add_column_wire invocation) into list of circuit components
obj_adder = FullAdder(self.add_column_wire(column=col, bit=0), self.add_column_wire(column=col, bit=1), self.add_column_wire(column=col, bit=2), prefix=self.prefix+"_fa"+str(self.get_instance_num(cls=FullAdder)))
self.add_component(obj_adder)
# Update the number of current and next column wires
self.update_column_heights(curr_column=col, curr_height_change=-2, next_column=col+1, next_height_change=1)
# Update current and next column wires arrangement
# add fa's generated sum to the bottom of current column
# add fa's generated cout to the top of next column
self.update_column_wires(curr_column=col, next_column=col+1, adder=self.get_previous_component(1))
col += 1
# Output generation
# First output bit from single first pp AND gate
self.out.connect(0, self.add_column_wire(column=0, bit=0))
# Final addition of remaining bits
# 1 bit multiplier case
if self.N == 1:
self.out.connect(1, ConstantWireValue0())
# 2 bit multiplier case
elif self.N == 2:
obj_ha = HalfAdder(self.add_column_wire(column=1, bit=0), self.add_column_wire(column=1, bit=1), prefix=self.prefix+"_ha"+str(self.get_instance_num(cls=HalfAdder)))
self.add_component(obj_ha)
self.out.connect(1, obj_ha.get_sum_wire())
obj_ha = HalfAdder(self.get_previous_component().get_carry_wire(), self.add_column_wire(column=2, bit=0), prefix=self.prefix+"_ha"+str(self.get_instance_num(cls=HalfAdder)))
self.add_component(obj_ha)
self.out.connect(2, obj_ha.get_sum_wire())
self.out.connect(3, obj_ha.get_carry_wire())
# Final addition of remaining bits using chosen unsigned multi bit adder
else:
# Obtain proper adder name with its bit width (columns bit pairs minus the first alone bit)
adder_name = unsigned_adder_class_name(a=a, b=b).prefix + str(len(self.columns)-1)
adder_a = Bus(prefix=f"a", wires_list=[self.add_column_wire(column=col, bit=0) for col in range(1, len(self.columns))])
adder_b = Bus(prefix=f"b", wires_list=[self.add_column_wire(column=col, bit=1) for col in range(1, len(self.columns))])
final_adder = unsigned_adder_class_name(a=adder_a, b=adder_b, prefix=self.prefix, name=adder_name, inner_component=True)
self.add_component(final_adder)
[self.out.connect(o, final_adder.out.get_wire(o-1), inserted_wire_desired_index=o-1) for o in range(1, len(self.out.bus))]
class SignedWallaceMultiplier(MultiplierCircuit):
"""Class representing signed wallace multiplier.
Signed wallace multiplier represents fast N-bit multiplier which utilizes
the functionality of wallace tree reduction algorithm proposed by Chris Wallace and uses Baugh-Wooley algorithm
to perform signed multiplication.
First partial products are calculated for each bit pair that form the partial product multiplication rows/columns.
The csa implementation uses carry save adder components to efficiently implement reduction of partial products utilizing the parallelism of the carry save adders. At last the reduced pairs are inserted into chosen multi bit unsigned adder to execute their summation and obtain the final output bits, additional XOR gate serves the necessary sign extension. It presents a faster version of multiplier opposed to the conventional architectures that are composed of interconnected half/full adders.
Wallace tree algorithm is described more in detail here:
https://en.wikipedia.org/wiki/Wallace_tree
It presents a smaller circuit in area opposed to an array multiplier but is slightly bigger then dadda because of less reduction stages.
Description of the __init__ method.
Args:
a (Bus): First input bus.
b (Bus): Second input bus.
prefix (str, optional): Prefix name of signed wallace multiplier. Defaults to "".
name (str, optional): Name of signed wallace multiplier. Defaults to "s_wallace_cla".
use_csa (bool, optional): Choose whether to use carry save adder architecture (True) or fully interconnected half/full adders (False). Defaults to True.
unsigned_adder_class_name (str, optional): Unsigned multi bit adder used to obtain final sums of products. Defaults to UnsignedCarryLookaheadAdder.
"""
def __init__(self, a: Bus, b: Bus, prefix: str = "", name: str = "s_wallace_cla", use_csa: bool = True, unsigned_adder_class_name: str = UnsignedCarryLookaheadAdder, **kwargs):
self.N = max(a.N, b.N)
super().__init__(a=a, b=b, prefix=prefix, name=name, out_N=self.N*2, signed=True, **kwargs)
# Bus sign extension in case buses have different lengths
self.a.bus_extend(N=self.N, prefix=a.prefix)
self.b.bus_extend(N=self.N, prefix=b.prefix)
# CSA IMPLEMENTATION
if use_csa is True:
# Initialize all rows partial products forming AND gates matrix
self.rows = self.init_row_lengths()
# Zero extension of partial product rows
for i in range(0, len(self.rows)):
self.rows[i] = Bus(prefix=self.rows[i].prefix, wires_list=[ConstantWireValue0() for _ in range(0, i)] + self.rows[i].bus)
while len(self.rows) > 2:
# Gradual creation of signed csa adder components to reduce the pp rows to the total count of 2
pp_index = 0
while pp_index < len(self.rows) and (pp_index+2) < len(self.rows):
csa_reduction = CarrySaveAdderComponent(a=self.rows[pp_index], b=self.rows[pp_index+1], c=self.rows[pp_index+2], prefix=self.prefix+"_csa"+str(self.get_instance_num(cls=CarrySaveAdderComponent)), inner_component=True, signed=True)
self.add_component(csa_reduction)
# 3 pp rows have been reduced to 2
[self.rows.pop(pp_index) for i in range(3)]
# Append rows of sum and carry results from csa calculation
csa_sums_N = self.out.N if csa_reduction.sum_bits.N > self.out.N-1 else csa_reduction.sum_bits.N
csa_sums = Bus(prefix=self.prefix+"_csa_s"+str(self.get_instance_num(cls=CarrySaveAdderComponent)), N=csa_sums_N)
csa_sums.connect_bus(connecting_bus=csa_reduction.out, end_connection_pos=csa_sums_N)
csa_carries_N = self.out.N if csa_reduction.carry_bits.N > self.out.N-1 else csa_reduction.carry_bits.N
csa_carries = Bus(prefix=self.prefix+"_csa_c"+str(self.get_instance_num(cls=CarrySaveAdderComponent)), N=csa_carries_N)
csa_carries.connect_bus(connecting_bus=csa_reduction.out, start_connection_pos=int(csa_reduction.out.N/2), end_connection_pos=int(csa_reduction.out.N/2)+csa_carries.N, offset=int(csa_reduction.out.N/2))
self.rows.insert(pp_index, csa_carries)
self.rows.insert(pp_index, csa_sums)
# Update of the number of pp rows
pp_index += 2
# Output generation
# 1 bit multiplier case
if self.N == 1:
self.out.connect(0, self.get_previous_component().out)
self.out.connect(1, ConstantWireValue0())
return
# 2 bit multiplier case
elif self.N == 2:
self.out.connect(0, self.components[0].out)
obj_ha = HalfAdder(a=self.components[1].out, b=self.components[2].out, prefix=self.prefix+"_ha"+str(self.get_instance_num(cls=HalfAdder)))
self.add_component(obj_ha)
self.out.connect(1, obj_ha.get_sum_wire())
obj_fa = FullAdder(a=self.get_previous_component().get_carry_wire(), b=ConstantWireValue1(), c=self.components[3].out, prefix=self.prefix+"_fa"+str(self.get_instance_num(cls=FullAdder)))
self.add_component(obj_fa)
self.out.connect(2, obj_fa.get_sum_wire())
self.out.connect(3, obj_fa.get_carry_wire())
# Final addition of remaining bits using chosen unsigned multi bit adder
else:
# Obtain proper adder name with its bit width (columns bit pairs minus the first alone bit)
adder_name = unsigned_adder_class_name(a=a, b=b).prefix + str(self.rows[0].N)
adder_a = Bus(prefix="a", N=self.rows[0].N)
adder_b = Bus(prefix="b", N=self.rows[1].N)
[adder_a.connect(w, self.rows[0].get_wire(w)) for w in range(0, self.rows[0].N)]
[adder_b.connect(w, self.rows[1].get_wire(w)) for w in range(0, self.rows[1].N)]
final_adder = unsigned_adder_class_name(a=adder_a, b=adder_b, prefix=self.prefix, name=adder_name, inner_component=True)
self.add_component(final_adder)
[self.out.connect(o, final_adder.out.get_wire(o), inserted_wire_desired_index=o) for o in range(0, final_adder.out.N-1)]
# Final XOR to ensure proper sign extension
obj_xor = XorGate(ConstantWireValue1(), self.out.get_wire(self.out.N-1), prefix=self.prefix+"_xor"+str(self.get_instance_num(cls=XorGate)), parent_component=self)
self.add_component(obj_xor)
self.out.connect(self.out.N-1, obj_xor.out)
# FULLY INTERCONNECTED HAs/FAs IMPLEMENTATION
else:
# Initialize all columns partial products forming AND/NAND gates matrix based on Baugh-Wooley multiplication
self.columns = self.init_column_heights()
# Not used for 1 bit multiplier
if self.N != 1:
# Adding constant wire with value 1 to achieve signedness based on Baugh-Wooley multiplication algorithm
# (adding constant value bit to last column (with one bit) to combine them in XOR gate to get the correct final multplication output bit at the end)
self.columns[self.N].insert(1, ConstantWireValue1())
self.update_column_heights(curr_column=self.N, curr_height_change=1)
# Perform reduction until all columns have 2 or less bits in them
while not all(height <= 2 for (height, *_) in self.columns):
col = 0
while col < len(self.columns):
# If column has exactly 3 bits in height and all previous columns has maximum of 2 bits in height, combine them in a half adder
if self.get_column_height(col) == 3 and all(height <= 2 for (height, *_) in self.columns[0:col-1]):
# Add half adder and also AND/NAND gates if neccesarry (via add_column_wire invocation) into list of circuit components
obj_adder = HalfAdder(self.add_column_wire(column=col, bit=0), self.add_column_wire(column=col, bit=1), prefix=self.prefix+"_ha"+str(self.get_instance_num(cls=HalfAdder)))
self.add_component(obj_adder)
# Update the number of current and next column wires
self.update_column_heights(curr_column=col, curr_height_change=-1, next_column=col+1, next_height_change=1)
# Update current and next column wires arrangement
# add ha's generated sum to the bottom of current column
# add ha's generated cout to the top of next column
self.update_column_wires(curr_column=col, next_column=col+1, adder=self.get_previous_component(1))
# If column has more than 3 bits in height, combine them in a full adder
elif self.get_column_height(col) > 3:
# Add full adder and also AND/NAND gates if neccesarry (via add_column_wire invocation) into list of circuit components
obj_adder = FullAdder(self.add_column_wire(column=col, bit=0), self.add_column_wire(column=col, bit=1), self.add_column_wire(column=col, bit=2), prefix=self.prefix+"_fa"+str(self.get_instance_num(cls=FullAdder)))
self.add_component(obj_adder)
# Update the number of current and next column wires
self.update_column_heights(curr_column=col, curr_height_change=-2, next_column=col+1, next_height_change=1)
# Update current and next column wires arrangement
# add fa's generated sum to the bottom of current column
# add fa's generated cout to the top of next column
self.update_column_wires(curr_column=col, next_column=col+1, adder=self.get_previous_component(1))
col += 1
# Output generation
# First output bit from single first pp AND gate
self.out.connect(0, self.add_column_wire(column=0, bit=0))
# Final addition of remaining bits
# 1 bit multiplier case
if self.N == 1:
self.out.connect(1, ConstantWireValue0())
return
# 2 bit multiplier case
elif self.N == 2:
obj_ha = HalfAdder(self.add_column_wire(column=1, bit=0), self.add_column_wire(column=1, bit=1), prefix=self.prefix+"_ha"+str(self.get_instance_num(cls=HalfAdder)))
self.add_component(obj_ha)
self.out.connect(1, obj_ha.get_sum_wire())
obj_fa = FullAdder(self.get_previous_component().get_carry_wire(), self.add_column_wire(column=2, bit=0), self.add_column_wire(column=2, bit=1), prefix=self.prefix+"_fa"+str(self.get_instance_num(cls=FullAdder)))
self.add_component(obj_fa)
self.out.connect(2, obj_fa.get_sum_wire())
self.out.connect(3, obj_fa.get_carry_wire())
# Final addition of remaining bits using chosen unsigned multi bit adder
else:
# Obtain proper adder name with its bit width (columns bit pairs minus the first alone bit)
adder_name = unsigned_adder_class_name(a=a, b=b).prefix + str(len(self.columns)-1)
adder_a = Bus(prefix=f"a", wires_list=[self.add_column_wire(column=col, bit=0) for col in range(1, len(self.columns))])
adder_b = Bus(prefix=f"b", wires_list=[self.add_column_wire(column=col, bit=1) for col in range(1, len(self.columns))])
final_adder = unsigned_adder_class_name(a=adder_a, b=adder_b, prefix=self.prefix, name=adder_name, inner_component=True)
self.add_component(final_adder)
[self.out.connect(o, final_adder.out.get_wire(o-1), inserted_wire_desired_index=o-1) for o in range(1, len(self.out.bus))]
# Final XOR to ensure proper sign extension
obj_xor = XorGate(ConstantWireValue1(), self.out.get_wire(self.out.N-1), prefix=self.prefix+"_xor"+str(self.get_instance_num(cls=XorGate)), parent_component=self)
self.add_component(obj_xor)
self.out.connect(self.out.N-1, obj_xor.out)
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