# Copyright 2013-2015 Robert Jordens <jordens@gmail.com>
#
# This file is part of pdq2.
#
# pdq2 is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# pdq2 is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
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# You should have received a copy of the GNU General Public License
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from math import atan, atanh, log, sqrt, pi
from migen.fhdl.std import *
[docs]class TwoQuadrantCordic(Module):
"""Coordinate rotation digital computer
Trigonometric, and arithmetic functions implemented using
additions/subtractions and shifts.
http://eprints.soton.ac.uk/267873/1/tcas1_cordic_review.pdf
http://www.andraka.com/files/crdcsrvy.pdf
http://zatto.free.fr/manual/Volder_CORDIC.pdf
The way the CORDIC is executed is controlled by `eval_mode`.
If `"iterative"` the stages are iteratively evaluated, one per clock
cycle. This mode uses the least amount of registers, but has the
lowest throughput and highest latency. If `"pipelined"` all stages
are executed in every clock cycle but separated by registers. This
mode has full throughput but uses many registers and has large
latency. If `"combinatorial"`, there are no registers, throughput is
maximal and latency is zero. `"pipelined"` and `"combinatorial"` use
the same number of shifters and adders.
The type of trigonometric/arithmetic function is determined by
`cordic_mode` and `func_mode`. :math:`g` is the gain of the CORDIC.
* rotate-circular: rotate the vector `(xi, yi)` by an angle `zi`.
Used to calculate trigonometric functions, `sin(), cos(),
tan() = sin()/cos()`, or to perform polar-to-cartesian coordinate
transformation:
.. math::
x_o = g \\cos(z_i) x_i - g \\sin(z_i) y_i
y_o = g \\sin(z_i) x_i + g \\cos(z_i) y_i
* vector-circular: determine length and angle of the vector
`(xi, yi)`. Used to calculate `arctan(), sqrt()` or
to perform cartesian-to-polar transformation:
.. math::
x_o = g\\sqrt{x_i^2 + y_i^2}
z_o = z_i + \\tan^{-1}(y_i/x_i)
* rotate-hyperbolic: hyperbolic functions of `zi`. Used to
calculate hyperbolic functions, `sinh, cosh, tanh = cosh/sinh,
exp = cosh + sinh`:
.. math::
x_o = g \\cosh(z_i) x_i + g \\sinh(z_i) y_i
y_o = g \\sinh(z_i) x_i + g \\cosh(z_i) z_i
* vector-hyperbolic: natural logarithm `ln(), arctanh()`, and
`sqrt()`. Use `x_i = a + b` and `y_i = a - b` to obtain `2*
sqrt(a*b)` and `ln(a/b)/2`:
.. math::
x_o = g\\sqrt{x_i^2 - y_i^2}
z_o = z_i + \\tanh^{-1}(y_i/x_i)
* rotate-linear: multiply and accumulate (not a very good
multiplier implementation):
.. math::
y_o = g(y_i + x_i z_i)
* vector-linear: divide and accumulate:
.. math::
z_o = g(z_i + y_i/x_i)
Parameters
----------
width : int
Bit width of the input and output signals. Defaults to 16. Input
and output signals are signed.
widthz : int
Bit with of `zi` and `zo`. Defaults to the `width`.
stages : int or None
Number of CORDIC incremental rotation stages. Defaults to
`width + min(1, guard)`.
guard : int or None
Add guard bits to the intermediate signals. If `None`,
defaults to `guard = log2(width)` which guarantees accuracy
to `width` bits.
eval_mode : str, {"iterative", "pipelined", "combinatorial"}
cordic_mode : str, {"rotate", "vector"}
func_mode : str, {"circular", "linear", "hyperbolic"}
Evaluation and arithmetic mode. See above.
Attributes
----------
xi, yi, zi : Signal(width), in
Input values, signed.
xo, yo, zo : Signal(width), out
Output values, signed.
new_out : Signal(1), out
Asserted if output values are freshly updated in the current
cycle.
new_in : Signal(1), out
Asserted if new input values are being read in the next cycle.
zmax : float
`zi` and `zo` normalization factor. Floating point `zmax`
corresponds to `1<<(widthz - 1)`. `x` and `y` are scaled such
that floating point `1` corresponds to `1<<(width - 1)`.
gain : float
Cumulative, intrinsic gain and scaling factor. In circular mode
`sqrt(xi**2 + yi**2)` should be no larger than `2**(width - 1)/gain`
to prevent overflow. Additionally, in hyperbolic and linear mode,
the operation itself can cause overflow.
interval : int
Output interval in clock cycles. Inverse throughput.
latency : int
Input-to-output latency. The result corresponding to the inputs
appears at the outputs `latency` cycles later.
Notes
-----
Each stage `i` in the CORDIC performs the following operation:
.. math::
x_{i+1} = x_i - m d_i y_i r^{-s_{m,i}},
y_{i+1} = y_i + d_i x_i r^{-s_{m,i}},
z_{i+1} = z_i - d_i a_{m,i},
where:
* :math:`d_i`: clockwise or counterclockwise, determined by
`sign(z_i)` in rotate mode or `sign(-y_i)` in vector mode.
* :math:`r`: radix of the number system (2)
* :math:`m`: 1: circular, 0: linear, -1: hyperbolic
* :math:`s_{m,i}`: non decreasing integer shift sequence
* :math:`a_{m,i}`: elemetary rotation angle: :math:`a_{m,i} =
\\tan^{-1}(\\sqrt{m} s_{m,i})/\\sqrt{m}`.
"""
def __init__(self, width=16, widthz=None, stages=None, guard=0,
eval_mode="iterative", cordic_mode="rotate",
func_mode="circular"):
# validate parameters
assert eval_mode in ("combinatorial", "pipelined", "iterative")
assert cordic_mode in ("rotate", "vector")
assert func_mode in ("circular", "linear", "hyperbolic")
self.cordic_mode = cordic_mode
self.func_mode = func_mode
if guard is None:
# guard bits to guarantee "width" accuracy
guard = int(log(width)/log(2))
if widthz is None:
widthz = width
if stages is None:
stages = width + min(1, guard) # cuts error below LSB
# input output interface
self.xi = Signal((width, True))
self.yi = Signal((width, True))
self.zi = Signal((widthz, True))
self.xo = Signal((width, True))
self.yo = Signal((width, True))
self.zo = Signal((widthz, True))
self.new_in = Signal()
self.new_out = Signal()
###
a, s, self.zmax, self.gain = self._constants(stages, widthz + guard)
stages = len(a) # may have increased due to repetitions
if eval_mode == "iterative":
num_sig = 3
self.interval = stages + 1
self.latency = stages + 2
else:
num_sig = stages + 1
self.interval = 1
if eval_mode == "pipelined":
self.latency = stages
else: # combinatorial
self.latency = 0
# inter-stage signals
x = [Signal((width + guard, True)) for i in range(num_sig)]
y = [Signal((width + guard, True)) for i in range(num_sig)]
z = [Signal((widthz + guard, True)) for i in range(num_sig)]
# hook up inputs and outputs to the first and last inter-stage
# signals
self.comb += [
x[0].eq(self.xi<<guard),
y[0].eq(self.yi<<guard),
z[0].eq(self.zi<<guard),
self.xo.eq(x[-1]>>guard),
self.yo.eq(y[-1]>>guard),
self.zo.eq(z[-1]>>guard),
]
if eval_mode == "iterative":
# We afford one additional iteration for in/out.
i = Signal(max=stages + 1)
self.comb += [
self.new_in.eq(i == stages),
self.new_out.eq(i == 1),
]
ai = Signal((widthz + guard, True))
self.sync += ai.eq(Array(a)[i])
if range(stages) == s:
si = i - 1 # shortcut if no stage repetitions
else:
si = Signal(max=stages + 1)
self.sync += si.eq(Array(s)[i])
xi, yi, zi = x[1], y[1], z[1]
self.sync += [
self._stage(xi, yi, zi, xi, yi, zi, si, ai),
i.eq(i + 1),
If(i == stages,
i.eq(0),
),
If(i == 0,
x[2].eq(xi),
y[2].eq(yi),
z[2].eq(zi),
xi.eq(x[0]),
yi.eq(y[0]),
zi.eq(z[0]),
)]
else:
self.comb += [
self.new_out.eq(1),
self.new_in.eq(1),
]
for i, si in enumerate(s):
stmt = self._stage(x[i], y[i], z[i],
x[i + 1], y[i + 1], z[i + 1], si, a[i])
if eval_mode == "pipelined":
self.sync += stmt
else: # combinatorial
self.comb += stmt
def _constants(self, stages, bits):
if self.func_mode == "circular":
s = range(stages)
a = [atan(2**-i) for i in s]
g = [sqrt(1 + 2**(-2*i)) for i in s]
#zmax = sum(a)
# use pi anyway as the input z can cause overflow
# and we need the range for quadrant mapping
zmax = pi
elif self.func_mode == "linear":
s = range(stages)
a = [2**-i for i in s]
g = [1 for i in s]
#zmax = sum(a)
# use 2 anyway as this simplifies a and scaling
zmax = 2.
else: # hyperbolic
s = []
# need to repeat some stages:
j = 4
for i in range(stages):
if i == j:
s.append(j)
j = 3*j + 1
s.append(i + 1)
a = [atanh(2**-i) for i in s]
g = [sqrt(1 - 2**(-2*i)) for i in s]
zmax = sum(a)*2
# round here helps the width=2**i - 1 case but hurts the
# important width=2**i case
cast = int
if log(bits)/log(2) % 1:
cast = round
a = [cast(ai*2**(bits - 1)/zmax) for ai in a]
gain = 1.
for gi in g:
gain *= gi
return a, s, zmax, gain
def _stage(self, xi, yi, zi, xo, yo, zo, i, ai):
dir = Signal()
if self.cordic_mode == "rotate":
self.comb += dir.eq(zi < 0)
else: # vector
self.comb += dir.eq(yi >= 0)
dx = yi>>i
dy = xi>>i
dz = ai
if self.func_mode == "linear":
dx = 0
elif self.func_mode == "hyperbolic":
dx = -dx
stmt = [
xo.eq(xi + Mux(dir, dx, -dx)),
yo.eq(yi + Mux(dir, -dy, dy)),
zo.eq(zi + Mux(dir, dz, -dz))
]
return stmt
[docs]class Cordic(TwoQuadrantCordic):
"""Four-quadrant CORDIC
Same as :class:`TwoQuadrantCordic` but with support and convergence
for `abs(zi) > pi/2 in circular rotate mode or `xi < 0` in circular
vector mode.
"""
def __init__(self, **kwargs):
TwoQuadrantCordic.__init__(self, **kwargs)
if self.func_mode != "circular":
return # no need to remap quadrants
cxi, cyi, czi = self.xi, self.yi, self.zi
self.xi = xi = Signal.like(cxi)
self.yi = yi = Signal.like(cyi)
self.zi = zi = Signal.like(czi)
###
q = Signal()
if self.cordic_mode == "rotate":
self.comb += q.eq(zi[-2] ^ zi[-1])
else: # vector
self.comb += q.eq(xi < 0)
self.comb += [
If(q,
Cat(cxi, cyi, czi).eq(Cat(-xi, -yi,
zi + (1 << flen(zi) - 1)))
).Else(
Cat(cxi, cyi, czi).eq(Cat(xi, yi, zi))
)
]