# Linear Phase Lowpass Filter Design (FDESIGN)#

`fdesign.py`

```"""
## GAMSSOURCE: https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_fdesign.html
## MODELTYPE: QCP
## KEYWORDS: quadratic constraint programming, second order cone programming, engineering, finite impulse response filter designment

Linear Phase Lowpass Filter Design (FDESIGN)

This model finds the filter weights for a finite impulse response
(FIR) filter. We use rotated quadratic cones for the constraints.

This model is the minimax linear phase lowpass filter design from Lobo
et. al (Section 3.3) We model the nonlinear term 1/t in the model as
follows: introduce variables u,v, where v = 2 (and u = 1/t). Then 1/t
can be modeled as the quadratic cone

||[v, u-t]|| <= u+t,   u,t >=0

Contributed by Michael Ferris, University of Wisconsin, Madison

Lobo, M S, Vandenberghe, L, Boyd, S, and Lebret, H, Applications of
Second Order Cone Programming. Linear Algebra and its Applications,
Special Issue on Linear Algebra in Control, Signals and Image
Processing. 284 (November, 1998).
"""

from __future__ import annotations

import math
import os

import gamspy.math as gams_math
from gamspy import Card
from gamspy import Container
from gamspy import Equation
from gamspy import Model
from gamspy import Ord
from gamspy import Parameter
from gamspy import Problem
from gamspy import Sense
from gamspy import Set
from gamspy import Sum
from gamspy import Variable

def main():
m = Container(
system_directory=os.getenv("SYSTEM_DIRECTORY", None),
delayed_execution=int(os.getenv("DELAYED_EXECUTION", False)),
)

# Set
i = Set(m, name="i", records=[str(idx) for idx in range(0, 181)])
omega_stop = Set(
m,
name="omega_stop",
domain=i,
records=[str(idx) for idx in range(120, 181)],
)
omega_pass = Set(
m,
name="omega_pass",
domain=i,
records=[str(idx) for idx in range(0, 91)],
)
k = Set(m, name="k", records=[str(idx) for idx in range(0, 11)])

# Parameter
beta = Parameter(m, name="beta", records=0.01)
step = Parameter(m, name="step", records=math.pi / 180)
n = Parameter(m, name="n", records=20)
omega = Parameter(m, name="omega", domain=i)
omega[i] = (Ord(i) - 1) * step

# Variable
h = Variable(m, name="h", domain=k)
t = Variable(m, name="t")
v2 = Variable(m, name="v2", description="for conic variable u - t")
v3 = Variable(
m, name="v3", type="Positive", description="for conic variable u + t"
)
u = Variable(m, name="u", type="Positive")
v = Variable(m, name="v", type="Positive")

# Equation
passband_up_bnds = Equation(m, name="passband_up_bnds", domain=i)
cone_lhs = Equation(m, name="cone_lhs")
cone_rhs = Equation(m, name="cone_rhs")
so = Equation(m, name="so")
passband_lo_bnds = Equation(m, name="passband_lo_bnds", domain=i)
stopband_bnds = Equation(m, name="stopband_bnds", domain=i)
stopband_bnds2 = Equation(m, name="stopband_bnds2", domain=i)

passband_up_bnds[i].where[omega_pass[i]] = (
2
* Sum(
k.where[Ord(k) < Card(k)],
h[k] * gams_math.cos((Ord(k) - 1 - (n - 1) / 2) * omega[i]),
)
<= t
)

cone_rhs[...] = v2 == u - t
cone_lhs[...] = v3 == u + t
so[...] = v3**2 >= v**2 + v2**2

passband_lo_bnds[i].where[omega_pass[i]] = u <= 2 * Sum(
k.where[Ord(k) < Card(k)],
h[k] * gams_math.cos((Ord(k) - 1 - (n - 1) / 2) * omega[i]),
)

stopband_bnds[i].where[omega_stop[i]] = -beta <= 2 * Sum(
k.where[Ord(k) < Card(k)],
h[k] * gams_math.cos((Ord(k) - 1 - (n - 1) / 2) * omega[i]),
)

stopband_bnds2[i].where[omega_stop[i]] = (
2
* Sum(
k.where[Ord(k) < Card(k)],
h[k] * gams_math.cos((Ord(k) - 1 - (n - 1) / 2) * omega[i]),
)
<= beta
)

t.lo[...] = 1
v.fx[...] = 2

fir_socp = Model(
m,
name="fir_socp",
equations=m.getEquations(),
problem=Problem.QCP,
sense=Sense.MIN,
objective=t,
)
fir_socp.solve()

assert math.isclose(fir_socp.objective_value, 1.0465, rel_tol=0.001)

if __name__ == "__main__":
main()
```