/usr/share/tcltk/tcllib1.18/math/fourier.tcl is in tcllib 1.18-dfsg-3.
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# Package for discrete (ordinary) and fast fourier transforms
#
# Author: Lars Hellstrom (...)
#
# The two top-level procedures defined are
#
# dft data-list
# inverse_dft data-list
#
# which take a list of complex numbers and apply a Discrete Fourier
# Transform (DFT) or its inverse respectively to these lists of numbers.
# A "complex number" in this case is either (i) a pair (two element
# list) of numbers, interpreted as the real and imaginary parts of the
# complex number, or (ii) a single number, interpreted as the real
# part of a complex number whose imaginary part is zero. The return
# value is always in the first format. (The DFT generally produces
# complex results even if the input is purely real.) Applying first
# one and then the other of these procedures to a list of complex
# numbers will (modulo rounding errors due to floating point
# arithmetic) return the original list of numbers.
#
# If the input length N is a power of two then these procedures will
# utilize the O(N log N) Fast Fourier Transform algorithm. If input
# length is not a power of two then the DFT will instead be computed
# using a the naive quadratic algorithm.
#
# Some examples:
#
# % dft {1 2 3 4}
# {10 0.0} {-2.0 2.0} {-2 0.0} {-2.0 -2.0}
# % inverse_dft {{10 0.0} {-2.0 2.0} {-2 0.0} {-2.0 -2.0}}
# {1.0 0.0} {2.0 0.0} {3.0 0.0} {4.0 0.0}
# % dft {1 2 3 4 5}
# {15.0 0.0} {-2.5 3.44095480118} {-2.5 0.812299240582} {-2.5 -0.812299240582} {-2.5 -3.44095480118}
# % inverse_dft {{15.0 0.0} {-2.5 3.44095480118} {-2.5 0.812299240582} {-2.5 -0.812299240582} {-2.5 -3.44095480118}}
# {1.0 0.0} {2.0 8.881784197e-17} {3.0 4.4408920985e-17} {4.0 4.4408920985e-17} {5.0 -8.881784197e-17}
#
# In the last case, the imaginary parts <1e-16 would have been zero in
# exact arithmetic, but aren't here due to rounding errors.
#
# Internally, the procedures use a flat list format where every even
# index element of a list is a real part and every odd index element is
# an imaginary part. This is reflected in the variable names by Re_ and
# Im_ prefixes.
#
namespace eval ::math::fourier {
#::math::constants pi
namespace export dft inverse_dft lowpass highpass
}
# dft --
# Return the discrete fourier transform as a list of complex numbers
#
# Arguments:
# in_data List of data (either real or complex)
# Returns:
# List of complex amplitudes for the Fourier components
# Note:
# The procedure uses an ordinary DFT if the number of data is
# not a power of 2, otherwise it uses FFT.
#
proc ::math::fourier::dft {in_data} {
# First convert to internal format
set dataL [list]
set n 0
foreach datum $in_data {
if {[llength $datum] == 1} then {
lappend dataL $datum 0.0
} else {
lappend dataL [lindex $datum 0] [lindex $datum 1]
}
incr n
}
# Then compute a list of n'th roots of unity (explanation below)
set rootL [DFT_make_roots $n -1]
# Check if the input length is a power of two.
set p 1
while {$p < $n} {set p [expr {$p << 1}]}
# By construction, $p is a power of two. If $n==$p then $n is too.
# Finally compute the transform using Fast_DFT or Slow_DFT,
# and convert back to the input format.
set res [list]
foreach {Re Im} [
if {$p == $n} then {
Fast_DFT $dataL $rootL
} else {
Slow_DFT $dataL $rootL
}
] {
lappend res [list $Re $Im]
}
return $res
}
# inverse_dft --
# Invert the discrete fourier transform and return the restored data
# as complex numbers
#
# Arguments:
# in_data List of fourier coefficients (either real or complex)
# Returns:
# List of complex amplitudes for the Fourier components
# Note:
# The procedure uses an ordinary DFT if the number of data is
# not a power of 2, otherwise it uses FFT.
#
proc ::math::fourier::inverse_dft {in_data} {
# First convert to internal format
set dataL [list]
set n 0
foreach datum $in_data {
if {[llength $datum] == 1} then {
lappend dataL $datum 0.0
} else {
lappend dataL [lindex $datum 0] [lindex $datum 1]
}
incr n
}
# Then compute a list of n'th roots of unity (explanation below)
set rootL [DFT_make_roots $n 1]
# Check if the input length is a power of two.
set p 1
while {$p < $n} {set p [expr {$p << 1}]}
# By construction, $p is a power of two. If $n==$p then $n is too.
# Finally compute the transform using Fast_DFT or Slow_DFT,
# divide by input data length to correct the amplitudes,
# and convert back to the input format.
set res [list]
foreach {Re Im} [
# $p is power of two. If $n==$p then $n is too.
if {$p == $n} then {
Fast_DFT $dataL $rootL
} else {
Slow_DFT $dataL $rootL
}
] {
lappend res [list [expr {$Re/$n}] [expr {$Im/$n}]]
}
return $res
}
# DFT_make_roots --
# Return a list of the complex roots of unity or of -1
#
# Arguments:
# n Order of the roots
# sign Whether to use 1 or -1 (for inverse transform)
# Returns:
# List of complex roots of unity or -1
#
proc ::math::fourier::DFT_make_roots {n sign} {
set res [list]
for {set k 0} {2*$k < $n} {incr k} {
set alpha [expr {2*3.1415926535897931*$sign*$k/$n}]
lappend res [expr {cos($alpha)}] [expr {sin($alpha)}]
}
return $res
}
# Fast_DFT --
# Perform the fast Fourier transform
#
# Arguments:
# dataL List of data
# rootL Roots of unity or -1 to use in the transform
# Returns:
# List of complex numbers
#
proc ::math::fourier::Fast_DFT {dataL rootL} {
if {[llength $dataL] == 8} then {
foreach {Re_z0 Im_z0 Re_z1 Im_z1 Re_z2 Im_z2 Re_z3 Im_z3} $dataL {break}
if {[lindex $rootL 3] > 0} then {
return [list\
[expr {$Re_z0 + $Re_z1 + $Re_z2 + $Re_z3}] [expr {$Im_z0 + $Im_z1 + $Im_z2 + $Im_z3}]\
[expr {$Re_z0 - $Im_z1 - $Re_z2 + $Im_z3}] [expr {$Im_z0 + $Re_z1 - $Im_z2 - $Re_z3}]\
[expr {$Re_z0 - $Re_z1 + $Re_z2 - $Re_z3}] [expr {$Im_z0 - $Im_z1 + $Im_z2 - $Im_z3}]\
[expr {$Re_z0 + $Im_z1 - $Re_z2 - $Im_z3}] [expr {$Im_z0 - $Re_z1 - $Im_z2 + $Re_z3}]]
} else {
return [list\
[expr {$Re_z0 + $Re_z1 + $Re_z2 + $Re_z3}] [expr {$Im_z0 + $Im_z1 + $Im_z2 + $Im_z3}]\
[expr {$Re_z0 + $Im_z1 - $Re_z2 - $Im_z3}] [expr {$Im_z0 - $Re_z1 - $Im_z2 + $Re_z3}]\
[expr {$Re_z0 - $Re_z1 + $Re_z2 - $Re_z3}] [expr {$Im_z0 - $Im_z1 + $Im_z2 - $Im_z3}]\
[expr {$Re_z0 - $Im_z1 - $Re_z2 + $Im_z3}] [expr {$Im_z0 + $Re_z1 - $Im_z2 - $Re_z3}]]
}
} elseif {[llength $dataL] > 8} then {
set evenL [list]
set oddL [list]
foreach {Re_z0 Im_z0 Re_z1 Im_z1} $dataL {
lappend evenL $Re_z0 $Im_z0
lappend oddL $Re_z1 $Im_z1
}
set squarerootL [list]
foreach {Re_omega0 Im_omega0 Re_omega1 Im_omega1} $rootL {
lappend squarerootL $Re_omega0 $Im_omega0
}
set lowL [list]
set highL [list]
foreach\
{Re_y0 Im_y0} [Fast_DFT $evenL $squarerootL]\
{Re_y1 Im_y1} [Fast_DFT $oddL $squarerootL]\
{Re_omega Im_omega} $rootL {
set Re_y1t [expr {$Re_y1 * $Re_omega - $Im_y1 * $Im_omega}]
set Im_y1t [expr {$Im_y1 * $Re_omega + $Re_y1 * $Im_omega}]
lappend lowL [expr {$Re_y0 + $Re_y1t}] [expr {$Im_y0 + $Im_y1t}]
lappend highL [expr {$Re_y0 - $Re_y1t}] [expr {$Im_y0 - $Im_y1t}]
}
return [concat $lowL $highL]
} elseif {[llength $dataL] == 4} then {
foreach {Re_z0 Im_z0 Re_z1 Im_z1} $dataL {break}
return [list\
[expr {$Re_z0 + $Re_z1}] [expr {$Im_z0 + $Im_z1}]\
[expr {$Re_z0 - $Re_z1}] [expr {$Im_z0 - $Im_z1}]]
} else {
return $dataL
}
}
# Slow_DFT --
# Perform the ordinary discrete (slow) Fourier transform
#
# Arguments:
# dataL List of data
# rootL Roots of unity or -1 to use in the transform
# Returns:
# List of complex numbers
#
proc ::math::fourier::Slow_DFT {dataL rootL} {
set n [expr {[llength $dataL] / 2}]
# The missing roots are computed by complex conjugating the given
# roots. If $n is even then -1 is also needed; it is inserted explicitly.
set k [llength $rootL]
if {$n % 2 == 0} then {
lappend rootL -1.0 0.0
}
for {incr k -2} {$k > 0} {incr k -2} {
lappend rootL [lindex $rootL $k]\
[expr {-[lindex $rootL [expr {$k+1}]]}]
}
# This is strictly following the naive formula.
# The product jk is kept as a separate counter variable.
set res [list]
for {set k 0} {$k < $n} {incr k} {
set Re_sum 0.0
set Im_sum 0.0
set jk 0
foreach {Re_z Im_z} $dataL {
set Re_omega [lindex $rootL [expr {2*$jk}]]
set Im_omega [lindex $rootL [expr {2*$jk+1}]]
set Re_sum [expr {$Re_sum +
$Re_z * $Re_omega - $Im_z * $Im_omega}]
set Im_sum [expr {$Im_sum +
$Im_z * $Re_omega + $Re_z * $Im_omega}]
incr jk $k
if {$jk >= $n} then {set jk [expr {$jk - $n}]}
}
lappend res $Re_sum $Im_sum
}
return $res
}
# lowpass --
# Apply a low-pass filter to the Fourier transform
#
# Arguments:
# cutoff Cut-off frequency
# in_data Input transform (complex data)
# Returns:
# Filtered transform
#
proc ::math::fourier::lowpass {cutoff in_data} {
package require math::complexnumbers
set res [list]
set cutoff [list $cutoff 0.0]
set f 0.0
foreach a $in_data {
set an [::math::complexnumbers::/ $a \
[::math::complexnumbers::+ {1.0 0.0} \
[::math::complexnumbers::/ [list 0.0 $f] $cutoff]]]
lappend res $an
set f [expr {$f+1.0}]
}
return $res
}
# highpass --
# Apply a high-pass filter to the Fourier transform
#
# Arguments:
# cutoff Cut-off frequency
# in_data Input transform (complex data)
# Returns:
# Filtered transform (high-pass)
#
proc ::math::fourier::highpass {cutoff in_data} {
package require math::complexnumbers
set res [list]
set cutoff [list $cutoff 0.0]
set f 0.0
foreach a $in_data {
set ff [::math::complexnumbers::/ [list 0.0 $f] $cutoff]
set an [::math::complexnumbers::/ $ff \
[::math::complexnumbers::+ {1.0 0.0} $ff]]
lappend res $an
set f [expr {$f+1.0}]
}
return $res
}
#
# Announce the package
#
package provide math::fourier 1.0.2
# test --
#
proc test_dft {points {real 0} {iterations 20}} {
set in_dataL [list]
for {set k 0} {$k < $points} {incr k} {
if {$real} then {
lappend in_dataL [expr {2*rand()-1}]
} else {
lappend in_dataL [list [expr {2*rand()-1}] [expr {2*rand()-1}]]
}
}
set time1 [time {
set conv_dataL [::math::fourier::dft $in_dataL]
} $iterations]
set time2 [time {
set out_dataL [::math::fourier::inverse_dft $conv_dataL]
} $iterations]
set err 0.0
foreach iz $in_dataL oz $out_dataL {
if {$real} then {
foreach {o1 o2} $oz {break}
set err [expr {$err + ($i-$o1)*($i-$o1) + $o2*$o2}]
} else {
foreach i $iz o $oz {
set err [expr {$err + ($i-$o)*($i-$o)}]
}
}
}
return [format "Forward: %s\nInverse: %s\nAverage error: %g"\
$time1 $time2 [expr {sqrt($err/$points)}]]
}
# Note:
# Add simple filters
if { 0 } {
puts [::math::fourier::dft {1 2 3 4}]
puts [::math::fourier::inverse_dft {{10 0.0} {-2.0 2.0} {-2 0.0} {-2.0 -2.0}}]
puts [::math::fourier::dft {1 2 3 4 5}]
puts [::math::fourier::inverse_dft {{15.0 0.0} {-2.5 3.44095480118} {-2.5 0.812299240582} {-2.5 -0.812299240582} {-2.5 -3.44095480118}}]
puts [test_dft 10]
puts [test_dft 16]
puts [test_dft 100]
puts [test_dft 128]
puts [::math::fourier::dft {1 2 3 4}]
puts [::math::fourier::lowpass 1.5 [::math::fourier::dft {1 2 3 4}]]
}
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