```
clear all
% create the "price" vector
period = input( 'Enter period between 10 & 50 for underlying sine: ' ) ;
underlying_sine = sinewave( 500 , period ) ; % create underlying sinewave function with amplitude 2
trend_mult = input( 'Enter value for trend (a number between -4 & 4 incl.) to 4 decimal places: ' ) ;
trend = ( trend_mult / period ) .* (0:1:499) ;
noise_mult = input( 'Enter value for noise multiplier (a number between 0 & 1 incl.) to 2 decimal places: ' ) ;
noise = noise_mult * randn(500,1)' ;
smooth_price = underlying_sine .+ trend ;
price = underlying_sine .+ trend .+ noise ;
super_smooth = super_smoother( price ) ;
smooth = price ;
mean_value = price ;
mean_smooth = price ;
for ii = period+1 : length( price )
mean_value(ii) = mean( price( ii-period : ii ) ) ;
fft_input = price( ii-(period-1) : ii ) .- mean_value(ii) ;
N = length( fft_input ) ;
data_in_freq_domain = fft( fft_input ) ;
if mod( N , 2 ) == 0 % N even
% The DC component is unique and should not be altered
% adjust value of element N / 2 to compensate for it being the sum of frequencies
data_in_freq_domain( N / 2 ) = abs( data_in_freq_domain( N / 2 ) ) / 2 ;
[ max1 , ix1 ] = max( abs( data_in_freq_domain( 2 : N / 2 ) ) ) ; % get the index of the first max value
ix1 = ix1 + 1 ;
[ max2 , ix2 ] = max( abs( data_in_freq_domain( N / 2 + 1 : end ) ) ) ; % get index of the second max value
ix2 = N / 2 + ix2 ;
non_zero_values_ix = [ 1 ix1 ix2 ] ; % indices for DC component, +ve freq max & -ve freq max
else % N odd
% The DC component is unique and should not be altered
[ max1 , ix1 ] = max( abs( data_in_freq_domain( 2 : ( N + 1 ) / 2 ) ) ) ; % get the index of the first max value
ix1 = ix1 + 1 ;
[ max2 , ix2 ] = max( abs( data_in_freq_domain( ( N + 1 ) / 2 + 1 : end ) ) ) ; % get index of the second max value
ix2 = ( N + 1 ) / 2 + ix2 ;
non_zero_values_ix = [ 1 ix1 ix2 ] ; % indices for DC component, +ve freq max & -ve freq max
end
% extract the constant term and frequencies of interest
non_zero_values = data_in_freq_domain( non_zero_values_ix ) ;
% now set all values to zero
data_in_freq_domain( 1 : end ) = 0.0 ;
% replace the non_zero_values
data_in_freq_domain( non_zero_values_ix ) = non_zero_values ;
inv = ifft( data_in_freq_domain ) ; % reverse the FFT
recovered_sig = real( inv ) ; % recover the real component of ifft for plotting
smooth( ii ) = recovered_sig( end ) + mean_value( ii ) ;
mean_smooth( ii ) = mean( smooth( ii-period : ii ) ) ;
end
smooth = smooth .+ ( mean_value .- mean_smooth ) ;
plot( smooth_price , 'r' ,price , 'c' , smooth , 'm' , super_smooth , 'g' )
legend( 'smooth price' , 'price' , 'fft smooth' , 'super smooth' )
```

This script prompts for terminal input for the values for an underlying sine wave period, a trend multiplier and a noise multiplier component to construct the "price" time series. Here is a plot of output with the settings- period = 20
- trend multiplier = 1.3333
- noise multiplier = 0.0

where the underlying smooth "true" signal can now be seen in red. I think the performance of the magenta FFT smoother in recovering this "true" signal speaks for itself. My next post will show its performance on real price data.

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