Cosine window function & signal with nonzero mean

Question

I'm using cosine window function of order $p$:

$$ f(t) = \frac{2^p (p!)^2}{(2 p!)} \left(1 + \cos(\pi t)\right)^p $$

for frequency identification in real signals with several frequency components.

For a simple test signal $ s(t) = 0.1 + \sin(2 \pi t \times 0.2134567)$ with $t = 1,2,\dots,4096 $ an 'artifact' is observed at low frequencies in Fourier amplitude spectra.

First, the signal mean is removed (computed numerically), then Fourier is performed and log10 of its' absolute value is plotted for different window orders.

For signals with zero mean no 'artifact' is observed. Do I understand correctly that this is due to the amplification of low-frequency components by window function? How this effect is called? And how it can be reduced?

On the figure: log10 of Fourier amplitude spectra low-frequency components (left) and main peak (right) for different window orders.

Edit

Added Mathematica code used to generate figure in the original post together with 'true' mean subtraction with reduces this effect (4 orders in this example).

My main concern was that given a signal with low-frequency main peak, it can be shifted by this window effect.

(* def window function *)
ClearAll[window] ;
window[order_,length_] := 2^order Factorial[order]^2/Factorial[2 order] (1 + Cos[2 Pi ((N[Range[length]]-1)/length - 1/2)])^order ;
(* plot window function *)
ListPlot[
    Map[Curry[window][4096],Range[4]],
    AspectRatio -> 1,
    PlotTheme -> "Detailed",
    PlotStyle -> List[Red, Blue, Black, Magenta],
    ImageSize -> 400
]

(* def test signal *)
sig = 0.1 + Sin[2 Pi Range[4096] 0.2134567] ;

(* input data (signal with removed mean value) *)
dat = sig - Mean[sig] ;

(* compute spectra *)
fou1 = Fourier[ dat window[1,4096],FourierParameters->{-1,1}] ; fou1 = Log10[Abs[fou1]] ; fou1 = Take[fou1,2048] ;
fou2 = Fourier[ dat window[2,4096],FourierParameters->{-1,1}] ; fou2 = Log10[Abs[fou2]] ; fou2 = Take[fou2,2048] ;
fou3 = Fourier[ dat window[3,4096],FourierParameters->{-1,1}] ; fou3 = Log10[Abs[fou3]] ; fou3 = Take[fou3,2048] ;
fou4 = Fourier[ dat window[4,4096],FourierParameters->{-1,1}] ; fou4 = Log10[Abs[fou4]] ; fou4 = Take[fou4,2048] ;

(* plot spectra parts *)
Legended[
    Grid[
        List[
            List[
                ListPlot[List[fou1,fou2,fou3,fou4], PlotRange -> List[List[0,10],List[0,-10]], PlotTheme -> "Detailed", Joined -> True, PlotMarkers -> Automatic, AspectRatio -> 1, PlotStyle -> List[Red, Blue, Black, Magenta], ImageSize -> 400],
                ListPlot[List[fou1,fou2,fou3,fou4],PlotRange -> List[List[860,890],List[0,-5]], PlotTheme -> "Detailed", Joined -> True, PlotMarkers -> Automatic, AspectRatio -> 1, PlotStyle -> List[Red, Blue, Black, Magenta], ImageSize -> 400]
            ]
        ]
    ],
    Placed[LineLegend[{Red,Blue,Black,Magenta},{1,2,3,4},LegendLayout->"Row"],Top]
] 

(* 'true' mean *)
ClearAll[fun] ;
fun[cor_?NumericQ] := Block[
    {},
    mea = Mean[sig] ;
    dat = sig - mea + cor ;
    dat = dat window[1,4096] ;
    fst = First[Abs[Fourier[dat,FourierParameters->{-1,1}]]]
] ;
add = cor /. Last[FindMinimum[fun[cor],{cor,0.0}]]

(* substruct true mean *)
dat = sig - (Mean[sig] - add) ;

(* compute spectra *)
fou1 = Fourier[ dat window[1,4096],FourierParameters->{-1,1}] ; fou1 = Log10[Abs[fou1]] ; fou1 = Take[fou1,2048] ;
fou2 = Fourier[ dat window[2,4096],FourierParameters->{-1,1}] ; fou2 = Log10[Abs[fou2]] ; fou2 = Take[fou2,2048] ;
fou3 = Fourier[ dat window[3,4096],FourierParameters->{-1,1}] ; fou3 = Log10[Abs[fou3]] ; fou3 = Take[fou3,2048] ;
fou4 = Fourier[ dat window[4,4096],FourierParameters->{-1,1}] ; fou4 = Log10[Abs[fou4]] ; fou4 = Take[fou4,2048] ;

(* plot spectra parts *)
Legended[
    Grid[
        List[
            List[
                ListPlot[List[fou1,fou2,fou3,fou4], PlotRange -> List[List[0,10],List[0,-10]], PlotTheme -> "Detailed", Joined -> True, PlotMarkers -> Automatic, AspectRatio -> 1, PlotStyle -> List[Red, Blue, Black, Magenta], ImageSize -> 400],
                ListPlot[List[fou1,fou2,fou3,fou4],PlotRange -> List[List[860,890],List[0,-5]], PlotTheme -> "Detailed", Joined -> True, PlotMarkers -> Automatic, AspectRatio -> 1, PlotStyle -> List[Red, Blue, Black, Magenta], ImageSize -> 400]
            ]
        ]
    ],
    Placed[LineLegend[{Red,Blue,Black,Magenta},{1,2,3,4},LegendLayout->"Row"],Top]
]

Answer 1

By definition, your signal has a zero-frequency component (DC component) with amplitude 0.1. So, it should have peak at 0. Because of windowing, the peak will have mainlobe. It will spread on bins near bin-0, i.e bin-1, bin-2. The higher p, the mainlobe will be wider (See Window function). (OK, your plot seems to have no bin-0 value?).

You can remove the DC component by substracting signal with its mean. Or for realtime signal, you can do highpass filtering (See DC bias).

For the first method, you should compute the window-weighted mean to give more accurate mean.

$$ \mu = \frac{\sum _{n} w[n] s[n]}{\sum_{n} w[n]} $$