Calculate windowing correction factor

When using windows on my signal, I want to use a correction factor to correct either the energy or the amplitude for the window. When using the ratio mean(rectangular window) / mean(modified window), where mean(rectwin) = 1, to calculate the amplitude correction factor (ACF), it works well for both the Hanning and Hamming window, if I compare the results with the table in this link. But for the flattop window and the Blackman window, the results are not the same. Same goes for the energy correction factor (ECF) if I use the ratio rms(rectangular window) / rms(modified window), where rms(rectwin) = 1.

This is the code I use to calculate the correction factors

N = 2^10;
w = hanning(N);

rms_w = rms(w);
ECF = 1/rms_w

mean_w = mean(w);
ACF = 1/mean_w

This gives me ECF = 1.6322 and ACF = 1.9980 for the Hanning window, which is the same values as the table (ECF = 1.63 and ACF = 2). So I might say I'm doing the calculation right. But for the flat top window, using w = flattopwin(N), the results are ECF = 2.3901 and ACF = 4.6432, which should be ECF = 2.26 and ACF = 4.18. But when facing the flat top window, it has negative values as well. I am not sure if I have to use this negative values or the absolute values for calculating the ACF. When using w = abs(w), ACF = 3.9810, which is still not the same as the table. And besides that, I cannot substantiate to use the absolute value. The values for the ECF and ACF when using the blackman window by using the command w = blackman(N) gives ECF = 1.8128 and ACF = 2.3833, which should be ECF = 1.97 and ACF = 2.80.

Are the flat top window and blackman window used in MATLAB, the same as used in this table?

What am I doing wrong here?

Can someone help me please?

[acegi bdegi]

I have written some other code to calculate both the energy and amplitude factors and ratio. See here:

% signal
dt = 0.001;
fs = 1/dt;
x = 0:dt:10;
N = length(x);
y = 3*sin(2*pi*x*52) + 5.2*sin(2*pi*x*53) + 14*randn(size(x));

figure(1)
subplot(3,1,1)
plot(x,y)
title('signal')

rms_y = rms(y);

% window
w = hanning(N);   w = w';

rms_w = rms(w);
mean_w = mean(w);

ECF = 1/rms_w; % Energy correctionfactor
ACF = 1/mean_w; % Amplitude correctionfactor

% windowed signal
ws = (w.*y);

rms_ws = rms(ws); 

% fft signal
Z = 100*N; % add zeros
f = fs*(0:Z/2)/Z;

Yfft = fft(y,Z);
Y2 = abs(Yfft)/N;
Y = Y2(1:Z/2+1);
Y(2:end-1) = 2*Y(2:end-1);

figure(1)
subplot(3,1,2)
plot(f,Y)
title('signal frequency spectrum')
xlim([48 57]);

rms_Y = rms(Y);

% fft windowed signal
Yfftws = fft(ws,Z);
Y6 = abs(Yfftws)/N;
Y6 = ECF*Y6;                 % apply correctionfactor
Y5 = Y6(1:Z/2+1);
Y5(2:end-1) = 2*Y5(2:end-1);

figure(1)
subplot(3,1,3)
plot(f,Y5)
title('windowed frequency spectrum with correction factor')
xlim([48 57]);

rms_Yws = rms(Y5);

amplitude_Y = max(Y);
amplitude_Y_windowed = max(Y5);

ratio_amplitude = amplitude_Y/amplitude_Y_windowed
ratio_energy = rms_Y/rms_Yws

If I change ECF to ACF at the apply correction factor, it corrects the amplitude factor. If I change the window to any window, like blackman or flattopwin, the ratio of either the energy or the amplitude is approximately 1, as it should. With the ACF applied, both peaks are the same value for normal FFT plot and windowed FFT plot, as it should.

But still, for the flattopwin, ECF = 2.3891 and ACF = 4.6391, which is not the same as in the table. Same for blackman: ECF and ACF does not match.

Do I calculate both the ECF and ACF right this way?

[acegi bdegi]

As I've seen that almost no post about window correction factor is answered, can I state that there is a lack of knowledge about this subject?

[Christoph F.]

After a few experiments, my guess would be that the values given in the table are not correct. The site doesn't state how they were calculated.

Generate a few test signals, fft them with and without windows, and see if the factors given in the table to what they are supposed to do.

[acegi bdegi]

Thanks, but it's not an evidence but a guess. A guess is not enough for my final report. I showed in my script that the rms value of the FFT of the signal is nearly the same (~1.0 ratio) as the rms value of the corrected windowed FFT of the signal. Since rms value stands for the energy of the graph, can I state that the correction factor is calculated correctly? If yes, the values in the table are indeed not correct and I have proved that the energy (rms) is the same after applying the windowed correction factor and therefore the window correction factor is correct.

Answer 1

[Ameen Muhammed]

According to [1] eq. 9.23 amplitude correction of window effect is give by

Check [1] for proof.

[1] Brandt, A., 2011. Noise and vibration analysis: signal analysis and experimental procedures. John Wiley & Sons.

This simply evaluates to

ns = 1000;
w = hann(ns);
Aw=length(w)/sum(w);

This equation gives slightly different results than values reported in Siemens PLM blog

The equation above gives the following values

Window | Amplitude Correction

------------------------------------------------------

Hanning | 2.0020

Flattop | 4.6433

blackman | 2.3833

Hamming | 1.8534

[Michael]

Good answer.

Just be aware in relation to reference [1] that one of the formulae shown later for power correction of windowing contains a typo that has not yet been picked up in the errata for the book.

Equation 9.33 of [1] wrongly defines the window normalised equivalent noise bandwidth as:

which should of course be

ie the window coefficients in the denominator are not squared before summing, only after.

1. Brandt, A, 2011. Noise and vibration analysis: signal analysis and experimental procedures. John Wiley & Sons

Answer 2

[Christian Hildebrand]

I stumbled across this (old) question when doing research on window functions, hope someone finds this information helpful:

You have to correct the amplitude of the windowed time-signal because the window function introduces a constant offset (this is what the equation from Ameen does). Can be read in [p. 195, 1]. If you look at the help pages from matlab, e.g. https://de.mathworks.com/help/signal/ref/flattopwin.html for the Flat Top Window, you see that there is the constant coefficient , in this case 0.21557895. So for correct amplitudes you just have to multiply the time signal with .

The help pages always state the algorithm used for the window, so you should look there what constant value is used and use that in your code.

Example for Flat-Top Window:

%Create Signal
t=0:0.0001:0.1
y=0.5*square(2*pi*30e3*t) + 0.5;
%Create Flat-Top Window and apply
w = flattopwin(length(y))';
y = w.*y;
%Amplitude-Correction
y = y.*1/0.21557895;

[1]: M. Meyer. Signalverarbeitung: Analoge und digitale Signale, Systeme und Filter. Vieweg+Teubner Verlag, 2014.

Answer 3

[David Goodmanson]

Hi acegi,

I don't think Christoph was guessing. Most of these window functions have adjustable parameters, subject to change. I would say that the differences are due to incomplete description of what you are operating on. Plus, you get to adjust the number of points you are doing the calculation with.

For example, the Blackman window on the Siemens site is probably the Blackman-Nuttall window (see 'window function wiki') which for a large number of points gives

acf = 2.7507 ...... ecf = 1.9567

but with 64 points this can be coaxed up to

acf = 2.7940 ...... ecf = 1.9720

Answer 4

[Sudhar Ekanayake]

Math Work has very poor documentation on this. Few things I am still having issues.

1) First you multiply your PSD by factor of 2.00 to take into account the effect of Nyquist Frequency. (I believe for pwelch, we don't need to multiply by 2.00, it is already accounted in the function ???)

2)Secondly if we use Hann window, we need to multiply by 1.5 (ENBW - Equivalent Noise Band Width) to accommodate the amplitude error OR multiply by 2.0 to accommodate the amplitude error ( see the figure)

3)Do we need to multiply 1.63 energy correction factor when calculating RMS ?

Are the ACF or ECF factors are the same despite the total signal length ? Or it is depended on the signal length and the window size ? Have I understood this correctly ?