Evaluation of Various Window Functions using Multi-Instrument

By  Dr. Wang Hongwei

REV: 01
May 8, 2009

Note: VIRTINS TECHNOLOGY reserves the right to make modifications to this document at any time without notice. This document may contain typographical errors.

TABLE OF CONTENTS

1. Spectral Leakage and Window Function
2. How to use Multi-Instrument to evaluate a Window Function
2.1 Method 1: Analyze the window function WAV file with no window function
2.2 Method 2: Analyze a unit DC signal with the window function to be evaluated
2.3 Window Function Parameters to Be Evaluated
3. Window Functions
3.1 Rectangle Window
3.2 Triangle Window
3.3 Hanning Window
3.4 Hamming Window
3.5 Blackman Window
3.6 Exact Blackman Window
3.7 Blackman-Harris (4 terms) Window
3.8 Blackman-Nuttall Window
3.9 Flat Top Window
3.10 Exponential Window (a = 0.1)
3.11 Gaussian Window (a = 2.5)
3.12 Gaussian  Window (a = 3.0)
3.13 Gaussian  Window (a = 3.5)
3.14 Welch (Riesz) Window
3.15 Cosine Window (a = 1)
3.16 Cosine Window (a = 3)
3.17 Cosine Window (a = 4)
3.18 Cosine Window (a = 5)
3.19 Riemann Window
3.20 Parzen (De La Valle-Poussin) Window
3.21 Tukey (Tapered Cosine) Window (a = 0.25)
3.22 Tukey (Tapered Cosine) Window (a = 0.50)
3.23 Tukey (Tapered Cosine) Window (a = 0.75)
3.24 Bohman Window
3.25 Poisson Window (a = 2)
3.26 Poisson Window (a = 3)
3.27 Poisson Window (a = 4)
3.28 Hanning-Poisson Window (a = 0.5)
3.29 Hanning-Poisson Window (a = 1.0)
3.30 Hanning-Poisson Window (a = 2.0)
3.31 Cauchy Window (a = 3.0)
3.32 Cauchy Window (a = 4.0)
3.33 Cauchy Window (a = 5.0)
3.34 Bartlett-Hann Window
3.35 Kaiser-Bessel Window  (a = 0.5)
3.36 Kaiser-Bessel Window  (a = 1.0)
3.37 Kaiser-Bessel Window  (a = 2.0)
3.38 Kaiser-Bessel Window  (a = 3.0)
3.39 Kaiser-Bessel Window  (a = 4.0)
3.40 Kaiser-Bessel Window  (a = 5.0)
3.41 Kaiser-Bessel Window  (a = 6.0)
3.42 Kaiser-Bessel Window  (a = 7.0)
3.43 Kaiser-Bessel Window  (a = 8.0)
3.44 Kaiser-Bessel Window  (a = 9.0)
3.45 Kaiser-Bessel Window  (a = 10.0)
3.46 Kaiser-Bessel Window  (a = 11.0)
3.47 Kaiser-Bessel Window  (a = 12.0)
3.48 Kaiser-Bessel Window  (a = 13.0)
3.49 Kaiser-Bessel Window  (a = 14.0)
3.50 Kaiser-Bessel Window  (a = 15.0)
3.51 Kaiser-Bessel Window  (a = 16.0)
3.52 Kaiser-Bessel Window  (a = 17.0)
3.53 Kaiser-Bessel Window  (a = 18.0)
3.54 Kaiser-Bessel Window  (a = 19.0)
3.55 Kaiser-Bessel Window  (a = 20.0)
4. Summary of Parameters of Window Functions

1. Spectral Leakage and Window Function

Spectral leakage is the result of the assumption in the FFT algorithm that the time record in a FFT segment is exactly repeated throughout all time and that signals contained in a FFT segment are thus periodic at intervals that correspond to the length of the FFT segment. If the time record in a FFT segment has a non-integer number of cycles, this assumption is violated and spectral leakage occurs. Spectral leakage distorts the measurement in such a way that energy from a given frequency component spreads to adjacent frequency lines or bins. In most cases, you cannot guarantee that you are sampling an integer number of cycles. Choosing a window function correctly to suppress the spectral leakage for a certain measurement is thus critical.

To choose a window function, you must guess the signal frequency content. If the signal contains strong interfering frequency components distant from the frequency of interest, choose a window with a high side lobe roll-off rate. If there are strong interfering signal near the frequency of interest, choose a window with a low highest side lobe level. If the frequency of interest contains two or more signals very near to each other, then frequency resolution is very important. It is best to choose a window with a very narrow main lobe. If the amplitude accuracy of a single frequency component is more important than the exact location of the component in a given frequency bin, choose a window with a wide main lobe. If the signal spectrum is rather flat or broadband in frequency content, use the Rectangle Window. In general, the Hanning Window has good frequency resolution and reduced spectral leakage. It is satisfactory in 95% of cases.

2. How to use Multi-Instrument to evaluate a Window Function

2.1 Method 1: Analyze the window function WAV file with no window function 

Multi-Instrument supports 55 window functions. A 1024-point 24-bit WAV file of each window function is provided in the “WAV\window” directory of the software and can be used to evaluate the behavior of each window function in the frequency domain. The sampling rate of these WAV files is 44100 Hz.

To evaluate a window function using the provided WAV file, the following changes to the system default settings for the Spectrum Analyzer are required after loading the WAV file:

The following changes to the system default settings for the Spectrum Analyzer are recommended:

This method will be used in this article to evaluate all the window functions used in Multi-Instrument.

2.2 Method 2: Analyze a unit DC signal with the window function to be evaluated

The unit DC signal can be generated by following the steps below:

To evaluate a window function using the unit DC signal, the following changes to the system default settings for the Spectrum Analyzer are required:

The following changes to the system default settings for the Spectrum Analyzer are recommended:

The advantage of this method is that the real window function used in the software is evaluated. The drawback is that the shape of the window function is not shown in the Oscilloscope.

2.3 Window Function Parameters to Be Evaluated

The spectrum of a window function is continuous with a main lobe and several side lobes (see figure below). The width of the main lobe limits the frequency resolution of a windowed signal. The ability to distinguish two closely spaced frequency components increases as the width of the main lobe of the window function decreases.  The width of the main lobe can be described by its width at –3 dB and –6 dB below the peak of the main lobe. The side lobes are characterized by the Highest Side Lobe Level and the Side Lobe Fall Off Rate.

There are some more abstract parameters for a window functions, such as Scallop Loss, Coherent Gain, Equivalent Noise Bandwidth, etc. Readers are recommended to reference to the classic paper “On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform”, Fredric J. Harris, Proceedings of The IEEE, Vol. 66, No. 1, January 1978.
    

The following parameters of each window function can be measured with the help of the cursor reader provided in Multi-Instrument. (see figure above)

The following parameters of each window function can be obtained via the mean and RMS values measured in Multi-Instrument.

3.1 Rectangle Window

w(n) = 1,     n = 0, 1, …, N-1

w(n) = n/(N/2),     n = 0,1,…,N/2;
w(n) = w(N-n),     n = N/2,…,N-1;

w(n) = sin2(np/N) = 0.5-0.5cos(2np/N),     n = 0,1,…,N-1;

3.4 Hamming Window

w(n) = 0.54 – 0.46cos(2np/N),     n = 0,1,…,N-1;

3.5 Blackman Window
w(n) = 0.42 – 0.5cos(2np/N) + 0.08cos(4np/N),     n = 0,1,…,N-1;

w(n) = 7938/18608 – 9240/18608 ´ cos(2np/N) + 1430/18608 ´ cos(4np/N)

n = 0,1,…,N-1;

w(n) = 0.35875 – 0.48829 ´ cos(2np/N) + 0.14128 ´ cos(4np/N) –0.01168 ´ cos(6np/N)

n = 0,1,…,N-1;

w(n) = 0.3635819 – 0.4891775 ´ cos(2np/N) + 0.1365995 ´ cos(4np/N) –0.0106411 ´ cos(6np/N)

n = 0,1,…,N-1;

w(n) = 0.21557895 – 0.41663158 ´ cos(2np/N) + 0.277263158 ´ cos(4np/N) – 0.083578947 ´ cos(6np/N) + 0.006947368 ´ cos(8np/N)

n = 0,1,…,N-1;

w(n) = a n/N,     0<a<1,     n = 0,1,…,N-1;

a = 0.1

w(n) = e-0.5[a (2n/N – 1)] [a (2n/N – 1)],      a > =2,     n = 0,1,…,N-1;

a = 2.5

w(n) = e-0.5[a (2n/N – 1)] [a (2n/N – 1)],     a > =2,     n = 0,1,…,N-1;

a = 3.0

w(n) = e-0.5[a (2n/N – 1)] [a (2n/N – 1)],     a > =2,     n = 0,1,…,N-1;

a = 3.5

3.14 Welch (Riesz) Window

w(n) = 1 - [2n/N-1]2,     n = 0,1,…,N-1;

w(n) = sina(np/N),     n = 0,1,…,N-1;

a = 1

w(n) = sina(np/N),     n = 0,1,…,N-1;

a = 3

w(n) = sina(np/N),    n = 0,1,…,N-1;

a = 4

w(n) = sina(np/N),    n = 0,1,…,N-1;

a = 5

w(n) = sin[(2n/N-1) p]/[(2n/N-1) p],    n = 0,1,…,N-1;

w = 1 - 6(2n/N-1)2(1-|2n/N-1|)                          0 <= |n-N/2| <= (N/4)
w = 2(1-|2n/N-1|)3                                      (N/4) <= |n-N/2| <= (N/2)

n = 0,1,…,N-1;

w = 0.5{1 - cos[2pn/(aN)]}                                                          n < (aN/2)
w = 1.0                                                                         (aN/2) <= n <= N-(aN/2)
w = 0.5{1 - cos[2p/a - 2pn/(aN)]}                                               n > N-(aN/2) 

n = 0,1,…,N-1;

a = 0.25

w = 0.5{1 - cos[2pn/(aN)]}                                                          n < (aN/2)
w = 1.0                                                                         (aN/2) <= n <= N-(aN/2)
w = 0.5{1 - cos[2p/a - 2pn/(aN)]}                                               n > N-(aN/2) 

n = 0,1,…,N-1;

a = 0.50

w = 0.5{1 - cos[2pn/(aN)]}                                                          n < (aN/2)
w = 1.0                                                                         (aN/2) <= n <= N-(aN/2)
w = 0.5{1 - cos[2p/a - 2pn/(aN)]}                                               n > N-(aN/2) 

n = 0,1,…,N-1;

a = 0.75

3.24 Bohman Window

w = (1 - |2n/N - 1|)cos(p|2n/N - 1|) + sin(p|2n/N - 1|)/p,     n = 0,1,…,N-1;

w = e(-a|2n/N-1|),     n = 0,1,…,N-1;

a = 2

w = e(-a|2n/N-1|),     n = 0,1,…,N-1;

a = 3

w = e(-a|2n/N-1|),    n = 0,1,…,N-1;

a = 4

w = [0.5-0.5cos(2np/N)]e(-a|2n/N-1|),     n = 0,1,…,N-1;

a = 0.5

w = [0.5-0.5cos(2np/N)]e(-a|2n/N-1|),    n = 0,1,…,N-1;

a = 1.0

w = [0.5-0.5cos(2np/N)]e(-a|2n/N-1|),    n = 0,1,…,N-1;

a = 2.0

w = [1 + a(2n/N - 1)2]-1,    n = 0,1,…,N-1;

a = 3.0

w = [1 + a(2n/N - 1)2]-1,    n = 0,1,…,N-1;

a = 4.0

w = [1 + a(2n/N - 1)2]-1,     n = 0,1,…,N-1;

a = 5.0

3.34 Bartlett-Hann Window

w = 0.62-0.48|n/N-0.5|-0.38cos(2np/N),     n = 0,1,…,N-1;

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 0.5

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 1.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 2.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 3.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 4.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where: bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 5.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 6.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 7.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 8.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 9.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 10.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 11.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 12.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 13.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 14.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 15.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 16.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 17.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 18.0

3.54 Kaiser-Bessel Window  (a = 19.0)

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 19.0

w=bessi0[ap(1 - (2n/N - 1)2)0.5 ]/bessi0[ap]

where:
bessi0 is the zero-order modified Bessel function of the first kind.
bessel0(x) = S [(x/2)k / k!]2, where k= 0 ~ ¥.
n = 0,1,…,N-1;

a = 20.0

4. Summary of Parameters of Window Functions

The following steps can be used to compare the characteristics of different window functions graphically (see figure below):

1. Use “File Open” to open the WAV file of Rectangle window.
2. Use “File Combine” to open the WAV file of another window.
3. Set the settings for the Spectrum Analyzer properly as mentioned before
4. Push both curves in the Oscilloscope and Spectrum Analyzer to reference
5. Use “File Combine” to import another two WAV files
6. Push both curves in the Oscilloscope and Spectrum Analyzer to reference
7. Repeat 5.