Windows Connections. Preliminaries

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1 Windows Connections Dale B. Dalrymple Next Annual comp.dsp Conference Corrections Preliminaries The approach in this presentation Take aways Window types Window relationships Windows tables of information Windows references What are windows? Real, Finite, Possess DFT How do we look at windows? Literally: DFT Figuratively: Parameters 1

2 Window Parameters Parameter Units Definition References Coherent Gain ratio sum of normalized window harris divided by window length EN Effective Noise Bandwidth bins sum of the squares of the window coefficients divided by the square of the sum harris Scalloping Loss minimum possible value of the maximum gain of the adjacent bins for a tone not on bin harris WCPL Worst Case Processing Loss -3 Bandwidth -6 Bandwidth -6 Bandwidth 1 st Zero Crossing 1 st Sidelobe Level 1 st Sidelobe Frequency Maximum Sidelobe Level Maximum Sidelobe Bandwidth SLRO Sidelobe rolloff center scalloping loss minus the EN (in ) harris bins bandwidth containing highest frequency response exceeding -3 harris bins bandwidth containing highest harris frequency response exceeding -6 bins bandwidth containing highest frequency response exceeding -6 bins frequency of first zero crossing defining the edge of the mainlobe re: DC response at peak of first sidelobe fraction of frequency of peak of first Fs sidelobe re: DC harris highest sidelobe level harris fraction of Fs /octave width of main lobe at level of, parameter b in G&Y harris sidelobe falloff minus the parameter d in G&Y harris G&Y 2

3 Parameter Units Definition References G&Y Bin 1 Intercept similar to a2 in G&Y, G&Y altered to bin 1 using SLRO Window Pedistal value at first sample, even FFT -3 Efficiency fractional (%/1) fraction of total energy of response within -3-6 Efficiency fractional (%/1) MRE Mainlobe Remaining Energy fractional (%/1) Gecklili, N. C., and Yarns, D., "Some Novel Windows and a Concise Tutorial Comparison of Window Families", IEEE Trans. Acoust. Speech Sig. Proc, ASSP-26, pp , Dec bandwidth fraction of total energy of response within -6 bandwidth harris, f.j., "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform Proc. IEEE, 66, pp , January A portion of this is available (2144) at: fraction of energy outside of mainlobe 3

4 Rectangular Window Figure 1 Window: Rectangular 256 points DFT: 256 bins EN: 1 Coherent Gain: 1 Pedistal: Amplitude Time Window in Samples Linear in Time 21 Dale B. Dalrymple Full Main Lobe: 2 Max.SideLobe: SideLobe Slope: 6 /Oct. Response in 5 1 Response in Mainlobe and Adjacent Region, Linear in Frequency, Negative values in red WCAE: :.89 6 : 1.22 Scalloping Loss: Center of Mainlobe to First Half Bin, Linear in Frequency 21 Dale B. Dalrymple 1 This is what you get with no window. Top plot: Time domain window Middle plot: Frequency domain plot (linear in frequency): the sinc function. Zeros at integer offsets from bin center. This is convenient for the most common DFT definition. Bottom plot: Bin top response, bin center to mid-point between bins shows scalloping loss. 4

5 Rectangular Window Frequency Response Figure 2 Rectangular EN: 1 Full ML: 2 Max.SL: Window: 256pt DFT: 256bin SL Slope: 6 /Oct. Bin 1 Int pt 2 4 Magnitude: red Remaining Energy: green: Rectangular Half Width in, Logarithmic in Frequency 21 Dale B. Dalrymple This is what you get with no window. This is a sinc function with a mainlobe width of 2 bins and zeros at (nonzero) integer offsets from zero frequency 5

6 Cosine Sum Windows von Hann Maximim sidelobe rolloff Odd Even 2 [1 1]/2 => in Freq Dom [3 1]/4 => infreqdom 3 [3 4 1]/8 [1 5 1]/16 4 [ ]/32 [ ]/64 5 [ ]/128 [ ]/256 6 [ ]/512 [ ]/124 7 [ ]/248 [ ]/496 8 [ ]/8192 [ ]/16384 Hamming Minimum sidelobe Odd 2 [ ] 3 [ e-2] 4 [ e-2] 5 [ e e-3] 6 [ e e e-4] 7 [ e e e e-5] 8 [ e e e e e-6] 6

7 von Hann Window Frequency Response Figure 3 VonHann EN: 1.5 Full ML: 4 Max.SL: Window: 256pt DFT: 256bin SL Slope: 18.1 /Oct. Bin 1 Int pt 2 4 Magnitude: red Remaining Energy: green: VonHann Half Width in, Logarithmic in Frequency 21 Dale B. Dalrymple 7

8 Maximum Sidelobe Rolloff Table 21 Dale B. Dalrymple (248 PtFFT) Coherent Gain EN Scalloping Loss WCPL -3 Param: /Oct 2, , , , , , , , , , , , , , Level SLRO Bin1 Intercept This table alternates odd and even cosine sum windows. 8

9 Maximum Sidelobe Rolloff Frequency Responses Figure 4 Maximum Rolloff 2 (Von Hann) to 8 Coefficients 5 1 Magnitude Half Beam Width in 21 Dale B. Dalrymple These are all odd cosine sum maximum sidelobe rolloff windows. (Median filter has been used to remove clutter of negative tails.) 9

10 Hamming Window Frequency Response: Exact Hamming Figure 5 HammingEx EN: Full ML: 4 Max.SL: Window: 256pt DFT: 256bin SL Slope: 6 /Oct. Bin 1 Int pt 2 4 Magnitude: red Remaining Energy: green: HammingEx Half Width in, Logarithmic in Frequency 21 Dale B. Dalrymple 1

11 Minimim Sidelobe Table 21 Dale B. Dalrymple (248 Pt FFT) Coherent Gain EN Scalloping Loss WCPL -3 Param: /Oct Level SLRO Bin1 Intercept 11

12 Minimum Sidelobe Frequency Responses Figure 6 Minimum Sidelobe Cosine Sum Windows 2, 3, 4, 5, 6, 7, 8, 9, 1, 11 Coefficients 5 1 Magnitude Half Beam Width in 21 Dale B. Dalrymple These are all odd cosine sum windows. (Median filter has been used to remove clutter of negative tails.) 12

13 How can we reduce sidelobes by adding additional components to the sinc function? Blackman Contribution: zero sidelobe centers, odd number of frequency domain terms Solve a linear system of equations to normalize and place zeros at centers of highest sidelobes to reduce the height of the sidelobes. R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra, New York: Dover, 1958 Malocha and Bishop Contribution: even and odd terms Even numbers of terms provide frequency response intermediate between odd term responses. The even derived responses represent frequencies between original bin centers. D. C. Malocha and C. D. Bishop, The classical truncated cosine series functions and applications to SAW filters, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. UFFC-34, pp 75-85, Jan Kulkarni and Lahiri Contribution: sidelobe peak zeros The sinc function is the product of a sine wave and a hyperbola. The hyperbolic function moves the peaks away from the center of the sidelobes. Solve a linear system of equations to normalize and place zeros at peaks of highest sidelobes. (Only examined first sidelobe response.) R. G. Kulkarni and S. K. Lahiri, Improved sidelobe performance of cosine series functions, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 46, pp , Mar

14 Generalized Blackman Table 21 Dale B. Dalrymple (248 PtFFT) Param: Coherent Gain EN Scalloping Loss WCPL st SL Level Level SLRO /Oct Bin1 Intercept 2,1,B ,1,K ,1,B ,1,K ,1,B ,1,K ,1,B ,1,K ,1,B ,1,K ,,B ,,K ,,B ,,K ,,B ,,K ,,B ,,K ,,B ,,K This table includes odd (at top) and even (at bottom) cosine sum windows. Window parameter B indicates Blackman (lobe center zero), K indicates Kulkarni (sinc lobe peak zero). 14

15 - 2, 3 and 4 Term Cosine Sum Table 21 Dale B. Dalrymple (248 PtFFT) Param: Coherent Gain EN Scalloping Loss WCPL -3-6 Level SLRO /Oct Bin1 Intercept Rectangular von Hann HammingEx HammingRnd CS CS3C1D CS3D3D CS3min CS3minharris CS CS4C1D CS4C3D CS4C5D CS4min CS4minharris Blackman3Ex BlackmanNVSP Blackman3Rnd Notes on bold areas: CS3C1D this is an improved replacement if someone told you is called Blackman and you like it is the best sidelobe rejection for a 4 term window, is the best that 4-term Blackman-harris can do. 4-term Blackman-harris is not a minimum sidelobe window is the correct sidelobe rejection for 2 digit rounded 3-term Blackman which Blackman and Tukey called Blackman s not very serious proposal. The harris paper incorrectly gives this as -51 which may have unduly popularized the rounded version. Use CS3C1D from Nuttall instead of the rounded version for -18 /octave sidelobe rolloff and better minimum sidelobe rejection. 15

16 Other Optimizations Any time you hear optimum or optimized be sure you have heard with respect to what. What: Mini-max stopband error. What window gives the smallest maximum error outside the passband? Dolph-Chebychev Approximation to Dolph-Chebychev: Taylor (two parameter) What: Minimum stopband energy. What window gives the smallest total energy outside the passband? Prolate Spheroidal Functions implemented as Discrete Prolate Spheroidal Sequences (DPSS) Approximation to Prolate spheroidal functions: Io(a): Modified Bessel Function of the First Kind of Order Zero also know as Taylor 1 parameter and Kaiser-Bessel An improvement to Kaiser -Bessel! Modified Kaiser Bessel 16

17 Dolph-Chebychev 4 Plots Figure 7 Window: DolphCheb4 248 points DFT: 248 bins EN: Coherent Gain:.5893 Pedistal: Amplitude Time Window in Samples Linear in Time 21 Dale B. Dalrymple Full Main Lobe: Max.SideLobe: 4 SideLobe Slope: /Oct. Response in Mainlobe and Adjacent Region, Linear in Frequency, Negative values in red WCAE: : : 1.67 Scalloping Loss: 2.69 Response in Center of Mainlobe to First Half Bin, Linear in Frequency 21 Dale B. Dalrymple High peak at end of time window. Zero sidelobe rolloff. 17

18 Dolph-Chebychev 4 Frequency Response Figure 8 DolphCheb4 EN: Full ML: 3.53 Max.SL: 4 Window: 248pt DFT: 248bin SL Slope: /Oct. Bin 1 Int pt 2 4 Magnitude: red Remaining Energy: green: DolphCheb4 Half Width in, Logarithmic in Frequency 21 Dale B. Dalrymple 18

19 Dolph-Chebychev Frequency Responses Figure 9 DolphC 2(red), 3.5, 3, 3.5, 4(red), 4.5, 5, 6, 7(red) to 8(black) Half Beam Width in 21 Dale B. Dalrymple 19

20 Dolph-Cheb Table 21 Dale B. Dalrymple (248 PtFFT) Param: Coherent Gain EN Scalloping Loss WCPL -3-6 Level SLRO /Oct Bin1 Intercept Window Pedestal e e e e e e e-8 2

21 Taylor (2 Parameter) Window Maximum nbar for Monotonic Time Domain Window Tabulated for Sidelobe Rejection vs Transform Size Taylor Window with nbar in bold should be used to replace Dolph-Cheb. 21 Dale B. Dalrymple XformSize SLRR

22 Taylor Window Table 21 Dale B. Dalrymple (248 Coherent EN Scalloping WCPL -3-6 SLRO Bin1 Window PtFFT) Gain Loss Level Intercept Pedestal Param: /Oct 2, , , , , , , , , , , e-5 13, e-5 14, e-6 15, e-6 16, e-7 17, e-7 18, e-8 19, e-8 2, e-8 Villeneuve,A. T., Taylor Patterns for Discrete Arrays Antennas and Propagation, Transactions on; vol. AP-32, no. 1, October

23 Taylor 4 Plots Figure 1 Window: Taylor4 248 points DFT: 248 bins EN: Coherent Gain:.5716 Pedistal: Amplitude Time Window in Samples Linear in Time 21 Dale B. Dalrymple Full Main Lobe: Max.SideLobe: SideLobe Slope: 6 /Oct. Response in Mainlobe and Adjacent Region, Linear in Frequency, Negative values in red WCAE: : : 1.73 Scalloping Loss: Response in Center of Mainlobe to First Half Bin, Linear in Frequency 21 Dale B. Dalrymple 23

24 Taylor 4 Frequency Response Figure 11 Taylor4 EN: Full ML: 3.64 Max.SL: Window: 248pt DFT: 248bin SL Slope: 6 /Oct. Bin 1 Int pt 2 4 Magnitude: red Remaining Energy: green: Taylor4 Half Width in, Logarithmic in Frequency 21 Dale B. Dalrymple 24

25 DPSS Table 21 Dale B. Dalrymple (248 PtFFT) Coherent Gain EN Scalloping Loss WCPL -3 Param: /Oct Level SLRO Bin1 Intercept

26 Kaiser Window Table 21 Dale B. Dalrymple (248 PtFFT) Param: Coherent Gain EN Scalloping Loss WCPL -3-6 Level SLRO /Oct Bin1 Intercept NaN NaN 26

27 KaiserMod Table 21 Dale B. Dalrymple (248 PtFFT) Param: Coherent Gain EN Scalloping Loss WCPL -3-6 Level SLRO /Oct Bin1 Intercept NaN NaN How to modify Kaiser-Bessel? Subtract 1. from numerator and denominator. 27

28 Other Windows Gaussian The infinite continuous Gaussian window is the only window to achieve the minimum time-frequency uncertainty product. While this is not achieved for truncated Gaussians, it seems to be a frequently referenced attraction. The Gaussian also has interesting properties for frequency estimation of tones. The parabolic interpolator is very accurate for the Gaussian log power spectrum. There are also other techniques: McEachern, Robert H, Ratio detection precisely characterizes signals' amplitude and frequency, EDN, March 3, 1994 Available (2144) at: Modified Bessel Functions of the First Kind I(order,alpha) Order = Kaiser, Taylor (1953) Order =,1 Prabu Order = n, alpha > Reddy n = -2, -3/2, -1, -1/2, Order real, alpha real Nuttall 1,2,3 dimensions Kaiser, J. F. and R.W. Schafer, "On the Use of the lo-sinh Window for Spectrum Analysis", IEEE Trans.Acoust., Speech, Signal Proc., ASSP-28, pplos,198. Prabhu, K.M.M.; Bagan, K.B.; Variable parameter window families for digital spectral analysis Acoustics, Speech and Signal Processing, IEEE Tran. on Volume: 37 Issue:6, June 1989, page(s): Reddy, A.R.; Design of SAW bandpass filters using new window functions Ultrasonics, Ferroelectrics and Frequency Control, IEEE Tran. on, Volume: 35, Issue: 1,Publication Year: 1988, Page(s): 5-56 Nuttall, A.; A two-parameter class of Bessel weightings for spectral analysis or array processing--the ideal weighting-window pairs This paper appears in: Acoustics, Speech and Signal Processing, IEEE Transactions on, Volume: 31 Issue:5, page(s):

29 Gaussian Table 21 Dale B. Dalrymple (248 Coherent EN Scalloping WCPL -3-6 SLRO Bin1 PtFFT) Gain Loss Level Intercept Param: /Oct NaN NaN 29

30 Extended Bessel Function Io(alpha,order) Table 21 Dale B. Dalrymple (248 Coherent EN Scalloping WCPL -3-6 SLRO Bin1 PtFFT) Gain Loss Level Intercept Param: /Oct 1, , , , , , , , , Kaiser 2, Kaiser 3, Kaiser 4, Kaiser 1, , , , , , , , , , , ,

31 References on the net harris, f.j., "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform Proc.IEEE, 66, pp , January A portion of this is available (2144) at: harris, f.j., " WINDOWS: Finite Aperture Effects and Applications in Signal Processing This document is available (2144) from: McEachern, Robert H, Ratio detection precisely characterizes signals' amplitude and frequency, EDN, March 3, 1994 Available (2144) at: B&K Technical Review (viewed 2145) No. 3, 1987, Windows to FFT Analysis (Part I) No. 4, 1987, Windows to FFT Analysis (Part II) 31

Chapter 5 Window Functions. periodic with a period of N (number of samples). This is observed in table (3.1).

Chapter 5 Window Functions. periodic with a period of N (number of samples). This is observed in table (3.1). Chapter 5 Window Functions 5.1 Introduction As discussed in section (3.7.5), the DTFS assumes that the input waveform is periodic with a period of N (number of samples). This is observed in table (3.1).