BACKGROUND AND EXPLANATION OF WINDOWING METHOD?

What is windowing in dsp?

In digital signal processing windowing is defined as the procedure in which a small portion of dataset is assumed for analysis and processing purposes.Window is basically the weighting function of  limited amount that is multiplied with fourier series coefficients to make infinite and continuous fourier coefficients discontinuous.

windowing configuaration

Truncation of the infinite frequency response?

An FIR filter approximates the infinite frequency response of the digital systems.Normally,frequency response of digital system is infinite.If we have desire to expand somehow this infinite duration impulse response of the digital system well renowned Fourier series can be employed for this purpose.The recent situation is providing us infinite duration impulse response of IIR filters,but we need FIR filters and we are going  to design these finite duration impulse response filters using various techniques.To obtain the required consequences truncation of the infinite fourier series coefficients is done.This truncation gives birth to a new phenomenon known as Gibbs phenomenon or Gibbs effect.

truncation of Fourier coefficients

State Gibbs phenomenon or effect?

Gibbs phenomenon is stated as the generation of the ripples in the desired frequency response due to discontinuity produced during truncation of the continuous fourier coefficients.The ripples produced in the frequency response or waveform  of our interest remain always present there.In fact the ripples seen at the corners of the wave that is referred as Gibbs effect or phenomenon.

Which window is optimum for optimum filter design?

The criterion for the optimum window is that the most  of the energy of the window should be in its fourier transform's main lobe.And side lobes should be small in size.

Hamming's window(main characteristic only)

For this window 99.96% of the energy is confined in the main lobe of the window's fourier transform.
Energy for side lobes is less in amount as  compares with the main lobe energy.Here main lobe width is twice to that of rectangular window.

A list of window function

N represents the width, in samples, of a discrete-time, symmetrical window function   {\displaystyle w[n],\ 0\leq n\leq N-1.} w[n],\ 0\leq n\leq N-1.  When N is an odd number, the non-flat windows have a singular maximum point. When N is even, they have a double maximum.
It is sometimes useful to express   {\displaystyle w[n]} w[n]  as a sequence of samples of the lagged version of a zero-phase function:
{\displaystyle w[n]=\ w_{0}\left(n-{\frac {N-1}{2}}\right),\ 0\leq n\leq N-1.} w[n]=\ w_{0}\left(n-{\frac {N-1}{2}}\right),\ 0\leq n\leq N-1.
Each figure label includes the corresponding noise equivalent bandwidth metric (B),[note 1] in units of DFT bins.

B-spline windows

B-spline windows can be obtained as k-fold convolutions of the rectangular window. They include the rectangular window itself (k = 1), the triangular window (k = 2) and the Parzen window (k = 4). Alternative definitions sample the appropriate normalized B-spline basis functions instead of convolving discrete-time windows. A kth order B-spline basis function is a piece-wise polynomial function of degree k−1 that is obtained by k-fold self-convolution of the rectangular function.

Rectangular window

Rectangular window; B = 1.0000.
The rectangular window (sometimes known as the boxcar or Dirichlet window) is the simplest window, equivalent to replacing all but N values of a data sequence by zeros, making it appear as though the waveform suddenly turns on and off:

{\displaystyle w(n)=1.} w(n)=1.
Other windows are designed to moderate these sudden changes, which reduces scalloping loss and improves dynamic range, as described above (Window function#Spectral analysis).

The rectangular window is the 1st order B-spline window as well as the 0th power cosine window.

Triangular window

Triangular window (with L=N-1) or equivalently the Bartlett window; B = 1.3333.[15]
Triangular windows are given by:

{\displaystyle w(n)=1-\left|{\frac {n-{\frac {N-1}{2}}}{\frac {L}{2}}}\right|,} w(n)=1-\left|{\frac {n-{\frac {N-1}{2}}}{\frac {L}{2}}}\right|,
where L can be N,[11][16] N+1,[17] or N-1.[18] The last one is also known as Bartlett window or Fejér window. All three definitions converge at large N.

The triangular window is the 2nd order B-spline window and can be seen as the convolution of two N/2 width rectangular windows. The Fourier transform of the result is the squared values of the transform of the half-width rectangular window.

Parzen window

Parzen window; B = 1.92.
Not to be confused with Kernel density estimation.
The Parzen window, also known as the de la Vallée Poussin window,[11] is the 4th order B-spline window given by:

{\displaystyle w(n)=\left\{{\begin{array}{ll}1-6\left({\frac {n}{N/2}}\right)^{2}\left(1-{\frac {|n|}{N/2}}\right),&0\leqslant |n|\leqslant {\frac {N}{4}}\\2\left(1-{\frac {|n|}{N/2}}\right)^{3},&{\frac {N}{4}}<|n|\leqslant {\frac {N}{2}}\\\end{array}}\right.} {\displaystyle w(n)=\left\{{\begin{array}{ll}1-6\left({\frac {n}{N/2}}\right)^{2}\left(1-{\frac {|n|}{N/2}}\right),&0\leqslant |n|\leqslant {\frac {N}{4}}\\2\left(1-{\frac {|n|}{N/2}}\right)^{3},&{\frac {N}{4}}<|n|\leqslant {\frac {N}{2}}\\\end{array}}\right.}Courtesy of wikipedia.........