There are several methods of FIR filter, for example: window function design method, optimization design method, frequency sampling design method. Window function design technique is one of the main FIR filter design methods, because of its simple operation and easy physical meaning, window function method has become a method for widely use in engineering practice.

There are six kinds of basic window function; they are Rectangular window, Triangular window, Han window, Hamming window, Blackman window and Kaiser Window.

The basic idea of all window function design method is to select the filter on the basis of suitable and ideal frequency characteristics, and then its impulse response was truncated to obtain a FIR filter of linear-phase and cause and effect. Therefore the focus of this method is to select an appropriate window function and a suitable ideal filter [10] [11] .

Suppose the ideal response of desired filter is

$h_d\left({\text{e}}^{j\omega }\right)=\underset{n=-\infty }{\overset{\infty }{{\displaystyle \sum }}}{h}_d\left(n\right){\text{e}}^{-jn\omega }$

The design of FIR filter lies in finding a transfer function as shown in Figure 1 .

$H\left({\text{e}}^{j\omega }\right)=\underset{n=0}{\overset{N-1}{{\displaystyle \sum }}}{h}_d\left(n\right){\text{e}}^{-jn\omega }$ (1)

To approximate $H_{ad}\left({\text{e}}^{j\omega }\right)$, suppose

$h_d\left(n\right)=\frac{1}{2\pi }\underset{-x}{\overset{x}{{\displaystyle \int }}}{H}_d\left({\text{e}}^{j\omega }\right){\text{e}}^{j\omega }\text{d}\omega$ (2)

The rectangular frequency characteristics of $H_d\left({\text{e}}^{j\omega }\right)$ so $h_d\left(n\right)$ must be an infinite sequence and non-causal. The $h\left(n\right)$ of FIR filter to be designed is inevitable finite; infinite $h_d\left(n\right)$ was approximated by using finite $h\left(n\right)$.

The most effective way is to cut off $h_d\left(n\right)$, or $h_d\left(n\right)$ was intercepted by using finite window function sequence $w\left(n\right)$, i.e.

$h\left(n\right)=h_d\left(n\right)\omega \left(n\right)$ (3)

So the shape and length of Window function sequence were very critical. In the design process, window function $w\left(n\right)$ was selected according to requirements of transition bandwidth and stop-band attenuation of FIR filter [10] [11] .

FIR filter design methods are based on a direct approximation of the desired frequency response of the discrete-time system. For this method, the filter coefficients correspond to the impulse response of the filter to be designed. To be able to process different signals, it is necessary that these signals be finite. However, this is not always the case for all signals. The window method is often used to create these types of finite signal sequences from infinite sequences. This “cutting” of an infinite sequence to create a finite sequence, however, affects the frequency range. Non-ideal effects, which are observed due to the finite number of filter coefficients, can be mitigated by using a weighting window. The principle is that the filter coefficients in the middle are weighted more heavily than the coefficients at the beginning and end. There are several window functions that define the maximum achievable stopband attenuation.

The limited knowledge of the signal is equivalent to multiplying it by a time window function called a rectangular window represented by:

$w_R\left(n\right)=\{\begin{array}{ll}1\hfill & 0\le n\le N-1\hfill \\ 0\hfill & \text{elsewhere}\hfill \end{array}$ (4)

The function $H_R\left(\Omega \right)$, is the Fourier transform of the rectangular window and called spectral window.

$H_R\left(\Omega \right)=\underset{n=0}{\overset{N}{{\displaystyle \sum }}}{\text{e}}^{-j\Omega n}$ (5)

This truncation of infinite signals to obtain signals of finite duration leads to undesirable effects. A truncated signal is therefore the multiplication of a signal of infinite duration by a rectangular window, that is to say [Adel Jalal]:

$h_N\left(n\right)=h\left(n\right){\omega }_R\left(n\right)=\{\begin{array}{ll}h\left(n\right)\hfill & 0\le n\le N-1\hfill \\ 0\hfill & \text{Elsewhere}\hfill \end{array}$ (6)

The Bartlett window provides a maximum attenuation of 25 dB and a transition band of width

$\Delta \omega =\frac{6.1\pi }{N}$

${\omega }_R\left(n\right)=\{\begin{array}{ll}\frac{2n}{N-1}\hfill & 0\le n\le \frac{N-1}{2}\hfill \\ 2-\frac{2n}{N-1}\hfill & \frac{N-1}{2}\le n\le N-1\hfill \\ 0\hfill & \text{Elsewhere}\hfill \end{array}$ (7)

The Blackman window provides a limiting attenuation of 74 dB and a transition band of width

$\Delta \omega =\frac{11\pi }{N}$

${\omega }_R\left(n\right)=\{\begin{array}{ll}0.42-0.5\cos \left(\frac{2\pi n}{N-1}\right)+0.08\cos \left(\frac{4\pi n}{N-1}\right)\hfill & 0\le n\le N-1\hfill \\ 0\hfill & Elsewhere\hfill \end{array}$ (8)

The Hamming window provides a maximum attenuation of 53 dB and a transition band of width

$\Delta \omega =\frac{6.1\pi }{N}$ (9)

${\omega }_R\left(n\right)=\{\begin{array}{ll}0.54-0.46\cos \left(\frac{2\pi n}{N-1}\right)\hfill & 0\le n\le N-1\hfill \\ 0\hfill & \text{Elsewhere}\hfill \end{array}$ (10)

The hanning window provides a limiting attenuation of 44 dB and a transition band of width

$\Delta \omega =\frac{6.2\pi }{N}$ (11)

${\omega }_R\left(n\right)=\{\begin{array}{ll}\frac{1}{2}\left[1-\cos \left(\frac{2\pi n}{N-1}\right)\right]\hfill & 0\le n\le N-1\hfill \\ 0\hfill & \text{Elsewhere}\hfill \end{array}$ (12)

The width of the main lobe is inversely proportional to the length of the filter. The attenuation in the side lobe is, however, independent of the length and is function of the type of the window. A complete review of many window functions and their properties was presented by Harris [12] [13] . Therefore the length of the filter must be increased considerably to reduce the main lobe width and to achieve the desired transition band. Kaiser has chosen a class of windows having properties closely approximating those of the prolate spheroidal wave functions. This family of windows, known as the Kaiser windows is defined by

$w_k\left(\beta ,n\right)=\{\begin{array}{ll}\frac{I_o\left\{\beta {\left[1-{\left(\frac{2n}{N-1}\right)}^2\right]}^{\frac{1}{2}}\right\}}{I_o\left(\beta \right)}\hfill & -\frac{N-1}{2}\le InI\le \frac{N-1}{2}\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$ (13)

where N is window length and $I_o\left(x\right)$ is the modified Bessel function of the first kind of order zero, given by:

$I_o\left(x\right)=\underset{k=0}{\overset{x}{{\displaystyle \sum }}}{\left[\frac{{\left(\frac{x}{2}\right)}^2}{k!}\right]}^2$ (14)

The Kaiser window provides the designer considerable flexibility in meeting the filter specifications.

Finding the value of β in the Kaiser window is a tuning process that involves adjustments to achieve the desired characteristics of the window for signal processing applications. This is often done using practical or theoretical methods to select a suitable β value based on the requirements of the task or signal analysis. [14]

$\beta =\{\begin{array}{ll}0.1102\left(A_{sb}-8.7\right)\hfill & \text{for}{A}_s>50\hfill \\ 0.5842{\left(A_{sb}-21\right)}^{0.4}+0.07886\left(A_{sb}-21\right)\hfill & \text{for}21\le A_s\le 50\hfill \\ 0\hfill & \text{for}{A}_s<21\hfill \end{array}$ (15)

Determining the filter order in the Kaiser window is a crucial step in designing a digital filter or other filters. The calculation of the filter order often involves specifying the frequency response characteristics desired for the filter

$N=\frac{2.056A_{sb}-16.4}{2.285\left(\Delta \omega \right)}$ (16)

Kaiser Window β = 4.54

The Kaiser window provides a maximum attenuation of 50 dB and a transition band of width

$\Delta \omega =\frac{5.8\pi }{N}$ (17)

${\omega }_K\left(n\right)=\{\begin{array}{ll}\frac{I_o\beta \sqrt{N^2-4{\left(n-\frac{N-1}{2}\right)}^2}}{I_o\left(\beta \right)}\hfill & \text{for}0\le n\le N-1\hfill \\ 0\hfill & \text{Elsewhere}\hfill \end{array}$ (18)

where $I_o$ is the Bessel function modifying the zero order and β the parameter characterizing the energy exchange between the main lobe and the secondary lobes.

$I_o\left(x\right)=1+\underset{m=1}{\overset{\infty }{{\displaystyle \sum }}}{\left[\frac{1}{n!}{\left(\frac{x}{2}\right)}^n\right]}^2$ (19)

${\omega }_{c_i}=\left({\omega }_{p_i}+{\omega }_{a_i}\right)/2$ $\Rightarrow$ ${\omega }_{c_1}=0.275\pi$ rad/s and ${\omega }_{c_2}=0.725\pi$ rad/s

And $f_{c_i}=\frac{{\omega }_{c_i}}{2\pi }$ $\Rightarrow$ $f_{c_1}=0.14\text{Hz}$ and $f_{c_2}=0.36\text{Hz}$

$f_{p_i}=\frac{{\omega }_{p_i}}{2\pi }$ and $f_{a_i}=\frac{{\omega }_{a_i}}{2\pi }$ with

$\{\begin{array}{l}{f}_{p_1}=0.175\text{Hz}\\ f_{p_2}=0.325\text{Hz}\end{array}$ and $\{\begin{array}{l}{f}_{a_1}=0.1\text{Hz}\\ f_{a_2}=0.4\text{Hz}\end{array}$

$\Delta {\Omega }_{c_i}=2\pi \frac{\Delta F_{c_i}}{F_e}$ with $\Delta F_{c_i}=\left|f_{a_i}-f_{p_i}\right|$

Since we have two different stop bands, the calculations are done with the stop band with the highest attenuation. Similarly, the width of the transition band chosen is the one with the smallest value.

$h_{pB}\left(n\right)=\frac{\sin \left({\omega }_{c_2}n\right)-\sin \left({\omega }_{c_1}n\right)}{\pi n}$

So

$h_{pB}\left(n\right)=\frac{\sin \left(0.725\pi n\right)-\sin \left(0.275\pi n\right)}{\pi n}$

$h\left(n\right)=\frac{\sin \left(0.725\pi n\right)-\sin \left(0.275\pi n\right)}{\pi n}w\left(n\right)$

The Hamming window provides a maximum attenuation of 53 dB and a transition band of width

$\Delta \Omega =\frac{6.6\pi }{N}\Rightarrow N=\frac{6.6\pi }{\Delta \Omega }$

Figure 2 shows the evolution of the amplitude and pulsation of the bandpass filter as a function of frequency for N = 45. We see that the level of the side lobes of the filter is around 50 dB. The width of the main lobe is 0.6 Hz.

As shown in Figure 1 we have two pulsations respectively whose attenuation band ${\omega }_{s1}=0.2\pi$ et ${\omega }_{s1}=0.8\pi$ and the cut-off pulse is ${\omega }_c=0.5\pi$ what shows a perfect agreement between the simulation and the specifications.

The Kaiser window provides a maximum attenuation of -50 dB and a transition band of width

$\Delta \omega =\frac{5.8\pi }{N}$.

This window is one of the most effective $N=\frac{5.8\pi }{\Delta \Omega }$

Figure 3 shows the evolution of the amplitude and pulsation of the bandpass filter as a function of frequency For N = 42.

Figure 3(a) illustrates the evolution of the amplitude of the N = 42 order bandpass filter as a function of frequency.

As shown in Figure 3(b) , we have two pulsations respectively in the attenuation band ${\omega }_{s1}=0.2\pi$ et ${\omega }_{s1}=0.8\pi$, ${\omega }_{p1}=0.35\pi$ and ${\omega }_{p2}=0.65\pi$ in the bandwidth with the cut-off pulse ${\omega }_c=0.35\pi$.

Figure 4 shows a comparison of the bandpass filter with both Hamming and Kaiser windows.

The two hamming (blue) and kaiser (red) curves are the same at the passband level and slightly different at the lower and upper attenuation band level.