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Window based Finite Impulse Response filters have the problem that in order to obtain better performance from these filters in terms of minimum stopband attenuation cost has to be paid for half main-lobe width and vice-versa. A solution of this contradictory behavior is to increase the length of the window which in turn requires more hardware hence increasing the cost of system. This paper proposes a novel window based on two shifted hyperbolic tangent functions. The proposed window contains an adjustable parameter, with the help of which desired time and frequency domain characteristics may be achieved for relatively shorter window length. The characteristics of the proposed window are compared with those of the two well-known adjustable windows namely Cosh window and Exponential window. MATLAB simulation results show that for the same value of window length, the proposed window provides improved output, and thus it makes a good compromise between minimum stopband attenuation and half main-lobe width compared to the windows mentioned previously.

Ideal filters have perfectly flat passband with infinite attenuation in the stopband. Unfortunately these filters are practically unrealizable because of their infinite number of co-efficients in the time domain. Hence in order to implement them, the co-efficients must be truncated [_{d}(n) by a finite length sequence W(n) whose values range from 0 to (N − 1) to get a Finite Impulse Response h(n) having a length of N. However direct truncation results in unwanted ripples in the passband commonly known as “Gibbs phenomenon” as shown in

This is because multiplication in the time domain (

Windows may be fixed or adjustable. Fixed windows have only one parameter i.e. window length. In order to get better performance from fixed windows, the only way is to increase the window length which in terns increases the cost of the resultant system. On the other hand adjustable windows have two or more adjustable parameters that provide extra degree of freedom to control time and frequency domain characteristics [

Several windows whether fixed or adjustable, have been introduced in the field of digital signal processing [

Stopband attenuation is an important spectral parameter. For filter design applications, it is required to have good stopband attenuation in order to adequately block the unwanted frequencies. Better minimum stopband attenuation increases the ability of detecting weak signals in the presence of strong narrowband signals. Main-lobe width is another important parameter for some applications, such as frequency spectral analysis of signal. A narrower main-lobe width means that it distinguishes two closely spaced frequencies more accurately [

As mentioned previously, it has been proposed that for the same window length and main-lobe width, the exponential window provide better side roll-off ratio but worse ripple ratio and Cosh window provides better minimum stopband attenuation compared to Kaiser window [

A novel window based on hyperbolic tangent function is proposed in this article. For some values of adjustable parameter, this proposed window provides increased stopband attenuation while the half main lobe width remains unchanged or even gets better compared to some well known windows. In order to show the efficiency of our proposed window, we compare the performance of proposed window with that of Cosh window and Exponential window. Simulation results show that for smaller values of adjustable parameter of both Cosh and Exponential windows, the proposed window provides greater minimum stopband attenuation for the same window length and equal or narrower main-lobe half width. Hence, our proposed window makes a good compromise between stopband attenuation and main lobe width. Design relationship among adjustable parameter, window length and stopband attenuation are shown and two important design equations are provided. These design equations would be helpful in order to select the appropriate value of adjustable parameter of our proposed window for desired window length and stopband attenuation. The filtering action of the proposed window is also analyzed using MATLAB simulator. Significant suppression of unwanted high frequency components in the frequency domain by our proposed window has been observed.

Two shifted hyperbolic functions as hyperbolic window has been used for the development of frequency selective filters in the frequency domain. Using the scaling and shifting properties of Fourier Transform, it has been showed that within a certain limit, these two shifted hyperbolic tangent function approximates a Sinc function in the time domain, hence close to ideal filter [

In this paper to derive our proposed window we combine two hyperbolic tangent functions in the time domain rather than the frequency domain. The proposed window is defined as

where N is the length of window and β is the adjustable parameter (β ≠ 0).

Different performance characteristics and the corresponding data are obtained using MATLAB software. For simplicity we have considered small values of window length for analyzing the response of our proposed window. But the performance of our proposed window would be the same for larger window length.

_{R}) for β = 12, 16, 20 are tabulated in

From

In the frequency domain, stopband attenuation is one of the most important parameter

N | β | As (dB) | W_{R} |
---|---|---|---|

30 | 12 | −43.19 | 0.171 |

30 | 16 | −28.40 | 0.154 |

30 | 20 | −26.18 | 0.140 |

of a designed filter.

For fixed values of N, a general approximate design relation between β and As can be achieved using quadratic polynomial curve fitting method which is given by Equation (2). This equation will be very helpful for the designers who want to use this window for a desired value of N and As.

Values of co-efficients a, b, c, d, e in Equation (2) are given in

N | As (in dB) |
---|---|

10 | −34.10 |

20 | −75.07 |

30 | −85.81 |

40 | −104.27 |

50 | −110.05 |

60 | −144.64 |

70 | −145.87 |

80 | −155.44 |

90 | −156.56 |

100 | −182.58 |

dB, −40 dB, −45 dB, −50 dB, −55 dB, −60 dB, −65 dB, −70 dB, −75 dB, −80 dB, −85 dB, −90 dB, −95 dB, −100 dB and obtained the corresponding values of β using Equation (2). From

In some applications window length N is kept constant. To obtain an approximate value of β for a given N and As, windows of length N = 20, 30, 40, 50, 60, 70, 80, 90, 100 were designed to cover the range −25 dB ≤ As ≤ −80 dB.

N | a | b | c | d | e |
---|---|---|---|---|---|

30 | 8.85134E−06 | 0.0023954 | 0.24565 | 10.945 | 187.36 |

40 | 5.9548E−06 | 0.0016918 | 0.17729 | 8.2839 | 157.78 |

50 | 5.1347E−06 | 0.0014723 | 0.15769 | 7.6756 | 157.13 |

60 | 7.0379E−06 | 0.0020263 | 0.21601 | 10.314 | 203.08 |

70 | 9.612E−06 | 0.0027062 | 0.2815 | 13.084 | 249.42 |

80 | 1.122E−05 | 0.003149 | 0.32538 | 14.984 | 283.43 |

90 | 1.303E−05 | 0.0037011 | 0.38619 | 17.867 | 335.64 |

100 | 1.3884E−05 | 0.00397048 | 0.41821 | 19.529 | 370.09 |

window length N and adjustable parameter β. Using quadratic polynomial curve fitting technique, an approximate design equation between β and N takes the general form,

Values of co-efficients x, y, z in Equation (3) for a range of As are presented in

In order to show the efficiency of proposed window, it is compared with well-known Cosh window and Exponential window. Although adjustable parameters of these two windows posses different characteristic values compared to the adjustable parameter of our proposed window, for the sake of comparison, β of the proposed window is chosen carefully that results a slightly better performance than the Cosh window and Exponential window.

Kaiser window has been derived by modifying the zero order Bessel function of first

As (in dB) | x | y | z |
---|---|---|---|

−80 | 0.00045 | 0.11 | 2.1 |

−75 | 0.00031 | 0.16 | 0.99 |

−70 | 0.00045 | 0.15 | 1.7 |

−65 | 0.00031 | 0.19 | 1.2 |

−60 | 0.00014 | 0.24 | 0.65 |

−55 | 4.5E−05 | 0.27 | 0.69 |

−50 | −0.00084 | 0.39 | −1.3 |

−45 | −0.001418 | 0.4585 | −1.681 |

−40 | 0.00022 | 0.4109 | 1.0119 |

−35 | 4.329E−05 | 0.47481 | 0.938 |

−30 | 5.411E−05 | 0.5517 | 2.0413 |

−25 | 0.0044 | 0.23 | 24 |

kind. This modified Bessel function exhibits the same shape characteristics as the Cosh function. Thereby Cosh window has been defined as [

It is clearly observed from the Figures 6(a)-(c) that, for all pairs of α and β, our proposed window results better minimum stopband attenuation and half mainlobe width simultaneously which is very rare for most of the existing window filters. For small system (i.e., small window length N) the difference between characteristic values obtained by Cosh and proposed window may not be significant enough but for larger values of N it would be more advantageous to use our proposed window other than Cosh window.

The shape characteristics of zero order Bessel function is also similar to that of exponential function. Hence Exponential window has been defined as [

Window | N | As (in dB) | W_{R} |
---|---|---|---|

Cosh (α = 0) | 20 | −19.59 | 0.20 |

Proposed (β = 50) | 20 | −20.58 | 0.20 |

Cosh (α = 1) | 20 | −25.32 | 0.22 |

Proposed (β = 14) | 20 | −26.05 | 0.22 |

Cosh (α = 2) | 20 | −36.00 | 0.25 |

Proposed (β = 8) | 20 | −36.26 | 0.26 |

namely stopband attenuation and mainlobe width. From

From the data placed in

Window | N | As (in dB) | W_{R} |
---|---|---|---|

Exponential (α = 0) | 20 | −24.65 | 0.154 |

Proposed (β = 90) | 20 | −25.24 | 0.151 |

Exponential (α = 1) | 20 | −30.52 | 0.174 |

Proposed (β = 13) | 20 | −32.14 | 0.174 |

Exponential (α = 2) | 20 | −36.50 | 0.200 |

Proposed (β = 10) | 20 | −39.15 | 0.195 |

length N, our proposed window remains superior in terms of both stopband attenuation and half mainlobe width. So our proposed can be a potential replacement for Cosh window and Exponential window for many important signal processing application where the length of the system cannot be increased (because this will increase the cost of the system) but better performance compared to the existing adjustable windows is required.

In order to show the efficient performance of our proposed window, A MATLAB simulator is used to generate a signal 10cos (2πn) mixed with high frequency component 2cos (20πn + π/4) shown in

In this paper we have proposed an adjustable window function and its performance characteristics are evaluated using MATLAB simulator. For most of the windowed filters, obtaining better minimum stopband attenuation is not possible without affecting the value of mainlobe half width. This contradictory behavior somehow limits the potential application of windowed filters. However our proposed window shows significant improvements in this regard. Within a particular range, this proposed window

provides better stopband attenuation without compromising the value of mainlobe width. This improved performance makes it suitable for practical applications such as medical imaging systems.

Sarker, O. and Khan, R.H. (2016) Simulation Based Design Analysis of an Adjustable Window Function. Journal of Signal and Information Processing, 7, 214-226. http://dx.doi.org/10.4236/jsip.2016.74019