Where, H0 represents the maximum pass band gain and H1 represents the minimum pass band gain. But, if we want to define Amax at another voltage gain value, consider 1dB, or 1.1220 (1dB = 20logAmax) then the value of Оµ can be found by: If we define Amax at cut-off frequency -3dB corner point (Ж’c), then Оµ will be equal to one and thus Оµ2 will also be equal to one. Where вЂnвЂ™ indicates the filter order, вЂП‰вЂ™ = 2ПЂЖ’, Epsilon Оµ is maximum pass band gain, (Amax).

The frequency response of the nth order Butterworth filter is given as If the Butterworth filter order increases, then the cascaded stages within the Butterworth filter design increases and also the brick wall response & filter gets closer as shown in the above figure. Butterworth Filter Ideal Frequency Response The low pass Butterworth filter standard approximations for various filter orders along with the ideal frequency response which is termed as a вЂњbrick wallвЂќ are shown below. The Butterworth filter changes from pass band to stop-band by achieving pass band flatness at the expense of wide transition bands and it is considered as the main disadvantage of Butterworth filter. The quality factor for this type is just Q=0.707 and thus, all high frequencies above the cut-off point band rolls down to zero at 20dB per decade or 6dB per octave in the stop band. The Butterworth filter has frequency response as flat as mathematically possible, hence it is also called as a maximally flat magnitude filter (from 0Hz to cut-off frequency at -3dB without any ripples). The corner frequency or cutoff frequency is given by the equation: Cutoff Frequency

The filters are used for shaping the signalвЂ™s frequency spectrum in communication systems or control systems. There are various types of Butterworth filters such as low pass Butterworth filter and digital Butterworth filter.

Hence, this type of filter named as Butterworth filter. In 1930 physicist and the British engineer Stephen Butterworth described about a Butterworth filter in his вЂњon the theory of filter amplifiersвЂќ paper for the first time. The signal processing filter which is having a flat frequency response in the passband can be termed as Butterworth filter and is also called as a maximally flat magnitude filter.

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