Chebyshev low pass Approximations

It is commonly used standard approximations.  It uses Chebyshev Polynomial which produces equi-ripple in passband (variation in passband). Outside passband, gain decrease monotonically but at faster rate than Butterworth. It has ripples at the passband but a smooth increase in stopband. To understand Chebyshev Response, basic understanding about Lissajous Figure is needed which acts as a supporting concept.

fig: Deflection Plate 

Here, Horizontal deflecting plate is maintained at constant frequency

Vertical deflecting plate is maintained with variable frequency. 

When adjustable frequency is multiple of fixed frequency, then that stationary figure appears at CRO is known as Lissajous figure.

When adjustable frequency is not multiple of fixed frequency then the cosine wave of inconsistent motion appears on CRO. 

fig: Lissajous Figure

 

In the above figure, ripples extend from -1 to 1.

Analysis,

When on frequency is exact multiple of other, stationary figure appears.

Let ‘x’ be the deflection due to voltage on horizontal plates be

\[
x = \cos(KT) \tag{1}
\]

\[
\text{Where,} \quad K = 2\pi f,\quad T = \text{sampling period}
\]

\[
\Rightarrow KT = \cos^{-1}(x)
\]

Let ‘y’ be the deflection due to voltage on vertical plates will be then,

\[
y = \cos^n(KT)
\]

\[
\text{Where } n \text{ is an integer and represents multiple frequencies, i.e., } n = 1, 2, 3, \dots
\]

\[
\text{From equation (1),}
\quad y = \cos^n(\cos^{-1}(x)) = C_n(x)
\]

\[
C_n(x) = \cos^n(\cos^{-1}(x))
\]

which is the equation for Lissagious figures

\[
\text{Vertical Deflection} = C_n(x) \times \text{Horizontal Deflection}
\]