Butterworth denormalized approximation functions

Assume that it is necessary to define a low pass Butterworth filter with cut-off frequency rad/s. In order to obtain such a filter we have to substitute each symbol of normalized low pass filter transfer function by . The resulting low pass filter will possesses with the cut-off frequency equal to . In this case, the low pass Butterworth filter of the n-th order and cut-off frequency is given by following magnitude response

. (3)

The order of the Butterworth filter is dependent on the specifications provided by the user. These specifications include the edge frequencies and gains. The standard formula for the Butterworth order calculation is given by

(n needs to be round to integer value). (4)

Chebyshev filters

There are two types of Chebysheshev filters. Type I Chebyshev filters are all-pole filters that exhibit equiripple behaviour in the pass band and a monotonic characteristic in the stop band. On the other hand, the family type II Chebyshev filters contains both poles and zeros and exhibits a monotonic behaviour in the pass band and equiripple behaviour in the stop band. The zeros of this class of filters lie on the imaginary axis in the s- plane.

The magnitude squared of frequency response characteristic of a type I Chebyshev filter is given as

, (5)

where δ is a parameter of the filter related to the ripple in the pass band and T_N(ω) is the N-th order Chebyshev polynomial defined as

, .

The Chebyshev polynomial can be generated via recursive equation

, N=1, 2, … (6)

where and . From equation (6) we obtain

and so on.

Main properties of Chebyshev type I polynomials

1. for all .

2. for all N.

3. All the roots of the polynomials T_N(ω) occur in the interval .

The filter parameter δ is related to the ripple in the pass band, as illuctrated in Figure 3, for N odd and N even. For N odd, and hence . On the other hand, for N even and hence . At the cut-off frequency , we have , so that

or, equivalently,

, where δ is the value of the pass band ripple.

For a given set of specifications, it is possible to determine the order of the filter from equation

, (7)

where n needs to be round to integer value.

Figure 3 Type I Chebyshev Filter Characteristics

STANDARD TASK FOR LABORATORY WORK

1. Define transfer function of Butterworth low pass filter of the n-th order with cut-off frequency . Values for cut-off frequency and filter order are given in Table 1.

Table 1

BUTTERWORTH LOW PASS CHARACTERISTICS

Variants	1	2	3	4	5	6
ωс, rad/s	0.25	7	1·103	5	0.75	0.85
n, order	2	1	1	2	1	2

2. A Butterworth low pass filter is given by magnitude response function

.

Define an order of Butterworth low pass filter that satisfies the following requirements (see eq. 4)

1. maximum attenuation in pass band, R_p=1 dB;

2. pass band frequency is equal to rad/s;

3. stop band frequency, rad/s;

4. attenuation in stop band, R_s=40 dB.

3. Define a magnitude frequency response of low pass Chebyshev filter of the n-th order if the pass band is equal to 1 rad/s and pass band ripple, δ is equal to 0.5. The order of the filter is given in Table 2.

Table 2

CHEBYSHEV FILTER ORDER

Variants	1	2	3	4	5	6
n, order	3	8	4	5	7	4
Variants	7	8	9	10	11	12
n, order	6	4	5	6	7	3

4. Construct an analog low pass Butterworth filter within Simulink environment and investigate its properties. The filter structure is shown in Figure 4.