dsp butterworth filter

8/9/2019 DSP Butterworth Filter

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DIGITAL SIGNAL PROCESSING

Lecture by

R. Sivarajan, Assistant Professor

Adhiparasakthi Engineering College


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Magnitude Response

The magnitude function of the butterworth low pass filter isgiven by

---------------(1)

where, N is the order of the filter, and

.

Lecture by R. Sivarajan, AP/ECE/APEC


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Magnitude Response



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Magnitude Response

As shown in the figure above, the function is monotonicallydecreasing, where the maximum response is unity at0.

The ideal response is shown by dashed line.

t can e seen t at t e magn tu e response approac es t eideal low pass characteristics as the order N increases.

For values, () 1, For values,

() .

, 0.0

3 .



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Normalized Butterworth Filter From equation (1), we can get the magnitude square function

of a normalized butterworth filter as (1 /)

(1)

. , /,



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Normalized Butterworth Filter

The above equation tells us that this function has poles in the

LHS as well as in RHS because of the presence of two factorH(s) and H(-s).

If H(s) has roots in the LHS, then the H(-s) has corresponding

.

These roots we can get by equating denominator to zero, i.e.,

For N odd,

-------------------(2)



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For N odd,

------------------(3)

We know that for N odd, the roots can be obtained fromequation (2).

For N = 3,



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All the poles are located in the s plane as shown in the figure

below.



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It is found that the angular separation between the poles is

given by 360/2N, which in this case is equal to 60o and all thepoles lie on the circle.

To ensure stability, considering only the poles that lie in the

,

transfer function H(s) as

Therefore the transfer function of a third order butterworth

filter for cut off frequency1 /

---------------(4)



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As we interested on a poles that lie in the left half of the s

plane, the same can be found by usingwhere

----------(5)

Now for N=3,



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Hence the denominator of the transfer function is

So, the transfer function for normalized butterworth filter is

Similarly, for N=4,



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Hence the denominator of the transfer function is

So, the transfer function for normalized butterworth filter is

Similarly we can find the transfer function of a normalizedbutterworth filter for any order of a filter as shown in table

below.



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Normalized Butterworth FilterN Denominator of H(s)

1

2

3

4

5

6

7



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Unnormalized Butterworth Filter

Equation 5 gives us the pole locations of butterworth filter for

1 / .

,

---------------------(6)

The transfer function of such butterworth filter can be obtained

by substituting s s/

.



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Order of the Filter

In equation (1a), the filter was restricted to -3 dB attenuation at

.

(3 )

.



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Order of the Filter



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Order of the Filter

From the figure above, we can find that at ,

,

--------------(7)Lecture by R. Sivarajan, AP/ECE/APEC


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Order of the Filter

Referring to the above figure, at,



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Order of the Filter



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Order of the Filter ,

()

, .

()



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Order of the Filter

,


dsp butterworth filter

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