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APPLIED INDUSTRIAL CONTROL SOLUTIONS

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Digital Filter Design
Writing Difference Equations For Digital Filters

Brian T. Boulter

© ApICS ® LLC 2000

ABSTRACT

Difference equations are presented for 1st, 2nd, 3rd, and 4th order low pass and high pass filters, and 2nd, 4th and 6th order band-pass, band-stop and notch filters along with a resonance compensation (RES_COMP) filter. The low pass, high pass, bandpass and bandstop difference equations are obtained from the normalized butterworth continuous time filter descriptions given below.

1st. order normalized Butterworth low pass filter:

(1.0a)

2nd. order normalized Butterworth low pass filter:

(1.0b)

3rd. order normalized Butterworth low pass filter:

(1.0c)

4th. order normalized Butterworth low pass filter:

(1.0d)

2nd. order normalized Notch filter

1.1a)

4th order normalized Notch filter

(1.1b)

6th. order normalized Notch filter

(1.1c)

Normalized Res_Comp filter

(1.1d)

The above low pass filter (not the notch or RES_COMP filters) equations are mapped to the appropriate filter type using the following mappings.

To map (1.0x) to a lowpass filter use:

; = desired low-pass 3 [d.b.] cutt-off frequency. (1.2)

To map (1.0x) to a highpass filter use:

; = desired high-pass 3 [d.b.] pass frequency. (1.3)

To map (1.0x) to a bandpass filter use:

(1.4)

where:

B = A number that controls the bandpass depth/width (equivelant to 1/Q).

= The desired continuous time center frequency.

To Map (1.0x) to a band-stop filter use:

(1.5)

where:

B = A number that controls the notch depth/width (equivelant to 1/Q).

= The desired continuous time center frequency.

For example if we want to obtain a fourth order band-stop filter we use equation (1.0d) and use the mapping of equation (1.5) after some algebra we obtain the following continuous filter description:

To create a digital filter we normalize the above filter by setting = 1, and map to the z-domain using the bi-linear transformation:

(1.6)

The c co-efficient in the above equation is used to accomplish frequency warping. That is, to compensate for an inherent inaccuracy in the bi-linear transformation method that is a function of frequency and sample rate. (see any D.S.P. text for a more complete explanation)

Once the above mapping is performed by substituting (1.6) into the continuous time s-domain filter equation we obtain a z-domain transfer function of the form:

(1.7)

Where, for any filters obtained using the normalized low-pass butterworth filter as the basis for the digital filter, n = d = order of the filter. For low-pass to low-pass and low-pass to high-pass mapping the order of the digital filter is the same as the order of the starting normalized butterworth filter. For low-pass to band-pass and low-pass to band-stop mapping the order of the digital filter is twice the order of the starting normalized butterwoth low-pass filter.

Let:

the normalized notch center frequency. (i.e. 1)

= the desired notch center frequency

= Sample time in sec.

The warping equation is given as:

(1.8a)

The RES_COMP filter which uses the same mapping as above, requires that the

denominator resonant frequency be post-warped with (1.8b) below.

(1.8b)

After some algebra we obtain the following co-efficients in (1.7) for low-pass, high-pass,

band-pass and band-stop filters for the indicated order of the filter.

Low-Pass Filter

n=1

n=2

n=3

n=4

High-Pass Filter

n=1

n=2

n=3

n=4

Band-pass filter

n=2

n=4

n=6

Note:

1) B in the above equations is equavelant to 1/Q; where Q is the depth factor in notch filter

designs.

Band-stop filter

n=2

n=4

n=6

Note:

1) B in the above equations is equavelant to 1/Q; where Q is the depth factor in notch filter

designs.

Notch filter

n=2

n=4

Proprietary

n=6

Proprietary

RES_COMP filter

Proprietary

Notes:

1) is the zero compensation frequency with damping equal to /2.

2) is the pole compensation frequency with damping equal to /2.

CREATING THE DIFFERENCE EQUATIONS:

From the z-domain transfer function we obtain the difference equation as follows:

Know from (1.7)

(1.9)

Therefore (1.9) becomes

In difference equation form:

or in terms of the latest output:

(1.10)

Where for each filter described above the appropriate co-efficients are used in (1.10).

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