8. Digital Filters

In digital signal processing a system is something that operates on one or more inputs to produce one or more outputs. A digital filter is defined as a system (in the case of Clampfit, a software algorithm) that operates on digitized data to either pass or reject a defined frequency range. The objective of digital filtering is to remove undesirable frequency components from a digitized signal with minimal distortion of the components of interest.

There will be instances when it is necessary to filter experimental data after they have been digitized. For example, you might want to remove random noise or line frequency interference from the signal of interest. To this end, Clampfit offers several types of digital filters.

The lowpass filters include Bessel (8-pole), boxcar, Butterworth (8-pole), Chebyshev (8pole), Gaussian, a single-pole RC and an 8-coincident-pole RC. The highpass filters include Bessel (8-pole) and 8-coincident-pole RC. The Gaussian and boxcar filters are finite impulse response (FIR) filters while the Bessel, Butterworth, Chebyshev and RC

filters are infinite impulse response (IIR) filters (see following section: “Finite vs. Infinite

Impulse Response Filters”).

A notch filter is available to reject a narrow band of frequencies and an electrical interference filter is provided to reject 50 or 60 Hz line frequencies and their harmonics.

FINITE VS. INFINITE IMPULSE RESPONSE FILTERS

Digital filters can be broadly grouped into finite impulse response (FIR) filters and infinite impulse response (IIR) filters. FIR filters are also referred to as nonrecursive filters while

IIR filters are referred to as recursive filters.

The output of FIR filters depends only on the present and previous inputs. The general

“recurrence formula” for an FIR filter, which is used repeatedly to find successive values of

y

, is given by:

y n

=

M

∑

k

= 0

b k x n

–

k

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8. Digital Filters

where

y n

is the output value for the

nth

point

x

and

b k

is the

kth

of

M

filter coefficients. In the case of the Gaussian and boxcar filters in Clampfit, the

M

points ahead of the current point are also used, giving a general recurrence formula of:

y n

=

k

M

=

∑

–

M b k x n

–

k

where the filter width is 2(

M +

1

)

points.

The disadvantage of FIR filters is that they can be computationally inefficient as they might require several tens, hundreds or even thousands of coefficients depending on the filter characteristics.

The advantages are that FIR filters are inherently stable because there is no feedback and they possess ideal linear phase characteristics, exhibiting no phase distortion. That is, all frequency components passing through the filter are subject to the same pure time delay.

On the other hand, the output of IIR filters depends on one or more of the previous output values as well as on the input values. That is, unlike FIR filters, IIR filters involve feedback. The general recurrence formula for an IIR filter is given by:

y n

=

N

∑

j

= 1

a j y n

–

j

+

M

∑

k

= 0

b k x n

–

k

where

a

and

b

are the

N

and

M

filter coefficients, where

a

represents the feedback coefficients. Note that the value of

y

for a given point

n

depends on the values of previous outputs

y n–1

to

y n–N

as well as the input values

x

.

The major advantage of IIR filters is that they are computationally more efficient, and therefore much faster, than FIR filters. The disadvantages are that IIR filters can become unstable if the feedback coefficients are unsuitable, and recursive filters cannot achieve the linear phase response that is characteristic of FIR filters. Therefore, all IIR filters introduce a phase delay to the filtered data.

The problem of potential instability of IIR filters is solved in Clampfit by limiting the cutoff frequencies for all filter types to a range where the response is always be stable (see

“Cutoff Frequency Limitations” on page 154). However, the phase delay is not corrected.

The Nyquist rate (see “The Sampling Theorem in Clampfit” on page 15) has important

consequences for digital filtering in that the maximum analog frequency that a digital system can represent is given by:

f h

=

1

2T where

T

is the minimum sampling interval and

f h

is the Nyquist frequency.

pCLAMP 10 User Guide — 1-2500-0180 Rev. A