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AVR223: Digital Filters with AVR

Features

• Implementations of Simple Digital Filters

• Coefficient and Data Scaling

• Fast Implementation of 2nd Order FIR Filter

• Compact Implementation Of 8th Order FIR Filter

• Fast Implementation of 2nd Order IIR Filter

• Compact Implementation of 8th Order IIR Filter

8-bit

Microcontroller

Application

Note

Introduction

Applications involving processing of data from external analog sources (sensors)

often require some kind of digital filtering of the data prior to using the data to respond

to some external event. In applications where extremely high filter performance is

required Digital Signal Processors (DSP) are usually chosen, but in many applications

these are too expensive to use. In these cases 8- and 16-bit Microcontrollers (MCU)

comes into the picture: They are inexpensive, efficient and have all the required I/O

features and communication modules that the DSP often does not have.

The Atmel AVR microcontrollers are excellent for signal processing applications due to

their powerful architecture, strong instruction set and built-in multi-channel 10-bit Ana-

log to Digital Converter (ADC). The megaAVR ® series further have a hardware

multiplier, which is important in signal processing applications.

This document focus on the use of the AVR hardware multiplier, the use of the general

purpose registers for accumulator functionality, how to scale coefficients when imple-

menting algorithms on fixed point architectures, the actual implementation examples

and finally possible ways to optimize/modify the implementations suggested.

If the reader has a need for understanding digital filters, transfer functions, frequency

response, and other topics that are briefly mentioned here, most literature on the topic

of digital signal processing can provide this information. A literature list is enclosed

last in this document.

General Digital Filters

Two filter types are referred to when discussing digital filters, the Finite Impulse

Response (FIR) filters and Infinite Impulse Response (IIR) filters. The output of FIR fil-

ters is based on filter input only and the filter output is therefore calculated using a

non-recursive algorithm (feed-forward). The output of the IIR filter is based on both the

input and a feedback of the filter output. The IIR filter output is thus calculated using a

recursive algorithm.

The description of digital filters, covering both FIR and IIR filters, can be generalized

to the discrete differential equation seen in Equation 1.

Rev. 2527A–AVR–9/02

[ ]

⋅

−

⋅

−

Equation 1

The filter coefficients related to the feedback and the feedforward part of the generalized

digital filter are respectively denoted by a m and b k . The filter input is x[n], where n refers

to the number of a given sample – similarly y[n] is the output corresponding to sample n.

The ranges of the summations M and N are referred to as the filter order or the filter

length.

The right side of the equation describes the feedforward in the filter function and the left

side the feedback in the filter function. A filter function can consist of either or both sides

of the differential equation. If only the right side of the differential equation is used, the

filter is a FIR filter (N 0 and M = 0). If both sides are used the filter is an IIR filter (N

≠

0). A special case is when only the left side of the differential equation is used

(N = 0 and M

≠

0). In that case the filter is called an “all-pole” filter. The all-pole filter is

actually also an IIR type filter since all filters that involve feedback are having an infinite

impulse response. In this document the IIR filter is however defined to be where ranges

of the two summations are equal (M = N). The all-pole IIR filter implementation will

therefore not be described separately in this context.

≠

Often digital filters are described in the Z-domain. The Z-transform of Equation 1 is seen

from Equation 2.

[ ]

[ ] ()

()

⋅

−

⋅

−

←

→

⋅

−

⋅

−

Equation 2

The topic of Z-transforms is left for the reader to study further, recommended reading

could be [1]. For now it is sufficient to know that Z X in the Z-domain represent a delay

element and that the exponent determines the number of discrete delays generated.

The reason that the Z-transform of the differential equation is provided is that character-

istics of the digital filters are described by their transfer function, H(z), in the Z-domain.

The transfer function of a filter (or any system) describes the relation between the input

and output of the filter (or system). See Figure 1.

Figure 1. System

H(Z)

The general form of the relation between the filter input and the filter output, the transfer

function of the filter, can be seen in Equation 3.

⋅

−

(

)

()

(

)

⋅

−

Equation 3

The numerator describes the feedforward part of the filter function and the denominator

the feedback of the filter function.

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2527A–AVR–9/02

and M

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Filter Stability

From the transfer function of the filter it is possible to determine if the filter is stable. A fil-

terissaidtobestableifaboundedinputgenerates abounded output, meaning that the

output of the filter will die out when the filter input ceases. The criterion of stability is not

that the output becomes zero if the input becomes zero, but that the magnitude of the

output will decrease continuously toward zero (or becomes zero) if the input to the filter

becomes zero. Without elaborating on the topic of filter stability, it should be known that

the roots of the denominator polynomial, the poles that is, should be within the unit-circle

for the filter to be stable. Further information on the topic of filter stability can be found in

e.g. [1] and [2].

FIR Filters

FIR filters are characterized by being feed-forward filters, and there will thus only be an

output from the filter as long a data are fed into the filter input tap. The duration from the

input to the filter have ended to the output is “silent” (zero) depends on the order of the

filter, since the order of the filter determines the number of delay elements that would be

found in the filter tap delay line. An important feature of the FIR filter is that it is always

stable, since a bounded input generates a bounded output.

The FIR filter is the special case of the general digital filter, where M = 0 and N

≠

0. The

mathematical description of the FIR filter can be seen in Equation 4.

[ ]

⋅

−

⋅

−

;

for

and

≠

[] [ ]

⋅

−

Equation 4

2527A–AVR–9/02

The transfer function of the FIR in the Z-domain is seen in Equation 5.

⋅

−

(

)

()

⋅

−

(

)

Equation 5

From both Equation 4 and the FIR transfer function (Equation 5) it can be seen that the

filter output, y[n], sole rely on the input samples, x[n-k]. To make this even more obvious

one can study the illustration of the FIR filter structure, seen in Figure 2. This filter struc-

ture is referred to as “Direct Form I”.

Figure 2. FIR Filter, Direct Form I Structure

x[n]

Z -1

b 0

b 1

b 2

b n1

b n

∑

y[n]

ThearrowsinthefilterstructureseeninFigure2representthedataflowinthefilter.

Boxes are functions, such as multiplications between signal and coefficients or discrete

signal delays.

The FIR filter can be used for both High Pass (HP) and Low Pass (LP) filter – this is a

matter of choosing the coefficient to match the desired frequency response. However,

the transfer function is only containing “zeros” – root of the numerator polynomial, and

since the zeros cause attenuation of the amplitude of signal filtered, the FIR is efficient

to attenuate selected frequencies. The FIR filters are good at removing undesired fre-

quency components of a signal. In Figure 3 a FIR filter magnitude response is

illustrated. From this figure it is obvious that this specific filter is attenuating the normal-

ized frequency (1) 0.5 by approximately 50 dB (by a factor 316). These filter coefficients

thus generates a notch filter.

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2527A–AVR–9/02

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Figure 3. Magnitude Response for FIR Filter, b K = {0.25, 0.25, 0.25, 0.25}

-20

-40

-60

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Normalized Angular Frequency (x

rads/sample)

FIR filters are often used to smoothen signals and minimizing fluctuations in the signal

due to background noise. This can be done e.g., by using an averaging filter, where all

the coefficients equals 1/(N+1), where N is the filter order. By doing this the filter will pro-

duce the average of the last N+1 samples feed into the filter.

Note: 1. When working with discrete signals, frequencies are always normalized. When sam-

pling a signal it is not possible to get valid information about the frequencies above

half the sampling frequency, fs. Refer to Shannon's sampling theorem for more infor-

mation on this. The normalization is thus done relative to fs/2.

IIR Filters

The IIR filter is a combination of both feedforward and feedback. The IIR filter is more

efficient than the FIR filter, but have due to the feedback the disadvantage that the filter

can be unstable. Following common guidelines for designing IIR filters this is not in gen-

eral a problem however. If considering computation complexity with respect to the filter

order the IIR filter is more computational heavy than the FIR filter. An IIR filter is approx-

imately twice as complex as a FIR of the same order. The IIR filter is defined as the

general form of the digital filter, with the assumption N = M

≠

Often the scaling of the coefficients is made so that the a 0 coefficient equals 1. The filter

differential equation (Equation 1) can therefore be rearranged as follows from Equation

6. Presented in this way it is easier to understand what the filter output is generated

from.

[ ]

⋅

−

⋅

−

;

for

[] [ ] [ ]

⋅

−

⋅

−

Equation 6

2527A–AVR–9/02

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