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Digital Filters: An Introduction

In signal processing, the function of a filter is to remove
unwanted parts of the signal, such as random noise, or to
extract useful parts of the signal, such as the components
lying within a certain frequency range.

Figure 1:  A block diagram of a basic filter

There are two main kinds of filter, analog and digital. They are quite different in their physical makeup
and in how they work.

An analog filter uses analog electronic circuits made up
from components such as resistors, capacitors and op amps to
produce the required filtering effect. Such filter circuits are
widely used in such applications as noise reduction, video
signal enhancement, graphic equalisers in hi-fi systems, and
many other areas.

There are well-established standard techniques for designing
an analog filter circuit for a given requirement. At all
stages, the signal being filtered is an electrical voltage or
current which is the direct analog of the physical quantity
(for example, a sound or video signal or transducer output)
involved.

A digital filter uses a digital processor to perform
numerical calculations on sampled values of the signal. The
processor may be a general-purpose computer such as a PC, or a
specialized DSP (Digital Signal Processor) chip.

The analog input signal must first be sampled and digitized
using an ADC (analog-to-digital converter). The resulting
binary numbers, representing successive sampled values of the
input signal, are transferred to the processor, which carries
out numerical calculations on them. These calculations
typically involve multiplying the input values by constants and
adding the products together. If necessary, the results of
these calculations, which now represent sampled values of the
filtered signal, are output through a DAC (digital-to-analog
converter) to convert the signal back to analog form.

Note that in a digital filter, the signal is represented by
a sequence of numbers, rather than a voltage or current.

Figure 2:  A block diagram of a basic digital filter

Advantages of Digital Filters

The following list gives some of the main advantages of
digital over analog filters:

  1. A digital filter is programmable, in other words, its
    operation is determined by a program stored in the
    processor's memory. This means the digital filter can easily
    be changed without affecting the circuitry (hardware). An
    analog filter can only be changed by redesigning the filter
    circuit.
  2. Digital filters are easily designed, tested and
    implemented on a general-purpose computer or
    workstation.
  3. The characteristics of analog filter circuits
    (particularly those containing active components) are subject
    to drift and are dependent on temperature. Digital filters do
    not suffer from these problems, and so are extremely stable
    with respect both to time and temperature.
  4. Unlike their analog counterparts, digital filters can
    handle low frequency signals accurately. As the speed of DSP
    technology continues to increase, digital filters are being
    applied to high frequency signals in the RF (radio frequency)
    domain which, in the past, was the exclusive preserve of
    analog technology.
  5. Digital filters are very much more versatile in their
    ability to process signals in a variety of ways. This
    versatility includes the ability of some types of digital filter to adapt
    to changes in the characteristics of the signal.

Fast DSP processors can handle complex combinations of
filters in parallel or cascade (series), making the hardware
requirements relatively simple and compact in comparison with
the equivalent analog circuitry.

Operation of Digital Filters

In the next few sections, we will develop the basic theory of
the operation of digital filters. This is essential to an
understanding of how digital filters are designed and used.
First of all, we need to introduce a basic notation.

Suppose the “raw” signal that is to be digitally filtered is
in the form of a voltage waveform described by the function

V = x (t)

where t is time.

This signal is sampled at time intervals h (the
sampling interval). The sampled value at time t = ih
is

xi = x (ih)

Thus the digital values transferred from the ADC to the
processor can be represented by the sequence

x0 , x1 , x2 ,
x3 , …

corresponding to the values of the signal waveform at times
t = 0, h, 2h, 3h, … (where t = 0 is the instant
at which sampling begins).

At time t = nh (where n is some positive
integer), the values available to the processor, stored in
memory, are

x0 , x1 , x2 ,
x3 , … , xn

Note that the sampled values xn+1 ,
xn+2 , and so on are not available as they haven't
happened yet!

The digital output from the processor to the DAC consists of
the sequence of values

y0 , y1 , y2 ,
y3 , … , yn

In general, the value of yn is calculated
from the values x0 , x1 , x2 ,
x3 , … , xn . The way in which the y values are calculated from the x values determines the filtering action of the digital filter.

Examples of Simple Digital Filters

The following examples illustrate the essential features of
digital filters.

  1. UNITY GAIN FILTER: yn = xn

    Each output value yn is exactly the same
    as the corresponding input value xn :

    y0 = x0
    y1 = x1
    y2 = x2
    … etc

    This is a trivial case in which the filter has no
    effect on the signal.

  2. SIMPLE GAIN FILTER: yn = Kxn
    (K = constant)

    This simply applies a gain factor K to each
    input value:

    y0 = Kx0
    y1 = Kx1
    y2 = Kx2
    … etc

    K > 1 makes the filter an amplifier,
    while 0 < K < 1 makes it an attenuator. K < 0 corresponds to an inverting amplifier. Example 1 above is the special case where K = 1.

  3. PURE DELAY FILTER: yn = xn-1

    The output value at time t = nh is simply the
    input at time t = (n-1)h, in other words, the signal is
    delayed by time h:

    y0 = x-1
    y1 = x0
    y2 = x1
    y3 = x2
    … etc

    Note that as sampling is assumed to commence at t =
    0, the input value x-1 at t =
    -h is undefined. It is usual to take this (and any
    other values of x prior to t = 0) as
    zero.

  4. TWO-TERM DIFFERENCE FILTER: yn =
    xn – xn-1

    The output value at t = nh is equal to the
    difference between the current input xn
    and the previous input xn-1 :

    y0 = x0 – x-1
    y1 = x1 – x0
    y2 = x2 – x1
    y3 = x3 – x2
    … etc

    in other words, the output is the change in the input
    over the most recent sampling interval h. The effect
    of this filter is similar to that of an analog
    differentiator circuit.

  5. TWO-TERM AVERAGE FILTER: yn =
    (xn + xn-1 ) / 2

    The output is the average (arithmetic mean) of the
    current and previous input:

    y0 = (x0 + x-1 ) /
    2
    y1 = (x1 + x0 ) /
    2
    y2 = (x2 + x1 ) /
    2
    y3 = (x3 + x2 ) /
    2
    … etc

    This is a simple type of low-pass filter as it tends to smooth out high-frequency variations in a signal. (We will look at more effective low-pass filter designs later).

  6. THREE-TERM AVERAGE FILTER: yn =
    (xn + xn-1 + xn-2 ) / 3

    This is similar to the previous example, with the
    average being taken of the current and two previous
    inputs:

    y0 = (x0 + x-1 +
    x-2 ) / 3
    y1 = (x1 + x0 +
    x-1 ) / 3
    y2 = (x2 + x1 +
    x0 ) / 3
    y3 = (x3 + x2 +
    x1 ) / 3
    … etc

    As before, x-1 and
    x-2 are taken to be zero.

  7. CENTRAL DIFFERENCE FILTER: yn =
    (xn – xn-2 ) / 2

    This is similar in its effect to Example 4 . The output is equal to half the change in the input signal over the current value and value two time intervals prior:

    y0 = (x0 – x-2 ) /
    2
    y1 = (x1 – x-1 ) /
    2
    y2 = (x2 – x0 ) /
    2
    y3 = (x3 – x1 ) /
    2
    … etc

Order of a Digital Filter

The order of a digital filter can be defined as the number
of previous inputs (stored in the processor's memory) used
to calculate the current output.

This is illustrated by the filters given as examples in the
previous section.

Example 1:  yn = xn

This is a zero-order filter, since the current
output yn depends only on the current input
xn and not on any previous inputs.

Example 2:  yn = Kxn

The order of this filter is again zero, since no
previous outputs are required to give the current output
value.

Example 3:  yn = xn-1

This is a first-order filter, as one previous
input (xn-1 ) is required to calculate
yn . (Note that this filter is classed as
first-order because it uses one previous input, even
though the current input is not used).

Example 4:  yn = xn
xn-1

This is again a first-order filter, since one
previous input value is required to give the current
output.

Example 5:  yn = (xn +
xn-1 ) / 2

The order of this filter is again equal to 1 since it
uses just one previous input value.

Example 6:  yn = (xn +
xn-1 + xn-2 ) / 3

To compute the current output yn , two
previous inputs (xn-1 and
xn-2 ) are needed; this is therefore a
second-order filter.

Example 7:  yn = (xn
xn-2 ) / 2

The filter order is again 2, since the processor must
store two previous inputs in order to compute the current
output. This is unaffected by the absence of an explicit
xn-1 term in the filter expression.

The order of a digital filter may be any positive integer. A
zero-order filter (such as those in Examples 1 and
2
above) is possible, but somewhat trivial, since it does
not really filter the input signal in the accepted sense.

Digital Filter Coefficients

All of the digital filter examples given in the previous
section can be written in the following general forms:

Zero order: yn = a0 xn
First order: yn = a0 xn +
a1 xn-1
Second order: yn = a0 xn +
a1 xn-1 +
a2 xn-2

Similar expressions can be developed for filters of any
order.

The constants a0 , a1 ,
a2 , … appearing in these expressions are
called the filter coefficients. The values of these
coefficients determine the characteristics of a particular
filter.

The table below gives the values of the coefficients of
each of the filters given as examples in the previous
section.

Example Order
a0

a1

a2
1 0 1
2 0 K
3 1 0 1
4 1 1 -1
5 1 1 /2 1 /2
6 2 1 /3 1 /3 1 /3
7 2 1 /2 0 1 /2

Recursive and Non-Recursive Filters

For all the examples of digital filters discussed so far, the
current output (yn ) is calculated solely from
the current and previous input values (xn ,
xn-1 , xn-2 , …). This type of filter
is said to be non-recursive.

A recursive filter is one which in addition to input
values also uses previous output values. These, like the
previous input values, are stored in the processor's
memory.


Note: FIR and IIR filters


Some people prefer an alternative terminology in which
a non-recursive filter is known as an FIR (or
Finite Impulse Response) filter, and
a recursive filter as an IIR (or Infinite
Impulse Response) filter. These terms refer
to the differing “impulse responses” of the two types
of filter

The impulse response of a digital filter is the output
sequence from the filter when a unit impulse is
applied at its input. (A unit impulse is a very simple
input sequence consisting of a single value of 1 at
time t = 0, followed by zeros at all subsequent
sampling instants). An FIR filter is one whose impulse
response is of finite duration. An IIR filter is one
whose impulse response (theoretically) continues forever, because the recursive (previous output) terms
feed back energy into the filter input and keep it
going. The term IIR is not very accurate, because the
actual impulse responses of nearly all IIR filters
reduce virtually to zero in a finite time.
Nevertheless, these two terms are widely used.

The word recursive literally means “running back”, and
refers to the fact that previously-calculated output values go
back into the calculation of the latest output. The expression
for a recursive filter therefore contains not only terms
involving the input values (xn , xn-1 ,
xn-2 , …) but also terms in yn-1 ,
yn-2 , …

From this explanation, it might seem as though recursive
filters require more calculations to be performed, since there
are previous output terms in the filter expression as well as
input terms. In fact, the reverse is usually the case. To
achieve a given frequency response characteristic using a
recursive filter generally requires a much lower order filter,
and therefore fewer terms to be evaluated by the processor,
than the equivalent non-recursive filter. This will be
demonstrated later.

Example of a Recursive Filter

A simple example of a recursive digital filter is given
by

yn = xn + yn-1

In other words, this filter determines the current output
(yn ) by adding the current input
(xn ) to the previous output
(yn-1 ).

Thus:

y0 = x0 + y-1
y1 = x1 + y0
y2 = x2 + y1
y3 = x3 + y2
… etc

Note that y-1 (like x-1 )
is undefined, and is usually taken to be zero.

Let us consider the effect of this filter in more detail.
If in each of the above expressions we substitute for
yn-1 the value given by the previous
expression, we get the following:

y0 = x0 + y-1 =
x0
y1 = x1 + y0 =
x1 + x0
y2 = x2 + y1 =
x2 + x1 + x0
y3 = x3 + y2 =
x3 + x2 + x1 + x0
… etc

Thus we can see that yn , the output at
t = nh, is equal to the sum of the current input
xn and all the previous inputs. This filter
therefore sums or integrates the input values, and so has a
similar effect to an analog integrator circuit.

This example demonstrates an important and useful feature of
recursive filters: the economy with which the output values are
calculated, as compared with the equivalent non-recursive
filter. In this example, each output is determined simply by
adding two numbers together.

For instance, to calculate the output at time t =
10h, the recursive filter uses the expression

y10 = x10 + y9

To achieve the same effect with a non-recursive filter
(in other words, without using previous output values stored in memory)
would entail using the expression

y10 = x10 + x9 +
x8 + x7 + x6 + x5
+ x4 + x3 + x2 +
x1 + x0

This would necessitate many more addition operations, as
well as the storage of many more values in memory.

Order of a Recursive (IIR) Digital Filter

The order of a digital filter was defined earlier as the
number of previous inputs which have to be stored in order to
generate a given output. This definition is appropriate for
non-recursive (FIR) filters, which use only the current and
previous inputs to compute the current output. In the case of
recursive filters, the definition can be extended as
follows:

The order of a recursive filter is the largest
number of previous input or output values required to
compute the current output.

This definition can be regarded as being quite general: it
applies both to FIR and IIR filters.

For example, the recursive filter discussed above, given
by the expression

yn = xn + yn-1

is classed as being of first order, because it uses one
previous output value (yn-1 ), even though no
previous inputs are required.

In practice, recursive filters usually require the same
number of previous inputs and outputs. Thus, a first-order
recursive filter generally requires one previous input
(xn-1 ) and one previous output
(yn-1 ), while a second-order recursive filter
makes use of two previous inputs (xn-1 and
xn-2 ) and two previous outputs
(yn-1 and yn-2 ); and so on,
for higher orders.

Note that a recursive (IIR) filter must, by definition, be
of at least first order; a zero-order recursive filter is an
impossibility.

Coefficients of Recursive (IIR) Digital Filters

From the above discussion, we can see that a recursive filter
is basically like a non-recursive filter, with the addition of
extra terms involving previous outputs (yn-1 ,
yn-2 , and so on).

A first-order recursive filter can be written in the general
form

yn = (a0 xn +
a1 xn-1 – b1 yn-1 )
/ b0

Note the minus sign in front of the “recursive” term
b1 yn-1 , and the factor (1/b0 )
applied to all the coefficients. The reason for expressing the
filter in this way is that it allows us to rewrite the
expression in the following symmetrical form:

b0 yn + b1 yn-1
= a0 xn +
a1 xn-1

In the case of a second-order filter, the general form
is

yn = (a0 xn +
a1 xn-1 + a2 xn-2
b1 yn-1 – b2 yn-2 )
/ b0

An alternative “symmetrical” form of this expression is

b0 yn + b1 yn-1
+ b2 yn-2 = a0 xn +
a1 xn-1 +
a2 xn-2

Note the convention that the coefficients of the inputs (the
x's) are denoted by a's, while the coefficients
of the outputs (the y's) are denoted by b's.

The Transfer Function of a Digital Filter

In the last section, we used two different ways of expressing
the action of a digital filter: a form giving the output
yn directly, and a “symmetrical” form with all
the output terms (y's) on one side and all the input
terms (x's) on the other.

In this section, we introduce what is called the transfer
function of a digital filter. This is obtained from the
symmetrical form of the filter expression, and it allows us to
describe a filter by means of a convenient, compact expression.
The transfer function of a filter can be used to determine
many of the characteristics of the filter, such as its
frequency response.

The Unit Delay Operator
First of all, we must introduce the unit delay operator,
denoted by the symbol

z-1

When applied to a sequence of digital values, this operator
gives the previous value in the sequence. Therefore, it introduces a delay of one sampling interval.

Applying the operator z-1 to an input
value (say xn ) gives the previous input
(xn-1 ):

z-1 xn = xn-1

Suppose we have an input sequence

x0 = 5
x1 = -2
x2 = 0
x3 = 7
x4 = 10

Then

z-1 x1 = x0 = 5
z-1 x2 = x1 = -2
z-1 x3 = x2 = 0

and so on. Note that z-1 x0
would be x-1 which is unknown (and usually
taken to be zero, as we have already seen).

Similarly, applying the z-1 operator to an
output gives the previous output:

z-1 yn = yn-1

Applying the delay operator z-1 twice
produces a delay of two sampling intervals:

z-1 (z-1 xn ) =
z-1 xn-1 = xn-2

We adopt the (fairly logical) convention

z-1 z-1 = z-2

in other words, the operator z-2
represents a delay of two sampling intervals:

z-2 xn = xn-2

This notation can be extended to delays of three or more
sampling intervals, the appropriate power of
z-1 being used.

Let us now use this notation in the description of a
recursive digital filter. Consider, for example, a general
second-order filter, given in its symmetrical form by the
expression

b0 yn + b1 yn-1
+ b2 yn-2 = a0 xn +
a1 xn-1 +
a2 xn-2

We will make use of the following identities:

yn-1 = z-1 yn

yn-2 = z-2 yn

xn-1 = z-1 xn

xn-2 = z-2 xn

Substituting these expressions into the digital filter
gives

(b0 + b1 z-1 +
b2 z-2 ) yn = (a0 +
a1 z-1 + a2 z-2 )
xn

Rearranging this to give a direct relationship between the
output and input for the filter, we get

yn / xn = (a0 +
a1 z-1 + a2 z-2 ) /
(b0 + b1 z-1 +
b2 z-2 )

This is the general form of the transfer function for a
second-order recursive (IIR) filter.

For a first-order filter, the terms in z-2
are omitted. For filters of order higher than 2, further terms
involving higher powers of z-1 are added to
both the numerator and denominator of the transfer
function.

A non-recursive (FIR) filter has a simpler transfer function
which does not contain any denominator terms. The coefficient
b0 is regarded as being equal to 1, and all
the other b coefficients are zero. The transfer function
of a second-order FIR filter can therefore be expressed in the
general form

yn / xn = a0 +
a1 z-1 + a2 z-2

Transfer Function Examples

  1. The three-term average filter, defined by the
    expression

    yn = 1 /3
    (xn + xn-1 + xn-2 )

    can be written using the z-1 operator
    notation as

    yn = 1 /3
    (xn + z-1 xn +
    z-2 xn )

    = 1 /3 (1 + z-1 +
    z-2 ) xn

    The transfer function for the filter is therefore

    yn / xn =
    1
    /3 (1 + z-1 +
    z-2 )

  2. The general form of the transfer function for a
    first-order recursive filter can be written

    yn / xn = (a0 +
    a1 z-1 ) / (b0 +
    b1 z-1 )

    Consider, for example, the simple first-order recursive
    filter

    yn = xn + yn-1

    which we discussed earlier. To derive the transfer
    function for this filter, we rewrite the filter expression
    using the z-1 operator:

    (1 – z-1 ) yn = xn

    Rearranging gives the filter transfer function as

    yn / xn = 1 / (1 –
    z-1 )

  3. As a further example, consider the second-order IIR
    filter

    yn = xn + 2xn-1 +
    xn-2 – 2yn-1 + yn-2

    Collecting output terms on the left and input terms on
    the right to give the “symmetrical” form of the filter
    expression, we get

    yn + 2yn-1 – yn-2 =
    xn + 2xn-1 + xn-2

    Expressing this in terms of the z-1
    operator gives

    (1 + 2z-1 – z-2 ) yn =
    (1 + 2z-1 + z-2 ) xn

    and so the transfer function is

    yn / xn = (1 + 2z-1 +
    z-2 ) / (1 + 2z-1
    z-2 )

Acknowledgments
This article originally appeared on Dr. Iain A. Robin's DSP site (www.dsptutor.freeuk.com).

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