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

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.

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

x_i = 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, ... , x_n

Note that the sampled values x_n+1, x_n+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, ... , y_n

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

1. UNITY GAIN FILTER: y_n = x_n

   Each output value y_n is exactly the same as the corresponding input value x_n:

   
   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: y_n = Kx_n (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: y_n = x_n-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: y_n = x_n - x_n-1

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

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

Example 1:  y_n = x_n

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

Example 2:  y_n = Kx_n

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

Example 3:  y_n = x_n-1

This is a first-order filter, as one previous input (x_n-1) is required to calculate y_n. (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:  y_n = x_n - x_n-1

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

Example 5:  y_n = (x_n + x_n-1) / 2

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

Example 6:  y_n = (x_n + x_n-1 + x_n-2) / 3

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

Example 7:  y_n = (x_n - x_n-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 x_n-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.

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.

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.

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.

In other words, this filter determines the current output (y_n) by adding the current input (x_n) to the previous output (y_n-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 y_n-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 y_n, the output at t = nh, is equal to the sum of the current input x_n 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.

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

y_n = x_n + y_n-1

is classed as being of first order, because it uses one previous output value (y_n-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 (x_n-1) and one previous output (y_n-1), while a second-order recursive filter makes use of two previous inputs (x_n-1 and x_n-2) and two previous outputs (y_n-1 and y_n-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.

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

Note the minus sign in front of the "recursive" term b₁y_n-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:

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

An alternative "symmetrical" form of this expression is

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.

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

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 x_n) gives the previous input (x_n-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:

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

We adopt the (fairly logical) convention

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

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

We will make use of the following identities:

Substituting these expressions into the digital filter gives

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

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

Transfer Function Examples

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

   y_n = ¹/₃ (x_n + x_n-1 + x_n-2)

   can be written using the z^-1 operator notation as

   y_n = ¹/₃ (x_n + z^-1x_n + z^-2x_n)

   = ¹/₃ (1 + z^-1 + z^-2) x_n

   The transfer function for the filter is therefore

   y_n / x_n = ¹/₃ (1 + z^-1 + z^-2)

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

   y_n / x_n = (a₀ + a₁z^-1) / (b₀ + b₁z^-1)

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

   y_n = x_n + y_n-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) y_n = x_n

   Rearranging gives the filter transfer function as

   y_n / x_n = 1 / (1 - z^-1)

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

   y_n = x_n + 2x_n-1 + x_n-2 - 2y_n-1 + y_n-2

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

   y_n + 2y_n-1 - y_n-2 = x_n + 2x_n-1 + x_n-2

   Expressing this in terms of the z^-1 operator gives

   (1 + 2z^-1 - z^-2) y_n = (1 + 2z^-1 + z^-2) x_n

   and so the transfer function is

   y_n / x_n = (1 + 2z^-1 + z^-2) / (1 + 2z^-1 - z^-2)