The unit delay operator

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 y_n 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. Tthe 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. It therefore in effect 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):

z^-1 x_n = 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:

z^-1 y_n = y_n-1

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

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

We adopt the (fairly logical) convention

z^-1 z^-1 = z^-2

i.e. the operator z^-2 represents a delay of two sampling intervals:

z^-2 x_n = x_n-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

b₀y_n + b₁y_n-1 + b₂y_n-2 = a₀x_n + a₁x_n-1 + a₂x_n-2

We will make use of the following identities:

y_n-1 = z^-1 y_n

y_n-2 = z^-2 y_n

x_n-1 = z^-1 x_n

x_n-2 = z^-2 x_n

Substituting these expressions into the digital filter gives

(b₀ + b₁z^-1 + b₂z^-2) y_n = (a₀ + a₁z^-1 + a₂z^-2) x_n

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

y_n / x_n = (a₀ + a₁z^-1 + a₂z^-2) / (b₀ + b₁z^-1 + b₂z^-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 ofz^-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

y_n / x_n = a₀ + a₁z^-1 + a₂z^-2

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)