Audio Engineering

470 Chapter 15

frequencies forces the number of stages to be very high. You can appreciate this through
recognizing that the FIR fi lter response is determined by the number and value of the
coeffi cients applied to each of the taps in the delayed signal stages. The value of these
coeffi cients is an exact copy of the fi lter’s impulse response. Thus an impulse response
intended to be effective at low frequencies is likely to require a great many stages. This
places pressure on the hardware that has to satisfy the demand to perform the necessary
large number of multiplications within the time allotted for processing each sample value.
In many situations a suffi ciently accurate response can be obtained with less circuitry by
feeding part of a fi lter’s output back to the input ( Figure 15.16 ).

Single sample delay units

a 5

7/96

x(n)

xaverage(n)

a 1

7/96

a 4

24/96

a 3

34/96

a 2

24/96



(e)

Figure 15.15(e) : A useful way of showing the process being carried out in (d) is to

draw a block diagram in which each time that a sample value is read it is loaded into a

form of memory while the previous value is moved on to the next memory stage. We

take the current value of the input sample and the output of each of these memory

stages and multiply them by the weighting factor before summing them to produce the

output average. The operation can also be expressed in an algebraic form in which the

numerical values of the weighting coeffi cients have been replaced by an algebraic symbol:

xnax ax ax axaverage ().11 2 2 3 3 4 4nn n n This is a simple form of a type of

digital fi lter known as a fi nite impulse response or transversal fi lter. In the form shown

here it is easy to see that the delay of the fi lter is constant and thus the fi lter will show

linear phase characteristics. If the input to the fi lter is an impulse, the values you should

obtain at the output are identical, in shape, to the profi le of the weighting values used. This

useful property can be used in the design of fi lters, as it illustrates the principle that the

characteristics of a system can be determined by applying an impulse to it and observing the

resultant output.