Finite Impulse Response Filters - Digital Signal Processing and Applications with the C6713 and C6416 DSK - page 126

Different techniques are available for the design of FIR filters, such as a com-
monly used technique that utilizes the Fourier series, as discussed in Section 4.4.
Computer-aided design techniques such as that of Parks and McClellan are also
used for the design of FIR filters [5,6].
The convolution equation (4.23) is very useful for the design of FIR filters, since
we can approximate it with a finite number of terms, or

N -

 0

1

() = () ( ) yn hkxn k - (4.24)

k =

If the input is a unit impulse x ( n ) =d (0), the output impulse response will be y ( n )

= h ( n ). We will see in Section 4.4 how to design an FIR filter with N coefficients

h (0), h (1),..., h ( N

- 1), and N input samples x ( n ), x ( n - 1),..., x ( n - ( N - 1)). The

input sample at time n is x (n), and the delayed input samples are x ( n - 1),...,

x ( n 1)). Equation (4.24) shows that an FIR filter can be implemented with

knowledge of the input x ( n ) at time n and of the delayed inputs x ( n

- ( N -

k ). It is non-
recursive, and no feedback or past outputs are required. Filters with feedback
(recursive) that require past outputs are discussed in Chapter 5. Other names used
for FIR filters are transversal and tapped-delay filters.
The z -transform of (4.24) with zero initial conditions yields

-

() = () () + () () + () () +◊◊◊+ ( )

( )

( )

Yz hXzhzXzhzXz 0 1

- 1

2

- 2

hN z - 1

-- N 1

Xz

(4.25)

Equation (4.24) represents a convolution in time between the coefficients and
the input samples, which is equivalent to a multiplication in the frequency domain,
or

() = () () Yz HzXz (4.26)

where H ( z ) = ZT [ h ( k )] is the transfer function, or

N - 1

Â

() = () () + () () ( )

( ) - k - 1 - 2 -- N 1

Hz hkz = h 0 h 1 z + h 2 z +◊◊◊+ hN - 1 z

k = 0

()

( )

() () ( )

N - 1 N - 2 N - 3

hz 0 + hz 1 + hz 2 +◊◊◊+ hN - 1

= (4.27)

z

N - 1

which shows that there are N 1 poles, all of which are located at the origin. Hence,

this FIR filter is inherently stable, with its poles located only inside the unit circle.
We usually describe an FIR filter as a filter with “no poles.” Figure 4.2 shows an FIR
filter structure representing (4.24) and (4.25).
A very useful feature of an FIR filter is that it can guarantee linear phase .The
linear phase feature can be very useful in applications such as speech analysis, where
phase distortion can be critical. For example, with linear phase, all input sinusoidal

-