Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 378

1

x ()

vn

k ()

yn

k ()

z

1

W N k

vn

k (

1

2 k

N

2 cos

1

z

1 vn

k (

2 )

FIGURE 8.53

Second-order Goertzel IIR filter.

v k ðnÞ¼ 2 cos 2 pk
N

v k ðn 1 Þv k ðn 2 ÞþxðnÞ (8.71)

y k ¼ v k W N v k n 1 n n

(8.72)

with initial conditions v k ð 2 Þ¼ 0, v k ð 1 Þ¼ 0
Then the DFT coefficient XðkÞ is given as

XðkÞ¼y k ðNÞ (8.73)

The squared magnitude x(k) is computed as

k ðN 1 Þ 2 cos 2 pk

2 2 2

jXðkÞj ¼ v

k ðNÞþv

v k ðNÞv k ðN 1 Þ (8.74)

N

We show the derivation of Equation (8.74) as follows. Note that Equation (8.72) involves complex
algebra, since the equation contains only one complex number, a factor

¼ cos 2 pk
N

j sin 2 pk
N

2 pk
N

W N ¼ e j

discussed in Chapter 4. If our objective is to compute the spectrum value, we can substitute n ¼ N into
Equation (8.72) to obtain XðkÞ and multiply XðkÞ by its conjugate X ðkÞ to achieve the squared

magnitude the DFT coefficient. It follows (Ifeachor and Jervis, 2002) that

2

¼ XðkÞX ðkÞ

jXðkÞj

Since

X k ¼ y k N W N v k N 1
X

¼ y k W N v k N 1

k N