Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 380

y 1 ð1Þ¼v 1 ð1Þþjv 1 ð0Þ¼2 þ j 1 ¼ 2 þ j

v 1 ð2Þ¼v 1 ð0Þþxð2Þ¼1 þ 3 ¼ 2

y 1 ð2Þ¼v 1 ð2Þþjv 1 ð1Þ¼2 þ j 2 ¼ 2 þ j2

v 1 ð3Þ¼v 1 ð1Þþxð3Þ¼2 þ 4 ¼ 2

y 1 ð3Þ¼v 1 ð3Þþjv 1 ð2Þ¼2 þ j 2 ¼ 2 þ j2

v 1 ð4Þ¼v 1 ð2Þþxð4Þ¼2 þ 0 ¼2

y 1 ð4Þ¼v 1 ð4Þþjv 1 ð3Þ¼2 þ j 2 ¼2 þ j2

Then the DFT coefficient and its squared magnitude are determined as

X ð1Þ¼y 1 ð4Þ¼2 þ j2

jX ð1Þj 2 ¼ v 1 4 þ v 1 3 ¼ð2Þ 2 þð2Þ 2 ¼ 8

Thus, the two-sided amplitude spectrum is computed as

r

jX ð1Þj 2

A 1 ¼ 1
4

¼ 0:7071

and the corresponding single-sided amplitude spectrum is A 1 ¼ 2 0:707 ¼ 1:4141.

From this simple illustrative example, we see that the Goertzel algorithm has the following
advantages:

1. We can apply the algorithm for computing the DFT coefficient XðkÞ for a specified frequency
bin k ; unlike the fast Fourier transform (FFT) algorithm, all the DFT coefficients are computed
once it is applied.
2. If we want to compute the spectrum at frequency bin k , that is, jXðkÞj , Equation (8.71) shows that
we need to process v k ðnÞN þ 1 times and then compute jXðkÞj

2 . The operations avoid complex

algebra.

If we use the modified Goertzel filter in Figure 8.54 , then the corresponding transfer function is
given by

G k ðzÞ¼ V k ðzÞ

1
1 2 cos 2 pk
N

XðzÞ ¼

(8.77)

z 1 þ z 2