## A.4 Pole frequency calculation

This annex describes the algorithm used to determine whether the pole frequency for a second order analysis of the signal frame is less than 385 Hz.

The filter coefficients for a second order synthesis filter are calculated from the first two unquantized reflection coefficients rc[1..2] obtained from the speech encoder. This is done using the routine described in clause 3.10.4. If the filter coefficients a[0..2] are defined such that the synthesis filter response is given by:

H(z) = 1 / (a[0] + a[1]z‑1 + a[2]z‑2 ) (1)

Then the positions of the poles in the Z‑plane are given by the solutions to the following quadratic:

a[0]z2 + a[1]z + a[2] = 0, a[0] = 1 (2)

The positions of the poles, z, are therefore:

z = re  j*sqrt(im), j2 = ‑1 (3)

where:

re = ‑ a[1] / 2 (4)

im = (4*a[2] ‑ a[1]2 ) / 4 (5)

If im is negative then the poles lie on the real axis of the Z‑plane and the signal is not a tone and the algorithm terminates. If re is negative then the poles lie in the left hand side of the Z‑plane and the frequency is greater than 2 000 Hz and the prediction error test can be performed.

If im is positive and re is positive then the poles are complex and lie in the right hand side of the Z‑plane and the frequency in Hz is related to re and im by the expression:

freq = arctan (sqrt(im)/re ) * 4 000 /  (6)

Having ensured that both im and re are positive, the test for a dominant frequency less than 385 Hz can be derived by substituting Equations 4 and 5 into Equation 6 and re‑arranging:

(4*a[2] ‑ a[1]2 ) / a[1]2 < (tan(*385/4 000))2 (7)

or

(4*a[2] ‑ a[1]2 ) / a[1]2 < 0.0973 (8)

If this test is true then the signal is not a tone and the algorithm terminates, otherwise the prediction error test is performed.

Annex B (normative):
Test sequences

The test vectors are described in the present document are supplied in archive en_300965v080001p0.zip which accompanies the present document. The files contained in this archive are listed in clause A.2.

The full rate test vectors apply to both GSM Phase 1 and Phase 2. However, the files pole1.* pole2.* pred1.* pred2.* and freq_sw.* are not required for Phase 1 (uplink and downlink) and Phase 2 uplink implementations.

Annex C (informative):
Change Request History

 Change history SMG No. TDoc. No. CR. No. Section affected New version Subject/Comments SMG#09 4.0.5 ETSI Publication SMG#17 4.2.1 ETSI Publication SMG#23 4.3.1 ETSI Publication SMG#23 5.0.3 Release 1996 version SMG#27 6.0.0 Release 1997 version SMG#29 7.0.0 Release 1998 version 7.0.1 Version update to 7.0.1 for Publication SMG#31 8.0.0 Release 1999 version 8.0.1 Update to Version 8.0.1 for Publication