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Dual-tone Multifrequency (DTMF) Signals 629
FIGURE 12.15
Realization of two-pole resonator for computing the DFT
The desired output is X (k) =y k (N). To perform this computation, we
can compute once and store the phase factor W −k
N .
The complex multiplications and additions inherent in (12.46) can
be avoided by combining the pairs of resonators possessing complex conjugate
poles. This leads to 2-pole filters with system functions of the
form
1 − WN k H k (z) =
z−1
1 − 2 cos (2πk/N) z −1 + z −2 (12.47)
The realization of the system illustrated in Figure 12.15 is described by
the difference equations
v k (n)=2 cos 2πk
N v k (n − 1) − v k (n − 2) + x(n) (12.48)
y k (n)=v k (n) − W k Nv k (n − 1) (12.49)
with initial conditions v k (−1) = v k (−2) = 0. This is the Goertzel algorithm.
The recursive relation in (12.48) is iterated for n =0, 1,...,N, but the
equation in (12.49) is computed only once, at time n = N. Each iteration
requires one real multiplication and two additions. Consequently, for a real
input sequence x(n), this algorithm requires N +1 real multiplications to
yield not only X (k) but also, due to symmetry, the value of X (N − k).
We can now implement the DTMF decoder by use of the Goertzel
algorithm. Since there are eight possible tones to be detected, we require
eight filters of the type given by (12.47), with each filter tuned to one of the
eight frequencies. In the DTMF detector, there is no need to compute the
complex value X (k); only the magnitude |X(k)| or the magnitude-squared
value |X(k)| 2 will suffice. Consequently, the final step in the computation
of the DFT value involving the numerator term (feedforward part of the
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