# Efficient Detection Approach for DTMF Signal Detection
## 2 Related Works  
In this section, two conventional single point detection approaches, the one-point DFT and Goertzel approaches, will be discussed. First, the DFT of a signal \(x[n]\) is defined as:  
\[\begin{array}{ll}X[k]&=\sum\limits_{n=0}^{N-1}x[n]W_{N}^{nk}=\sum\limits_{n= 0}^{N-1}x[n]e^{-j\frac{2\pi nk}{N}}\\ &=\sum\limits_{n=0}^{N-1}x[n]\Big{\{}\cos\left(\frac{2\pi nk}{N}\right)-j\sin \left(\frac{2\pi nk}{N}\right)\Big{\}}\end{array} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Symbols** & **1209 Hz** & **1336 Hz** & **1477 Hz** & **1633 Hz** \\ \hline
697 Hz & 1 & 2 & 3 & A \\
770 Hz & 4 & 5 & 6 & B \\
852 Hz & 7 & 8 & 9 & C \\
941 Hz & * & 0 & \# & D \\ \hline \hline \end{tabular}
\end{table}
Table 1: Relations between the paired frequencies and the corresponding symbols in dual-tone multi-frequency (DTMF) signals.  
and is used to the one-point DFT approach along the frequency bin \(k\). During a tone detection, \(X[k]\) is first evaluated corresponding to each tone frequency, and then the tone corresponding to the input signal \(x[n]\) can be identified. As indicated in Equation (1), a computational load of \(2N\) real multiplications and (\(2N-2\)) additions are required to evaluate \(X[k]\) at a specified \(k\). That is, such computational load is required when the one-point DFT approach is performed at a specified \(k\).  
On the other hand, the direct-form realization of a Goertzel algorithm is presented in Figure 1[13], and the operation thereof is expressed as:  
\[v_{k}[n]=x[n]+2\cos(\frac{2\pi k}{N})v_{k}[n-1]-v_{k}[n-2] \tag{2}\]  
\[X[k]=y_{k}[N]=v_{k}[N]-W_{N}^{k}v_{k}[N-1] \tag{3}\]  
where \(X[k]\) is the DFT of a signal \(x[n]\). In this manner, (2) is iterated \(N\) times until \(v_{k}[N]\) is acquired, and then \(X[k]\) is obtained by (3). In addition, \(k\) is constant when (2) and (3) are evaluated. In this context, evaluation of \(X[k]\) at a specified \(k\) requires a computational load of (\(N+2\)) real multiplications and (\(2N+1\)) additions. Obviously, this approach is found to have the same tone detection procedure as the above one-point DFT approach.  
To further analyze the valid DTMF tones in ITU specification, an example of tone detection at a frequency of 1477 Hz with a sampling rate (SR) of 8000 Hz and a DFT size of 512 is illustrated in Figure 2. In this case, frequencies over the range of 1455 to 1499 Hz must be accepted, whereas those below 1425 or beyond 1529 Hz must be rejected. Hence, tone detections must be performed over the frequency bin \(k=92\) to 97.  
Figure 1: Direct-form realization of a Goertzel algorithm.  
Figure 2: An example of tone detection at a frequency of 1477 Hz.  
Subsequently, a comparison is made between the one-point DFT approach and the Goertzel algorithm as follows. Taking Figure 2 as an instance, the former requires \(2N\) real multiplications and \((2N-2)\) additions to evaluate \(X[k]\), given in (1), for frequency bin \(k\) ranging between 92 and 97, that is, the above-stated evaluations are performed for six times in total. In contrast, the latter requires \((N+2)\) real multiplications and \((2N+1)\) additions to evaluate \(X[k]\), given in (2) and (3), for \(k\) in the same range as in the former case, but the latter is found to have a 25% approximately computational load reduction in comparison with the former.  
As described above, one thing that these two detection approaches have in common is that a tone detection task is performed at a single point each time, meaning that there is an inevitable and significant rise in the required computational load.