Comparative Analysis of Methods Used in the Design of DTMF Tone Detectors

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1 Proceedings of the 2007 IEEE International Conference on Telecommunications and Malaysia International Conference on Communications, May 2007, Penang, Malaysia Comparative Analysis of Methods Used in the Design of DTMF Tone Detectors Suvad Selman Department of Electrical Engineering, University of Malaya, Raveendran Paramesran Department of Electrical Engineering, University of Malaya, Abstract Dual-Tone Multi-Frequency (DTMF) Signals are used in touch-tone telephones as well as many other areas. Since analog devices are rapidly being replaced with digital devices, digital DTMF decoders with superior quality have become the ideal replacement. In this paper a comparative analysis on the four DTMF detector designs are studied. They are namely, filter design approach, Discrete Fourier Transform(DFT), Goertzel Algorithm and Sub-Band on-uniform DFT (SB-DFT). The algorithms are developed in the MatLab environment and the implementation is done on a Digital Signal Processor, TMS320 C6713.The figure of merit and computational complexity are used as a performance measure and in both cases the SB-DFT is found to be the best. I. ITRODUCTIO The Dual-Tone Multi-Frequency dialing system traces its roots to a technique AT&T developed in the 950s called Multi- Frequency (MF)[1][2]. It was deployed within the AT&T telephone network to direct calls between switching facilities using in-band signaling. In the early 1960s, a derivative technique was offered by AT&T through its Bell System telephone companies as a modern way for network customers to place calls[3][4][5]. In AT&T s Compatibility Bulletin o. 105, AT&T described the product as a method for pushbutton signaling from customer stations using the voice transmission path[2][3][4]. The consumer product was marketed by AT&T under the registered trade name Touch-Tone. Other vendors of compatible telephone equipment called this system Tone dialing or Dual-Tone Multi-Frequency. Therefore, Dual- Tone Multi-Frequency (DTMF) is an international signaling standard for telephone digits (number buttons)[3][4]. These signals are used in touch-tone telephone call signaling as well as many other areas such as interactive control applications, telephone banking and pager systems. Since analog devices are rapidly replaced with digital devices, digital DTMF decoders have become important. Digital implementation has many advantages over analog implementation such as better accuracy, stability, re-programmability and a lower chip count; that is, instead of using several analog chips for decoding multi-channel DTMF tones, a digital signal processor (DSP) chip can be used. The DTMF system uses eight different signals, in which two signals are used to decode an entry request executed by means of pressing a push-button as shown in Figure 1. The frequencies used were chosen to prevent any harmonics from being incorrectly detected by the receiver Fig. 1. Touch-Tone Telephone Keypad: A Row and a Column Tone is Associated With Each Digit part[1][2]. The transmitter of a Dual-Tone Multi-Frequency signal simultaneously sends one frequency from the highgroup and one frequency from the low-group. This pair of signals represents the digit or symbol shown at the intersection of row and column in the table. For example, when digit 4 is pressed, the two signals, 770HZ and 1290HZ are selected and transmitted. Tone detection circuitry which forms an important part of the DTMF system detects the pressed key by analysing the two transmitted signals of 1290Hz and 770Hz. The common techniques used in the design and analysis of the detection circuitry are Filtering Design Approach Discrete Fourier Transform Goertzel Algorithm Sub-Band on-discrete Fourier Transform In this paper we investigate the above four methods in terms of their speed and complexity. Speed is of utmost importance in Dual-Tone Multi-Frequency decoding. Since multi-channel Dual-Tone Multi-Frequency decoding is performed by timemultiplexing the channels on a digital signal processor, a lower decoding time imply that a larger number of Dual-Tone Multi-Frequency channels can be decoded simultaneously. Whenever a button is pressed, a Dual-Tone Multi-Frequency /07/$ IEEE. 335

2 signal containing two frequencies is generated. II. TOE DETECTIO Decoding a Dual-Tone Multi-Frequency signal involves extracting the two tones in the signal and determining from their values the pressed Dual-Tone Multi-Frequency digit. Tone detection by analog circuits are implemented by counting the number of zero-crossing of the input signal. Alternatively, the tone detection is also done by analog filters. ow, with the advancement of digital technology and widespread use of PCM systems, some digital realizations based on the use of digital FIR and IIR filters ave been proposed[7][8][9]. One of the most used detection technique is based on an algorithm for efficient computing Discrete Fourier Transform, known as the Goertzel s algorithm[12][13]. This algorithm has been further improved in detecting the tones. Finally, for the detection of Dual-Tone Multi-Frequency a new transform denoted as Sub- Band on-uniform Discrete Fourier Transform (SBDFT) was introduced, which permits most efficient computation and reliable signal detection in the presence of noise. A. Filtering Approach The filtering approach is the conceptually the simplest method for multi-frequency detection. The composite signal is usually first filtered by a pair of lowpass / highpass filters and then by a set of bandpass filters as shown in Figure 2[9]. The lowpass / highpass group filters divide the composite signal into low and high group in the case of Dual-Tone Multi- Frequency. The number of bandpass blocks depends on the number of frequencies in each of the ranges. The filters used in B. Discrete Fourier Transform Several techniques for digital Dual-Tone Multi-Frequency detection have been used, but most designers have settled on either digital filtering or Discrete Fourier Transform (DFT). In digital filtering, Dual-Tone Multi-Frequency signals are passed through digital bandpass filters centered at the signaling frequencies (shades of analog). The power at each frequency is then measured repeatedly to detect the Dual-Tone Multi- Frequency tones. A DSP then interprets and translates them for the proper switching. The DFT also measures signal power at the signaling frequencies but has the additional need to check for signals oome minimum duration. The problem addressed in this subsection is how, given a digital singlechannel data stream, to detect the presence of valid Dual- Tone Multi-Frequency tones. The algorithm must be capable of accurately determining which of the eight Dual-Tone Multi-Frequency frequencies are present, the relative signal power at each of the frequencies, and the duration that these signals are present. Several methods exist for determining the component of a signal at a given frequency[11]. The DFT allows us to represent non periodic discrete signal as the algebraic sum of its frequency components. The DFT has the form, X(k) 1 2πkn j x[n]e 1 x[n]w kn (1) where x[n] is a finite duration sequence of length and n 0, 1, 2, 1[10]. If you consider signal, x(t) A sin(ω 0 t) (2) the sampled signal than will be x[n] A sin( ω 0n ) A sin( 2πfn ) (3) The Discrete Fourier Transform (DFT) of the signal is than given by, X(k) 1 2πn j x[n]e for n 0, 1, 2,, 1. 1 A sin( 2πfn )e 2πn (4) Fig. 2. Dual-Tone Multi-Frequency detection using digital filters. this approach can be realized using IIR [9], or FIR filters[7][8]. In the case of IIR realization, the group filters are usually realized as 4th or 6th order elliptic filters, while bandpass filters can be simpler 2nd or 4th order Butterworth filters. In the case of FIR realization, the group filters are not used, but bandpass filters should be of high order, typically greater than 30. Due to very high computational complexity, these filters are usually implemented in hardware. Modern tone detection schemes do not use this approach, since it is not suitable for detection of tones in multiple channels. C. Goertzel Algorithm The DFT approach is conceptually very simple which could be seen from the previous section. A block oamples are taken, and its Discrete Fourier Transform is found for a set of frequencies used in signalization. If x[n] denotes receiving signal, the Discrete Fourier Transform (DFT) of samples of x[n] will be simply computed as in Eq. 1. Since the number of frequencies is small, the use of Fast Fourier Transform algorithms is not suitable. The most efficient way to find DFT for small number of frequencies is the use of Goertzel s algorithm[12][13]. The Goertzel s algorithm is based on the use opecial second-order cell, which uses 336

3 signal samples as input and gives the DFT coefficients as its output. The Goertzel algorithm calculates the frequency domain magnitude at the specified frequency. Goertzel is not considered a Fast Fourier Transform (FFT) because it is order n 2, nor order n log 2 n. The Goertzel algorithm consists of a group of a second-order (IIR) filters that extracts the energy present at a specific frequency. It is more efficient than an FFT when log2 or fewer coefficients of the DFT are needed. The filter bank implementation has the tremendous advantage that it can process the input data as it arrives. The FFT has to wait until the entire sample window has arrived. Therefore, the Goertzel algorithm reduces the data memory required significantly. As it can be seen that Goertzel Algorithm is more efficient way of Dual-Tone Multi-Frequency tone detection we will have inside look of the algorithm. Consider the twiddle factor property, W k e j 2πk 1 (5) If both sides of Eq. 5 are multiplied with W k, we will have X(k) W k 1 1 x[n]w kn x[n]w k( n) (6) Above DTF can be related to the bandpass filter by defining new sequence y k [n] 1 x[m]w k(n m) x[n] W kn u[n] (7) The transfer function reveals that the coefficient W k is complex[12], thus requiring complex multiplication and addition. Replacing the complex coefficient with the real coefficient can reduce computational load [13]. After reduction transfer function looks like H k(z) 1 W k z cos( 2πk )z 1 + z 2 (8) From Figure 3 real coefficients are used in the feedback, and the recursive computation of y k [n] is expressed as v k [n] x[n] + 2 cos( 2πfk )v k (n 1) v k (n 2) (9) The complex computation can be eliminated by computing a magnitude square of X(k) for tone detection. The square magnitude is computed as, 2 ( ) 2πk X(k) vk( 1) 2 2 cos v k ( 1)v k ( 2)+v 2 k( 2) (10) This equation does not contain any complex coefficients. The second-order recursive computation of the DFT by means of the Goertzel algorithm as shown in Fig. 3 computes a new y k [n] output for every new input sample x[n]. The DFT result, X(k), is equivalent to y k [n] when n, i.e., X(k) y k (). Since any other value of y k [n], in which n, does not contribute to the end result X(k), there is no need to compute y k [n] until n. This implies that the Goertzel algorithm is functionally equivalent to a secondorder IIR filter, except that the one output result of the filter is generated only after input samples have occurred. D. Sub-Band on-uniform DFT To decode a Dual-Tone Multi-Frequency signal, its energy at the eight Dual-Tone Multi-Frequency frequencies are determined by evaluating the samples of the DFT at these frequencies. In the proposed SB-DFT algorithm, these DFT samples are computed by decomposing the input signal into two sub-bands. Since Dual-Tone Multi-Frequency signals occupy the low-frequency part of the telephone bandwidth, the higher sub-band can be discarded for a fast, approximate computation. This algorithm is based on the fact that all useful frequencies in Dual-Tone Multi-Frequency algorithms are below 2 KHz, what permits sub-band decomposition and the use of two times lower sample rate (4 KHz). The nonuniform discrete Fourier transform (DFT) of a sequence x[n] of length is defined [14] as: Fig. 3. Block diagram of the Goertzel algorithm without complex coefficient X(x k ) 1 x[n]z k (11) where k 0, 1, 2,, 1. The problem of determining the energies present, in the signal at the DTMF frequencies is, therefore, equivalent to computing the DFT samples located at the eight DTMF frequencies. This is also equivalent to modified Goertzel algorithm[15]. The subband DFT (SB-DFT) is a method for computing the DFT based on a sub-band decomposition of the input sequence[16]. Consider a sequence x[n] with an even number oamples. We first decompose x[n] into two subsequences, g L [n] and g H [n], of length /2 each: 337

4 g L [n] 1 (x[2n] + x[2n + 1]) 2 g H [n] 1 (12) 2 (x[2n] x[2n + 1]) for, n 0, 1, 2, These sequences are simply lowpass/highpass filtered and down sampled versions of x[n]. By substituting for x[n] in the DFT expression in Eq. 11, it can be obtained X(z k ) (1 + z 1 k )G L(zk) 2 + (1 z 1 k )G H(zk) 2 (13) where G L (zk 2) and G H(zk 2 ) are the DFTs of the sequences, g L [n] and g H [n], evaluated at z zk 2. Therefore, we can find the DFT of x[n] at the point z zk 2 by computing the DFTs of the smaller sequences, g L [n] and g H [n],at the point z zk 2 and then combining them. For Dual-Tone Multi-Frequency decoding, we only need the squared magnitudes of the approximate DFT samples. X(z k ) 2 4 X(z k ) 2 8G L (z 2 (1 + cos ω k ) 2 k ) 2 (14) 1 + cos ω k The factor of eight in Eq.14 can be neglected in the calculation, because only the relative squared magnitudes are of importance. The value of G L (zk 2 ) 2 is computed by using the modified Goertzel algorithm [15]as shown in Figure 4. Fig. 4. DTMF decoding algorithm using the SB-DFT. In the computation the two coefficients for each frequency indicated as, in Figure4,a k and b k can be obtained a k 2 cos 2ω k (15) b k cos ω k (16) III. COMPARATIVE AALYSIS The DTMF decoding algorithm based on the SB-DFT has a much lower computational complexity compared with algorithms based on the DFT (Goertzel Algorithm) and DFT[15]. Let, be the number oamples required to properly detect all DTMF digits under the receiving conditions specified by the AT&T standard, Touch-Tone[6]. Since the minimum duration of a DTMF signal is 40 ms, and the sampling rate is 8 khz, the number oamples available for decoding each DTM digit is 0.04 x The value of required depends on the detection algorithm used. The algorithms based on the DFT and DFT require ( + 4) real multiplications and (2 +2) real additions for each frequency, leading to a total of (8 + 32) real multiplications and ( ) real additions for the eight DTMF frequencies. Therefore, the SB-DFT algorithm reduces computation by nearly half as compared with those based on the DFT and DFT. We can also have a DTMF decoding algorithm based on the SB-DFT. This is similar to the SB-DFT algorithm described here, except that ω k 2πk/. Eight integer values are chosen for k such that the corresponding frequencies k / are closest to the DTMF frequencies. Although the computational complexity of this algorithm is the same as for the SB-DFT, the sampling error caused by uniform sampling leads to a worse performance. A. Figure of Merit To compare the performance of the proposed algorithm with those of other approaches, we define a new measure. Assuming that the dialed DTMF digit, was decoded successfully, the figure of merit corresponds to the minimum among the differences between the db-levels of the highest and second highest row and column tones: where, F min(d r, D c ) (17) D r 10 log X r 10 log X rs D c 10 log X c 10 log X cs (18) Here, X r, and X c, are the maximum values among the energies detected at the row and column tones, respectively; X rs, and X cs, are the second-highest values among the energies detected at the row and column tones, respectively. Clearly, F > 0 for successful decoding. A higher figure of merit is desirable so that the operational tones are detected at levels reliably higher than the non-operational tones. Typically, the number of input samples is chosen so that the worst-case figure of merit is about 6 db. TABLE I MIIMUM UMBER O SAMPLES REQUIRED BY DTMF DECODIG ALGORITHMS TO ATTAI VARIOUS FIGURES OF MERIT. B. Performance Comparison F 3 db 6 db 9 db DFT SB-DFT A performance comparison between four DTMF decoding algorithms is presented here. We determine the figure of merit F attained by each algorithm for a given number of input samples. For each value of, the worst-case figure of merit is determined by considering all the 16 DTMF digits, each under three twist conditions (no twist, maximum reverse twist, and maximum normal twist), and then taking the lowest value of F obtained. The DTMF signal is generated by adding 338

5 two sinusoids at the operational row and column frequencies, and six sinusoids with 30 db lower amplitudes at the other six frequencies. To simulate the receiving conditions specified in the AT&T Touch-Tone standard, we vary the two operational frequencies within tolerance bands, ±1.5% about the standard frequencies, by taking f to be multiples of 5 Hz. We compare the four algorithms quantitatively by determining the minimum number oamples required to attain a certain figure of merit. This is shown in Table 1 for three values of the figure of merit. The entry indicates that the particular figure of merit is never attained. As shown in Table 1, we cannot attain a high figure of merit using the DFT and SB- DFT algorithms. Finally, we determine the total computation required to detect the energies present in the input signal at the eight DTMF frequencies, using the minimum number of samples given in Table 1. The number of real multiplications M, and the number of real additions A, are shown in Table 2. Clearly, the SBDFT algorithm requires the least computation to attain a certain figure of merit. The gain is substantial when compared to the other methods. [7] F.G. Braun, onrecursive digital filters for detecting multifrequency coded signals, IEEE Trans. Acoust. Speech, Signal Process., Vol. 23, o. 3, pp , June [8] G. Thyssens, L. Verbist, Digital multifrequency receivers and senders, Electr. Commun., Vol. 54, o. 4, pp , [9] S. Kikuchi, H. Kitano, J. Matsuda, A study on digital pushbutton signaling receiver, Review of Electr. Commun. Lab., Vol. 27, o. 7-8, pp , Aug [10] Marple, S. Lawrence Jr. Digital Spectral Analysis. Prentice-Hall, Inc., 1987, pp [11] Oppenheim, Alan V. and Schafer, Ronald W. Digital Signal Processing. Prentice-Hall of India, [12] I. Koval, G. Gara, Digital MF receiver using Discrete Fourier transform, IEEE Trans. Commun., Vol. 21, o. 12, pp , Dec [13] P. Mock, Add DTMF generation and decoding to DSP P designs, ED, Vol. 30, pp , March [14] S.K. Mitra, S. Chakrabarti, and E. Abreu, onuniform discrete Fourier transform and its applications in signal processing, in Proc. EU-SIPCO 92, pp , August [15] S.L. Gay, J. Wartung, and G.L. Smith, Algorithms for multi-channel DTMF detection for the WE DSP32 family, in Proc. IEEE Int. Conf. on ASSP, pp , May [16] S. Bagchi, The onuniform Discrete Fourier Transform and its Applications in Signal Processing. Ph.D Dissertation, University of California at Santa Barbara, TABLE II COMPUTATIO REQUIRED BY DTMF DECODIG ALGORITHMS TO DETECT EERGIES PRESET AT THE EIGHT DTMF FREQUECIES USIG THE MIIMUM UMBER OF SAMPLES. 3 db 6 db 9 db F M A M A M A DFT SB-DFT IV. COCLUSIO These algorithms utilize the fact that DTMF frequencies reside in the low-frequency part of the telephone bandwidth. A good approximation is obtained to the energies present at the DTMF frequencies by discarding calculations in the highfrequency band of a two-band decomposition. Simulations are done to compare the performances of two DTMF decoding algorithms based on the DFT and SB-DFT. It is found that the SB-DFT algorithm requires much lower computation than the others to attain a desired level of performance. REFERECES [1] Dual-Tone Multifrequency Receiver Generic Requirements for End-to- End Signaling Over Tandem- Switched Voice Links. Bell Communications Research Technical Reference TR-TSY , Issue 1, March [2] International Telecommunication Union, ITU-T Recommendation Q.23, Technical features of push-button telephone sets, Fascicle VI.1, Blue Book, [3] International Telecommunication Union, ITU-T Recommendation Q.24, Multifrequency push-button signal detection, Fascicle VI.1, Blue Book, [4] International Telecommunication Union, ITU-T Recommendation Q.454, The sending part of the multifrequency signaling equipment, Fascicle VI.4, Blue Book, [5] International Telecommunication Union, ITU-T Recommendation Q.455, The receiving part of the multifrequency equipment, Fascicle VI.4, Blue Book, [6] ITU, CCITT Red Book, October Volume VI, Fascicle VI.l. 339

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