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DFT Estimation with the Goertzel Algorithm

This example shows how to use the Goertzel function to implement a DFT-based DTMF detection algorithm.

Dual-tone Multi-Frequency (DTMF) signaling is the basis for voice communications control and is widely used worldwide in modern telephony to dial numbers and configure switchboards. It is also used in systems such as in voice mail, electronic mail and telephone banking.

Generating DTMF Tones

A DTMF signal consists of the sum of two sinusoids - or tones - with frequencies taken from two mutually exclusive groups. These frequencies were chosen to prevent any harmonics from being incorrectly detected by the receiver as some other DTMF frequency. Each pair of tones contains one frequency of the low group (697 Hz, 770 Hz, 852 Hz, 941 Hz) and one frequency of the high group (1209 Hz, 1336 Hz, 1477Hz) and represents a unique symbol. The frequencies allocated to the push-buttons of the telephone pad are shown below:

                        1209 Hz   1336 Hz   1477 Hz
                       _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
                      |         |         |         |
                      |         |   ABC   |   DEF   |
              697 Hz  |    1    |    2    |    3    |
                      |_ _ _ _ _|_ _ _ _ _|_ _ _ _ _|
                      |         |         |         |
                      |   GHI   |   JKL   |   MNO   |
              770 Hz  |    4    |    5    |    6    |
                      |_ _ _ _ _|_ _ _ _ _|_ _ _ _ _|
                      |         |         |         |
                      |   PRS   |   TUV   |   WXY   |
              852 Hz  |    7    |    8    |    9    |
                      |_ _ _ _ _|_ _ _ _ _|_ _ _ _ _|
                      |         |         |         |
                      |         |         |         |
              941 Hz  |    *    |    0    |    #    |
                      |_ _ _ _ _|_ _ _ _ _|_ _ _ _ _|

First, let's generate the twelve frequency pairs

symbol = {'1','2','3','4','5','6','7','8','9','*','0','#'};
lfg = [697 770 852 941]; % Low frequency group
hfg = [1209 1336 1477];  % High frequency group
f  = [];
for c=1:4,
    for r=1:3,
        f = [ f [lfg(c);hfg(r)] ];
    end
end
f'
ans =

         697        1209
         697        1336
         697        1477
         770        1209
         770        1336
         770        1477
         852        1209
         852        1336
         852        1477
         941        1209
         941        1336
         941        1477

Next, let's generate and visualize the DTMF tones

Fs  = 8000;       % Sampling frequency 8 kHz
N = 800;          % Tones of 100 ms
t   = (0:N-1)/Fs; % 800 samples at Fs
pit = 2*pi*t;

tones = zeros(N,size(f,2));
for toneChoice=1:12,
    % Generate tone
    tones(:,toneChoice) = sum(sin(f(:,toneChoice)*pit))';
    % Plot tone
    subplot(4,3,toneChoice),plot(t*1e3,tones(:,toneChoice));
    title(['Symbol "', symbol{toneChoice},'": [',num2str(f(1,toneChoice)),',',num2str(f(2,toneChoice)),']'])
    set(gca, 'XLim', [0 25]);
    ylabel('Amplitude');
    if toneChoice>9, xlabel('Time (ms)'); end
end
set(gcf, 'Color', [1 1 1], 'Position', [1 1 1280 1024])
annotation(gcf,'textbox', 'Position',[0.38 0.96 0.45 0.026],...
    'EdgeColor',[1 1 1],...
    'String', '\bf Time response of each tone of the telephone pad', ...
    'FitBoxToText','on');

Playing DTMF Tones

Let's play the tones of phone number 508 647 7000 for example. Notice that the "0" symbol corresponds to the 11th tone.

audid = audiodevinfo(0,Fs,16,1);
if  audid ~= -1
    for i=[5 11 8 6 4 7 7 11 11 11],
        p = audioplayer(tones(:,i),Fs,16,audid);
        play(p)
        pause(0.5)
    end
end

Estimating DTMF Tones with the Goertzel Algorithm

The minimum duration of a DTMF signal defined by the ITU standard is 40 ms. Therefore, there are at most 0.04 x 8000 = 320 samples available for estimation and detection. The DTMF decoder needs to estimate the frequencies contained in these short signals.

One common approach to this estimation problem is to compute the Discrete-Time Fourier Transform (DFT) samples close to the seven fundamental tones. For a DFT-based solution, it has been shown that using 205 samples in the frequency domain minimizes the error between the original frequencies and the points at which the DFT is estimated.

Nt = 205;
original_f = [lfg(:);hfg(:)]  % Original frequencies
original_f =

         697
         770
         852
         941
        1209
        1336
        1477

k = round(original_f/Fs*Nt);  % Indices of the DFT
estim_f = round(k*Fs/Nt)      % Frequencies at which the DFT is estimated
estim_f =

         702
         780
         859
         937
        1210
        1327
        1483

To minimize the error between the original frequencies and the points at which the DFT is estimated, we truncate the tones, keeping only 205 samples or 25.6 ms for further processing.

tones = tones(1:205,:);

At this point we could use the Fast Fourier Transform (FFT) algorithm to calculate the DFT. However, the popularity of the Goertzel algorithm in this context lies in the small number of points at which the DFT is estimated. In this case, the Goertzel algorithm is more efficient than the FFT algorithm.

Plot Goertzel's DFT magnitude estimate of each tone on a grid corresponding to the telephone pad.

figure,
for toneChoice=1:12,
    % Select tone
    tone=tones(:,toneChoice);
    % Estimate DFT using Goertzel
    ydft(:,toneChoice) = goertzel(tone,k+1); % Goertzel use 1-based indexing
    % Plot magnitude of the DFT
    subplot(4,3,toneChoice),stem(estim_f,abs(ydft(:,toneChoice)));
    title(['Symbol "', symbol{toneChoice},'": [',num2str(f(1,toneChoice)),',',num2str(f(2,toneChoice)),']'])
    set(gca, 'XTick', estim_f, 'XTickLabel', estim_f, 'XLim', [650 1550]);
    ylabel('DFT Magnitude');
    if toneChoice>9, xlabel('Frequency (Hz)'); end
end
set(gcf, 'Color', [1 1 1], 'Position', [1 1 1280 1024])
annotation(gcf,'textbox', 'Position',[0.28 0.96 0.45 0.026],...
    'EdgeColor',[1 1 1],...
    'String', '\bf Estimation of the frequencies contained in each tone of the telephone pad using Goertzel', ...
    'FitBoxToText','on');

Detecting DTMF Tones

The digital tone detection can be achieved by measuring the energy present at the seven frequencies estimated above. Each symbol can be separated by simply taking the component of maximum energy in the lower and upper frequency groups.

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