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# Signal Processing Toolbox

## 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)),']'])
xlim([0 25]);
ylabel('Amplitude');
if toneChoice>9
xlabel('Time (ms)');
end
end
hgcf = gcf;
hgcf.Color = [1 1 1];
hgcf.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)),']'])
hgca = gca;
hgca.XTick = estim_f;
hgca.XTickLabel = estim_f;
xlim([650 1550]);
ylabel('DFT Magnitude');
if toneChoice>9
xlabel('Frequency (Hz)');
end
end
hgcf = gcf;
hgcf.Color = [1 1 1];
hgcf.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.