Dual-Tone Multi-Frequency (DTMF)

• Dual-tone multi-frequency (DTMF) - The basis for voice communications control and is used worldwide in modern telephony to dial numbers and configure switch board

• DTMF tone - Commonly known as a digit, is a signal consisting the sum of two sinusoids or tones with frequencies from two exclusive groups (low and high group frequency)

  A DTMF signal is expressed as $d_N[n] = sin(\omega_on) + sin(\omega_1n)$, where $d_N[n]$ is the digit of keypad of a discrete time index $n$, and $\omega_o$ and $\omega_1$ are the low and high DTMF in radians/sample.

  By using $ω_{o/1} = 2 \times \pi \times f_n$ and the normalized frequency $f_n = \frac {f_{L/H}}{F_s}$, a DTMF signal is expressed as $d_N[n] = sin(2 \times \pi \times \frac {f_{L}}{F_s} \times n) + sin(2 \times \pi \times \frac {f_{H}}{F_s} \times n)$ in Hz.

• Frequencies are chosen to prevent any harmonic from being incorrectly detected by the receiver as some other DTMF tone

DTFT¹ Frequencies for Digits Sampled at F_s = 8192 Hz

	1209 Hz	1336 Hz	1477 Hz
697 Hz	1	2	3
770 Hz	4	5	6
852 Hz	7	8	9
941 Hz	*	0	#

• Ex: Digit 1 is represented by the signal $d_1[n] = sin(2 \times \pi \times \frac {697}{F_s} \times n) + sin(2 \times \pi \times \frac{1209}{F_s} \times n)$

DFT Based Implementation

DTMF Tones Corresponding to Digits 0-9 When Pressed on a Telephone Keypad²

• Digits 0-9 are defined in a matrix called dtmf over interval $0 \le n \le 999$, where $n$ is the sample indices
  • The matrix stores each DTMF tone as a pair of frequencies defined as as $[f_L, f_H]$
• Task: Listen to each DTMF tone using the MATLAB function sound

Results

Note: First beep is Digit 0

dtmf_sound.mp4

Corresponding Index $k$ for DTMF Digits²³

• Task: Compute 2048 samples of $X(e^{j\omega})$ to determine its corresponding index $k$ using the MATLAB function fft⁴

Results

Frequency (Hz)	Index $k$
697	175
770	193
852	214
941	236
1209	303
1336	335
1477	370

Energy Spectrum of DTMF for Digit 8²

Energy Spectrum and Digit 8 Frequencies

• $|X(e^{j\omega _k})|^2$ - Energy in a signal at frequency $\omega_k$
• Digit 8's DTMF - $f_L$ = 852 Hz and $f_H$ = 1336 Hz
• Task: Compute Digit 8's energy $|D_8(e^{j\omega _k})|^2$ using the MATLAB function ftt

Results

DFT Based Implementation's Magnitude & Energy

Frequency (Hz)	Magnitude	Energy
697	1.9029	3.6209
770	13.015	169.39
852	500.51	2.5051e+05
941	7.1501	51.124
1209	6.9656	48.519
1336	500.26	2.5026e+05
1477	3.0512	9.3098

Energy Spectrum of Digit 8 Using DFT Based Implementation

ttdecode(x) Function³

• ttdecode - MATLAB function that accepts a touch-tone signal as the input with 1000 samples for each digit and is separated by 100 samples of silence, and decodes the input to return it as a phone number
• Task: Test the MATLAB function ttdecode on the signals

Results

Test output to ensure the code can output all digits: 1 2 3 4 5 6 7 8 9 0

Output that satisfies the scope of the project: 5 5 5 7 3 1 9

ttdecode2(x) Function³

• ttdecode2 - MATLAB function that accepts a touch-tone signal as the input, where digits and silence can have varying lengths, and decodes the input to return it as a phone number
  • MATLAB code loads touch.mat to decode two input signals stored as vectors named hardx1 and hardx2
• Task: Test the MATLAB function ttdecode2 on the two input signals in touch.mat

Results

Digits from hardx1: 4 9 1 5 8 7 7

Digits from hardx2: 2 5 3 1 0 0 0

Goertzel Algorithm Based Decoder Implementation

DFT Magnitude and Energy Spectrum of DTMF for Digit 8²

• Goertzel algorithm - An efficent method to compute the spectrum of a signal when only a small number of spectral values or frequency bins needs computing
  • Full length of DFT does not need to be computed
• Task: Compute Digit 8's DFT magnitude $|D_8[k]|$ and energy $|D_8[k]|^2$ using the MATLAB function goertzel⁵

Results

Goertzel Algorithm's DFT Implementation's Magnitude & Energy

Frequency (Hz)	Magnitude	Energy
697	5.4727e-12	2.9951e-23
770	8.1237e-13	6.5995e-25
852	1024	1.0486e+06
941	1.7206e-12	2.9604e-24
1209	1.419e-12	2.0135e-24
1336	1024	1.0486e+06
1477	6.9756e-13	4.8659e-25

DFT Magnitude and Energy Spectrum of Digit 8 Using Goertzel Algorithm

Spectrogram

Digit 012 Spectrogram²

• pspectrum - MATLAB function that computes an FFT-based spectral estimate over each sliding window and visualizes how the frequency content of the signal changes over time⁶
  • Signals are divided into segments, known as windows, affecting spectrogram depending on the window length due to the inverse relationship between frequency and time, expressed as $T = \frac {1}{F_s}$
• Digit 012 is chosen as a three-digit encoded test telephone number
• Task: Compute the spectrogram of a three-digit encoded test telephone number

Windowing and Resolution

• Short window: Good time resolution and choppy transitions

• Longer window: Good frequency resolution and smooth transitions

  For a rectangular window, the frequency resolution is the width of the main lobe, expressed as $\Delta \omega \approx \frac {4\pi}{M+1}$ in radians or $\Delta F = \frac {\Delta \omega}{2\pi} \times Fs$ in Hz.

  The time resolution of a signal is expressed as $\Delta t = MT$, where $M$ is the number of samples in the window.

Results

Note: A rectangular window (Leakage = 1) is used in pspectrum to improve frequency resolution

Digit 012 Signal in Time and Frequency Domain

Three-Digit DTMF Peak Frequencies

Digit	Low Frequency (Hz)	High Frequency (Hz)
0	941	1336
1	697	1209
2	697	1336

Power Spectra of Digits 0-2

Digit 012 DTMF Spectrograms

Note: Spectrograms with different pspectrum parameters⁷

• Ex: Spectrogram 1 (Default Balance of $\Delta F$ & $\Delta t$)

% Spectrogram 1: No ΔF & 0% overlap, 'pspectrum' will find a good balance bet ΔF & Δt according to the 3-digit signal length
pspectrum(full_signal,Fs,"spectrogram",Leakage=1,OverlapPercent=0, ...      % 0% overlap to see the signal duration and locations in time
    MinThreshold=-10,FrequencyLimits=[650, 1500]);                          % MinThreshold = -10dB to visualize the main freq components

Footnotes

1. Discrete-Time Fourier Transform (DTFT). DTFT {x[n]} ⇔ DTFT^-1 {X(e^jω)} ↩

2. Code by eoommaa (dtmf_a.m, dtmf_f.m, dtmf_g.m, dtmf_goertzel_alg.m, & dtmf_spectrogram.m) ↩ ↩² ↩³ ↩⁴ ↩⁵

3. Code by TeddyDo915K (dtmf_f.m, dtmf_h.m, & dtmf_i.m) ↩ ↩² ↩³

4. MATLAB function fft documentation ↩

5. MATLAB function goertzel documentation ↩

6. Practical Introduction to Time-Frequency Analysis MATLAB documentation ↩

7. pspectrum Parameters (Spectrogram 1, 2, 3, & 4) ↩