https://github.com/eoommaa/dtmf

DTMF project using DFT, Goertzel algorithm, and Spectrogram for Signal Processing class
https://github.com/eoommaa/dtmf

dft digital-signal-processing discrete-fourier-transform discrete-time-fourier-transform dsp dtft dtmf dtmf-tones dual-tone-multi-frequency fast-fourier-transform fft goertzel goertzel-algorithm matlab signal-processing

Last synced: 2 months ago
JSON representation

DTMF project using DFT, Goertzel algorithm, and Spectrogram for Signal Processing class

Host: GitHub
URL: https://github.com/eoommaa/dtmf
Owner: eoommaa
License: mit
Created: 2025-06-25T20:56:54.000Z (9 months ago)
Default Branch: main
Last Pushed: 2025-07-24T19:50:02.000Z (8 months ago)
Last Synced: 2025-07-24T22:51:59.565Z (8 months ago)
Topics: dft, digital-signal-processing, discrete-fourier-transform, discrete-time-fourier-transform, dsp, dtft, dtmf, dtmf-tones, dual-tone-multi-frequency, fast-fourier-transform, fft, goertzel, goertzel-algorithm, matlab, signal-processing
Language: MATLAB
Homepage:
Size: 531 KB
Stars: 0
Watchers: 0
Forks: 0
Open Issues: 0
Metadata Files:
- Readme: README.md
- License: LICENSE

Awesome Lists containing this project

README

          # 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

***DTFT[^1] 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 Keypad[^2]

- 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*

https://github.com/user-attachments/assets/d005f1db-a5e3-4964-b408-3e9c4b268e8b

## Corresponding Index $k$ for DTMF Digits[^2][^3]

- **Task:** Compute 2048 samples of $X(e^{j\omega})$ to determine its corresponding index $k$ using the MATLAB function `fft`[^4]

  

### 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 8[^2]

**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***

![image](./plots/dtmf_g.png)

## `ttdecode(x)` Function[^3]

- `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)` Function[^3]

- `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`](https://github.com/eoommaa/DTMF/blob/main/DTF%20Based%20Implementation/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 8[^2]

- 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 `goertzel`[^5]

### 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***

![image](./plots/dtmf_goertzel_alg.png)

# Spectrogram

## Digit 012 Spectrogram[^2]

- `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 time[^7]

  - 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***

![image](./plots/spectrogram_1.png)

***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***

![image](./plots/spectrogram_2.png)

***Digit 012 DTMF Spectrograms***

*Note: Spectrograms with different [`pspectrum` parameters](https://github.com/eoommaa/DTMF/blob/23265028efedc3723cc8f73f632df4f716a25a4b/Spectrogram/dtmf_spectrogram.m#L125-L132)[^8]*

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

```matlab

% 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

```

![image](./plots/spectrogram_3.png)

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

[^2]: Code by [`eoommaa`](https://github.com/eoommaa) ([`dtmf_a.m`](https://github.com/eoommaa/DTMF/blob/main/DTF%20Based%20Implementation/dtmf_a.m), [`dtmf_f.m`](https://github.com/eoommaa/DTMF/blob/main/DTF%20Based%20Implementation/dtmf_f.m),

[`dtmf_g.m`](https://github.com/eoommaa/DTMF/blob/main/DTF%20Based%20Implementation/dtmf_g.m), [`dtmf_goertzel_alg.m`](https://github.com/eoommaa/DTMF/blob/main/Goertzel%20Algorithm/dtmf_goertzel_alg.m),

& [`dtmf_spectrogram.m`](https://github.com/eoommaa/DTMF/blob/main/Spectrogram/dtmf_spectrogram.m))

[^3]: Code by [`TeddyDo915K`](https://github.com/TeddyDo915K) ([`dtmf_f.m`](https://github.com/eoommaa/DTMF/blob/main/DTF%20Based%20Implementation/dtmf_f.m), [`dtmf_h.m`](https://github.com/eoommaa/DTMF/blob/main/DTF%20Based%20Implementation/dtmf_h.m), & [`dtmf_i.m`](https://github.com/eoommaa/DTMF/blob/main/DTF%20Based%20Implementation/dtmf_i.m))

[^4]: [MATLAB function `fft` documentation](https://www.mathworks.com/help/matlab/ref/fft.html)

[^5]: [MATLAB function `goertzel` documentation](https://www.mathworks.com/help/signal/ref/goertzel.html?searchHighlight=Goertzel&s_tid=srchtitle_support_results_1_Goertzel)

[^6]: [DFT Estimation with the Goertzel Algorithm](https://www.mathworks.com/help/signal/ug/dft-estimation-with-the-goertzel-algorithm.html)

[^7]: [Practical Introduction to Time-Frequency Analysis MATLAB documentation](https://www.mathworks.com/help/signal/ug/practical-introduction-to-time-frequency-analysis.html)

[^8]: `pspectrum` Parameters (Spectrogram [1](https://github.com/eoommaa/DTMF/blob/95b5676741261cf0e957216341d240d338d8a714/Spectrogram/dtmf_spectrogram.m#L115-L116),

[2](https://github.com/eoommaa/DTMF/blob/95b5676741261cf0e957216341d240d338d8a714/Spectrogram/dtmf_spectrogram.m#L137-L138),

[3](https://github.com/eoommaa/DTMF/blob/95b5676741261cf0e957216341d240d338d8a714/Spectrogram/dtmf_spectrogram.m#L146-L147), &

[4](https://github.com/eoommaa/DTMF/blob/95b5676741261cf0e957216341d240d338d8a714/Spectrogram/dtmf_spectrogram.m#L155-L156))

ecosyste.ms

Data

Tools

Indexes

Applications

Experiments

Awesome

https://github.com/eoommaa/dtmf

Awesome Lists containing this project

README