1	2	3	→	697 Hz
4	5	6	→	770 Hz
7	8	9	→	852 Hz
*	0	#	→	941 Hz
↓	↓	↓
1,209 Hz	1,336 Hz	1,477 Hz

To be more precise, one can take the DTMF signal corresponding to a given key k to have the form x_k(t) = cos(2π f_L t) + cos(2π f_H t), where f_L and f_H are the low and high frequencies associated with the key k. For key 6, for example, x₆(t) = cos(2π770t) + cos(2π1477t). The composite DTMF signal for a sequence of keys, i.e., to a telephone number, is obtained as the superposition of the individual, truncated and time-shifted dual-tones corresponding to the numbers. As would be expected from everyday experience with telephones, truncation and time shifting of individual dual-tones is done in such a way that there is no overlapping of successive dual-tones and there is a short silence between them. So each dual-tone is heard distinctly.

When it comes to constructing the sampled representation of a composite DTMF signal in the lab, the superposition task described above reduces to concatenation of sampled representations of individual finite-duration dual-tones. To represent silence between numbers, one simply inserts an appropriate number of zeros between the sampled dual-tones.

Music synthesis: A musical piece is a sequence of notes, which are essentially signals at various frequencies. An octave covers a range of frequencies from a base pitch f₀ to twice the base pitch, 2f₀. In the western musical scale, there are 12 notes per octave, A through G^# (or A^b), and they are logarithmically equispaced over the octave. That is, the frequency of the k-th note in a given octave with the base pitch of f₀ is equal to f_k = 2^k/12f₀, k = 0, 1, 2, ..., 11. This leads to the following integer encoding of notes within a given octave:

Notes:	A	A# or Bb	B	C	C# or Db	D	D# or Eb	E	F	F# or Gb	G	G# or Ab
Integer k:	0	1	2	3	4	5	6	7	8	9	10	11

As for the general form of the signal for a given note, one can consider a pure tone as the simplest form. That is to say, for note D, for example, the form is x₅(t) = cos(2π f₅t + θ). Pure tones, however, will not sound as pleasant as one would like. To add some color to the sound, consider the improved form x_k(t) = α(t)cos(2π f_kt + θ), where α(t) shapes the otherwise flat envelope of the pure tone signal. This helps simulate the attack-sustain-release characteristics of different instruments, and gives rise to overtones (harmonics) thus enriching the sound. The following figure exemplifies the envelope forms α(t) and simulated notes for woodwind- and string/keyboard-type instruments, where the duration of the note is normalized to unity.

Can you tell from this figure the longer sustain characterictics of woodwind intruments as opposed to the pronounced release (decay) characteristics of string or keyboard intruments?

General: Recall, from the previous assignment, that MATLAB's sound function accepts the reconstruction sampling frequency as an optional second argument. If you do not pass this information to sound explicitly, the default sampling frequency of 8,192 Hz is assumed. In this experiment, you would do fine with this default value, without risking aliasing. The highest frequency involved in the DTMF dialing case is 1,477 Hz, which is less than half of 8,192 Hz. In the case of music synthesis, a base pitch of f₀ = 440 Hz is the standard choice that you might use, and even if you go as high as three octaves above this value, your highest frequency will be 2³·440 = 3,520 Hz, which is still less than half of 8,192 Hz.

Also recall that sound clips off portions of the input signal exceeding unity in absolute amplitude. Make sure that your signals are normalized so that all sample values lie in the interval [-1.0,1.0].