EE 451

Lab 4: Digital Oscillators on the 56002

A touchtone telephone uses two frequencies for each key pressed. The frequencies are shown in the diagram below:

Thus, when you press the ``1'' key you get a tone

$$y(n) = \frac{1}{2} ( \cos(2 \pi f_L t) + \cos(2 \pi f_H t))$$

where f_L = 697 Hz and f_H = 1209 Hz. These tones are called dual-tone multi-frequency (DTMF) signals. In this lab we want to use the 56002 to generate these DTMF signals. We can do this by implementing a discrete-time system with an impulse response

$$h(n) = \frac{1}{2} ( \cos(\omega_L n) + \cos(\omega_H n))$$ (1)

This is a fourth-order system and can be written in the z-domain as

$$H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2} + b_3 z^{-3} + b_4 z^{-4}} {1 + a_1 z^{-1} + a_2 z^{-2} + a_3 z^{-3} + a_4 z^{-4}}$$ (2)

This can be implemented in several ways. It can be implemented as a fourth-order system by the difference equation:

$$\begin{array}{lcl} y(n) & = & b_0 x(n) + b_1 x(n-1) + b_2 x(n... ... a_1 y(n-1) - a_2 y(n-2) - a_3 y(n-3) - a_4 y(n-4) \end{array}$$ (3)

It can be implemented as a cascade of second-order systems - in the z-domain this would look like:

$$H(z) = \left(\frac{b_{00} + b_{01} z^{-1} + b_{02} z^{-2}} {a... ...b_{12} z^{-2}} {a_{10} + a_{11} z^{-1} + a_{12} z^{-2}}\right)$$ (4)

It can be implemented in parallel form - in the z-domain this would look like:

$$H_1(z) = \frac{b_{10} + b_{11} z^{-1} + b_{12} z^{-2}} {1 + a_{11} z^{-1} + a_{12} z^{-2}} \nonumber \\$$

$$H_2(z) = \frac{b_{20} + b_{21} z^{-1} + b_{22} z^{-2}} {1 + a_{21} z^{-1} + a_{22} z^{-2}} \nonumber \\$$

H(z) = H₁(z) + H₂(z) (5)

For reasons of stability in a finite word length processor the form in Equation 5 is the best for implementing this system.

1.

Find the z-transform of Equation 1. Write it in the form of Equation 5.

2.

Write a MATLAB m-file which will calculate the a_jk and b_jkcoefficients when given the two frequencies $\omega_L$ and $\omega_H$, and will calculate the first few values of h₁(n) and h₂(n): h₁(0), h₁(1), h₁(2), h₂(0), h₂(1), and h₂(2).

Also, your MATLAB program should plot the impulse responses so you can see what the signals should look like.

3.

Run your MATLAB program to find the a_jk's and initial values needed for several different DTMF tones.

4.

Write a program for the 56002 which will implement the difference equation. Modify last week's lab to do this. Perhaps you can calculate H₁(z) in Accumulator a and H₂(z) in Accumulator b, and add the two together for the final result.

Note that your program needs no input. For n>2 the difference equation for H₁ is simply

h₁(n) = -a₁₁ h₁(n-1) - a₁₂ h₁(n-2)

Thus, if you start your system at n = 3, with h₁(1) and h₁(2)initialized in memory, you do not need to use the b_jk coefficients.

5.

Test your program. Listen to the different outputs and see if they sound like touch-tone tones. Look at the outputs on an oscilloscope to determine if the two correct frequencies are present.