“Sine Waves ”

1. “Sine Waves” Texas Instruments University Programme Teaching Materials

2. Sine Waves

3. Introduction • DSP can be used to generate sine waves • Sine waves can be used in audio to: • Generate musical tones and complex waveforms • Generate tones for touch phones (DTMF) • Modulate audio signals (alien voices) • Control audio effects (chorus/phasing/flanging).

4. Objectives • To generate sine waves from 10Hz to 16000Hz. • To introduce the Texas Instruments library of DSP functions DSPLIB.

5. Knowledge Required • Some understanding of fixed-point and floating-point numbers is required. • Details of two useful articles from www.cnx.org are given in the References Section.

6. Sine Wave and FFT • A sine wave is a pure tone. It only contains one frequency:

7. Complex Waveform and FFT • A complex waveform has several frequency components:

8. Generating Sine Waves • There are 3 main ways to generate sine waves: • Look-up Table • Recursive Equation • Taylor Expansion.

9. Look-up Table • This is the simplest way to generate a sine wave. • Put known values into a table: • Values are read using an offset e.g. sinetable[3];

10. About Look-up Tables • Advantages: • Fast to implement • Values are always accurate • Disadvantages: • Can only be used for exact divisions of sampling frequency.

11. Recursive Equation • Uses the following mathematical equation: • The next sine output is derived from the previous values • We shall look at this in more detail in Chapter 7, Infinite Impulse Response (IIR) filters.

12. Taylor Series • A sine function can be implemented as a geometric series: where x is the input in radians. • This method is used by the Texas Instruments DSP Library DSPLIB.

13. About Taylor Series • Advantages: • Can generate any frequency • Disadvantages: • Not as accurate as look-up table because there are rounding errors • Care needs to be taken to avoid overflow during multiplications.

14. C Code Implementation

15. Sine Function in C • As standard, C comes with the function sin(x) in “math.h”. • This uses floating-point maths. • It is not efficient for real-time applications. • A better way is to use DSPLIB.

16. Introducing DSPLIB

17. About DSPLIB • Texas Instruments provides a library containing a whole range of useful functions used in DSP: • Fast Fourier Transform (FFT) • Sine, Cosine and Tangent • Exponentials and logs. • Each function is optimised for the processor, in this case the TMS320C55xx.

18. DSP LIB Headers • When using DSPLIB, you need to add the two following #include statements to your code:

19. DSPLIB Library • The library file 55xdsph.lib must be present in the build. DSPLIB for TMS320C5505 USB Stick.

20. DSPLIB Sine Function • Is written in TMS320C55xx assembly language. • The function takes 3 parameters: • Parameter 1. Address of location containing the frequency • Parameter 2. Address of location to store calculated sine • Parameter 3. Always 1.

21. Scaling the sine() function • Need to convert frequency in Hz to value for sine() function. • Use a scaling factor of 22368.

22. Magic Numbers • Where did the magic number 22368 come from? • The TMS320C5505 is a 16-bit fixed-point processor that uses: • 32767 to represent 1.000 • –32767 to represent –1.000 • Here 22368 represents 0.682 decimal. • We shall now look at how this magic number was obtained.

23. DSPLIB sine() function • The DSPLIB function sine() calculates the sine of an angle. • The input to the function is a fixed-point number that represents an angle: • 0 => 0o • 16383 => 90o • 32767 => 180o • 2 * 32767 => 360o

24. Sine 90o • To generate a waveform using 4 values we use: • sin 0o • sin 90o • sin 180o • sin 270o. • If Fs = 48000 Hz, the frequency generated will be: • 48000/4 = 12000 Hz.

25. Sine 45o • To generate a waveform using 8 value use: • sin 0o • sin 45o • sin 90o • sin 135o etc. • If Fs = 48000 Hz, the frequency generated will be: • 48000/8 = 6000 Hz.

26. Generate 1 Hz Sine Wave • To generate a 1 Hz sine wave we work backwards: • 48000/value = 1 Hz • value = 1/48000 • There corresponding angle will be: • 360o/48000 = 0.0075o • To implement a 1 Hz sine wave we use: • 0o, 0.0075o, 0.015o, 0.0225o, 0.030o etc.

27. Fixed-Point Implementation • For 1 Hz we require each angle to be multiples of: • 360o/48000 = 0.0075o • For 1 Hz using fixed-point using DSPLIB we require: • 2 * 32767 / 48000

28. Scaling Factor • We can use the value for 1 Hz as a scaling factor to calculate other frequencies: • SCALING FACTOR = 360o/48000 = 0.0075o • For 2 Hz: • 2 * SCALING FACTOR = 2 * 360o/48000 = 0.015o • For 10 Hz: • 10 * SCALING FACTOR = 10 * 360o/48000 = 0.075o

29. Scaling Factor Calculation • The fixed-point scaling factor is: • In fixed-point maths, to divide by 48000 is awkward • However, to divide by 32768 is easy because 32768 = 215 • Example: To divide 3FFFFFFFh by 32768d shift right 15 places. Result = 7FFFh • In C code, divide by 32768 is implemented as >> 15.

30. Scaling Factor Calculation • The fixed-point scaling factor is derived as follows: • The divide by 32768 is implemented as >>15 • Here 2/32768 is implemented as >>14. • The scaling factor used is therefore 22368.

31. Introduction to Laboratory

32. USB Stick Setup TMS320C5505 USB to PC USB Stick Headphones

33. Installing the Application • Use the code given in Application 4, Sine Waves • Follow the steps previously given in Chapter 1 to set up the new project.

34. Create New Project

35. Files Used in Project

36. Console • Sampling frequency and Gain are shown in the Console window.

37. Experiments

38. Change the Headphone Volume • Reduce gain from 10000 to 5000.

39. Change the Frequencies • Rather than 200 Hz and 500 Hz, use two musical notes: A = 440 Hz C = 523 Hz

40. Change the Sampling Frequency • Change the sampling frequency to 24000 Hz. • The output frequencies will have changed. • You will need to alter the scaling factor in sinewaves.c

41. Questions • What are 3 ways to generate sine waves? • Which method is best suited to the TMS320C5505 USB Stick? • What are 3 applications of sine waves?

42. References • TMS320C55xx DSP Library Programmer’s Reference. SPRU 422. • Digital Signal Processing with C and the TMS320C30 by Rulph Chassaing. ISBN 0-471-55780-3. • www.cnx.org Fixed Point Arithmetic and Format (m10919) by Hyeokho Choi. • www.cnx.org Fixed Point Arithmetic (m11054) by Hyeokho Choi.