Chapter 4 Sine Waves

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2010 Texas Instruments Inc

Practical Audio Experiments using the TMS320C5505 USB Stick

Sine Waves

Texas Instruments University Programme

Teaching Materials


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Chapter 4 - Slide 2 2010 Texas Instruments Inc

Sine Waves


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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).


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Objectives

To generate sine waves from 10Hz to 16000Hz.

To introduce the Texas Instruments library of DSPfunctions DSPLIB.


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Knowledge Required

Some understanding of fixed-point and floating-pointnumbers is required.

Details of two useful articles from www.cnx.org are

given in the References Section.
http://www.cnx.org/http://www.cnx.org/


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Sine Wave and FFT

A sine wave is a pure tone. It only contains onefrequency:


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Complex Waveform and FFT

A complex waveform has several frequencycomponents:



Generating Sine Waves

There are 3 main ways to generate sine waves:

Look-up Table

Recursive Equation

Taylor Expansion.



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];



About Look-up Tables

Advantages:

Fast to implement

Values are always accurate

Disadvantages:

Can only be used for exact divisions of sampling

frequency.



Recursive Equation

Uses the following mathematical equation:

The next sine output is derived from the previousvalues

We shall look at this in more detail in Chapter 7,Infinite Impulse Response (IIR) filters.

21

1

.cos21

.sin

)(

)()(

zzT

zT

nx

nyzH



Taylor Series

A sine function can be implemented as a geometricseries:

where x is the input in radians.

This method is used by the Texas Instruments DSPLibrary DSPLIB.

!7!5!3)sin(

753 xxx

xx



About Taylor Series

Advantages:

Can generate any frequency

Disadvantages:

Not as accurate as look-up table because thereare rounding errors

Care needs to be taken to avoid overflow during

multiplications.



C Code Implementation



Sine Function in C

As standard, C comes with the function sin(x) inmath.h.

This uses floating-point maths.

It is not efficient for real-time applications.

A better way is to use DSPLIB.



Introducing DSPLIB


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About DSPLIB

Texas Instruments provides a library containing awhole 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 thiscase the TMS320C55xx.


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DSP LIB Headers

When using DSPLIB, you need to add the twofollowing #include statements to your code:


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DSPLIB Library

The library file55xdsph.lib

must be presentin the build.

DSPLIB forTMS320C5505

USB Stick.


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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.


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Scaling the sine() function

Need to convert frequency in Hz to value for sine()function.

Use a scaling factor of 22368.


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Magic Numbers

Where did the magic number 22368 come from?

The TMS320C5505 is a 16-bit fixed-point processorthat 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 wasobtained.


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DSPLIB sine() function

The DSPLIB function sine() calculates the sine of anangle.

The input to the function is a fixed-point number that

represents an angle: 0 => 0o

16383 => 90o

32767 => 180o

2 * 32767 => 360o


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Sine 90o

To generate a waveform using 4 values we use: sin 0

o

sin 90o

sin 180o

sin 270o.

If Fs = 48000 Hz, the frequency generated will be:

48000/4 = 12000 Hz.


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Sine 45o

To generate a waveform using 8 value use: sin 0

o

sin 45o

sin 90o

sin 135o

etc.

If Fs = 48000 Hz, the frequency generated will be:

48000/8 = 6000 Hz.


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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 = 0.0075

o

To implement a 1 Hz sine wave we use:

0o, 0.0075

o, 0.015

o, 0.0225

o, 0.030

oetc.


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Fixed-Point Implementation

For 1 Hz we require each angle to be multiples of: 360

o/48000 = 0.0075

o

For 1 Hz using fixed-point using DSPLIB we require:

2 * 32767 / 48000


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Scaling Factor

We can use the value for 1 Hz as a scaling factortocalculate other frequencies:

SCALING FACTOR = 360o/48000 = 0.0075

o

For 2 Hz:

2 * SCALING FACTOR = 2 * 360o/48000 = 0.015

o

For 10 Hz:

10 * SCALING FACTOR = 10 * 360o/48000 = 0.075

o


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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 right15 places. Result = 7FFFh

In C code, divide by 32768 is implemented as >> 15.

48000

32767*2


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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.

32768

22368*2

48000

32767*2


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Introduction to Laboratory


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USB Stick Setup TMS320C5505

USB to PC

Headphones

USB Stick


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Installing the Application

Use the code given in Application 4, Sine Waves

Follow the steps previously given in Chapter 1 toset up the new project.


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Create New Project


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Files Used in Project


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Console

Sampling frequency and Gain are shown in theConsole window.


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Experiments


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Change the Headphone Volume

Reduce gain from 10000 to 5000.


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Change the Frequencies

Rather than 200 Hz and 500 Hz, use two musicalnotes:

A = 440 Hz C = 523 Hz


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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 insinewaves.c


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Questions

What are 3 ways to generate sine waves?

Which method is best suited to the TMS320C5505USB Stick?

What are 3 applications of sine waves?


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References

TMS320C55xx DSP Library ProgrammersReference. SPRU 422.

Digital Signal Processing with C and theTMS320C30 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) byHyeokho Choi.
http://www.cnx.org/http://www.cnx.org/http://www.cnx.org/http://www.cnx.org/

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Chapter 4 Sine Waves