Problem Set 2.1 Spring 2009
Composition and Decomposition of Complex Waveforms

This problem set is designed to help students develop an intuitive understanding of how sine waves may be combined to produce complex sounds using additive synthesis and how complex sounds can be decomposed into individual sine waves.  Each of the following problems requires manipulation of the various parameters of the sine function in combination with other sine functions.  Don't limit yourself to the suggested explorations.  If you need to see more examples to get a firm grasp of synthesis and decomposition, make up your own functions and note which parameters you change and what the resulting effects are.  As you vary the parameters of the sine functions, look for patterns in the resulting waveforms.

1. Use Excel to define the following pairs of functions and to graph their sum.  Study the graphs you obtain until you are able to describe the relationship between the amplitudes and periods of the original sine waves and the various properties of the composite (complex) sound.  In a clear paragraph or two, describe and explain this relationship.  Feel free to use sketches in your explanation. (NOTE: you need not turn in graphs of all the functions below.  The purpose of graphing the three functions, y₁(x), y₂(x),  and y₃(x) in Excel is to understand the relationship among parameters.)

   a.  y1(x) = 4sin(x)	y2(x) = sin(3x)	y3(x) = y1 + y2
   b.  y1(x) = 0.5sin(x)	y2(x) = 2sin(4x)	y3(x) = y1 + y2
   c.  y1(x) = 3sin(2x)	y2(x) = sin(1x)	y3(x) = y1 + y2
   d.  y1(x) = sin(6x)	y2(x) = 3sin(4x)	y3(x) = y1 + y2
   e.  y1(x) = 3sin(6x)	y2(x) = sin(9x)	y3(x) = y1 + y2

   f.  Make up your own examples to confirm your ideas about the relationship.
   

2. Find a series of sine functions that sum to approximate the sawtooth wave below.  Write equations representing the first four sine functions of the series. Experiment and test your series in Excel by graphing the sum of the first four sine functions.

   

3. When added together, the pairs of sine functions below produce a complex periodic waveform.  Determine the period of each of the four complex signals produced by summing the pairs below.
   1. 100 Hz and 200 Hz
   2. 150 Hz and 400 Hz
   3. 110 Hz and 660 Hz
   4. 200 Hz and 700 Hz

4. Refer to the following four figures.  Decompose each complex waveform into the sum of two simple sine waves.  Use the results of your explorations in problems (1) and (3) to help you.  You may wish to graph the above pairs of functions with various amplitude factors to gain a deeper understanding of the composition and decomposition of complex waveforms.
   

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