Plotting waveforms Scale Axes Origin Time axis Amplitude Frequency Negative axes Scope time base
The sound of waveforms The note A above Middle C is defined to be 440 Hz Here is a pure sine wave at 440Hz Time in seconds-> Volts->
Other waveforms Here is a Square wave at 440Hz (A above Middle C) Time in seconds-> Volts->
Fundamentals and Harmonics In general fundamental frequencies are sine waves. Any waveform can be broken down into a fundamental sine wave and its harmonics. Harmonics are 2,3,4 etc (i.e. integer) times the fundamental frequency. A square wave can be shown to consist of a fundamental (of the same frequency) plus only odd harmonics.
The harmonic content of a square wave A square wave has a 3 rd harmonic of amplitude 1/3 plus a fifth of amplitude 1/5 etc. If it is a perfect square wave these go on forever. (Being a symmetrical waveform it has no even harmonics) We will add the harmonics one at a time and inspect them.
3 rd Harmonic Playing just the harmonic Time in seconds-> Volts->
Fundamental and odd harmonics up to 15 There are a total of 7 notes playing
Compare our original square wave A Square wave at 440Hz (A above Middle C)
And compare our original pure sine wave Here is our pure sine wave at 440Hz again
Linear and non linear systems We have seen that waveforms can be broken down and rebuilt by adding sine waves. This only works well for linear systems (i.e. if you can trust addition.) For example if in your system doubling the input signal doesn’t double the output signal you have a non-linear system.
Non linear systems In a linear system when you apply a sine wave of frequency F you just get a sine wave of frequency F out. In a nonlinear system you also get some harmonics at frequencies 2F, 3F etc. (only odd ones if its symmetrical) E.g. if you seriously overdrive an amplifier with a sine wave you will get something like a square wave.
Non linear systems In a linear system when you apply two sine waves of frequency F and G you just get frequencies F and G In a nonlinear system you also get sine waves at frequencies F+G and F-G. (You also get all the harmonics and all the sums and differences of the harmonics)
The ideal mixer Another day we will look at the electronics of mixers. An ideal mixer multiplies rather than adds waveforms. If you feed two sine waves at frequencies F and G into a multiplier you just get sine waves at frequencies F+G and F-G and no harmonics. Rather than prove this using maths this lets look and listen.
The output from the ideal mixer 200Hz 4200Hz and
Comparison sounds to check the output 200Hz 4200Hz
Some maths Did you notice the output waveforms were 90 degree phase shifted sine waves of half the amplitude? For many purposes this makes no difference Sin(f)* Sin(g) = Cos(f-g)/2 – Cos(f+g)/2 My last graphs allow for the phase shift. A mathematician would call them cosines but they are still sine waves.