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The Phase of a Sine Wave
Last time we looked at which properties of a Sine Wave we could play around with to help build up our signal. I said that there were 3 properties, Frequency, Amplitude and one other I would deal with in a later blog post. This is the blog post in which we are going to deal with the 3rd property: Phase.
So what is Phase? The phase of a wave is a measure of how much it is delayed in time. On the graph below are two Sine Waves, one in red and one in blue. The red image shows a classic Sine Wave which starts off from an amplitude of zero at time zero. The blue image however is shifted in time a bit. It still looks very similar to the one on the left (same frequency and amplitude), but the whole waveform has been shifted a little to the left. This is known as a shift in phase.
Strangely enough, in the world of sound, our ears just aren’t sensitive to the phase of the wave in the same way as they are to say its frequency or its amplitude. So both of the above waves would sound the same. Even when we add together two waves of differing frequencies, it doesn’t matter to our ears whether they are in phase (both waves have the same phase) or not. However, the Fourier Transform does not only apply to music and sound and as its brief is to provide as accurate a representation of the wave as possible, knowing the phase of each Sine Wave in the signal is important if we want to be able to reconstruct our signal from its component frequencies.
So how do we represent the above two waves mathematically.
The simplest way is to notice that the blue wave leads the red wave by 90 degrees. That is to say whatever value the blue wave is now, the red wave will be at that same value 90 degrees afterwards. This can be seen by the fact that the blue wave crosses the X-Axis at 90 degrees and the red wave crosses the X-Axis at 180 degrees.
So if the equation for the red wave is simply:
we could write the equation for the blue wave as:
The “-90°” being the amount by which the blue wave has been shifted.
That’s all well and good if you know exactly what the phase of your signal is, but in the real world, we generally don’t know and it is the phase we want to measure. We’ll get onto exactly how we do this in a later blog post when we talk about Convolution, but for now, here is a neat little way of changing the phase of your Sine Wave by adding it to another of the trigonometric functions: Cosine.
A Cosine wave is simply a shifted version of a Sine Wave. In fact, a Cosine wave leads a Sine Wave by precisely 90 degrees. The blue wave in the above diagram is indeed a Cosine wave.
This means that:
It just so happens that if we were to take our Sine Wave and Cosine wave and add them together:
…we could change the phase of our resultant wave (Y). In fact the phase of our resultant wave would be “x”.
Confused? Don’t worry, the following video illustrates what I’m talking about.
The top right hand graph shows the resultant wave. You can see, as the video runs, that the wave travels to the right, i.e. its phase is increasing.
The bottom right hand graph shows our two waves, Sine and Cosine. You can see how their amplitudes are changing. As the Cosine Wave’s amplitude shrinks, the Sine Wave’s amplitude grows and visa versa. It is by adding these two waves together that we obtain the resultant wave shown in the upper right hand graph.
The amplitudes of the Sine and Cosine waves shown in the bottom right hand graph are determined by the rotating line in the graph on the left. By rotating the line, we are changing it’s angle “x”. The rotating line is the Hypotenuse of a triangle. The height of the triangle gives us the amplitude of our Sine wave and the width of the triangle gives us the amplitude of our Cosine wave.
So by adding together a Sine Wave and a Cosine Wave of the same frequency, as we change the angle: “x”, we change the phase of the resultant wave.
So now we need some mathematical system that combines all the things we have learned so far. Sine Waves, Cosine Waves, amplitudes, frequencies and phases. This was the point at which I finally lost the plot back in my days at university as we are now about to enter a world where numbers are both real and imaginary and where the square root of minus one actually exists: The realm of complex numbers.
