Transforms: Part 2 - Fourier By Example

The Fourier transform is one of the most enduring and looking at the basics is a good place to start.

What M. Fourier discovered was that any periodic waveform can be replicated by adding harmonically related discrete frequencies together in various ways. Given the requirement for a harmonic relationship, there must be a component at a fundamental frequency, to which is added frequencies that are obtained by multiplying the fundamental by an integer, which is what a mathematician calls a whole number.

The requirement for a periodic waveform can neatly be circumvented as shown in Fig.1. A window is cut in the entire waveform and then the part of the waveform in the window is folded round until it is circular, which makes it periodic. If we are looking at rotating machinery, like helicopter blades, that happens automatically.

Fig.1 - A real-world waveform can be made periodic by cutting out a window and rolling it into a loop.

Fig.1 - A real-world waveform can be made periodic by cutting out a window and rolling it into a loop.

Although the frequencies we can use are restricted to being integer multiples, we have complete freedom over the amplitude and the phase of the various components. In order to make sense of this, it is necessary to be quite clear what we mean by terms like phase, because the word is used in various ways.

A sine wave is a mathematical concept, but in the real word, things can change sinusoidally, like the length of the day as Earth's seasons change, or like the voltage from the 1kHz tone generator in an audio mixer.

Fig.2a) shows where a sine wave comes from. Fundamentally it is the result of a rotation where one cycle corresponds to one rotation. The rotation is a constant angular frequency, which mathematicians describe in radians per second. A radian is a natural unit of angle, being unit circumference at unit radius, or about 57 degrees. As there are 2Pi radians in a circle, the frequency of the sine wave has to be multiplied by 2Pi to obtain the angular velocity.

The sine wave is one component of the rotation, which means what happens along the y-axis. Mathematically, if we want just a sine wave, we have to get rid of what is happening on the x-axis, which is a cosine wave. That is done as shown on Fig.2b). There are two opposed rotations. The components on the y-axis add up, whereas the components on the x-axis are in opposition and cancel one another out.

Fig.2 - At a) a constant rotation can be resolved as a sine wave and a cosine wave.  At b) using contra rotation, the cosine waves on the x-axis cancel out, leaving the sine waves on the y-axis to add.<br />

Fig.2 - At a) a constant rotation can be resolved as a sine wave and a cosine wave. At b) using contra rotation, the cosine waves on the x-axis cancel out, leaving the sine waves on the y-axis to add.

The result of the contra-rotation needed to obtain the sine wave alone is that all real sine waves contain equal amounts of positive and negative frequency. In the base band, the positive and negative frequency components in a sine wave can't be distinguished. If the sine wave is part of a modulation process, as in AM radio or in sampling, the negative frequency component shows up as the lower side band and the positive frequency component shows up as the upper side band.

It is also possible to see the negative frequency component using a stroboscopic light on rotating machinery such as the chuck of a lathe or a vinyl disk turntable. If the short duration of a photographer's flash gun can arrest motion, then a stroboscope flashing at the right frequency can arrest periodic motion. The flashes of light from the stroboscope sample the motion and if the flashing frequency is adjusted to be the same as the rotational frequency, the lower sideband of the sampling process goes down to zero Hz and the rotation appears to stop.

However, if the stroboscope frequency is reduced a little, the lower edge of the lower sideband of the sampling spectrum becomes a negative frequency and the rotating object will appear to be turning slowly in reverse.

Aliasing and stroboscopes are strongly related to how we actually perform a Fourier transform and those topics will come back to the surface in due course. It is also tied in strongly with how sampling works and so it is no exaggeration to say that these concepts form the foundations of just about everything that goes on in broadcasting.

We talk about pure sine waves as though they really exist, and quote incredibly low distortion figures from expensive signal generators. However, in the limit of precision and/or pedantry a pure sine wave is one that has always existed and will always exist. If the signal is ever terminated or faded down, the alteration of the waveform will require harmonics so the signal will no longer be pure. If that sounds far fetched, it's actually one of the reasons why the theoretical limit of sampling theory can't be reached in practice.

What we mean by phase in this context is the angle of the rotation that gave rise to the sine wave. That is an absolute phase. In other words if the frequency and the time are known, the instantaneous voltage of the wave can always be calculated. There is also relative phase, where two or more signals are involved. A sine wave and a cosine wave have a relative phase of Pi/2 or ninety degrees. Three-phase electrical distribution uses waves phased 2/3Pi or 120 degrees apart.

Let us try to synthesize a pulse. What we want to do is to "square up" the sine wave so it has sharper ends and a flat top. Adding some second harmonic doesn't work because it's going to lower the top instead of keeping it flat. Adding third harmonic seems to work. If we continue, we will find that no even harmonic helps; only the odd ones. Then look at Fig.3a) and there is the spectrum of a square wave. It's a function called sinx/x, which contains zero crossings where the even harmonics would be. That function is going to show up all over the place in anything to do with audiovisual technology and we will explore it one of these days.

Fig.3a) shows that the spectrum of a square wave is a sinx/x function. But it shows more than that, because if we reverse the columns, it shows that a filter having a rectangular frequency response (the famous brick wall filter) has an impulse response that is a sinx/x function.

Fig.3b) shows another result of transform duality. Transform duality is where doing something on one side of a transform has the opposite effect on the other. In the case of Fig.3b), as the pulse gets narrower, the spectrum and the number of harmonics increases.

Fig.3 - At a) the spectrum of a square wave is a sinx/x function, having no even harmonics. The dual is that the impulse response of a brick wall filter is a sinx/x waveform. At b) transform duality makes small things on one side large on the other and vice-versa. The narrower an impulse; the wider the bandwidth. c) A zero time pulse transforms to an infinite spectrum. d) The triangular MTF (Modulation Transfer Function) of diffraction limited optics.

Fig.3 - At a) the spectrum of a square wave is a sinx/x function, having no even harmonics. The dual is that the impulse response of a brick wall filter is a sinx/x waveform. At b) transform duality makes small things on one side large on the other and vice-versa. The narrower an impulse; the wider the bandwidth. c) A zero time pulse transforms to an infinite spectrum. d) The triangular MTF (Modulation Transfer Function) of diffraction limited optics.

Imagine a portion of a sine wave where the peak is in the center of the drawing. If we were to add a portion of twice that frequency, with the peak again in the center, the signal at the center would add linearly, whereas away from the center the phases would differ and the sum would be less. If we kept adding frequencies on that basis, the center peak of the sum would keep growing whereas elsewhere there would be as much cancellation as addition.

Having added an infinite number of harmonics, as in Fig.3c), the center peak now has shrunk to zero duration and the rest of the waveform has zero amplitude. The zero duration peak is known as a Dirac pulse or a delta function.

Delta functions can be useful. For example digital audio and video are obtained by amplitude modulating a train of delta functions. We call it sampling.

Suppose we swapped over the naming of the columns in Fig.3c), so that the frequency domain is now on the left. The relationship still holds, because now we see a waveform that remains flat for eternity, that in electricity we would call DC, contains no frequencies.

The sinx/x function will also be seen whenever there is an aperture effect and that includes optical apertures associated with lenses. A lens built to infinite accuracy still has finite resolution because of diffraction across the aperture. The lens acts like a spatial filter and the spatial impulse response in the image plane can only have zero width if the aperture becomes infinite. That's just another example of duality in action.

An electrical filter can deal with the bipolar nature of a sinx/x impulse, but an optical system can't because there is no such thing as negative light. The spatial impulse response of a lens becomes a sinx2/x2 function as shown in Fig.3d). The squaring process has rectifies the response so it as become positive only. A further result is that the spatial frequency response (also known as the modulation transfer function or MTF) of a diffraction-limited lens is triangular.

The example of Fig.3c) is one of extremes. Infinite spectra with zero duration and vice-versa. Suppose instead we looked for something less extreme? We would find it with the Gaussian function. The Gaussian function has the interesting characteristic that the Fourier transform of a Gaussian waveform is also Gaussian. Having moved away from the extremes, it is unsurprising to find that the Gaussian filter is free of overshoots when filtering a step.

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