Current Topic: 7.7.3.1. A brief excursion into Machine Learning by Vincent Randal
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Part 1.
In future email [or series of articles] I will illustrate a point I am hoping to make about Machine Learning when the data is a PWM (pulse width modulated) signal. My point, my claim, is that we can improve linear separability of the data in a PWM signal by sampling the signal faster. And that?s it for now. I?m not claiming anything new. It?s just an example that I hope is useful. The rest of this email [or article] is preparation for that discussion.
Assuming we know what we mean by linear separability, the only important preliminary concept is a basic principle from signal processing: time domain multiplication is equivalent to frequency domain convolution. We will apply this principle to a band-limited signal where the frequency spectrum is zero above some band-limit for baseband signals (or zero outside some band for bandpass signals).
For our purposes, by ?time domain multiplication? we mean the product of an analog waveform (or band-limited signal) and an ideal sampling function (Dirac Comb aka Sha function). By ?frequency domain convolution? we mean the convolution integral of the signal spectrum and the ideal sampling function spectrum. The result is a sampled signal with an interesting frequency spectrum; the spectrum is periodic. [By the way, when these statements are made precise and rigorous they outline a simple graphical proof of Shannon?s Sampling theorem and the Nyquist criterion.]
With this preliminary concept in hand we can then talk about the benefit of increasing the sample rate (called oversampling). Namely, oversampling dilates the spectrum which, for periodic spectra, increases the space between the spectral copies which in turn relaxes the filtering requirements. In the next email we will provide an example that uses PWM to illustrate the benefit of oversampling. [By the way, about 30 years ago oversampling was used to lower the cost and increase performance of audio CD players. Manufacturers of CD players would boast 8 or 16 times oversampling in their product.]
The next email will include an explanation of PWM as being the output of a comparator whose inputs are a signal and a triangle wave. The output of the comparator is a sequence of pulses of various widths. This variation in pulse width is called Pulse Width Modulation, and because it is created in this nonlinear way (with a comparator) there?s no easy or straightforward method (and perhaps still no known method) for writing down an analytic expression for the frequency spectrum of a PWM signal. But in the spirit of Machine Learning pioneers we won?t let that deter us. We will use tools like MATLAB and Python to numerically explore the frequency spectrum of PWM.
In the next email, using pictures (plots and graphs) we will show how the clock rate (and sample rate) affect the spectrum of a PWM signal. Then we can move on to the claim that we can improve linear separability of the data in a PWM signal by sampling the signal faster. It?s for the sake of discussion (in the context of digital signal processing) that we will be sampling a PWM signal. It?s not typical to sample PWM, but sampling allows us to numerically compute the discrete Fourier transform of the signal. The discrete Fourier transform produces a vector space with finite dimension which helps our discussion.
