# On Digital and Analog Signals

Have you ever wondered why some people claim that music stored on vinyl discs sounds so much better?  Is digital music really worse than analog?  There are many differences between the two mediums- the heart of the distinction lies in the difference between digital (discrete) and analog (continuous) data.

In mathematics, especially calculus and statistics, we often talk about discrete summations versus continuous functions.  Looking at the case of Expected Value from Probability Theory, we can see a glimpse of how they differ.

In the first case, the expected value is calculated by multiplication, addition and subsequent division- It’s basically calculating an average. In the second case, the actual probability function is used.  The expected value is calculated with integration- at times this can be more complex but it will be a truer computation since there is no estimation involved.

Going back to our audio signal case, we can see that these two procedures can describe the same occurrences, but they perform it in slightly different ways.  Analog functions describe smooth contours and by their nature encompass the entirety of a signal. It’s worth noting that your brain processes analog information, sound waves become brain waves.  In contrast, digital waveforms take samples at a given rate (the current standard is 44.1KHz) and use those samples for encoding and playback.  The thought is that since the human auditory system can resolve up to 20KHz (although this too degrades with age and hearing loss), then sampling at twice that rate should be sufficient to “trick” the brain into thinking it’s hearing a full signal.

However, you could “zoom in” on an analog signal infinitely and never find a perfectly flat line (except at the very tip of the peaks) but with digital signals, if you zoom in far enough, you will find points where the samples exist.  It’s by nature approximation of the signal.  Even if you sampled at 100MHz (over 1000x more data than 44.1KHz) you would still lose some data, although you would be losing much less of it.  Data stored on vinyl is stored in an analog medium- the entirety of the waveform is written on the physical disc whereas data stored digitally is a collection of samples.

Human hearing is extremely sensitive in general- we can tell when these tiny differences are present in the signal, even on the order of milliseconds.  This speed of processing is crucial for our ability to communicate with each other.  The brain doesn’t store information digitally- it’s a kind of analog computer (if not even a quantum computer) and digital information will never quite get the full signal.

As for the issue of vinyl versus mp3, there are even more differences in the mastering process, imperfections in the vinyl surface, extreme compression in mp3s (although more recent encoding standards are somewhat better) and the issue of playback systems to take into account.  The relative merits of digital and analog music can be argued ad infinitum, but now you should understand the basic difference between the two.

• Dr. Philips

To start I think this is very well thought out! But on the third paragraph where it says “The expected value is calculated with integration- at times this can be more complex but it will be a truer computation since there is no estimation involved.” This is not 100% correct because if you did the analog waves on a computer and added up the scenarios the equation is just not exact. That makes me a little disappointed because you should know this and shall not be giving false information to the viewers. Although I will not show this to anyone but I am very disappointed. Also in the seventh paragraph where it states ” The relative merits of digital and analog music can be argued ad infinitum, but now you should understand the basic difference between” The italicized word is unavailable because it that is not a word according to the law of Merriam dictionary.