1. How a CD Stores Sound as 44,100 Numbers Per Second
Sound is a vibration of air that changes smoothly and continuously. To turn this into a "notebook of numbers," we measure its size at regular intervals. This act of measuring at fixed intervals is called sampling. [6]
In the CD standard, sound is measured 44,100 times per second (44.1 kHz). Because CDs are stereo, this happens separately for the left and right channels. [6] These numbers are written using binary, a method that uses only 0s and 1s.
How many numbers does this create in one second? The size of each measurement is stored in 16 bits, which means 16 zeros and ones. If you calculate 44,100 measurements times 16 bits times 2 channels (left and right), you get 1,411,200 bits. So, one second of sound is about 1.41 million bits. [6] Since 8 bits make 1 byte, one second is about 176,000 bytes. For a 74-minute CD (4,440 seconds), that is about 780 million bytes. This is a rough estimate based on calculation. [6]
But can sound really return from a notebook that only has numbers? What happens to the sound during the times that were not measured?
2. How the Sampling Theorem Restores Sound Between Measurements
The sampling theorem states that if the highest frequency in the sound is B Hz, you can restore the wave shape between measurements in theory if you measure at intervals shorter than 1/(2B) seconds. [6] In other words, measuring more than twice as fast as the highest frequency is enough. However, CDs also round the size to 16-bit steps, so the original wave shape is not exactly the same (see section 4).
Humans can generally hear frequencies from about 20 Hz to 20 kHz, though this varies by age and person. [6] Twice 20 kHz is 40 kHz, and the CD's 44.1 kHz is slightly higher than that. It is explained that this allows the restoration of sounds up to half the sampling rate, or 22.05 kHz. [6]
If the measuring speed is not fast enough, high sounds become indistinguishable from low sounds. This is called aliasing (or folding noise). [6] The "twice as fast" rule is a theoretical minimum, so actual devices are built with extra margin.
Who first found this idea that "measuring twice as fast is enough"?
3. Who Discovered the Sampling Theorem: Ogura Kinnosuke and Whittaker
According to a historical research paper, the Japanese mathematician Ogura Kinnosuke stated this theorem as the inverse of Whittaker's theorem in a paper published in 1920. [6]
Other researchers also worked on this. Nyquist discussed the relationship with the speed of information transmission in 1928. Kotelnikov published the theorem in 1933, and Shannon published it in 1949. [6] Japanese explanations also state that these people worked on it independently. [6]
It is often said that Ogura provided the first rigorous proof. However, some historical research suggests Ogura only indicated the method for the proof. [6] Therefore, he is best described as one of the pioneers who wrote this idea in a paper in 1920.
The idea of sending sound as numbers, known as PCM (Pulse-Code Modulation), is said to have been conceived by British engineer Reeves in 1937, though it was not practical at the time. [6] So, how do we decide the "fineness" of the numbers?
4. Quantization: Rounding Measured Sound Values to Fixed Steps
The size measured by sampling is too detailed to keep as-is. So, we round it to the nearest value among fixed steps. This is called quantization. [6] It is like measuring length with a ruler marked in 1 cm steps: both 3.4 cm and 3.6 cm get rounded to a whole number.
CDs use 16 bits. This means the loudness is represented in 65,536 steps (2 to the power of 16). [6] If 14 bits were used, there would be 16,384 steps, so the 16-bit scale is four times finer.
Why was 16 bits chosen? According to an explanatory article, during development, Sony supported 16 bits citing dynamic range (the width from quiet to loud sounds), while Philips proposed 14 bits. [6] The CD was jointly developed by Philips and Sony and released in Japan in October 1982. [6]
Speed (sampling) and fineness (quantization) are the two main dials for turning things into numbers. But these alone do not decide all of the sound quality.
5. How Digital Photos Use Pixel Grids and Color Numbers
Digital photos divide the screen into a grid of squares. Each square (a pixel) has its color represented by numbers. If a photo has 12 million pixels with a 4:3 aspect ratio, it is 4,000 pixels wide by 3,000 pixels high, making 12 million squares. [6] Just as sound cuts time into fine pieces, photos cut space into fine pieces.
Colors are made using the three primary colors of light: Red, Green, and Blue (RGB). Each color's intensity is shown in steps. If each color has 256 steps (8 bits), then 256 x 256 x 256 equals 16,777,216, or about 16.77 million colors. If each color only had "on" or "off" (2 steps), there would be only 8 colors (2 x 2 x 2). [6]
Computers around 1984 could only show 16 colors, but now they can handle over 16 million colors. [6] This is the same as the fineness of quantization in sound.
6. Why Digital Copies Avoid the Quality Loss of Analog Copies
Analog copies degrade with each generation due to noise from cables and equipment. This is called generation loss. Digital copies, if the equipment works correctly, result in the same list of numbers, so there is no degradation from copying. [6] However, if you repeatedly use lossy compression like MP3, degradation does accumulate. So, it is accurate to say "it does not degrade when copied," not "it never degrades."
CDs use error correction called CIRC (Cross-Interleaved Reed-Solomon Code). This method, provided by Sony, is said to be strong against scratches and fingerprints. Under certain conditions, it is explained that errors lasting about 4,000 bits, or about 2.5 mm (0.1 inches) on the disc, can be corrected. [6] This is a rough upper limit; not every scratch can be fixed.
The choice to turn things into numbers made them strong against copying and scratches. This idea extends beyond sound and photos.
7. Zooming In on a Smartphone Photo to See Its Pixels
Open a photo taken with a smartphone and zoom in with your fingers. At a certain size, the smooth colors you saw before will separate into square grids. These are the pixels.
Look at how much the color changes between neighboring squares. You will only see the arrangement of squares and color differences. The system of representing colors as numbers is explained in this article and the references below. If you do not have a smartphone, you can use a magnifying glass on a printed photo or a newspaper photo.
Be careful not to bring your eyes too close to the screen and do not look at it for a long time. [6]
If you want to read more, try the digital image materials from the Nagoya City Science Museum [6] or the Japanese explanation of the sampling theorem [6].