Professional Sound - February 2018 - Page 56

SOUND ADVICE Fixed Point vs. Floating Point Illustration By Donny Chow D igital signal processing (DSP) can be separated into two catego- ries: fixed point and floating point. There are a lot of discussions and information available on the two counterparts, yet the concepts are still confusing to many in understanding their dif- ferences and how those differences affect real-world products. The main purpose of this article is not to go too deep technically, but instead to explore the difference of the numeric representations in a Ν₯΅Α±”ε•Π₯ΉΡΥ₯Ρ₯Ω”)݅丁M₯Ή”₯Ё₯́½Ή±δ™…₯ȁѼ½΅Α…Ι”…Έ…ΑΑ±”ΡΌ…Ή½Ρ‘•Θ…ΑΑ±”°έ—ŠeΙ”½₯ΉœΡΌ)ΥΝ”Ρ‘”Ν…΅”€ΜΘ΅‰₯Ё±•ΉΡ ™½Θ‰½Ρ Ή½Ρ…Ρ₯½ΉΜ₯ΈΡ‘₯́…ΙΡ₯±”…́͑½έΈ₯Έ₯œΈ€ΔΈ)%Έ€Θ)%Έ€Δ)½Θ₯ᕐA½₯ΉΠ9½Ρ…Ρ₯½Έθ)%ΉΡ••ΘY…±Υ”€τ€΄ΔM₯Έΰ ₯ΡΜ)M₯Έ€τ€ΐ€‘A½Ν₯Ρ₯Ω”Ω…±Υ”€½Θ€Δ€‘9•…Ρ₯Ω”Ω…±Υ”€) ₯ΡΜ€τ€Θ€ΜāΑ½ΝΝ₯‰±”Ω…±Υ•Μ)A½Ν₯Ρ₯Ω”5₯Έθ€Δ)A½Ν₯Ρ₯Ω”5…ΰ耬ΘΔΠάΠΰΜΨΠά)]₯Ρ Ν₯΅Α±”Ν…±₯Ήœ°™Ι…Ρ₯½Ή…°ΉΥ΅‰•Ί…Έ‰”Ι•ΑΙ•Ν•ΉΡ•…́ݕ±°ΈQ‘”Ι•ΑΙ”΄)Ν•ΉΡ•Ω…±Υ•Μ…Ι”•ΕΥ…±±δΝΑ…•…Ι½Ν́ё”έ‘½±”Ι…Ή”ΈQ‘”…Ά‰•Ρέ••Έ)…‘©…•ΉΠΩ…±Υ•Μ…Ι”…±έ…έё”Ν…΅”Έ)½Θ€ΜΘ΅‰₯Ё±½…Ρ₯ΉœA½₯ΉΠ€‘%€άΤΠ€θ)Y…±Υ”€τ€΄ΔM₯Έΰl Δ¬5…ΉΡ₯ΝΝ„€ΰ€Θ€‘αΑ½Ή•ΉΠ΄ΔΘά€t)M₯Έ€τ€ΐ€‘A½Ν₯Ρ₯Ω”Ω…±Υ”€½Θ€Δ€‘9•…Ρ₯Ω”Ω…±Υ”€)αΑ½Ή•ΉΠ€τ€ΔΡΌ€ΘΤΠ€ ΐ…Ή€ΘΤԁ…Ι”Ι•Ν•ΙΩ•™½ΘΝΑ•₯…°…Ν•Μ€ΈMΥ‰ΡΙ…Ρ₯Ήœ€$(ΔΘ܁₯Ω•Μ€΄ΔΘ؁Ѽ€ΔΘά)5…ΉΡ₯ΝΝ„€τ€ΐΡΌ€ΐΈδδδδδδΰΰā™½Θ…±°ƒŠ`ΑϊdΡΌ…±°ƒŠ`ΕϊdΈ‘‘₯Ήœ€Δ₯Ω•Μ€ΔΈΐΐΐΐΐΐΐΐΐ)ΡΌ€ΔΈδδδδδδΰΰΔ)A½Ν₯Ρ₯Ω”5₯Έθ€Δΰ€ΘƒŠ"HΔΘΨƒŠ& €ΔΈΘƒ\€ΔΐƒŠ"HΜΰ)A½Ν₯Ρ₯Ω”5…ΰθ€ΔΈδδδδδΰΰāΰ€Θ€ΔΘάƒŠ& €ΜΈΠƒ\€Δΐ€Μΰ)Q‘”Ι•ΑΙ•Ν•ΉΡ•Ω…±Υ•Μ…Ι”ΥΉ•ΕΥ…±±δΝΑ…•‰•Ρέ••ΈΡ‘•Ν”ΡέΌ•αΡΙ•΅•Μ°)ΝՍ Ρ‘…Ёё”…ΐ‰•Ρέ••Έ…‘©…•ΉΠΉΥ΅‰•Ί₯́΅Υ Ν΅…±±•Θ™½ΘΝ΅…±°Ω…±Υ•Μ)…Ή΅Υ ±…ɝ•Θ™½Θ±…ɝ”ΉΥ΅‰•ΙΜΈQ‘₯́Ή½Ρ…Ρ₯½ΈƒŠqΝΡ•…±ΟŠt•₯‘Ё‰₯Ρ́Ѽ‰•½΅”)Ρ‘”•αΑ½Ή•ΉΠ°έ‘₯ ₯ٕ́ё”•αΡ•ΉΝ₯Ω”‘εΉ…΅₯ŒΙ…Ή”°‰ΥЁѼ…₯ΈΡ‘₯Μ)‰•Ή•™₯Π°₯Ё±½Ν•ΜΡ‘”ƒŠqΝΡ½±•»Št•₯‘Ё‰₯ΡϊdΙ•Ν½±ΥΡ₯½Έ™½Θ…±°Ω…±Υ•ΜΈ)₯œΈ€Θ₯±±ΥΝΡɅѕ́‘½άΡ‘”ΡέΌΉ½Ρ…Ρ₯½ΉΜ‘₯™™•Θ₯ΈΡ‘•₯ȁΉΥ΅•Ι₯ŒΙ•ΑΙ•Ν•ΉΡ…Ρ₯½ΉΜΈ)%ΈΡ‘₯́₯±±ΥΝΡΙ…Ρ₯½Έ°έ”©ΥΝЁ™½Ύ½ΈΑ½Ν₯Ρ₯Ω”ΉΥ΅‰•ΙΜ€‘M₯Έ ₯Πτΐ€ΡΌ…Ω½₯)½΅Α±₯…Ρ₯½ΉΜΈQ‘•ΈΡ‘”•₯‘Ё΅½ΝЁΝ₯Ή₯™₯…ΉΠ‰₯Ρ́₯ΈΡ‘”ΝΡ…Ή‘…ɐ™₯ᕐ)Α½₯ΉΠ™½Ι΅…Ё…Ι”ΥΝ•…́ё”Ν…±₯Ήœ₯Ή‘•ΰΡΌ΅…э Ρ‘”ΉΥ΅‰•Θ½˜‰₯Ρ́½˜)Ρ‘”•αΑ½Ή•ΉΠ₯ΈΡ‘”™±½…Ρ₯ΉœΑ½₯ΉΠ™½Ι΅…ΠΈQ‘₯́±•…ٕ́ё”Ι•΅…₯Ή₯Ήœ€Θ́‰₯ΡΜ)ΡΌ‰”΅…э‘•™½Θ‰½Ρ ™½Ι΅…ΡΜΈ=Ή”Ρ‘”‰₯Π΅΅…э‘₯Ήœ₯́…½΅Α±₯Ν‘•°έ”)…ΈΡ‘•Έ•Ή•Ι…Ρ”΅•…Ή₯Ή™Υ°Ι•ΝΥ±ΡΜΈ(Τ؁AI=MM%=90M=U9)αΑ½Ή•ΉΠ”…Έ‰”…Ήδ…Ι‰₯ΡΙ…Ι䁹ա‰•Θ…́±½Ήœ…́…±°‰±½­Μ…Ι”έ₯Ρ‘₯Έ)Ρ‘”΅₯Έ½΅…ΰΙ…Ή”€ ΄ΔΘ؁Ѽ€ΔΘά€ΈQ‘”%Ή‘•ΰτā₯́΅…ΑΑ•ΡΌ”½Ή”Ρ‘”‘•Ν₯Ι•)‘…Ρ„Ι…Ή”₯́‘½Ν•Έ™½ΘΡ‘”™₯ᕐΑ½₯ΉΠ½΅ΑΥΡ…Ρ₯½ΈΈQ‘”Ι…Α Ν‘½έ́ё…Π)‰½Ρ Ή½Ρ…Ρ₯½ΉΜ‘…Ω”Ρ‘”Ν…΅”€ΘΜ΅‰₯Ёɕͽ±ΥΡ₯½Έ…Ё€τΔΈ½Θ€τΐ°Ρ‘”Ι•Ν½±ΥΡ₯½Έ)½˜™±½…Ρ₯ΉœΑ½₯ΉΠ₯́‰•ΡΡ•Θμ₯Έ™…Π°ΡΙ•΅•Ή‘½Υͱ䁉•ΡΡ•ΘΈQ‘•Ι”…Ι””¬ΔΘ؁™±½…Π΄)₯ΉœΑ½₯ΉΠ‰±½­Μ°έ‘₯±”Ρ‘•Ι”₯́½Ή±δ½Ή”™₯ᕐΑ½₯ΉΠ‰±½¬€‘€τΐ€°‰½Ρ ½Ω•Ι₯Ήœ)Ρ‘”Ν…΅”ΉΥ΅•Ι₯…°Ι…Ή”Έ)½Θ€Δρ€πΘΤΨ°Ρ‘”Ι•Ν½±ΥΡ₯½Έ½˜™₯ᕐΑ½₯ΉΠ₯́‰•ΡΡ•ΘΈ±Ρ‘½Υ ₯Ё͕•΅Μ)™±½…Ρ₯ΉœΑ½₯ΉΠ±½Ν•ΜΙ½ΥΉ…Ё‘₯‘•ΘΩ…±Υ•Μ°­••ΐ₯Έ΅₯ΉΡ‘…Ёё”™₯ᕐΑ½₯ΉΠ)½ΥΉΡ•ΙΑ…ΙЁ…Έ½Ή±δ‘…Ω”„Ρ½Ρ…°½˜€ΘΤΨ€ Θ€ΰ€€‰±½­Μέ₯Ρ €ΘΜ΅‰₯Ёɕͽ±ΥΡ₯½ΈΑ•Θ)‰±½¬Έ=Ή”Ρ‘”Ω…±Υ”½•Μ‰•ε½ΉΡ‘”ΥΑΑ•Θ±₯΅₯Π°Ρ‘”‘•Ω•±½Α•Θ₯́™½Ι•ΡΌ)͍…±”Υΐ™ΥΙΡ‘•Θ…Ή±₯Ω”έ₯Ρ „±½έ•ΘΙ•Ν½±ΥΡ₯½Έ₯Έ½Ι‘•ΘΡΌ‰”…‰±”ΡΌ…ΙΙδ)½ΥЁё”…ΝΝ₯Ή•΅…Ρ‘•΅…Ρ₯…°Ρ…Ν¬ΈQ‘₯́₯́ݑ䁙±½…Ρ₯ΉœΑ½₯ΉΠ₯́‘½΅₯Ή…ΉΠ)έ‘•ΈΡ‘”½΅ΑΥΡ…Ρ₯½ΈΩ…±Υ•Μ…Ι”Ν΅…±°°…Ή…Έ‘…Ή‘±”±…ɝ”ΉΥ΅‰•ΙΜ)έ‘•Έ™₯ᕐΑ½₯ΉΠ…ΉΉ½ΠΈQΌ΅…­”Ρ‘₯ΉΜ΅½Ι”½΅Α±₯…Ρ•™½Θ™₯ᕐΑ½₯ΉΠ)M@‘•Ω•±½Α•ΙΜ°Ρ‘•δ‘…Ω”ΡΌ‘…Ή‘±”½Ω•Ι™±½ά…ΉΡΙΥΉ…Ρ₯½Έ•ΙΙ½Ί…́ݕ±°Έ)%ΈMΥ΅΅…Ιδθ)‘Ω…ΉΡ…•Μ™½Θ±½…Ρ₯ΉœA½₯ΉΠθ+ŠˆαΡΙ•΅•±δ±…ɝ”‘εΉ…΅₯ŒΙ…Ή”+ŠˆΑΑ±₯…Ρ₯½ΉΜέ₯Ρ ₯ΉΡ•ΉΝ₯Ω”½΅ΑΥΡ…Ρ₯½ΉΜ+ŠˆQ‘”Ν΅…±±•ΘΡ‘”ΉΥ΅‰•Θ°Ρ‘”‘₯‘•ΘΡ‘”ΑΙ•₯Ν₯½Έ…Ή±½έ•ΘΕΥ…ΉΡ₯ι…Ρ₯½Έ€$)Ή½₯Ν”+ŠˆA½ΝΝ₯‰±”½΅ΑΥΡ…Ρ₯½Έ™½Θ±…ɝ”ΉΥ΅‰•ΙΜ°‰•ε½Ή™₯ᕐΑ½₯ΉΣŠé…Α…‰₯±₯Ρδ+ŠˆM₯΅Α±•Θ…Ή™…Νѕȁ‘•Ω•±½Α΅•ΉΠέ₯Ρ‘½ΥЁݽΙΙε₯Ήœ…‰½ΥЁ½Ω•Ι™±½ά…Ή€$)ΡΙΥΉ…Ρ₯½Έ•ΙΙ½ΙΜ)‘Ω…ΉΡ…•Μ½Θ₯ᕐA½₯ΉΠθ+Šˆ •Ρѕȁɕͽ±ΥΡ₯½Έέ₯Ρ‘₯Έ„Ή…Ιɽ܁ɅΉ”½˜€Δρ€πΘΤΨ€ ά½ΥЁ½˜€ΘΤ؁™±½…Ρ₯Ήœ€$)Α½₯ΉΠ•αΑ½Ή•ΉΠ‰±½­Μ€+ŠˆM₯΅Α±•ΘM@Ν₯±₯½Έ)½ΉΉδ ‘½ά₯́ё”½ΥΉ‘•Θ…Ή ‘₯•˜Ή₯Ή••Θ½˜a₯±₯„Έ½±±½έ₯Ήœ₯Ρ́΅½ΡΡΌ)½˜A…ΝΝ₯½ΈQ‘ɽ՝ A•Ι™½Ι΅…Ή”°a₯±₯„΅…ΉΥ™…ΡΥɕ́±…ΝΜ΅±•…‘₯ΉœΉ•Ρέ½Ι­•)…ΉΝΡ…Ή‘…±½Ή”MÁ™½Θ±₯Ω”…Ή₯ΉΝΡ…±±•Ν½ΥΉ…ΑΑ±₯…Ρ₯½ΉΜΈέέάΉα₯±₯„Ή½΄