{"id":27,"date":"1996-11-14T20:00:31","date_gmt":"1996-11-15T04:00:31","guid":{"rendered":"http:\/\/www.earlevel.com\/blog\/?p=27"},"modified":"2018-07-23T09:44:14","modified_gmt":"2018-07-23T16:44:14","slug":"the-fourier-series","status":"publish","type":"post","link":"https:\/\/www.earlevel.com\/main\/1996\/11\/14\/the-fourier-series\/","title":{"rendered":"The Fourier series"},"content":{"rendered":"

(Java is no longer supported by many popular browsers, and can be difficult to enable others…)<\/em><\/p>\n

Experiment with\u00a0harmonic (Fourier) synthesis<\/strong> with this Java applet! The\u00a0sliders<\/strong> represent the levels of the first eight harmonics in the harmonic series. The second harmonic is twice the frequency of the first, the third is three times that of the first, and so on. The\u00a0graph<\/strong> shows one cycle of the resulting waveform.<\/p>\n


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If you had a Java-equiped browser, you’d see as applet here that looks like this.<\/span>
\n<\/applet><\/center>Press the\u00a0Sawtooth<\/strong> button to get an eight-harmonic approximation of a sawtooth waveform. A sawtooth waveform contains all harmonics; the second harmonic is one-half the level of the first, the third harmonic is one-third the level of the first, and so on. (Continuing the series yields a more accurate sawtooth.)<\/p>\n

Similarly, press the\u00a0Square<\/strong> button for a square-wave approximation. A square wave is made of only odd-numbered harmonics, in the same relationship as those of the sawtooth.<\/p>\n

One way of looking at this is that the sliders represent the\u00a0frequency domain<\/strong> of a waveform (the level of its frequency components\u2014how we hear), and the graph represents its conversion to the\u00a0time domain<\/strong> (the signal as it is routed through audio equipment and speakers, only to be converted back to the frequency domain by our ears!).<\/p>\n","protected":false},"excerpt":{"rendered":"

(Java is no longer supported by many popular browsers, and can be difficult to enable others…) Experiment with\u00a0harmonic (Fourier) synthesis with this Java applet! The\u00a0sliders represent the levels of the first eight harmonics in the harmonic series. The second harmonic … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4,17,16,30],"tags":[],"_links":{"self":[{"href":"https:\/\/www.earlevel.com\/main\/wp-json\/wp\/v2\/posts\/27"}],"collection":[{"href":"https:\/\/www.earlevel.com\/main\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.earlevel.com\/main\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.earlevel.com\/main\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.earlevel.com\/main\/wp-json\/wp\/v2\/comments?post=27"}],"version-history":[{"count":2,"href":"https:\/\/www.earlevel.com\/main\/wp-json\/wp\/v2\/posts\/27\/revisions"}],"predecessor-version":[{"id":746,"href":"https:\/\/www.earlevel.com\/main\/wp-json\/wp\/v2\/posts\/27\/revisions\/746"}],"wp:attachment":[{"href":"https:\/\/www.earlevel.com\/main\/wp-json\/wp\/v2\/media?parent=27"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.earlevel.com\/main\/wp-json\/wp\/v2\/categories?post=27"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.earlevel.com\/main\/wp-json\/wp\/v2\/tags?post=27"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}