<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><br class=""><div><blockquote type="cite" class=""><div class="">On Nov 29, 2022, at 5:29 PM, Ian Fritz <<a href="mailto:ijfritz@comcast.net" class="">ijfritz@comcast.net</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><meta charset="UTF-8" class=""><span style="caret-color: rgb(0, 0, 0); font-family: Monaco; font-size: 11px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; float: none; display: inline !important;" class="">Thanks for the discussion! (Inline)</span><br style="caret-color: rgb(0, 0, 0); font-family: Monaco; font-size: 11px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br style="caret-color: rgb(0, 0, 0); font-family: Monaco; font-size: 11px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><blockquote type="cite" style="font-family: Monaco; font-size: 11px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">On Nov 29, 2022, at 10:12 AM, Donald Tillman <<a href="mailto:don@till.com" class="">don@till.com</a>> wrote:<br class=""><br class="">Phase Modulation is:<br class="">out = sin(kwt + f(w_m t))<br class=""><br class="">Frequency Modulation would be:<br class="">out = sin((kw + f(w_m t)) t)<br class=""></blockquote><br style="caret-color: rgb(0, 0, 0); font-family: Monaco; font-size: 11px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><span style="caret-color: rgb(0, 0, 0); font-family: Monaco; font-size: 11px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; float: none; display: inline !important;" class="">Ooopsy. You have to integrate the oscillating part, not just multiply it by t.</span><br style="caret-color: rgb(0, 0, 0); font-family: Monaco; font-size: 11px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><span style="caret-color: rgb(0, 0, 0); font-family: Monaco; font-size: 11px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; float: none; display: inline !important;" class="">Int(cos wt) = (1/w)sin wt.</span><br style="caret-color: rgb(0, 0, 0); font-family: Monaco; font-size: 11px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><span style="caret-color: rgb(0, 0, 0); font-family: Monaco; font-size: 11px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; float: none; display: inline !important;" class="">(Oh and how many times have I done the same thing!)</span><br style="caret-color: rgb(0, 0, 0); font-family: Monaco; font-size: 11px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><span style="caret-color: rgb(0, 0, 0); font-family: Monaco; font-size: 11px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; float: none; display: inline !important;" class="">It’s the (1/w_m factor that makes the FM waveshape (ideally) invariant over pitch.</span><br style="caret-color: rgb(0, 0, 0); font-family: Monaco; font-size: 11px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""></div></blockquote><div><br class=""></div><div>Did I screw up there?</div><div><br class=""></div><div>Lessee, f(w_m t) is intended to be the modulator output, an enveloped sin() at frequency w_m. Which would be:</div><div><br class=""></div><div> Mod = env_m(sin(w_m t))</div><div><br class=""></div><div>In the PM case I added it to the phase. In the FM case, I added it to the carrier frequency.</div><div><br class=""></div><div>Perhaps I should have written it:</div><div><br class=""></div><div> PM = sin(kwt + env_m(sin(w_m t))</div><div> FM = sin((kw + env_m(sin(w_m t)))t)</div><div><br class=""></div><div>Isn't that right?</div><div><br class=""></div><div>---</div><div><br class=""></div><div><br class=""></div><div>The confusion started with the first equation in Chowning's article:</div><div><br class=""></div><div><img apple-inline="yes" id="D9152CB2-D842-47A9-9519-DF380CA9FAB8" src="cid:F5BA8764-65AD-48A4-BF44-B8EAD0DC677E" class=""></div><div><br class=""></div><div><br class=""></div><div>I can't help but think that the guy really, really likes calling it frequency modulation.</div></div><div class=""><br class=""> -- Don<br class="">--<br class="">Donald Tillman, Palo Alto, California<br class=""><a href="https://www.till.com" class="">https://www.till.com</a><br class=""></div><br class=""></body></html>