double cosZ, sinZ;\n\nSinComplex(float freq, float *output, long samplesPerBlock)\n{\n\tlong sample;\n\tfloat freqAngle, angleReal, angleImag;\n\t\n\tfreqAngle = twoPi * freq;\n \/\/ calculate the phose angle for given frequency\n\tangleReal = cos(freqAngle);\n\tangleImag = sin(freqAngle);\n\tfor(sample = 0; sample<samplesPerBlock; sample++) \t\n { \t\t\n \/\/ complex multiply by phase angle for sin output\n *(output+sample) = angleReal * sinZ - angleImag * cosZ;\n \/\/ complex multiply by phase angle for cos output\n cosZ = angleReal * cosZ + angleImag * sinZ; \n sinZ = *(output+sample); \n \/\/ correction for accumulated computation inaccuracies\n if(sinZ < -1.0) sinZ = -1.0;\n if(sinZ > 1.0f) sinZ = 1.0f; \n } \n}<\/pre>\n<\/div>\n","protected":false},"excerpt":{"rendered":"Complex Sinusoids Complex sinusoids are two-dimensional signals whose value at any instant can be specified by a complex number; that is, a number with real\u00a0part and an imaginary\u00a0part. These two parts are in what is known as phase quadrature\u00a0with one another; i.e. the imaginary part is pi\/2 radians out of phase with the real part […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-116","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/synthnotes.ucsd.edu\/wp7\/index.php\/wp-json\/wp\/v2\/posts\/116","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/synthnotes.ucsd.edu\/wp7\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/synthnotes.ucsd.edu\/wp7\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/synthnotes.ucsd.edu\/wp7\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/synthnotes.ucsd.edu\/wp7\/index.php\/wp-json\/wp\/v2\/comments?post=116"}],"version-history":[{"count":5,"href":"https:\/\/synthnotes.ucsd.edu\/wp7\/index.php\/wp-json\/wp\/v2\/posts\/116\/revisions"}],"predecessor-version":[{"id":479,"href":"https:\/\/synthnotes.ucsd.edu\/wp7\/index.php\/wp-json\/wp\/v2\/posts\/116\/revisions\/479"}],"wp:attachment":[{"href":"https:\/\/synthnotes.ucsd.edu\/wp7\/index.php\/wp-json\/wp\/v2\/media?parent=116"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/synthnotes.ucsd.edu\/wp7\/index.php\/wp-json\/wp\/v2\/categories?post=116"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/synthnotes.ucsd.edu\/wp7\/index.php\/wp-json\/wp\/v2\/tags?post=116"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
A complex sinusoid is defined by the following mathematical representations:<\/p>\n
Cartesian Form: Polar Form: where A<\/em>\u00a0is amplitude, \u03c9<\/em>\u00a0is the angular frequency in radians per second (the frequency in Hertz scaled by 2\u03c0; 2\u03c0f<\/em>), \u03d5<\/em>\u00a0is the initial phase offset in radians, and t<\/em>\u00a0is time in seconds. The cosine portion of the sinusoid is the real component and the sine portion is the imaginary (phase quadrature) component. (In engineering, the variable j<\/em>\u00a0is typically denoted to mean the square root of negative 1, whereas pure math often uses the variable i<\/em>\u00a0to mean the same thing).<\/p>\n Recall the Pythagorean Identity: Reconfiguring the Cartesian form of a complex sinusoid with this in mind, we get: In other words, a complex sinusoid has a constant complex magnitude (A<\/em>). Since this is true, the projection of a complex signal (in complex form) onto the complex plane must lie on a circle; i.e. the radius (magnitude) remains the same while the angle changes). Recall the animation in the beginning of this section:<\/p>\n This animation shows the counter-clockwise motion of the real (cosine) component of the complex sinusoid projected onto the complex plane. We can also project the complex phasor onto orthogonal axes:<\/p>\n The green circle in the bottom right corner is the unit circle<\/a>\u00a0in the complex plane. We can rewrite the trigonometric form of the equation into polar form:
\n x(t) \\triangleq Acos(\\omega t + \\phi) + jAsin(\\omega t + \\phi)<\/span><\/p>\n
\n (t) = Ae^{j(\\omega t + \\phi)} <\/span><\/p>\nProjection<\/h2>\n
\n cos^2(\\phi) + sin^2(\\phi) = 1 <\/span><\/p>\n
\n |x(t)| \\triangleq \\sqrt{re^2\\{x(t)\\} + im\\{x(t)\\}} \\equiv A <\/span><\/p>\n![\"\"](\"https:\/\/synthnotes.ucsd.edu\/wp7\/wp-content\/uploads\/2019\/09\/Unfasor.gif\")
<\/a><\/p>\n<\/a><\/p>\n
\n cos\\theta + jsin\\theta = re^{j\\theta} <\/span><\/p>\n