{"id":393,"date":"2019-11-09T23:26:25","date_gmt":"2019-11-09T23:26:25","guid":{"rendered":"https:\/\/synthnotes.ucsd.edu\/wp7\/?p=393"},"modified":"2019-11-17T21:25:16","modified_gmt":"2019-11-17T21:25:16","slug":"second-order-allpass-filter","status":"publish","type":"post","link":"https:\/\/synthnotes.ucsd.edu\/wp7\/index.php\/2019\/11\/09\/second-order-allpass-filter\/","title":{"rendered":"Second-Order Allpass Filter"},"content":{"rendered":"\n

Introduction<\/h3>\n\n\n\n

In the section on phase shifters, 2 first order allpass filters are cascaded to obtain a phase shift of -\u03c0 in the middle of the frequency range. The -\u03c0 phase shift makes it possible to implement a band reject filter by adding the input to the phase shifted allpass output or a band pass by subtracting the allpass output from the input. <\/p>\n\n\n\n

There are many ways to formulate a second order allpass, but in this section we will be using a filter developed by professor fred harris at SDSU for an efficient parametric equalizer used in an early hybrid guitar amp<\/a>.<\/p>\n\n\n\n

y[n] = -c(x[n]) + (d - dc)x[n-1] + x[n-2] - (d - dc)(y[n-1]) + c(y[n-2]) <\/span><\/p>\n\n\n\n

<\/p>\n\n\n\n

Above is the difference equation for this allpass. The coefficients are c<\/em> and d<\/em>. The input parameters used are frequency and bandwidth. Frequency is the -\u03c0 phase shift point and bandwidth is the frequency distance between the -\u03c0\/2 and -3\u03c0\/2 points.<\/p>\n\n\n\n

The coefficients are calculated from frequency (f<\/em>) and bandwidth (bw<\/em>) by:<\/p>\n\n\n\n

\\begin{aligned}& d = -cos\\bigg(2\\pi \\frac{f}{sr} \\bigg) \\\\ & tf = tan\\bigg(\\pi \\frac{bw}{sr} \\bigg) \\\\ & c = \\frac{tf-1}{tf+1} \\end{aligned} <\/span><\/p>\n\n\n\n

Notice the d<\/em> is completely dependent on f<\/em> and c<\/em> is completely dependent on bw<\/em>. The decoupling of coefficients will allow easy optimization of this filter. <\/p>\n\n\n\n

\"\"
phase response of fred harris 2nd order allpass, f=2500, bw=1000<\/figcaption><\/figure>\n\n\n\n

The plot above shows that this filter reaches -\u03c0\/2 at 2000Hz, -\u03c0 at 2500Hz and -3\u03c0\/2 at 3000Hz. The C implementation is straightforward and can be derived directly from the difference equation above.<\/p>\n\n\n\n

#define SAMPLERATE 44100.0\n#define PI 3.141592653589793\ndouble x0 = 0.0; \/\/ input\ndouble x1 = 0.0; \/\/ delayed input\ndouble x2 = 0.0; \/\/ delayed input\ndouble y1 = 0.0; \/\/ delayed output\ndouble y2 = 0.0; \/\/ delayed output\n\nsoap(float *input, float *output, long samples, float cutoff, float bw) \n{   \n    double d = -cos(2.0 * PI * (cutoff\/SAMPLERATE));\n    double tf = tan(PI * (bw\/SAMPLERATE)); \/\/ tangent bandwidth   \n    double c = (tf - 1.0)\/(tf + 1.0); \/\/ coefficient     \n    for(int i = 0; i < samples; i++) \n    {  \n        x0 = *(input+i);   \n        *(output+i) = -c*x0 + (d - d*c)*x1 + x2 - (d - d*c)y1 + c*y2;\n        \/\/ move samples in delay for next sample\n        x2 = x1;\n        x1 = x0;\n        y2 = y1;\n        y1 = *(output+i);\n    }\n}<\/pre>\n\n\n\n
<\/p>\n\n\n\n