{"id":26,"date":"2011-08-15T18:37:52","date_gmt":"2011-08-15T23:37:52","guid":{"rendered":"http:\/\/www.graygoo.net\/blog\/?p=26"},"modified":"2011-08-19T14:57:10","modified_gmt":"2011-08-19T19:57:10","slug":"fun-with-filters","status":"publish","type":"post","link":"http:\/\/www.graygoo.net\/blog\/2011\/08\/fun-with-filters\/","title":{"rendered":"Fun with filters"},"content":{"rendered":"

In this post, I am going to review a number of Matlab functions useful for discrete-time signal processing.<\/p>\n

The transfer function<\/h2>\n
Suppose we have a discrete-time signal x[n] sampled with frequency Fs, and wish to pass it through a filter with output y[n]. The filter has impulse response h[n]. The output of the system y[n] is the convolution<\/a> of the input and the filter, x[n]*h[n]. Taking the z-transform<\/a> of the system takes convolution to multiplication, resulting in the equation<\/p>\n
$\"\"$ <\/a>H(z) is known as the transfer function of the filter, and is represented in the general case as a fraction B\/A of two polynomials in z.<\/p>\n
$\"\"$ <\/a>In practice, it is easier to work with powers of 1\/z than z, so we factor out an appropriate power of z to put H in terms of 1\/z.<\/p>\n

\n<\/a>This gives two vectors, B and A, of polynomial coefficients in ascending powers of 1\/z. These vectors characterize the filter in Matlab.<\/p>\n

A moving average filter<\/h2>\n

The basic Matlab filter command is filter.<\/code>filter<\/code> takes a filter [B,A] and an input vector x[n] and returns its output y[n]. Let’s create a 63-point moving average filter, with sampling frequency 1kHz. This filter “smooths” its output by averaging the last N=63 samples. This is a finite impulse response filter, so A is equal to 1. B is the average of the last 63 samples; i.e. a length 63 vector in 1\/z, divided by 63.<\/p>\n

a=1; b=ones(1,63)\/63;<\/pre>\nNow let’s look at its effect on a couple of sine waves. The output is an attenuated version of the input.<\/p>\n\r\nFs=1000; t=linspace(0,2*pi,Fs); x1=sin(t);x2=sin(10*t);\r\ny1=filter(b,a,x1);y2=filter(b,a,x2);\r\nsubplot(2,1,1); plot(t,y1); subplot(2,1,2); plot(t,y2);<\/pre>\n<\/a><\/p>\n
Matlab z-plane functions<\/h2>\nWe can look at the entire frequency response with the freqz<\/code> function. One invocation of freqz<\/code> takes our filter [B,A], along with the number of (uniformly spaced) frequencies and the sampling rate Fs, and returns a plot of the frequency response from 0 to Fs. Our discrete time moving average filter is a low pass filter, with some interesting oscillating behavior compared to an analog filter (to be explored later). Note that the response is symmetric with respect to Fs\/2, which is the Nyquist frequency<\/a> of the input signal.<\/p>\n
freqz(b,a,512,'whole',Fs)<\/code><\/p>\n
<\/a><\/p>\n
The corresponding impulse response (i.e. in the time domain) is computed by the impz<\/code> command. The impulse response for this uniformly weighted filter is pretty boring.<\/p>\n
impz(b,a)<\/pre>\n<\/a><\/p>\n
zplane<\/code> produces a pole-zero plot (where the poles and zeros are the roots of the polynomials B and A, obtained from the roots<\/code> command) on the complex plane.<\/p>\n
\r\nz=roots(b);\r\nzplane(b,a);\r\n<\/pre>\n<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"
In this post, I am going to review a number of Matlab functions useful for discrete-time signal processing. The transfer function Suppose we have a discrete-time signal x[n] sampled with frequency Fs, and wish to pass it through a filter … 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":[1],"tags":[11,13],"class_list":["post-26","post","type-post","status-publish","format-standard","hentry","category-ugoo","tag-dsp","tag-matlab"],"_links":{"self":[{"href":"http:\/\/www.graygoo.net\/blog\/wp-json\/wp\/v2\/posts\/26","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.graygoo.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.graygoo.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.graygoo.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.graygoo.net\/blog\/wp-json\/wp\/v2\/comments?post=26"}],"version-history":[{"count":0,"href":"http:\/\/www.graygoo.net\/blog\/wp-json\/wp\/v2\/posts\/26\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.graygoo.net\/blog\/wp-json\/wp\/v2\/media?parent=26"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.graygoo.net\/blog\/wp-json\/wp\/v2\/categories?post=26"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.graygoo.net\/blog\/wp-json\/wp\/v2\/tags?post=26"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}