Signal filter is electrical equipment that attenuates the unwanted signal characteristic wave. Filters can be analog or digital.

An analog filter processes analog signals. They are mainly arranged with capacitors and resistors.

Digital signal filters process digital signals which are quantized. Digital filters are arranged with solid state components to process the signals.

Let’s start from analog filters.

Analog filter is a circuit which filters unwanted frequencies. Filtering is done by choosing circuit transfer function.

Simplest analog low pass filter:

Then if we think about digital filtering, then U_in(t) and U_ex(t) we can change to x_n=x(nT) and y_n=y(nT). Then we can rewrite equation:

Then we get:

y_n=a₀x_n+a₁x_n-1-b₁y_n-1,

Where

a₀=1/(1+T/RC), a₁=-1/(1+T/RC), b₁=-1/(1+T/RC).

This equation now can be used to build digital filter.

Filter can be described using response function. Response function of analog filter is reaction to step function while digital filter response function is response to step function in samples:

u[n] = 1, if n>=0

u[n] = 0, otherwise

According to method described in previous article we can calculate filter transfer function h(n);

Results are in following table:

Filter	h at t/RC
	0	0.5	1	1.5	2
Analog	1	0.779	0.368	0.223	0.136
Digital
T=0.5RC	0.667	0.444	0.296	0.198	0.132
T=0.25RC	0.8	0.512	0.328	0.21	0.134
T=0.125RC	0.889	0.555	0.346	0.216	0.135

From this chart you can see, if time T between samples is smaller then digital filter trends to analog filter characteristic. This means that you may choose digital filter error by varying sample step.

Digital filters have some advantages against analog filters. For instance in some circumstances analog filter can’t be valid while digital can be valid.

Besides transfer function filter can be described using Laplace transform:

H(s)=y(s)/x(s).

And x transform:

H(z)=Y(z)/X(z).

Z transform can be done in following way:

Inverse Z transform:

I am not going too deep in it as there are tons of information about Z transform:

Using Z transform we can rewrite our filter equation in following form:

This is one Tap filter and its structure:

z­^-1 means delay by on sample, T – one clock period. a0, a1 and b1 are filter coefficients (multipliers).

We can rewrite filter equation in other way:

Now its easy to write filter transfer function:

H(z)=(a₀+a₁z^-1)/(1+b₁z^-1)

And filter frequency response:

Blogsphere: TechnoratiFeedsterBloglines
Bookmark: Del.icio.usSpurlFurlSimpyBlinkDigg
RSS feed for comments on this post  |  TrackBack URI for this post

This entry was posted on Monday, June 5th, 2006 at 3:43 pm and is filed under DSP Filtering. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.