Second Order DSP Filter explained

Second order DSP filter is more complex than first order filters. There is one more delay block (z^-1) added. Function diagram of second order DSP filter.

The equation for second order DSP Filter can be written as follows:

And transfer function expressed in Z transform would be:

There can be two particular cases of filter. If we can write general expression of filter like this:

Then:

1) When N=0 we get non recursive digital filter (FIR – Finite Impulse Response). This is simple sum of finite number of samples. In other words – averaging filter.

2) When N>0 we get recursive filter. This filter (IIR – Infinite Impulse Response) has a feedback where output samples are branched to input together with input samples. This type of filter has stability issues and so on.

General form of transfer function may look like this - applies to all filter types:

What about realization of filter? Using digital filters in embedded systems is a critical issue. Sampled signal is forwarded to filter input as x(nT) queue and output is y(nT) is a result of digital processing of signal. Speed of processing may be defined as minimal clock period T_min of sampled input signal x(nT) or by frequency band F_max=1/(2T_min) of signal (according to Nyquist identity). The lower T_min is – the higher F_max – speed of DSP processing. The main target of DSP processing usually is real time processing of wide band signals.

Universal microcontrollers are almost useless in higher speed signal processing. To make effective there are needed parallel modules in microcontrollers, but universal microcontrollers can perform one task at one time. Other situation is with DSP processors. They have ability to parallel some instructions and increase speed of critical parts of algorithms.

The most effective DSP can be done using FPGA. There all commands can be performed in parallel and high speed. I am not going to deep in this as this needs a separate analysis thread…

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