Digital systems
Digital filters

Digital filters have to modify frequency content of input signal in a particular way, e.g. to act as a low pass, passing low frequencies from specified band through, and attenuating higher frequencies. It is possible to realize high pass or band pass filter as well, using the very same structure, only using different parameters (coefficients in FIR or IIR [infinite impulse response] system interconnections).In certain applications, the requirements for digital filter are not imposed on frequency filtration, but rather on shaping of time behavior of processed value, e.g. to supress a noise or short time disturbance impulses. In digital image processing, a two dimensional filters are used. They are used for noise suppression, contrast manipulation, contour enhancement etc. Design of digital filter is with DFT and spectral analysis base for numerical control in signals. Describe the transfer, frequency response, impulse response and differential equation. These are algorithms or circuits, changing spectrum of discrete input signal. In real time the filter between the two samples have to calculate the convolution (FIR filters). Digital filters follow the passive and active analog filters and can be designed either directly (FIR), or by converting the analog prototype (IIR).

Filters are divided by impulse response:

- Filters with finite impulse response - FIR
- Filters with infinite impulse response – IIR

And by structure of scheme:

- Nonrecursive filters NRDF (haven’t feedback)
- Recursive filters (feedback) mostly IIR filters

## LTI

Linear time invariant systems.

If x(t) is input signal and y(t) output signal, then output signal lis given by transformation ofinput signal, thus y(t) = T{x(t)}.

Time invariance mean, that system responding for given input signal x(t) by the same output signal y(t). If system is excited by signal x(t) shifted in time, x(t−t0), then system responding y(t) with the same time shift, y(t−t0).

Linear system is system, where for multiply input signal x(t) by k, responding k-multiplication output signal y(t) and for sum of input signals responding by sum of responses .

These behavior are very important, because tend to simplify some math operations, and system understanding. „Digital variant“for LTI are named DLTI (discrete LTI). In digital systems instead of analogue signals we work with numerical sequentions. Input signal is done by sequentions {x (n T), n belongs Z – discrete state space}, output signal (response) numerical sequentions {y (n T), n belongs Z}. Note x (n T) mean, that this number can be size of signal in time n T, where T is period of samples. If we consider (input) signal as numerical sequentions without relation to time, we can samples write only with index n, x(n). For many cases of use numerical computation is time variable important (e.g. for signal filtration), and we’ll keep note x (n T).

Discrete time invariant system convert input signal (sequentions) {x (n t)} to output sequentions (signal) {y (n t)}, thus {y (n t)} = T{x (n T)}. Impulse response of digital system is own response to one input sample applied in time t = 0. Input signal (sequention) is {x (n T) = 1 for n = 0, x (n T) = 0 for n ≠ 0}. In determination of impulse response digital system we assume, that before application input impulse is system relaxed.

For linear time invariant systems is done superposition rule – input signal we decompose into suitable parts, find responses for these parts a responses compose to output. Response for input signal will be given. Input signal {x (n T)} we can decompose to system of samples supposed as impulses x size (i T) placed in time i · T. Response for this impulses is x (i T){h (n T − i T)}. Whole output sequention as response for input signal {x (n T)} is sum of all responses, responses for all i.