FIR Filter Design
Overview of filter design

          Filter are used to selectively change the wave shape, amplitude-frequency and phase frequency. The most common objectives of digital filters are to improve the quality of a signal, to extract information from signal, or to separate two or more signals previous combine. Nowadays, digital filters are widely used in different applications such as data compression, biomedical signal processing, digital audio, speech processing, image processing, etc. A digital filter, is a mathematical algorithm or routine implemented in hardware or/and software to perform filtering. Digital filters are mainly divided into two lasses; finite impulse response (FIR) and infinite impulse response (IIR). They are characteristic by their response to a unit impulse input sequence, h(k) where k=0,1,2…. The outputs can be obtained by convoluting the input signal and impulse response.
 

Finite impulse response (FIR) filter

FIR filter is characteristic by the following mathematical equation,
 
 

Where h(k) are impulse response of the filter; x and y are the input and output of the FIR filter respectively. According to the above equations, the impulse response of FIR has finite duration.

The FIR filter has three distinct properties:

• It is always stable
• It is always realizable, and
• It can always be designed to have a linear phase response.

             When FIR filter is represented in structure form, block or flow diagrams are often used to depict filter structures, which shows the computational procedures for implementing the digital filter. Generally, FIR filters are preferred when the system’s phase response is important.
 
 

Digital FIR filter design

There are 5 stages in the design of a digital filter:
1. Specification of the filter requirements
2. Calculation of suitable filter coefficients based on the desired specification.
3. Representation of the filter by a suitable structure (realization).
4. Analysis of the effects of finite word length on filter performance.
5. Implementation of filter in software and/or hardware.
 
 

IIR Filter Designs
Overview of IIR Filter

           Infinite impulse response (IIR) filter is characterised by the following mathematical equation,
 
 

 where h(k) us the impulse response of the filter and ak and bk are filter coefficients. According to the Equation (6), the impulse response of IIR filter could have infinite duration.

                One of the major advantages of the IIR filters is that it can achieve sharper transitions between passband and stopband than IFR filters of the same order because of the "peaked" responses created by poles. IIR filters have infinite-duration impulse responses. Therefore the basic technique of IIR filter design is by transforming well-known analog filters into digital filters using complex-valued mapping. The advantage of this technique lies in the fact that both analog filter design (AFD) tables and the mappings are available extensively in the literature. This basic technique is called the analog-to-digital filter transformation. However, the AFD tables are available only for lowpass filters. Other freuqency-selective filters (highpass, bandpass, bandstop) can be designed by applying frequency-band transformations to lowpass filters. The following approach is used in MATLAB for IIR lowpass filter design.

This IIR lowpass filter design technique involves the following steps:
 

•  Design analog lowpass filters
•  Study and apply filter transformations to obtain digital filters.

                The most common technique for designing IIR filters is to transform an existing analog filter into a digital IIR filter. Closed-form expressions for the transfer functions of analog filter designs with the more commonly required amplitude responses have been derived. The details of the IIR filter design can be found on tile lecture note. Basically, the design process of the IIR is the same as the FIR filter design.