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FIR filters are digital filters with finite impulse response. They are also known as **non-recursive digital filters** as they do not have the feedback (a recursive part of a filter), even though recursive algorithms can be used for FIR filter realization.

**Figure 2-1-1. Block diagrams of FIR and IIR filters**

FIR filters can be designed using different methods, but most of them are based on ideal filter approximation. The objective is not to achieve ideal characteristics, as it is impossible anyway, but to achieve sufficiently good characteristics of a filter. The transfer function of FIR filter approaches the ideal as the filter order increases, thus increasing the complexity and amount of time needed for processing input samples of a signal being filtered.

The resulting frequency response can be a monotone function or an oscillatory function within a certain frequency range. The waveform of frequency response depends on the method used in design process as well as on its parameters.

This book describes the most popular method for FIR filter design that uses window functions. The characteristics of the transfer function as well as its deviation from the ideal frequency response depend on the filter order and window function in use.

Each filter category has both advantages and disadvantages. This is the reason why it is so important to carefully choose category and type of a filter during design process.

FIR filters can have linear phase characteristic, which is not like IIR filters that will be discussed in Chapter 3. Obviously, in such cases when it is necessary to have a linear phase characteristic, FIR filters are the only option available. If the linear phase characteristic is not necessary, as is the case with processing speech signals, FIR filters are not good solution at all.

Figure 2-1-3 illustrates input and output signals of non-linear phase systems.

The system introduces a phase shift of 0 radians at the frequency of ω, and π radians at three times that frequency. Input signal consists of natural frequency ω and one harmonic with the same amplitude at three times that frequency. Figure 2-1-3. shows the block diagram of input signal (left) and output signal (right). It is obvious that these two signals have different waveforms. The power of signals is not changed, nor the amplitudes of harmonics, only the phase of the second harmonic is changed.

If we assume that the input is a speech signal whose phase characteristic is not of the essence, such distortion in the phase of the signal would be unimportant. In this case, the system satisfies all necessary requirements. However, if the phase characteristic is of importance, such a great distortion mustn’t be allowed.

In order that the phase characteristic of a FIR filter is linear, the impulse response must be symmetric or anti-symmetric, which is expressed in the following way:

h[n] = h[N-n-1] ; symmetric impulse response (about its middle element)

h[n] = -h[N-n-1] ; anti-symmetric impulse response (about its middle element)

One of the drawbacks of FIR filters is a high order of designed filter. The order of FIR filter is remarkably higher compared to an IIR filter with the same frequency response. This is the reason why it is so important to use FIR filters only when the linear phase characteristic is very important.

A number of delay lines contained in a filter, i.e. a number of input samples that should be saved for the purpose of computing the output sample, determines the order of a filter. For example, if the filter is assumed to be of order 10, it means that it is necessary to save 10 input samples preceeding the current sample. All eleven samples will affect the output sample of FIR filter.

The transform function of a typical FIR filter can be expressed as a polynomial of a complex variable z-¹. All the poles of the transfer function are located at the origin. For this reason, FIR filters are guaranteed to be stable, whereas IIR filters have potential to become unstable.