Test: FIR System Structures - 1

• There are several structures for implementing an FIR system, beginning with the simplest structure, called the direct form.
• There are several other methods like cascade form realization, frequency sampling realization and lattice realization which are used for implementing and FIR system.

The structure of the direct form realization, resembles a tapped delay line or a transversal system.

• The direct form realization follows immediately from the non-recursive difference equation given by 
• We observe that this structure requires M-1 memory locations for storing the M-1 previous inputs.

• When the FIR system has linear phase, the unit sample response of the system satisfies either the symmetry or asymmetry condition, h(n)= ±h(M-1-n)
• For such a system the number of multiplications is reduced from M to M/2 for M even and to (M-1)/2 for M odd. Thus for the structure given in the question, M is odd.

• We have to do 3 multiplications for every second order equation.
• So, we have to do 6 multiplications if we combine two second order equations and we have to perform 3 multiplications by directly calculating the fourth order equation.
• Thus the number of multiplications are reduced by a factor of 50%.

• To derive the frequency sampling structure, we specify the desired frequency response at a set of equally spaced frequencies, namely:
• ωk=2π / M(k+α) ,k=0,1…(M-1)/2 for M odd
• k=0,1….(M/2)-1 for M even α = 0 or 0.5.

• In frequency sampling realization, the system function H(z) is characterized by the set of frequency samples {H(k+ α)} instead of {h(n)}.
• We view this FIR filter realization as a cascade of two filters. One is an all-zero or a comb filter and the other consists of parallel bank of single pole filters with resonant frequencies.

Explanation: When the desired frequency response characteristic of the FIR filter is narrowband, most of the gain parameters {H(k+α)} are zero. Consequently, the corresponding resonant filters can be eliminated and only the filters with nonzero gains need be retained.

Explanation: The system function of the FIR filter according to the frequency sampling realization is given by the equation

The above system function can be represented in the cascade form as shown in the above block diagram.