FINITE IMPULSE-RESPONSE (FIR) DIGITAL FILTERS-I

EDITOR: B. SOMANATHAN NAIR

1. INTRODUCTION

Consider Fig. 1, which shows a long chain of delay lines interconnected using adders and multipliers in such a way that the system has only one input and one output. We can prove that this system can function as a digital filter. Form the figure, we get the relation

            y(n) = a_ox(n) + a₁x(n ‒ 1)+ a₂x(n ‒ 2)+ ∙∙∙ +a_mx(n ‒ m) (1)                   

where a_o, a₁, … , a_m are constants, and z ^‒1’s are delay lines. It can be seen from (1) that the output y(n) depends only on the present and past values of input x(n). In an FIR filter structure there are no feedback connections from the output to the input. In this context, we find that FIR filter structures are similar to combinational logic circuits, in which also there are no feedback connections.      

     An interesting feature to be noted in the case of FIR filters is that there are no equivalent structures of FIR filters in the analog regime; they are available only in the digital regime.

2. MEANING OF FINITE IMPULSE RESPONSE

Taking z transform of both sides of (1) yields

                        Y(z) = a_oX(z) + a₁z ^‒1X(z)+ a₂z ^‒2X(z)+ ∙∙∙ +a_mz ^‒mX(z)

                                   

                                = X(z)( a_o + a₁z ^‒1+ a₂z ^‒2 + ∙∙∙ +a_mz ^‒m) (2)

Rearranging (2), we obtain

                        H(z) = Y(z)/X(z) =  a_o + a₁z ^‒1+ a₂z ^‒2 + ∙∙∙ +a_mz ^‒m (3)                        

To find the impulse response, we apply the z transform of the impulse function d(n) in (3), where the Z [d(n)] = 1 = X(z), which gives

                                    H(z) = Y(z) =  a_o + a₁z ^‒1+ a₂z ^‒2 + ∙∙∙ +a_mz ^‒m (4)                     

Taking the inverse of (4), we get the impulse response

            h(n) =  a_o, a₁, a₂, ∙∙∙ a_m   (5)                                                           

For implementing the filter practically, the coefficients a_o, a₁, …, a_m must all be real numbers. Since the time-factor m contained in the impulse response is of finite value (which means that the impulse response is of finite duration), we call the filter governed by (4) as a finite-impulse response.