1 IIR filter analysis using VHDL. Allpass, multiple delay, and masking filters Anatolij Sergiyenko z-transform and transfer function z-transform is widely used in DSP due to the fact, that the signal x(n) = z n is the eigenfunction of any linear system which is invariant to the signal shift. The eigenfunction means that when such a function is processed by the system then the output signal has the same form but is delayed, and its magnitude is different. We can show that z n is the eigenfunction of the system which is described by some impulse response h(k). Consider the signal z n is inputted to the system. Then due to the properties of the linear shift invariant system the output signal is у(n) =   k=- h(k)z n–k =   k=- h(k)z –k z n = z n   k=- h(k)z –k . (1) We see that the function z n passes through the system invariably. Due to the formula (1), the transfer function of the system is H(z) =   k=- h(k)z –k . (2) The function H(z) is the complex function with the magnitude |H(z)| and phase arg(H(z)). The function H(z) has two sets of special points. The first set contains the points q 1 ,…,q M, for that |H(q і )| = 0. Therefore, these points are called zeros of the function H(z). The second set contains the points r 1 ,…,r N, for that |H(r і )| = , and therefore, they are called poles of the function H(z). To find the poles and zeros of H(z), the sum (2) is transferred to the fraction with the positive powers of z. Such a fraction is the following H(z) =z –L   M r=0 b r z М–r  N k=0 a k z N–k . (3) The numerator and denominator of the fraction (3) are polynomials with the positive powers of z, and a 0 = 1. The multiplier z –L represents the delay to L cycles, and can be taken off. The roots of these polynomials are equal to q 1 ,…,q M and r 1 ,…,r N , and are called as zeros sand poles, respectively. The function H(z) can be represented by the sets of zeros and poles as the following fractions H(z) =K (z – q 1 )…(z – q M ) (z – r 1 )…(z – r N ) = K (1 – q 1 z –1 )…(1 – q M z –1 ) (1 – r 1 z –1 )…(1 – r N z –1 ) , (4) where K is the transfer ratio or the amplification factor of the system.