A new type of second-order digital parametric equalizer is proposed whose frequency

response matches closely that of its analog counterpart throughout the Nyquist interval

and does not suffer from the prewarping effect of the bilinear transformation near the

Nyquist frequency. Closed-form design equations and direct-form and lattice realizations

are derived.

1. Introduction

Conventional bilinear-transformation-based methods of designing second-order digital parametric

equalizers [1–11] result in frequency responses that fall off faster than the corresponding analog

equalizers near the Nyquist frequency due to the prewarping nature of the bilinear transformation.

This effect becomes particularly noticeable when the peak frequencies and widths are relatively

high. Figure 1 illustrates this effect.

In this paper, we introduce an additional degree of freedom into the design, namely, the gain at

the Nyquist frequency, and derive a new class of digital parametric equalizers that closely match

their analog counterparts over the entire Nyquist interval and do not suffer from the prewarping

effect of the bilinear transformation.

The design specifications are the quantities {f , f , ?f, G , G , G, G }, namely, the sampling rate

s 0 0 1 B

f , the boost/cut peak frequency f , the bandwidth ?f , the reference gain G at DC, the gain G at

s 0 0 1

the Nyquist frequency fs /2, the boost/cut peak gain G at f0, and the bandwidth gain GB (that is, the

level at which the bandwidth ?f is measured.)

All previous methods of designing second-order equalizers assume G1 = G0 (usually set equal

to unity.) In these methods, the bilinear transformation is used to transform an analog equalizer

with equivalent specifications into the digital one. As remarked by Bristow-Johnson [9], all of these

designs are essentially equivalent to each other, up to a different definition of the bandwidth ?f and

bandwidth gain GB . For the equivalent analog equalizer, the quantity G0 = G1 represents the gain

at DC and at infinity, with the latter being mapped onto the Nyquist frequency f /2 by the bilinear

s

transformation.

In the method proposed here, we allow G to be different from G . In particular, we set G

1 0 1

equal to the gain an analog equalizer would have at f /2 if it were not bilinearly transformed. This

s

condition on G , together with the requirements that the gain at DC be G , that there be a peak

1 0

maximum (or minimum) at f0, that the peak gain be G, and that the bandwidth be ?f at level GB ,

provide five constraints that fix uniquely the five coefficients of the second-order digital filter.

The resulting digital filter matches the corresponding analog filter as much as possible, given

that there are only five parameters to adjust. The matching is exact at f = 0, f , f /2, and the two

0 s

filters have the same bandwidth ?f . These design goals are illustrated in Fig. 2.

†Presented at the 101st AES Convention, Los Angeles, November 1996, and published in JAES, vol.45, p.444, June 1997.

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