Digital Parametric Equalizer Design With Prescribed
Sunday, May 2, 2010 by sms
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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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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