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High-Order Digital Parametric Equalizer Design
Sophocles J. Orfanidis
Department of Electrical & Computer Engineering
Rutgers University, 94 Brett Road, Piscataway, NJ 08854-8058 Tel: 732-445-5017, e-mail:orfanidi@ece.rutgers.edu
November 15, 2005
Abstract
A family of digital parametric audio equalizers based on high-order Butterworth, Chebyshev, and elliptic analog prototype filters is derived that generalizes the con- ventional biquadratic designs and provides flatter passbands and sharper bandedges. The equalizer filter coefficients are computable in terms of the center frequency, peak gain, bandwidth, and bandwidth gain. We consider the issues of filter order and bandwidth selection, and discuss frequency-shifted transposed, normalized-lattice, and minimum roundoff-noise state-space realization structures. The design equa- tions apply equally well to lowpass and highpass shelving filters, and to ordinary bandpass and bandstop filters.
0. Introduction
Digital parametric audio equalizers are commonly implemented as biquadratic filters [1-15]. In some circumstances, it might be of interest to use equalizer designs based on high-order filters. Such designs can provide flatter passbands and sharper bandedges at the expense of higher computational cost. In this paper, we present a family of digital equalizers and shelving filters based on high- order Butterworth, Chebyshev, and elliptic lowpass analog prototypes and derive explicit design equations for the filter coefficients in terms of the desired peak gain, peak or cut frequency, bandwidth, and bandwidth gain. We discuss frequency-shifted transposed, normalized-lattice, and minimum roundoff-noise state-space realization structures, as well as structures that allow the independent control of center frequency, gain, and bandwidth. High-order equalizers have been considered previously by Moorer [3] who used a conformal mapping method based on elliptic functions to map a first-order lowpass digital shelving filter into a high-order elliptic equalizer, and by Keiler and Z
¨olzer [18] who obtained a fourth-order
equalizer based on a second-order analog Butterworth prototype. Our elliptic designs are essentially equivalent to Moorer's, but we follow a direct approach that closely parallels the conventional analog filter design methods and can be applied equally well to all three filter types, Butterworth, Chebyshev, and elliptic. Published inJ. Audio Eng. Soc., vol.53, pp. 1026-1046, November 2005. 1 We start by designing a high-order analog lowpass shelving filter that meets the given gain and bandwidth specifications. The analog filter is then transformed into a digital lowpass shelv- ing filter using the bilinear transformation. Finally, the digital shelving filter is transformed into a peaking equalizer centered at the desired peak frequency using a lowpass-to-bandpass z-domain transformation [16,17].
1. General Considerations
The design specifications for the digital equalizer are the quantities{G,G 0 ,G B ,f 0 ,Δf,f s }, that is, the peak or cut gain
G, the reference gainG
0 (usually set equal to unity), the bandwidth gain G B , the peak or cut frequencyf 0 in Hz, the bandwidthΔfmeasured at levelG B , and the sampling rate f s . These are illustrated in Fig. 1 for the Butterworth case. In the elliptic case, an additional gain, G s , needs to be specified, as discussed in Section 5. The bandwidth is related to the left and right bandedge frequencies f 1 ,f 2 byΔf=f 2 -f 1 . It is convenient to work with the normalized digital frequencies in units of radians per sample: 0
2πf
0 f s
2πΔf
f s 1
2πf
1 f s 2
2πf
2 f s (1) The starting point of the design method is an equivalent analog lowpass shelving filter, illustrated in Fig. 1, that has the same gain specifications as the desired equalizer, but with peak frequency centered at
Ω=0 and bandedge frequencies at±Ω
B The analog filter may be transformed directly to the desired digital equalizer by the bandpass transformation between the sandzplanes [16]: s=
1-2cosω
0 z -1 +z -2 1-z -2 (2)
The corresponding frequency mapping between
s=jΩandz=e jω is found from (2) to be: cosω 0 -cosω sinω (3) Fig. 1Specifications of high-order equalizer and the equivalent lowpass analog prototype. 2 whereω=2πf/f s andfis the physical frequency in Hz. The requirement that the bandedge frequencies 1 2 map onto±Ω B gives the conditions: cos 0 -cosω 1 sinω 1 B cosω 0 -cosω 2 sinω 2 B (4)
These may be solved for
0 andΩ B in terms ofω 1 andω 2 B =tan 2 tan 2 0 2 tan 1 2 tan 2 2 (5) where 2 1 . Equivalently, we have: cos 0 sin(ω 1 2 sinω 1 +sinω 2 (6)
Conversely, Eqs. (4) may be solved for
1 andω 2 in terms ofω 0 andΔω: e jω 1 =c 0 +j?Ω 2 B +s 20
1+jΩ
B ,e jω 2 =c 0 +j?Ω 2 B +s 20
1-jΩ
B (7) where
Δωenters throughΩ
B =tan(Δω/2). Extracting the real parts of Eq. (7), we obtain: cos 1 =c 0 B 2 B +s 20 2 B +1 ,cosω 2 =c 0 B 2 B +s 20 2 B +1(8) where we introduced the shorthand notation c 0 =cosω 0 ands 0 =sinω 0 . Eqs. (7) have the proper limits as 0 →0andω 0 →π, resulting in the cutoff frequencies (measured at levelG B of the digital lowpass and highpass shelving equalizers: 0 =0,ω 1 =0,ω 2 =Δω,(LP shelf) 0 1 2 =π,(HP shelf)(9) The magnitude responses of the high-order analog lowpass shelving Butterworth, Chebyshev, and elliptic prototype filters that we consider in this paper are taken to be: |H a 2 =G 2 +G 20 2 F 2 N (w)
1+ε
2 F 2 N (w) (10) where Nis the analog filter order,εis a constant, andF N (w)is a function of the normalized frequency w=Ω/Ω B given by: F N (w)=? ?w N ,Butterworth C N (w),Chebyshev, type-1 1 /C N (w -1 ),Chebyshev, type-2 cd (NuK 1 ,k 1 ), w=cd(uK,k),Elliptic(11) 3 whereC N (x)istheorder-NChebyshevpolynomial, thatis,C N (x)=cos(Ncos -1 x),andcd(x,k) is the Jacobian elliptic function cd with moduluskand real quarter-periodK. The parameters kandk 1 are defined in Section 5.
In all four cases, the function
F N (w)is normalized such thatF N (1)=1. The requirement that the bandwidth gain be equal to G B at the frequenciesΩ=±Ω B gives a condition from which the constant
εmay be determined. SettingΩ=Ω
B in Eq. (10), we obtain: |H a B 2 =G 2 +G 20 2
1+ε
2 =G 2 B G 2 -G 2 B G 2 B -G 20 (12)
The analog transfer function
H a (s)corresponding to Eq. (10) is constructed by finding the left-hand s-plane zeros of the numerator and denominator of (10) and pairing them in conjugate pairs. By construction, H a (s), and hence the equalizer transfer function, will have minimum phase. This is a desirable property because our designs imply that the transfer function of a cut by the same amount as a boost will be the inverse of the corresponding boost transfer function.
In terms of its
s-plane zeros and poles,H a (s)may be written in the factored form: H a (s)=H 0 1-s/z 0 1-s/p 0 rL i=1 1-s/z i )(1-s/z ?iquotesdbs_dbs14.pdfusesText_20
[PDF] High-Order Digital Parametric Equalizer Design† - Rutgers University