Digital graphic equalizer based on filters with finite impulse response

Аuthors

Popov L. N.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: polen39@mail.ru

Abstract

Graphic equalizer (EQ) main building block is a set of bandpass filters (BPF), allowing convert the original signal into a set of narrowband signals [1]. Controlling their gain with subsequent summing up we obtain the required type of equalizer Bode diagram.

The advantage of digital filters with finite impulse response (FIR filters) implementation consists in the possibility of designing and EQ, which does not introduce phase distortions into transmitted signal. However, the use of such filters operating in real time is impeded by the necessity of executing a great number of multiplication operators per one sample of digital signal. The purpose of this paper consists in demonstrating the way to decrease the number of necessary multiplications by a certain delay of signal passage through the equalizer.

The author suggests to form each BPF using two low-pass filters (LPF) by subtracting the output signal of the LPF with a low boundary frequency, Fg_k, fr om the output signal of the LPF with a high boundary frequency, Fg_k+1. The above said LPF boundary frequencies herewith remain the BPF boundary frequencies under condition that the filters gain at boundary frequencies is equal to the half of its maximum value. The problem of equalizer filters design herewith comes to realizing LPF with boundary frequencies of Fg₀, Fg₁ ... Fg_{N — 1}, where N is required number of BPFs. Each of these BPFs forms a signal with a delay increasing with boundary frequency decreasing. The need to align the delays requires implementation of first genus FIR filters, which delay is equal to integer number of sampling periods.

To compute the filters coefficients a weighting method, implementing Kaiser Window [2, 3]., was used. The necessity to obtain the line of LPFs with similar amplitude-frequency response, required Kaiser Window modification, defining it with the equation

Here parameter µ can take a fractional value, contrary to the classical definition. It is shown that obtaining an LPF filter with similar characteristics and boundary frequencies Fg₁ and Fg₂, requires fulfillment of the condition:

Thus, if the sel ected parameters β and µ for the most wideband LPF of a line with boundary frequency of Fg₀, then the similar parameters of a filter with boundary frequency of Fg_n will be β_n = β and µ = µRⁿ under similarity hypothesis, where R = Fg_k/Fg_{k + 1}. Design equations allowing compute coefficients for the n^th filter of the line with specified parameters β, µ and R. The paper shows that direct equalizer realization based on signal processor within operating frequency band of 20 Hz—20 kHz is impossible due to the great number of multiplication operators (no less than 15348) per one sampling period.

A known method of required multiplication operators’ reduction consists in implementing interpolated FIR (IFIR) filters with embedded structure [5, 6]. The author proposes equalizer filter functional scheme, where shaping sub-filters with higher boundary frequency act as masking sub-filters within IFIR filters with lower boundary frequency. Such structure is formed by cascade coupling of the blocks shown in the Figure below.

The first stage with parameter L = 1, generates the signals: , corresponding to the LPF with boundary frequencies  (assuming that the ratio R is a multiple of 1/3 of octave, which corresponds to the most common 15 and 30 channel equalizers). The coefficients of the corresponding FIR filters are denoted by the vectors     The lengths of these vectors according to similarity conditions equal respectively to . The block of the subsequent stages differ fr om the first one by the L parameter value, necessary to form the sub-filter with the corresponding boundary frequency. For 15-channel equalizer, wh ere R corresponds to 2/3 of octave, the value of L for each of the subsequent stages should be 4 times greater than for the previous one.

The equations allowing compute the equalizer frequency response for the specified parameters β and µ are presented. Based on numerical experiments the paper shows, that with ±12 dB control range the acceptable quality of 15-channel equalizer characteristics can be obtained with β = 4.5, µ= 6.92. The required number of multiplication operators herein equals 162 per one sampling period, which is 1000 times less than in the case of a direct scheme.

To evaluate the software realization complexity for the proposed equalizer structure, the storage space required to store constants and variables that define its current state was calculated. In the case of using 32-bit numbers, the storage space requires about 300 KB.

The methods stated above can be implemented either in allied fields (frequency analysis, adaptive filtering [7], etc.).

Keywords:

band pass filter, interpolated finite impulse response filters folded structure

References

1. Vologdin E.I. Metody i algoritmy obrabotki zvukovykh signalov (Audio processing methods and algorithms), Sankt-Peterburg, SPbGUT, 2009, 96 p.

2. U. Tietze, Ch. Schenk. Poluprovodnikovaya skhemotekhnika (Semiconductor circuit engineering), Moscow, Mir, 1983, 512 p.

3. Rabiner L.R., Gold B. Teoriya i primenenie tsifrovoi obrabotki signalov (Theory and application of digital signal processing), Moscow, Mir, 1978, 848 p.

4. Komarov A.V. Tsifrovye signal'nye protsessory: http://halyavchik.com/item/b6224842-dcbc-46ac-aa7e-88b8d2c312dd

5. Richard Lyons. Interpolated narrowband lowpass FIR filters, IEEE SIGNAL PROCESSING MAGAZINE, January 2003, pp. 50-57.

6. Richard Lyons. Tsifrovaya obrabotka signalov (Understanding Digital Signal Processing), Moscow, Binom-Press, 2006, 656 p.

7. Grubrin I. V., Lygina I. Yu. Trudy MAI, 2013, no.69: http://www.mai.ru/science/trudy/eng/published.php?ID=43335

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