Tuned deconvolution digital filter for elimination of loudspeaker output blurring

A FIR (finite impulse response) type digital filter operates on digital audio signals in modern sound reproduction systems. It is shown that this operation forces the loudspeaker to produce a sound pressure wave having the original signal waveform. Given a multi-driver speaker, its response to a known broad band analog signal (impulsive) is sampled at least as fast as the Nyquist rate. The result is used to construct a deconvolution filter which compacts, in the least-squares sense, the blurred signal (speaker output) back into its original waveform. Since this anti-blurring process is linear and time invariant, it can be applied to the speaker driving signal as a blur preventive. A fine-tuning procedure utilizing Lagrange's Method of Multipliers modifies the deconvolution process such that the blur-free speaker output achieves a degree of flatness in frequency response beyond what could be attained with a simple deconvolution filter.

FIELD OF THE INVENTION 
This invention pertains to high fidelity audio systems and more 
particularly to the waveshaping of audio signals before presentation to 
the speaker of the system. 
BACKGROUND OF THE INVENTION 
The loudspeaker as an energy conversion device exhibits its own motion 
characteristics under excitation. Its various modes of resonance at 
different frequencies depends on a multitude of mechanical and electrical 
design parameters. It remains a designer's dream to have flat 
magnitude-frequency and linear phase-frequency characteristics. 
A common technique for modifying the magnitude-frequency characteristic of 
the input electric signal and thus modifying the magnitude-frequency of 
the acoustic output is to filter the input in a selective manner. A band 
of pink noise 1/3 octave wide is fed into the loudspeaker for sound 
pressure measurement at a fixed distance from the loudspeaker. Signal gain 
in this particular band can then be changed accordingly. Obviously, this 
conventional method of "equalizing" is a very coarse adjustment--only the 
averaged deviation can be corrected. Two undesirable side effects 
occur--overlap in adjacent band pass filters and phase irregularities at 
the band edges. 
Ishii et al. (U.S. Pat. No. 4,015,089) disclosed a multi-driver speaker 
system wherein the the relative positions of the drivers along the 
radiation path helps to create a cancellation of sound waves at a 
particular frequency. This cancellation results in a favorable condition 
for a smooth phase characteristic when a particular crossover network is 
used. The claim to flat amplitude and linear phase response seems 
groundless in a strict sense. 
Berkovitz et al. (U.S. Pat. No. 4,458,362) uses an adaptive filter to 
equalize signals for room acoustic compensation. In the same patent it was 
shown that the same adaptive process can be used for loudspeaker 
performance improvement. While the adaptive process is desirable for room 
acoustic compensation, it does not represent what can be achieved 
ultimately for loudspeaker sound improvement. Though the advantage of the 
Widrow-Hoff adaptation algorithm is that prior knowledge of the speaker 
characteristic is not needed, the algorithm generates only approximate 
values for filter coefficients through stochastic approximation. In terms 
of loudspeaker sound improvement, an one-time operation, more accurate 
results can be obtained by the deteministic process of the current 
disclosure instead of stochastic approximation. 
Serikawa et al. (U.S. Pat. No. 4,751,739) corrects the speaker sound 
pressure frequency characteristic by multi-band digital filters with 
desired frequency reponses. The coefficients of these filters are 
generated by inverse Fourier transform of a transfer function resulting 
from repeated Hilbert transforms and modifications. However, while the 
Hilbert transforms render the resultant time sequence causal, phase 
linearity is lost. 
BRIEF DESCRIPTION OF THE INVENTION 
It is a general object of the invention to provide an improved high 
fidelity system. 
It is another object of the invention to provide a high fidelity system 
wherein the sound pressure wave produced by the speaker resembles the 
input electric audio signal in true high fidelity. 
It is a further object of the invention to provide a method and apparatus 
wherein both the amplitude and phase of the input electric signal are 
shaped to compensate for the inevitable blurring of the signal by the 
speaker. 
Briefly the invention contemplates a method and apparatus for improving the 
fidelity of an audio reproduction system by deconvolving the electric 
audio signal with respect to the known blurring effect of the loudspeaker. 
The deconvolution process is carried out in the form of a FIR type of 
digital filter. The filter coefficients are derived from the method of 
least sqares (in the time domain) and then fine-tuned for further 
enhancement in the frequency response of the speaker output.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT OF THE INVENTION 
I. Deconvolution Theory 
The term "deconvolution" is widely used in the literature when the input 
signal to a linear, time-invariant system is recovered from the system 
output. FIG. 1 shows a deconvolution filter with impulse response h(t) 
operating on the output of a linear, time-invariant system having the 
impulse reponse y(t). From the theory of linear systems the overall output 
is 
##EQU1## 
where s(t) is the arbitrary input. 
Deconvolution means the cancellation of the effect of y on s, i.e., if h 
satisfies 
##EQU2## 
EQU then 
EQU s*y*h=s*(y*h)=s*.delta.=s 
In general the existence of well-behaved inverse h(t) is questionable 
because of the difficulty of compacting the dispersed signal into an 
impulse. However, in the case of loudspeakers, it will be shown that a 
well-behaved h(t) exists in the form of sampled data. With modern digital 
technology the process of deconvolution can be readily carried out. 
The loudspeaker, as a band-limited device, can be represented by its 
response y(t) to the input x(t) which is a "band-limited" version of the 
impulse function .delta.(t): 
##EQU3## 
where f.sub.h is the upper limit of the hearing range. This response can 
be adequately represented by the sample data if the sampling period T is 
smaller than 1/2f.sub.h (Nyquist). For practical reasons the response y(t) 
is truncated at both ends so that only N+1 most significant samples are 
kept for processing: 
EQU y.sub.0,y.sub.1,y.sub.2 . . . y.sub.N 
II. Apparatus for Measurement of Speaker Impulse Response 
FIG. 2 depicts the generation of y's. At t=0 the function generator 10 
starts the signal x(t-LT/2) and ends the signal at t=LT. The excitation 
period LT is chosen to be sufficiently large such that the signal can be 
considered, in the engineering sense, as band limited. In response to this 
excitation, the loudspeaker 12 produces a sound pressure wave y(t-LT/2). 
Microphone 14 picks up the sound wave at t=t.sub.a where t.sub.a is the 
travelling time of the sound wave in the air. Starting at t=t.sub.a, 
sample and hold amplifier 16 feeds the signal to the A/D converter 18 
every T seconds until data samples fades into an insignificant level. 
Finally N+1 most significant, consecutive data samples y.sub.0,y.sub.1, . 
. . y.sub.N are chosen from the memory 20 to represent the band-limited 
impulse response. 
III. Method of Generaing Filter Coefficients 
To obtain the set of filter coefficients designated by 
EQU h.sub.0,h.sub.1,h.sub.2, . . . h.sub.M 
The following set of equations in matrix form represents the deconvolution 
in disrete form. Equivalently, the following matrix equation is the 
requirement that sound pressure wave follows the electric input signal 
with a delay of D sampling periods. Parameter D is to be determined later 
for best speaker performance in both time and frequency domains. 
##STR1## 
for convenience N is assumed to be even, and 
EQU [h]=COL[h.sub.0,h.sub.1, . . . h.sub.M ] 
EQU [x]=COL[x.sub.0,x.sub.1, . . . x.sub.N+M ] with x.sub.i =x[(i-N/2-D)T] 
This set of equations has no exact solution since the number of unknowns 
M+1 is smaller than the number of equations (M+1)+(N+1)-1=N+M+1. However, 
it is common engineering practice to seek least-squares solutions to 
overdetermined systems. In this case the set of "best" filter coefficients 
{h.sub.i ; i=0, 1, . . . M} satisfies 
EQU [Y][h]=[x] (1) 
with 
EQU [x]=COL[x.sub.0, x.sub.1, . . . x.sub.N+M ] 
representing the "nearly exact" replica of the input signal. The error 
vector e is the difference between the "exact" and the "nearly exact", 
i.e., 
EQU e.sub.i =x.sub.i -x.sub.i, i=0,1, . . . N+M (2) 
To minimize the sum of squares of these errors 
##EQU4## 
Define the (M+1).times.(M+1) sampled autocorrelation matrix as 
EQU [R]=[Y].sup.T [Y] (4) 
For minimum error the necessary conditions are 
##EQU5## 
Solving the resultant linear set of equations yields 
EQU [h]=[R].sup.-1 [Y].sup.T [x] (5) 
This is the untuned deconvolution filter. Since the matrix [R] is positive 
definite and of the "Toeplitz" form, it can be inverted very efficiently 
by the Levinson-Cholesky algorithm. For any output lag D the time domain 
speaker behavior (filtered) can be seen by computing E according to 
Eq.(3). Meantime the speaker frequency response is obtained by plotting 
##EQU6## 
The selection of optimum lag D.sub.opt, yielding the best performance, is 
as follows: 
The delay for the best "least-squares" error in time domain may or may not 
coincide with the delay for maximum flatness in frequency domain. However, 
in most cases these two delay values are close to each other. 
Selection of optimal delay should be biased in favor of best 
magnitude-frequency response at slight increase in time domain error. This 
is due to the fact that human ears are more sensitive to frequency content 
than phase linearity. 
IV. Method of Filter Tuning 
The choice of the filter order M+1 is governed by the desire to have M as 
small as possible so as to minimize computation in the implementation, 
while having M as large as possible so as to faithfully deconvolve away 
the speaker characteristic. In general, a small M flattens broad magnitude 
irregularities. As M increases, finer peaks and dips can be corrected. The 
mathematical manipulation discussed below "fine tunes" the filter 
coefficients so as to eliminate any local irregularity without increasing 
the filter length M. 
Consider the case in which a deconvolution filter leaves P+1 
magnitude-frequency irregularities at and near frequencies f.sub.0, 
f.sub.1, . . . f.sub.P. To mitigate the sonic effect of these anomalies 
the following set of quadratic constraints, based on the frequency 
response of the sequence x.sub.i, are imposed onto the original 
minimization problem: 
##EQU7## 
where constant K is the desired speaker output magnitude for all 
frequencies. 
Following Lagrange's Method of Multipliers, the error to minimize becomes 
##EQU8## 
where E is the sum defined in Eq. (3) and .lambda..sub.p 's are Lagrangian 
multipliers. Note that every term in Eq.(3') is a quadratic form of x. 
Given a set of .lambda..sub.p 's, this particular structure allows for an 
explicit expression for the filter coefficients 
EQU h'.sub.0, h'.sub.1, . . . h'.sub.M 
with all the constraints (which depend on .lambda..sub.p 's) automatically 
in effect. To show this, the partial derivatives are set to zero again 
##EQU9## 
which translates to the new set of linear equations to solve: 
##EQU10## 
where 
EQU [C.sub.p ]=COL[1, cos 2.pi.f.sub.p T, cos 4.pi.f.sub.p T, . . . cos 
2(N+M).pi.f.sub.p T] 
EQU [S.sub.p ]=COL[0, sin 2.pi.f.sub.p T, sin 4.pi.f.sub.p T, . . . sin 
2(N+M).pi.f.sub.p T] 
The (M+N+1).times.(M+N+1) matrix inside the brackets { }could be simplified 
to 
EQU [U]=[u.sub.ij ],i,j=0, 1, . . . N+M 
where 
##EQU11## 
In a manner similar to Eq.(4), the modified autocorrelation matrix is 
defined as: 
EQU [R']=[Y].sup.T [U][Y] (4') 
Thus, the tuned deconvolution filter is 
EQU [h']=[R'].sup.-1 [Y].sup.T [x] (5') 
It can readily be shown that [R'] is also positive definite and Toeplitz. 
The design procedure for the tuned deconvolution filter for any loudspeaker 
is summarized as follows: 
a. Sample speaker response to the band-limited impulse and digitize to 
obtain y.sub.i, i=0, 1, . . . N 
b. Compute [R] by Eq.(4). 
c. Compute untuned filter coefficients by Eq.(5) for different output time 
lags and compare performances for optimal delay. 
d. Use frequency response data to set the Lagrangian multipliers for fine 
tuning. 
e. Compute new filter coefficients by Eq.s (4') and (5'). 
Steps d and e can be repeated if the trial set of Lagrangian multipliers 
does not yield the satisfactory result. 
V. Preferred Embodiment of the Invention 
FIG. 3 is the diagram of one half of a stereo hi-fi system incorporating 
the invention. Analog input signal 30 (tuner, phonograph, analog tape 
etc.) of suitable level, say 1 volt rms, is first anti-aliased by low pass 
filter 32 and then digitized by the A/D converter 34. The output of the 
A/D converter or the direct digital input 36 (compact disc, digital audio 
tape, etc.) can be switch selected 38. The deconvolution filter 40 has in 
its ROM storage 41 a set of coefficients generated as described in section 
IV and based on the measurement as described in section II on the speaker 
80. Delay elements 42 can be implemented by shift registers, charge 
coupled devices, FIFO memories or ordinary RAM's with sequential access. 
Multipliers 43 and accumulator 44 are already commercially available. 
(e.g., device AM29510 made by Advanced Micro Devices, Inc., Sunnyvale, 
Calif.) It is also possible to construct the entire filter by programming 
a microprocessor. More importantly, since FIR type digital filter has been 
successfully fabricated in a single IC, (for example, the device YM3434 
made by Yamaha Corp. of Japan constitutes the interpolating filter 50 
depicted in FIG. 3) a special purpose LSI device can be designed to handle 
the entire deconvolution with internal or external coefficient memory 45. 
It is also noted that both digital filters 40 and 50 can be combined into 
one filter. If memory capacity permits, multiple sets of deconvolution 
filter coefficients for different loudspeakers can be stored and 
eventually switch-selected by the user. 
It is intended that all matter contained in the above description shall be 
illustrative and not limiting. For example, it should be apparent to those 
skilled in the art that a different deconvolution filter can be 
constructed by a different error criterion than Eq.(3) such as