Spline-wavelet signal analyzers and methods for processing signals

A linear spline function generator, comprising:

• a memory for storing parameters of a spline precursor function, said parameters preselected such that an analog signal based thereon, when integrated, will yield a linear B-spline;
• a multiplier, a first port of said multiplier receiving said parameters of said spline precursor function, a second port of said multiplier receiving a string of spline coefficients, an output of said multiplier outputting a plurality of current sets of multiplied parameters in response to receiving said parameters of said spline precursor function and said string of spline coefficients;
• a summer for summing said current multiplied parameters with stored prior multiplied precursor parameters and generating an intermediate sequence in response thereto;
• a digital to analog converter having an input port for receiving said intermediate sequence and an output port for outputting an intermediate signal;
• an integrator having an input port for receiving said intermediate signal and an output port, an integrated intermediate signal appearing at said output port in response to the receipt of said intermediate signal at said input port of said integrator;
• and
• a buffer having an input port coupled to said output port of said integrator, said linear spline function appearing at said output port of said buffer in response to said integrated intermediate signal being received at said input port thereof.

A linear spline wavelet function generator, comprising:

• a memory for storing parameters of a spline wavelet precursor function, said parameters preselected such that an analog signal based thereon, when integrated, will yield a linear B-wavelet;
• a multiplier, a first port of said multiplier receiving said parameters, a second port of said multiplier receiving a string of wavelet coefficients, an output of said multiplier outputting a plurality of current sets of multiplied parameters in response to receiving said parameters and said string of wavelet coefficients;
• a summer for summing said current multiplied parameters with stored prior multiplied precursor parameters and generating an intermediate sequence in response thereto;
• a digital to analog converter having an input port for receiving said intermediate sequence and an output port for outputting an intermediate signal;
• an integrator having an input port for receiving said intermediate signal and an output port, an integrated intermediate signal appearing at said output port in response to the receipt of said intermediate signal at said input port of said integrator;
• and
• a buffer having an input port coupled to said output port of said integrator and an output port, said linear wavelet function appearing at said output port of said buffer in response to said integrated intermediate signal being received at said input port thereof.

A wavelet generator for generating an mth order wavelet function of a sequence, comprising:

• a memory for storing parameters of a B-wavelet precursor function which, when passed-through a digital to analog converter and integrated (m-1) times and buffered (m-1) times, will yield a B-wavelet of the mth order;
• a multiplier, a first port of said multiplier receiving said B-wavelet precursor parameters, a second port of said multiplier receiving a string of wavelet coefficients derived from said signal, an output of said multiplier outputting a plurality of current multiplied precursor parameters in response to receiving said B-wavelet precursor parameters and said string of wavelet coefficients;
• a summer for summing said current multiplied precursor parameters with stored prior multiplied precursor function wavelet elements and generating an intermediate sequence in response thereto;
• a digital to analog converter having an input port for receiving the said intermediate sequence and an output port for outputting an intermediate signal;
• and
• (m-1) integrating and buffering stages, each of said integrating and buffering stages comprising an integrator having an input port and an output and a buffer having an input coupled to said output of said integrator and an output port thereof, said input port of a first of said integrating and buffering stages operable to receive said intermediate signal, said mth order wavelet function of said signal appearing at said output port of a last of said integrating and buffering stages in response to said input port of said first stage receiving said intermediate signal.

A spline generator for generating an mth order spline function of a sequence, comprising:

• a memory for storing parameters of a B-spline precursor function which, when passed through a digital to analog converter and integrated (m-1) times and buffered (m-1) times, will yield a B-spline of the mth order;
• a multiplier, a first port of said multiplier receiving said B-spline precursor parameters, a second port of said multiplier receiving a string of spline coefficients derived from said signal, an output of said multiplier outputting a plurality of current multiplied precursor parameters in response to receiving said B-spline precursor parameters and said string of spline coefficients;
• a summer for summing said current multiplied precursor parameters with stored prior multiplied precursor function spline elements and generating an intermediate sequence in response thereto;
• a digital to analog converter having an input port for receiving the said intermediate sequence and an output port for outputting an intermediate signal;
• and
• (m-1) integrating and buffering stages, each of said integrating and buffering stages comprising an integrator having an input port and an output and a buffer having an input coupled to said output of said integrator and an output port thereof said input port of a first of said integrating and buffering stages operable to receive said intermediate signal, said mth order spline function of said signal appearing at said output port of a last of said integrating and buffering stages in response to said input port of said first stage receiving said intermediate signal.

A method for processing an analog signal F(t), comprising the steps of:

• sampling the signal F to obtain a plurality of digitized samples {F.sub.n }, F.sub.n =F(nh), where h is the sampling period and n is the number of samples;
• generating a plurality of zero-level B-spline coefficients c.sup.0 (n) as a weighted moving average of the digitized samples, according to the formula:
• ##EQU37## where i.epsilon. ,
• m is the order of spline,
• w.sup.(m).sub.n+(m/2)-ilm is one of a plurality of predetermined weights, and l.sub.2x-1 =l.sub.2x =x, x.epsilon. ;
• for each derived order of resolution k=-1, -2, -3 . . . , performing the following steps:
• for k.ltoreq.-2, generating a plurality of (k+1)-level spline coefficients c.sup.k+1 (n) from the (k+2) level spline coefficients c.sup.k+2 (n) according to the formula:
• ##EQU38## where {a.sub.n } is a set of predetermined constants;
• and generating a plurality of (k) level wavelet coefficients d.sup.k (n) according to the formula:
• ##EQU39## where {b.sub.n } is a second set of predetermined constants, generating a spline S.sub.M f, where M is larger integer and l.sub.m /2.sup.M =h, according to the formula ##EQU40## where N.sub.m is a stored m.sup.th order of B-spline and j.epsilon. ;
• and generating a wavelet function g.sub.k (.chi.) from a plurality of wavelet coefficients d.sub.j.sup.k according to the formula ##EQU41## where k is an integer and .PSI..sub.m is a basic wavelet of order m whereby the analog signal is processed for analyzing frequency information of the analog signal.

A signal analyzer, comprising:

• a sampler for sampling an input analog signal at a predetermined sampling rate;
• a digitizer coupled to said sampler for digitizing a plurality of samples resulting from said input analog signal;
• a B-spline interpolator for generating a plurality of B-spline coefficients of the zero-level in response to receiving the digital signal samples, said interpolator including a multiplier for multiplying a plurality of predetermined stored weights to respective selected ones of the digitized signal samples and a summer for summing the weighted digitized signal samples together in a moving average operation to derive the zero-level B-spline coefficients;
• and
• a multiplier for multiplying the zero-level B-spline coefficients in a moving average operation by a plurality of predetermined B.sub.n -constants to derive a plurality of first-level wavelet coefficients;
• a spline function generator for generating at least one spline function from at least one respective set of spline coefficients;
• and
• a wavelet generator for generating at least one wavelet function from at least one respective set of wavelet coefficients whereby said input analog signal is processed for analyzing frequency information of said input analog signal.

A signal analyzer, comprising:

• means for receiving a plurality of digital signal samples;
• means for generating a plurality of spline coefficients of the zero level in response to receiving the digital spline sample, said means for generating including means for multiplying a plurality of predetermined stored weights to respective selected ones of the digitized signal samples and means for summing the weighted digitized signal samples together in a moving average operation to derive zeroth-level B-spline coefficients;
• means for multiplying the zeroth-level B-spline coefficients in a moving average operation by a plurality of a.sub.n -constants to derive a plurality of first-level B-spline coefficients for the signal;
• means for multiplying the zeroth-level B-spline coefficients in a moving average operation by a plurality of b.sub.n -constants to derive a plurality of first-level wavelet coefficients;
• means for generating at least one spline function from at least one respective set of spline coefficients;
• and
• means for generating a wavelet function from at least one respective set of wavelet coefficients whereby said signal is processed for analyzing frequency information of said signal.