Method and apparatus for zero-mixing spectrum analysis with Hilbert transform

An apparatus for analyzing a spectrum of an input signal (χ(t)) having at least one line with a center frequency ({circumflex over (ω)}x) at the center of the line includes: a mixer for zero-mixing the input signal (χ(t)) to produce a base band signal (z(t)) by sweeping a local oscillator frequency (ωs) generated by a local oscillator, a resolution filter for filtering the base band signal (z(t)) to produce a filtered base band signal (y(t) ), and an envelope reconstruction means for reconstructing the envelope (E(ω)) of the spectrum of the input signal (χ(t)) by using an estimated amplitude (ŷx) at an estimated center frequency ({circumflex over (ω)}x) of each line of the input signal (χ(t)). Further, only the real component (I) of the base band signal (z(t)) is filtered with the resolution filter and the imaginary component (Q) is generated from the filtered base band signal (y(t)) by performing a Hilbert transform in a Hilbert filter arranged downstream of the resolution filter.

BACKGROUND OF THE INVENTION

The invention concerns to a method and an apparatus for zero-mixing spectrum analysis.

For example from U.S. Pat. No. 5,736,845 a spectrum analyzer with two step down conversion to the base band is known. In a first down conversion stage with a variable first local oscillator and a first mixer the input signal is transferred to an intermediate frequency. At a second stage with a constant second local oscillator and a second mixer the intermediate frequency is transferred to the base band. This configuration of spectrum analyzer is widely used. However, the two mixer stages are costly and thus it is desirable to have a spectrum analyzer with only one mixer stage. Such a concept is known as zero-mixing concept, which means that the input signal is directly converted to the base band.

The problem in frequency spectrum analyzers using the zero-mixing concept is the problem that surging occurs when the local oscillator frequency approaches the center frequency of one of the lines within the input signal. The envelope of the spectrum of the input signal has to be reconstructed. The central peak, i.e. the amplitude of the signal beyond the resolution filter when the frequency of the local oscillator equals one of the center frequencies of the lines of the input signal, strongly depends on the phase difference between the input frequency component and the sweep signal of the local oscillator. Thus no linear interpolation can reconstruct the spectrum envelope near the central peak.

In WO 03/069359 A1 it is proposed to reconstruct the envelope of the spectrum of the input signal by the use of an estimated amplitude at an estimated center frequency of each line of the input signal, whereby the estimated amplitude and the estimated center frequency are calculated from the time of occurrence, the duration and the maximum value of several halfwaves (wavelets) of the filtered base band signal. In the vicinity of the center frequency of each of the lines of the input signal, the signal outputted from the resolution filter is a surging signal which can be divided into several halfwaves (wavelets). For several halfwaves (wavelets) in the vicinity of the center frequency (when the frequency of the local oscillator approaches or leaves the center frequency) the time of occurrence, the duration and the maximum value of each halfwave are evaluated. It has been found that from the time of occurrence, the duration and the maximum value of several halfwaves an estimated center frequency and an estimated amplitude at the center frequency can be calculated. If the center frequency and the amplitude at the center frequency of each line of the input signal are known, the envelope of the input signal can be reconstructed near the center frequency.

It is a disadvantage of the method and apparatus know from WO 03/069359 A1 that the method is only performed for the real component of the complex base band signal. Thus, the number of halfwaves (wavelets) is restricted. Thus, for a sufficient accuracy of this method the sweep velocity must be limited.

SUMMARY OF THE INVENTION

It is the object of the present invention to improve the known method and the known apparatus in a manner to provide further halfwaves (wavelets) in order to speed up the procedure of reconstructing the envelope.

The object is solved by the features of claim1as concerns the method and by the features of claim9as concerns the apparatus. The dependent claims comprise further developments of the invention.

According to the invention only the real component of the base band signal is filtered with the resolution filter and the imaginary component is generated from the real component of the filtered base band signal by performing a Hilbert transform.

The reconstruction of the envelope of the spectrum of the input signal can then preferably be done on the basis of the real component and on the basis of the imaginary component generated with the Hilbert transform.

Preferably the real component of the filtered base band signal is delayed by a delay time equivalent to the processing time of the Hilbert transform in order to enable synchronized further processing of the real component and of the imaginary component. Preferably the amplitude values of the imaginary component generated by performing the Hilbert transform are corrected by an estimation on the basis of the neighboring amplitude values of the real component.

The Hilbert transform can be performed in a Hilbert filter, which can be a digital finite impulse response filter (FIR-filter) having specific coefficients as outlined in the dependent claims.

For a better understanding of the differences and improvements of the present invention vis-à-vis the state of art known from WO 03/069359 A1FIG. 1shows a block diagram of the known apparatus1for zero-mixing spectrum analysis. An input signal x(t), which has to be analyzed, is provided to a mixer2. To facilitate the description it is assumed that the input signal is a sinus-signal having only one circle frequency ωx. The input signal can thus be expressed as
x(t)=sin(ωx·t+φ)  (1)

In general, however, the input signal is a superposition of several spectral lines. For each spectral line of the spectrum of the input signal there is a center frequency ωx,i. For the present simplified case there is only one center frequency ωx. φ is a phase shift with respect to the signal s(t) provided by a local oscillator3and fed to mixer2. The signal s(t) of the local oscillator3can be expressed as a function of time t as follows:
s(t)=sin(ωs(t)·t)  (2)

The circle frequency ωsof the local oscillator3is not constant but a function of time t as the local oscillator is swept from a start circle frequency ωstartto a stop circle frequency ωstopwithin the sweep time Tsweep. The actual circle frequency ωs(t) of the local oscillator3can be expressed as a function of time t as follows:

DESCRIPTION OF THE PREFERRED EMBODIMENT

As already explained in the opening part of the description, the concept is related to the zero-mixing concept. This means that mixer2does not convert the input signal x(t) to an intermediate frequency, but the input signal x(t) is directly converted to the base band. Base band signal z(t) is transferred to the resolution filter4having the resolution filter frequency response R(ω).

The output signal y(t) of the resolution filter4is transferred to absolute value means5, which outputs the absolute values |y(t)| of the filtered base band signal y(t). The absolute values of the filtered base band signal |y(t)| are transferred to envelope reconstruction means6for reconstructing the spectrum envelope E(ω) of the input signal.

However, only on the basis of the absolute values of a filtered base band signal |y(t)| would the reconstruction of the envelope be rather inaccurate in the vicinity of the center frequency ωx, which will be explained later on with respect toFIGS. 2 and 3. Thus, detector means7is connected to the output of the resolution filter4and is provided with the filtered base band signal y(t). Detector means7detects the time of occurrence ti(offset), the duration ΔTi(width) and the maximum absolute value y′i(peak) of several halfwaves (wavelets) of the filtered base band signal y(t). This data are fed to envelope reconstruction means6and used for spectrum envelope reconstruction in the vicinity of the center frequency ωxas will be explained in detail later on.

For a better understandingFIGS. 2A,2B,3A and3B show the absolute values |y| of the filtered base band signal y as a function of the actual frequency ωsof the sweep signal s. As, according to equation (3), the actual frequency ωs(t) of the sweep signal s is a linear function of time t,FIGS. 2A to 3Bat the same time show the signal y as a function of time of measurement t.

FIGS. 2A and 2Bshow a fast sweep measurement, i.e. the total sweep time Tsweepin equation (3) is rather short. The total sweep time Tsweepis the same forFIGS. 2A and 2B, but forFIG. 2Athe phase difference φ between the input signal x and the sweep signal s is φ=0.2π and forFIG. 2Bthe phase difference φ between the input signal x and the sweep signal s is φ=0.6π. It can be obtained fromFIGS. 2A and 2Bthat a surging occurs when the actual frequency ωs(t) approaches the center frequency ωxor leaves the center frequency ωxof the input signal. The surging frequency is the actual frequency {tilde over (ω)}yof the filtered base band signal y(t). The relation between the actual circle frequency {tilde over (ω)}y,ifor halfwave (wavelet) i, the actual frequency ωsat time of occurrence ti, when halfwave (wavelet) i occurs, and the center frequency ωxof the input signal can be expressed as follows:
ωx=ωs(t=ti)+{tilde over (ω)}y,isweep is approaching ωx(4)
ωx=ωs(t=ti)−{tilde over (ω)}y,isweep is leaving ωx(5)

The equation (4) is true when the sweep is approaching center frequency ωx(ωs<ωx). Equation (5) is true when the sweep is leaving ωx(ωs>ωx).

An attempt can be made to reconstruct the envelope of the spectrum of the input signal by fitting an envelope function e(ω) through the absolute maxima y′iof the absolute values |y| of the filtered base band signal y as shown inFIGS. 2A and 2B. This reconstruction method is successful in a frequency range FR1and FR5far from the center frequency ωxof the input signal. However, in the vicinity of the center frequency ωxthis approach is not successful as the peaks of the absolute values |y| of the filtered base band signal y near the center frequency ωxstrongly depend on the phase difference φ between the input signal x and the sweep signal s. This can be seen by comparingFIGS. 2AandFIG. 2B. As already mentioned the phase difference φ is φ=0.2π inFIG. 2Aand φ=0.6π inFIG. 2B. In the case ofFIG. 2Ba small maximum occurs at the center frequency ωxleading to a minimum of curve e(ω). Thus this approach is totally unreliable in the frequency range FR3and cannot be used as a single method in frequency ranges FR2and FR4. Only in frequency ranges FR1and FR5can this method be used.

The same effect appears to be true for the fast sweep situation shown inFIG. 3Afor phase difference φ=0.2π and shown inFIG. 3Bfor phase difference φ=0.6π.

An estimation of the center frequency ωxand of the amplitude yxat the center frequency ωxis made on the basis of the time of occurrence (“offset”) ti, the duration (“width”) ΔTiand the maximum absolute value y′iof several halfwaves (wavelets) i. i is the index of the halfwave ti, ΔTiand y′iare indicated for one of the halfwaves inFIG. 2A.

The duration (“width”) ΔTiof the halfwave (wavelet) is half of the period and thus is related to the actual frequency {tilde over (ω)}y,iof base band signal y by

Inserting equation (6) in equations (4) and (5) leads to

An estimated value for the center frequency {circumflex over (ω)}x,iis obtained by equation (7) from the time of occurrence (“offset”) tiand the duration (“width”) ΔTifor each evaluated halfwave (wavelet) i individually.

Since the resolution filter frequency response R(ω) is known, an estimated amplitude value ŷx,ican be obtained from the absolute maximum value (peak) y′iof the evaluated halfwave (wavelet) i and the duration (“width”) ΔTiof the evaluated halfwave i by

From formulas (7) and (8) an estimated value of the center frequency {circumflex over (ω)}x,iand an estimated amplitude value ŷx,iat the center frequency are obtained for each evaluated halfwave (wavelet) individually. The different values obtained from the different halfwaves have to be averaged. Preferably not all values of all evaluated halfwaves are included in the averaging procedure.

Only the estimated values of distinguished halfwaves (wavelets) are used for averaging. To select the suitable values, the frequency range is divided into intervals11a,11bas shown inFIG. 4. Preferably there is a first set of intervals11aand a second set of intervals11b, whereby the second set of intervals11bis overlapping the first set of intervals11amost preferably by 50%. Each estimated value for the center frequency {circumflex over (ω)}x,iis related to these intervals. Each time {circumflex over (ω)}x,iof a certain halfwave (wavelet) falls into a specific interval, a count is assigned to this interval, i.e. a counter for this interval is incremented. After having related all values {circumflex over (ω)}x,iof all evaluated halfwaves (wavelets) to the intervals11a,11b, distinguished intervals are selected which have a minimum number of counts. This threshold of minimum number of counts must be determined experimentally as a fraction of nominal wavelet count (NWC), which is the theoretical maximum of wavelets per classification interval11a,11b. A well working minimum number of counts appears to be around 0.9 of the nominal wavelet count.

InFIG. 4it is shown how the estimated values for the halfwaves (wavelets) are projected into the intervals. The number of counts is indicated for each interval11a,11b. In the present example only interval11chaving the highest number of 8 counts is selected for averaging. This means that all estimated values of the center frequency {circumflex over (ω)}x,iand all estimated amplitude values ŷx,iof the selected interval11care added and divided by the number of counts n of the selected interval11cas follows (in the example n=8):

The remaining step is to reconstruct the spectrum envelope E(ω). In the frequency range FR3around center frequency ωxand partly in the frequency ranges FR2and FR4the known resolution filter frequency response R(ω) is used.FIG. 5shows the resolution filter frequency response R(ω) which has the maximum ŷxobtained from formula (10) and the estimated center frequency {circumflex over (ω)}xobtained from formula (9) as fitting parameters. The resolution filter frequency response R(ω) is weighted by a weighting function WR(ω), which is also shown inFIG. 5. The weighting function WR(ω) equals 1 in the frequency range FR3, equals 0 in the frequency ranges FR1and FR5and descends from 1 to 0 in the frequency ranges FR2and FR4. The resulting function R(ω)·WR(ω) can be seen fromFIG. 5additionally.

In the frequency ranges FR1and FR5far from center frequency ωxthe envelope e(ω) of the spectrum is reconstructed by using the maximum absolute values y′iof the filtered base band signal y as shown inFIGS. 2A,2B,3A and3B. The respective weighting function for this “original” envelope e(ω) equals 1 in the frequency ranges FR1and FR5, equals 0 in frequency range FR3and descends from 1 to 0 in frequency ranges FR2and FR4. The resulting spectrum envelope E(ω) is obtained from
E(ω)=R(ω)·WR(ω)+e(ω)·We(ω)  (11)
and is additionally shown inFIG. 6.

If the input signal x(t) contains more than one spectral line the above described procedure has to be repeated for each spectral line.

In the above concept, polarity of a wavelet was assumed by comparing its peak with the peak of a previous wavelet. If it was greater, formula (4) for approaching was used, if it was smaller, formula (5) for leaving was used. The same information can be gained by comparing the width of the wavelet with the previous one. Preferably both of these heuristics are used to achieve the best possible noise resistance.

In the state of art embodiment the signal y(t) beyond resolution filter4is only the in-phase component I of the complex signal. The quadrature component Q of the base band signal is not available. This reduces the accuracy of the method.

The inventive improvement is related to the generation of the quadrature component Q to be additionally used for the envelope reconstruction. If not only the wavelets of the real in-phase component I of the filtered base band signal y(t) filtered by the resolution filter4but also the imaginary quadrature phase component Q of the filtered base band signal y(t) is available, the total number of wavelets which can be used for reconstructing the envelope is significantly increased. Thus, the accuracy of the method is increased. On the other hand, with a given accuracy which is to be reached the sweep velocity can be increased because even if the sweep velocity is doubled the same total amounts of useable wavelets is obtained compared to the state of the art method in which only the real component I is evaluated.

FIG. 12shows a first part of the inventive apparatus whereasFIG. 16shows a second part of the inventive apparatus. Only the components which are different from the state of the art apparatus shown inFIG. 1are shown inFIG. 12 and 16. This means that only the resolution filter4and all parts following the resolution filter4are shown inFIGS. 12 and 13and especially mixer2and local oscillator3, which are shown inFIG. 1, are not shown inFIG. 12but are also present in the inventive apparatus.

As shown inFIG. 12the inventive apparatus comprises a Hilbert filter21for performing a Hilbert transform of the filtered base band signal y(t) filtered by the resolution filter4. Thus, the input of the Hilbert filter21is connected to the output of resolution filter4. Also, an optional delay buffer20is connected with its input to the output of resolution filter4. The delay buffer20is preferred for the best mode apparatus but is optional. The output of the Hilbert filter21and of the delay buffer20are connected to a correction unit22.

As shown inFIG. 16the real in-phase component I and also the output with the imaginary quadrature corrected component Q′ are supplied to the envelope reconstruction unit27. The output of the envelope reconstruction unit27is connected with the input of a Fuzzy peak correction unit26which outputs the corrected envelope signal. The outputs of the in-phase component I and of the quadrature phase component Q′ of the correction unit22are also both connected with wavelet detector23. The output of wavelet detector23is connected with the wavelet translation unit24. The output of the wavelet translation unit24is connected with wavelet classification unit25. The output of the wavelet classification unit25is connected with Fuzzy peak correction unit26in order to control the Fuzzy peak correction in the Fuzzy peak correction unit26. The function of the block elements shown in block diagrams ofFIG. 12 and 16will be explained later on.

According to the invention the Hilbert transformation is used to generate the imaginary quadrature component Q from the real in-phase component I. The Hilbert transformation is carried out in Hilbert filter21. Preferably Hilbert filter21is realized as a finite impulse response digital filter FIR-filter.FIG. 12AandFIG. 12Bshow two possible embodiments of such a FIR-filter. It should be noted that also other embodiments of Hilbert filter21are possible within the scope of the present invention.

As shown inFIG. 12Athe filtered base band signal y filtered by the resolution filter4is inputted to the Hilbert filter21. The filtered base band signal y is supplied to a number of multipliers30−M. . .30M−2,30M−1,30M. The first multiplier30−Mis connected with a first delay element31−Mwhich is connected to a first adder not shown inFIG. 12A. The FIR-filter shown inFIG. 12Acomprises several such structure elements designated as tabs. In the shown embodiment there are 2M tabs. Each adder30M−2,30M−1,30Mreceives the value of the delay element of the proceeding tab and of a respective multiplier and outputs the added value to the next delay element. Each multiplier30−M. . .30Mmultiplies the digital input value of the filter base band signal y with a specific coefficient hn. As the index n runs from −M to +M there are 2M+1 different coefficients.

FIG. 12Bshows a different embodiment of a digital FIR-filter. The difference with respect to the embodiment shown inFIG. 12Amainly is that the delay elements31−M. . .31M−1are not located between the adders30−M+1. . .30Mbut in the input line32of the multipliers30−M. . .30M.

It should be noted that in the preferred embodiment resolution filter4and all elements downstream of the resolution filter4are in the digital regime and are realized by digital signal processing. Thus, as shown inFIG. 1an analog/digital-converter33is situated between mixer2and resolution filter4. However, in other embodiments of the inventive apparatus analog/digital-converter33can also be present before mixer2or between resolution filter4and Hilbert filter21for example.

The coefficients hnof Hilbert filter21shown inFIG. 12A and 12Bare defined by

Equation (12) shows that Hilbert transformation is not causal. Thus, an approximation has to be used. Preferably an approximation by FIR-filter with linear phase is used. The impulse response is symmetrically truncated to finite length and its delay is half of this length. As shown inFIG. 7a preferred filter is a 31 taps FIR-filter.FIG. 7shows the coefficients hnof the FIR-filter.

Choosing an odd number of taps causes half of the coefficients to be null. This optimizes the realization of this filter21. Some variables have to be defined for further description as follows: fris the frequency related to resolution bandwidth. Spp is the sampling frequency to resolution bandwidth ratio (typically 13.33).

FIG. 8shows the transfer function H(fr) of the Hilbert filter in the frequency domain. The figure shows that the Hilbert filter approximation does not have a homogenous gain. Thus, the output signal Q should be corrected. Correction is mainly needed for frequencies that are passed through the Gauss resolution bandwidth filter4. The Gauss curve of the transfer function of the resolution filter4is shown inFIG. 9.

For frequencies that are passed through the resolution filter4the Hilbert filter21has some ripple. Less taps of the Hilbert filter cause smaller ripple. Thus, a Hilbert filter21that has only 7 taps can be used. This helps to implement this filter for higher sampling rates and greater resolution bandwidths. The suppression of higher frequencies including noise is still insignificant.

FIG. 10shows the frequency characteristic of the transfer function H(fr) of a 7 taps Hilbert filter.FIG. 11shows traces of the I-component and created Q-component signal beyond the Hilbert filter21and the delay buffer20. What is shown is the absolute value in the logarithmic scale. It can be seen that the Q-component has to be corrected to get accurate envelope. The method of correction is described hereafter.

The following text contains some terms which are defined as follows: Resolution trace I is the waveform of the signal y(t) beyond the resolution filter4created by sweeping over a single frequency (see I-component inFIG. 11). Resolution trace Q is the waveform of the signal beyond the Hilbert filter21(see Q component inFIG. 11). Wavelet is the portion of the resolution trace (half-wave, section between two zero-crossings), described by its width (length in samples) and peak (maximum absolute value). Wavelet frequency offset is the frequency difference between the sweep signal and the input signal (assuming single frequency component in the input signal) at the time of the wavelet's occurrence, i.e. wx−ws(twavlet). Relative frequency offset is the frequency offset related to resolution bandwidth of the resolution filter4.

The block diagram of signal process with Q-correction is shown inFIG. 12. We have to insert the delay buffer20for the I-component to match correct Q samples at the output of the Hilbert filter21in order to compensate the runtime through Hilbert filter21by the delay buffer20. Q-correction is performed in correction unit22. The mixer2, local oscillator3and resolution filter4are identical with the embodiment ofFIG. 1and thus not shown inFIG. 12. The output of resolution filter4is provided to Hilbert filter21and delay buffer20.

The easiest method for Q-correction is to multiply the whole Q wavelet, so that its peak is the average of the I peaks beside it in logarithm scale, as shown inFIG. 13. This leads to formula (13).

Unfortunately signal processing is made in linear scale, so numerous mathematic operations are necessary. Thus, this method is not preferred for real-time signal processing.

Hilbert filter frequency response is known. As there is an exact relation between wavelet width and relative frequency offset, we can find a multiplying correction coefficient for each Q wavelet. It is

fris the relative frequency

Spp is the sampling frequency to resolution bandwidth ratio

n is the Q wavelet width.

coef is the multiplying coefficient

H(fr) is the Hilbert filter frequency response.

There are only a few possible valid wavelet widths (preferably 2–32). Thus, it is helpful to prepare a correction table with two items n and coef. It is not necessary to correct wavelets greater than 32 samples. These wavelets are close to the signal, where frequency nears zero (ωx−ωs→0). If it is sufficiently precise to compute Q-component by a Hilbert filter21with a small number of tabs, additional correction of the peak is needed as described later on.

Because there is a small set of wavelet widths, that means also a small set of multiplying correction coefficients, the accuracy of this method is not sufficient. For the best results we have to combine both methods described above.

Each wavelet is multiplied by a coefficient according to a second method. Then the difference between the actual maximum and expected maximum is computed in linear scale by

ACoef=MaxI1+MaxI22-MaxQ·Coef(16)
and finally this difference ACoef is added to all samples of the wavelet. This simplification is allowable because the difference is small and it has practically no effect on the outputted envelope E(ω). This method is illustrated inFIG. 14.

The next step of the signal processing is computing the envelope E(ω) of the complex signal (I, Q′) by the known cordic algorithm. This is accomplished in the envelope reconstruction unit27. The cordic algorithm computes the magnitude and phase angle of a complex signal with the real component I and the imaginary component Q′. The magnitude represents the envelone E(ω).

A block diagram of the second part of the signal processing is shown inFIG. 16. As described, envelope reconstruction is located in cordic unit22which receives the components I and Q′ from Q correction unit22shown inFIG. 12. Components I and Q′ are also received by wavelet detector23. The output of wavelet detector23is passed to wavelet translation unit24. The output of the wavelet translation unit24is passed to wavelet classification unit25. The function of units23to25is described later on in detail. The units work similar as wavelet detector7inFIG. 1.

FIGS. 15A and 15Bshow the signal beyond envelope reconstruction cordic unit22for a different initial phase. It can be seen that a peak correction is needed to get precise envelope trace, as it was mentioned before.

Peak correction is an algorithm detects frequency components from the resolution traces and then reconstructs respective parts of the envelope with the known shape of resolution filter R(ω).

For the following explanation the following terms have to be defined: Wavelet polarity is the sign of frequency offset. It is positive when the sweep approaches wx. It is negative when the sweep leaves wx. Wavelet detection range is the range of frequency offset at which wavelets are detected and taken valid for further processing. It matches the range of wavelet widths. Wavelet cumulation factor is the number of consequent half-waves in the resolution trace which compose the stream of wavelets. Nominal wavelet count is the theoretical number of wavelets that can be detected in an ideal resolution trace in a particular wavelet detection range. The k-factor is the relative sweeptime. The k-factor is defined by

FIG. 17shows the resolution trace of I-component (or Q-component respectively) where the frequency ωs(t) of the sweep signal s(t) is close to an input frequency component ωx.

Width of the wavelet relates to the relative frequency offset by equation (18):

Wavelets from a certain detection range have to be detected with wavelet detector23. This means that only wavelets are needed that have their width in defined boundaries. This wavelet validation is made by wavelet detector23on both traces I and Q′. For further processing it is useful to have the same number of wavelets over different sweep conditions (Spp, k-factor).

Average wavelet widthnover the detection range can be computed by:

The number of samples n12over the detection range is equal to:
n12=k·Spp·(fr2−fr1)  (20)

It follows that the nominal wavelet count NWC can be calculated by

It is not easy to find the boundaries fr1and fr2of the detection range to get the same nominal wavelet count NWC for different k. Best results are achieved by recursive computing as follows: The closest relative frequency of the detection range is chosen to be for example fr1=0.3. There are no wavelets or wrong wavelets for frequencies below this limit.

The frequency offset in samples is
n0=k·Spp·fr1(22)
nmwith

nm=nm-1+Spp2·k2·nm-1(23)
is counted M=NWC·Cumulation times. The frequency offset is

Now the minimum and maximum width can be computed by

For the next processing it is desirable to have enough wavelets that are not too small. The polarity is generated by a polarity predictor. This predictor compares the width of present and previous wavelet and uses the knowledge of the resolution filter frequency response to estimate whether the sweep signal approaches or leaves the frequency component ωxof the input signal.

Of course the amount of detected wavelets will be smaller in cases where the signal has a low s/n (signal/noise) ratio or where two close frequency components in comparison with resolution bandwidth are included.

Each wavelet can be transformed in wavelet translation unit24to give information about the input signal frequency component that caused its occurrence. The width of the wavelet relates to immediate frequency of resolution trace by equation (26):

fxis the searched frequency

fsis the immediate sweep frequency

It is needed to process positive and negative wavelets separately. Once we have the streams of translated wavelets, we need a mechanism that will sum up these pieces of information to produce reliable description of emerging frequency components.

This is accomplished by projecting the wavelets into discrete frequency classes in the form of intervals11a,11band by thresholding.FIG. 18illustrates this process. In a real situation two traces I and Q are processed in order to have more useable wavelets.

The frequency classes are composed of intervals11aand11bused in wavelet classification unit25which are preferably overlapped by 50% to preserve the homogeneity of the frequency axis. Class width and threshold must be determined experimentally. It is preferred that the width of each class in samples is equal to 2·Spp. Well working threshold appears to be more than 8 wavelets.

By this process one frequency class (e.g. interval11c) is selected. By averaging of all projected wavelet positions we can get the exact position where the peak correction should be applied. Together with this position we also have the count of projected positive or negative wavelets. This information is used to recognize whether there is an isolated peak or whether the peak is close to another peak and one type (approaching or leaving) of wavelets dominates or if the signal is noisy which results in less wavelets.

Now the Fuzzy peak correction performed in the correction unit26is described. At this part of the signal processing we have the envelope trace and accurate position of the needed correction. Now we have to compute the regular shape of the correction. The envelope trace is not replaced by this correction shape directly. Some fuzzy rules that improve the result can be used.

For the correction we need the levels of the envelope trace for the times when frequency offset is equal to ±0.5 RBW and ±1 RBW. RBW is the resolution bandwidth of resolution filter4. Difference 0.5 RBW in samples can be computed by:

Because the envelope trace is not ideal the levels in given times as the average of several samples have to be calculated. For example four numbers L(−1.0), L(−0.5), L(0.5), L(1.0) are received as shown inFIG. 19. Peak level L(0.0) is computed dependent on their differences and the number of wavelets. This is done in order to get a more stable amplitude value.

Once all five levels are obtained, the regular shape of the corrected envelope can be created. These five points divide the correction shape into four parts as shown inFIG. 20. In each part both points are interpolated by some simple curves. All interpolation curves are computed in linear scale.

For the best approximation of the envelope and correction shape and for the better interpolation of noisy signals preferably weighting functions as shown inFIG. 21are used.

Also other conditions have effect on the weights. Less than two negative wavelets decrease weight w−. On the other hand less than two positive wavelets decrease weight w+. Weight w0is computed as the maximum of both weights w−and w+computed above.

InFIGS. 22A to 22Cthere are some examples of corrected signal plotted in dark lines in comparison to an ideal envelope plotted in light lines.FIG. 22Ashows the result for a single frequency component ωxin the input signal.FIG. 22Bshows the result for a single frequency component in the input signal with a low signal/noise ratio.FIG. 22Cshows the result for two close frequency components in the input signal. It can be seen that the used peak interpolation is sufficient.