ORLOVSKY IGOR ANATOLYEVICH (RU)
PRASOLOV VICTOR ANDREEVICH (RU)
MURZINOV DMITRY DMITRIEVICH (RU)
| US6509728B1 | 2003-01-21 | |||
| SU1451617A1 | 1989-01-15 | |||
| US5038096A | 1991-08-06 | |||
| US4054785A | 1977-10-18 |
| WHAT IS CLAIMED I S: 1. A digital sweep type spectrum analyzer with up/down frequency conversion, which provides a frequency spectrum of a complex analog input signal, the analyzer comprising: (i) two three-level analog- to-digital converters (ADC), whose inputs are analog quadrature components Ia(t), Qa(t) of the complex input signal and whose digital quadrature output (IADC, QADC) is quantized into three levels (-1 ; 0; +1), wherein an ADC sampling frequency fs corresponds to a bandwidth Bjn of the complex input signal (fs Bjn); (ii) a quadrature local oscillator (LO), which generates a digital complex sinusoid with quadrature components (lw, Ql°) based on a sweep signal at its frequency control input; (iii) a MUX block comprising four 9-to-l multiplexers MUX 1 , MUX 2, MUX 3, MUX 4 whose selected inputs are connected to the outputs of the first and second ADC and whose data inputs connected to the quadrature components (Ii0, QLO), wherein the MUX block generates, without using multiplication, real numbers as follows: R// _ ^ADC c ji R2/ = Q^C * iLO- wherein the four 9-to-l multiplexers MUX 1, MUX 2, MUX 3, MUX 4 operate as follows: Select inputs Outputs jADC QADC MUX 1 MUX 2 MUX 3 MUX 4 R11 ROQ RiO ROi 1) 0 0 0 0 0 0 O 2) 0 + 1 0 QLO 0 jL 3) 0 -1 0 -QLO 0 _jLO 4) +1 0 ,LO 0 QLO 0 O 5) -1 jI. 0 0 -QLO 0 6) + 1 + 1 [LO QLO QLO jLO 7) -1 -1 rLO -QLO -QLO rLO 8) +1 -1 rLO -QLO QLO rLO 9) -1 + 1 .LO -QLO rLO (iv) a first block of registers comprising four registers connected to respective outputs of the MUX block; (v) a block of adders comprising four adders, whose inputs are connected to registers of the first block of registers, wherein the block of adders produces a complex up-converted signal with quadrature components (lu, Qu) and a complex down-converted signal with quadrature components (Γ°, QD) as follows: Qu = Rie + RQI , lD = Rn + RQQ , QD = -RIQ + RQI ; a second block of registers comprising four registers; a block of accumulators comprising four accumulators whose inputs are fed from the outputs of the block of adders, wherein every accumulator accumulates M samples (M » 1) respectively lu, Qu, ID, QD at a constant frequency of the LO, then the resulting values 'A , Q„ 'Α , Im 'A , Qm A are divided by a coefficient 2K (2K < M) by shifting by K bits toward a least significant bit (LSB) and resulting values ί„ Κ , Qm K , ^m ^ , are written into the four registers of the second block of registers, then all four accumulators are reset to zero and the process is repeated for the next frequency of the LO; (viii) a level detector that estimates vector levels Vu and VD with complex quadrature samples (I ' , Q ' ) and (I · , Q ' ) respectively of the second block of registers, wherein the level Vu is an estimate of the input signal spectrum at a frequency point -fio and the level VD is an estimate of the input signal spectrum at a frequency point +f£o; (ix) a data analysis and collection logic that receives spectrum data from the level detector, analyzes the spectrum data and sequentially stores the spectrum data within one scanning period, after which the spectrum data is updated; (x) a sweep and data collection controller that sends a sweep signal to frequency control input of the LO to control the LO by a stepwise change of frequency in a range (0, +fs/2), sequentially setting N » 1 of equidistant frequency values in = fs*(2«-l)/4N, where « = 1, 2, . . . , N, wherein each accumulator of the block of accumulators accumulates M samples at each frequency in: M = floor [&*N] , where constant k is selected from a range (0.5, . . . , 5.0), wherein the spectrum analyzer forms 2N spectrum points in a range (-fs/2, +fs/2) during each sweep period. The digital spectrum analyzer of claim 1 , wherein the data analysis and collection logic analyzes the spectrum data to detect presence of narrowband interference frequencies for interference rejection. The digital spectrum analyzer of claim 1, wherein the level detector produces a rough estimate of vector length for each input vector using the following successive steps: d) forming auxiliary vector with orthogonal components (X, Y), where X=max ( x\, \ \), Y-min (|J |, x, y - orthogonal components of the input vector, wherein the auxiliary vector is located in the (0, π/4) octant; e) finding in the (0, π/4) octant a preset sector, which contains the auxiliary vector; f) selecting a linear equation corresponding to the found sector; and g) solving the linear equation with preset coefficients. The digital spectrum analyzer of claim 1, further comprising displaying the spectrum data to a user. A digital sweep type spectrum analyzer with up/down frequency conversion, the analyzer comprising: (i) two analog-to-digital converters (ADCs), whose inputs are analog quadrature components Ia(t), Qa(t) of a complex input signal and whose digital quadrature . · rADC (-.ADC-, output is (1 , Q ); (ii) a quadrature local oscillator (LO), which generates a digital complex sinusoid with quadrature components (Ii0, QLO) based on a sweep signal; (iii) a MUX block comprising four multiplexers whose selected inputs are connected to the outputs of the two ADCs and whose data inputs receive the quadrature components (ILO, Qi0), wherei k generates real numbers as follows: (iv) a first block of registers connected to respective outputs of the MUX block; (v) a block of adders whose inputs are connected to the first block of registers, wherein the block of adders produces a complex up-converted signal with quadrature components (lu, Qu) and a complex down-converted signal with quadrature components (lD, Q°) as follows: lu = Rn - RQQ, Qu = R'e + RQI, QD = -RIQ + RQI; (vi) a second block of registers; (vii) a block of accumulators whose inputs are fed from the block of adders, wherein every accumulator accumulates M samples respectively of I , Q , I , Q TU A U A TD<A at a constant frequency of the LO, the resulting values i m , dm , L m , Jm are divided by 2 (2 < M) by shifting by K bits toward a least significant I U,K U K i D'K nD'K bit (LSB) and resulting values ί m , dm , L m , Sim are written into the second block of registers, and then the accumulators are reset to zero and the steps are repeated for the next frequency of the LO; (viii) a level detector that estimates vector levels Vu and VD with complex quadrature samples (I ' , Q ' ) and (I ' , Q · ) respectively of the second block of registers, where Vu is an estimate of an input signal spectrum at a frequency point -fLo and VD is an estimate of the input signal spectrum at a frequency point +tw, (ix) a controller that sends the sweep signal to the LO to control the LO by a change of frequency in a range (0, +fs/2), sequentially setting N » 1 of equidistant frequency values in = fs*(2«-l)/4N, where n = 1, 2, . . . , N, wherein each accumulator of the block of accumulators accumulates M samples at each frequency in: M = floor [&*N] , where A; is a constant, and wherein the spectrum analyzer forms 2N spectrum points in a range (-fs/2, +fs/2) during each sweep period. 6. The digital spectrum analyzer of claim 5, further comprising a data analysis and collection logic that receives spectrum data from the level detector, analyzes the spectrum data and stores the spectrum data within one scanning period; 7. The digital spectrum analyzer of claim 6, wherein the data analysis and collection logic analyzes the spectrum data to detect presence of narrowband interference frequencies for interference rejection. 8. The digital spectrum analyzer of claim 5, wherein the level detector produces a rough estimate of vector length for each input vector using the following successive steps: h) forming auxiliary vector with orthogonal components (X, Y), where X=max (|jc|, \y\), Y=min (|*|, \y\), x, y - orthogonal components of the input vector, wherein the auxiliary vector is located in the (0, π/4) octant; i) finding in the (0, π/4) octant a preset sector, which contains the auxiliary vector; j) selecting a linear equation corresponding to the found sector; and k) solving the linear equation with preset coefficients. 9. The digital spectrum analyzer of claim 5, further comprising displaying the spectrum data to a user. 10. The digital spectrum analyzer of claim 5, wherein output of the two ADCs is quantized into three levels (-1, 0, +1). 11. The digital spectrum analyzer of claim 5, wherein a sampling frequency fs of the ADCs corresponds to a bandwidth B;n of the complex input signal (fs >Bjn). 12. The digital spectrum analyzer of claim 5, wherein the four multiplexers (MUX 1, MUX 2, MUX 3, MUX 4) operate as follows: 13. The digital spectrum analyzer of claim 5, whereinin the constant A: is in a range (0.5 - 5.0). A digital sweep type spectrum analyzer with up/down frequency conversion, the analyzer comprising: (i) two analog-to-digital converters (ADCs), whose inputs are analog quadrature components of a complex input signal and that produce a digital quadrature output (IADC QADC) ; (ii) a quadrature local oscillator (LO), which generates a digital complex sinusoid with quadrature components (lLO, QLO); (iii) a MUX block that generates real numbers based on (IADC, QADC) and (lL0, QLO) as follows: R// _ ^ADC * jIO ^ QQ = Q^C * Qi05 (iv) a first block of registers connected to outputs of the MUX block; (v) a block of adders whose inputs are connected to the first block of registers, wherein the block of adders produces a complex up-converted signal (lu, Qu) and a complex down-converted signal (Γ°, Q°) as follows: Qu= RrQ + RQI, lD = Ru + R∞ QD = -RIQ + RQJ; (vi) a second block of registers; (vii) a block of accumulators receiving inputs from the block of adders, wherein each accumulator accumulates M samples respectively of lu, Qu, \D, QD at a frequency of the LO, the resulting values are divided by 2K (2K < M) by shifting by K bits toward a least significant bit (LSB) and resulting values are written into the second block of registers, and then the accumulators are reset to zero and the steps are repeated for the next frequency of the LO; (viii) a level detector that estimates a spectrum of the complex input signal at frequency T U ,K U, K T D,K ND, K points -iio and +fLO , using the values , Jm , l m , m ; (ix) a controller that sends a sweep signal to the LO to change a frequency of the LO in a range (0, +fs/2), sequentially setting N » 1 of frequency values f« = fs*(2n- l)/4N, where n = I, 2, . . . , N, wherein each accumulator accumulates M samples at each in: M = floor [&*N] , where k is a constant, and wherein the spectrum analyzer forms spectrum points in a range (-fs/2, +fs/2). |
FREQUENCY CONVERSION
BACKGROUND OF THE INVENTION
[0001] Communication systems and satellite navigation systems often need spectrum analyzers.
Swept spectrum analyzers have been in use for many years, because they are less complex compared to FFT (Fast Fourier Transform) spectrum analyzers. Some analyzers have both analog and digital signal processing means.
[0002] US Patent No. 6700366, entitled "Very fast swept spectrum analyzer", describes a system in which the signal spectrum is formed by mixing a ramping local oscillator and an input signal to generate an IF signal, the resulting IF (Intermediate Frequency) signal having a phase change with respect to the input signal, the phase change including a quadratic portion. For analysis, the IF signal is processed such that the quadratic component of the phase change is removed. IF processing circuitry includes a resolution bandwidth filter and an analog-to-digital converter.
[0003] US Patent No. 6275020, entitled "Frequency analysis method and sweep type spectrum analyzer", describes a sweep type spectrum analyzer, where analyzing the frequency components included in the signal to be measured are based on the correspondence relation between a power of each of the intermediate frequency signals and a frequency of the main swept frequency signal, an inverse swept signal is multiplied by each intermediate frequency signal, the inverse swept signal being frequency-swept in the reverse direction to the frequency transition direction of the associated intermediate frequency signal, and a constant frequency component obtained from the results of the multiplications is extracted as a frequency spectrum.
[0004] US Patent No. 4771232, entitled "Non-interruptive spectrum analyzer for digital modems", describes a system that provides extracting spectral data from the actual received modem signal, where multipliers are used in the digital frequency converter.
[0005] However, devices with analog components are not always stable and provide less accuracy compared to fully digital solutions. The present invention is intended as a simple system for full digital sweep type spectrum analysis with up/down frequency conversion, where analysis time is less than half compared to sweep type spectrum analyzers with only down frequency conversion. The system provides measurements of frequency spectrum of an input complex analog baseband signal without use of multiplication operations for frequency conversion. SUMMARY OF THE INVENTION
[0006] The present invention is intended for creating a system for digital sweep type spectrum analysis with up/down frequency conversion. The system provides measurements of frequency spectrum of complex analog baseband input signal, where analysis time is less than half compared to sweep type spectrum analyzers with only down frequency conversion. First, a complex analog input signal is quantized into three levels (- 1 ; 0; + 1 ) with a sampling frequency fs agreed with the bandwidth of the input signal. Then, a block of four 9-to-l multiplexers, a first block of registers and a block of adders perform operations that are equivalent to mathematical complex multiplication of the quadrature components of the digitized input signal and the quadrature components of the local oscillator, but without using real multiplications.
[0007] Two complex signals with up-shifted and down-shifted spectrum are formed at the output of the block of adders. Further, the quadrature components of these complex signals are inputted to the block of accumulators with reset, which act as low-pass filters (LPF) and accumulate several samples at a constant frequency ho of the local oscillator. Levels of two accumulated complex samples are estimates of the input signal spectrum in two frequency points +f £0 and -i L0 . A sweep controller changes the frequency stepwise of the local frequency-tunable oscillator from zero to fs/2 by steps that compared to the LPF bandwidth, where estimates of the input signal spectrum generated sequentially in the range from -fs/2 to +fs/2. These spectral estimates are analyzed and stored using a data analysis and storing logic, where they are periodically updated and may be used, e.g., for displaying.
[0008] Additional features and advantages of the invention will be set forth in the description that follows, and in part will be apparent from the description, or may be learned by practice of the invention. The advantages of the invention will be realized and attained by the structure particularly pointed out in the written description and claims hereof as well as the appended drawings.
[0009] It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are intended to provide further explanation of the invention as claimed. BRIEF DESCRIPTION OF THE ATTACHED FIGURES
[0010] The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification and together with the description serve to explain the principles of the invention.
[0011] In the drawings:
[0012] FIG. 1 is a functional block diagram for an embodiment of a digital sweep type spectrum analyzer with up/down frequency conversion.
[0013] FIG. 2 is a functional diagram for a MUX block.
[0014] FIG. 3a is an example of the spectrum of a digitized complex input signal.
[0015] FIG. 3b is an example of the spectrum of the signal of a quadrature local oscillator.
[0016] FIG. 3c is a spectrum of an up converted digitized complex input signal of FIG. 3a.
[0017] FIG. 3d is a spectrum of a down converted digitized complex input signal of FIG. 3a.
[0018] FIG. 4 is an example of the flowchart of a primitive algorithm of estimating for the case of only sector with borders (0, π/4).
[0019] FIG. 5 is an example of the flowchart of the estimation algorithm when octant (0, π/4) is divided into two sectors.
[0020] FIG. 6 is an example of the flowchart of the estimation algorithm when octant (0, π/4) is divided into three sectors.
[0021] FIG. 7 is an example of the flowchart of the estimation algorithm when octant (0, π/4) is divided into four sectors.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0022] Reference will now be made in detail to the preferred embodiments of the present invention, examples of which are illustrated in the accompanying drawings.
[0023] FIG. 1 shows a functional block diagram for a digital sweep type spectrum analyzer with up/down frequency conversion, which provides a frequency spectrum of an complex analog baseband input signal.
[0024] The analog quadrature components I' if) and QT if) of the complex input signal are pre-quantized with a sampling frequency fs by two analog-to-digital converters (ADC) 10, 11 into three levels (-1 ; 0; +1):
Qi DC = - if Q in (n*O<-u L -
Q; dc =0, [{ -U L ≤Q in (n*T s )≤+U L ;
Q DC = + x Q n (n*T s )>+U L ;
here
U i is the preset level;
T s —1/ f s is the sampling period,
n = 0, 1, 2,...;
[0025] where sampling frequency f s corresponds to the bandwidth Bj„ of the input signal (f s >Bj n ).
An example of the spectrum of a digitized input signal is shown in FIG.3a.
[0026] An example of the signal spectrum of a quadrature local oscillator (LO) 12 with the quadrature components (l L0 , Q LO ) is shown in FIG.3b. A complex up-converted signal with quadrature components (l u , Q u ) and a complex down-converted signal with quadrature components (1°, Q^) can be obtained through complex multiplication of the quadrature components (l ADC , Q ADC ) and the quadrature components (l LO , Q L0 ) of LO by formulas:
[0027] The implementation of equations (1) and (2) would require four real multiplications:
nil _ j ADC jLQO
n n n ·, (3) (4)
I' o QQ n lQ nQI
[0028] In the present invention, the values ^ , K N , K N , Λ η are produced without using multiplication by a MUX block 13 comprising four 9-to-l multiplexers operating in accordance with the combined logic table
[0029] An example of a logic diagram for MUX block is shown in FIG. 2.
[0030] Zero number is supplied to data inputs 1 , 2, 3 of the multiplexer 1 , to the data inputs 1 , 4, 5 of the multiplexer 2, to data inputs 1 , 2, 3 of the multiplexer 3 and to data inputs 1 , 4, 5 of the multiplexer 4.
[0031] Component 1 L0 is supplied to data inputs 4, 6, 8 of the multiplexer 1 and to the data inputs
2, 6, 9 of the multiplexer 4.
[0032] Component I L0 with the opposite sign is supplied to data inputs 5, 7, 9 of the multiplexer
1 and to the data inputs 3 , 7, 8 of the multiplexer 4. [0033] Component Q is supplied to data inputs 2, 6, 9 of the multiplexer 2 and to the data inputs 4, 6, 8 of the multiplexer 3.
[0034] Component Q LO with the opposite sign is supplied to data inputs 3, 7, 8 of the multiplexer
2 and to the data inputs 5, 7, 9 of the multiplexer 3.
[0035] The real numbers are written into the first block of registers 14, comprising four registers.
[0036] A block of adders 15 produces a complex up-converted signal with quadrature components (l n u , Q n u ) and a complex down-converted signal with quadrature components (I n D , Q n D ) by the following formulas:
= R" - R? e
(7)
[0037] A spectrum of the up-converted signal with the quadrature components (7) is shown in
FIG. 3c and a spectrum of the down converted signal with the quadrature components (8) is shown in FIG. 3d. Thus, a MUX block 13, a local quadrature oscillator 12, a first block of registers and a block of adders perform the role of a dual up/down frequency converter.
[0038] The complex samples (7) and (8) are inputted to a block of accumulators 16 comprising four accumulators with reset, which accumulates M samples (M » 1) at a constant frequency of the local oscillator:
jU,A nU,A T D,A riD,A .. . . „
[0039] The resultant values i m , m , 1 m , { m are divided by a coefficient 2 A (2 < M) by shifting in K bits toward LSBs and are written into four registers of the second block of registers 17, i.e.:
jU.K = I U,A /2 K
m 1
Q m uK =Q m uA l2 K
η υκ =Ο υ ' Α /2 Α
[0040] Then all four accumulators are reset to zero; wherein the secondary sampling frequency becomes equal to F M = fs/M (down sampling by an integer factor, M).
[0041] A level detector 18 produces an estimate of vector length with complex quadrature samples ) respectively of the second block of registers using an algorithm as described below. The level ft^ ls an estimate of the input signal spectrum at frequency point -fu) and level is an estimate of the input signal spectrum at jrU D
frequency point +fw- Exact values of * m , ' m can be calculated by formulas based on the Pythagorean th
[0042] Calculations based on the Pythagorean theorem require a lot of computationally intensive operations. However, an estimate of the vector length with acceptable accuracy can be obtained using relatively simple algorithms. In this case, the level detector produces a rough estimate of vector length for each input vector using the following successive steps:
a) forming auxiliary vector with orthogonal components (X, Y), where X=max (\x\, \y\), Y=min (|J |, y|), x, y - orthogonal components of the input vector V; where the auxiliary vector located in the (0, π/4) octant;
b) finding in the (0, π/4) octant a preset sector which contains the auxiliary vector; c) selecting a linear equation corresponding to the sector found;
d) solving the linear equation with preset coefficients.
[0043] This approach is illustrated using an example of preselected octant with borders (0, π/4).
The auxiliary vector with orthogonal components (X, Y), where X=max (\x\, \y{), Y=min (|jc|, \y\), x, y are orthogonal components of input vector, has the same length as the input vector and is located in octant (0, π/4).
[0044] FIG. 4 shows an example of the flowchart of a simple algorithm of estimating for the case of only sector (S=l) with borders (0, π/4). A Ve estimate of V vector length is:
[0045] here Cj and Di are the preset constant numbers.
[0046] For example, when Ci and Di are preset
C, = 21, Di=64,
[0047] the Ve estimate has an maximum error ε max = 100*|V - Ve|/V = 5.68%.
[0048] FIG. 5 shows an example of the flowchart of the estimation algorithm when octant (0,
7t/4) is divided into two (S=2) sectors:
θι with borders (0, arctan (B1/A1)) and
θ 2 with borders (arctan (B1/A1), π/4);
[0049] here Ai and Bi are the preset constant numbers.
[0050] And Ve estimate of V vector length is:
Ve = X + d*Y/D, for the Θ, sector, and
Ve = X + C 2 *Y/D 2 for the θ 2 sector;
[0051] here C \ , Di and C 2 , D 2 are the preset constant numbers.
[0052] For example, if the following parameters are preset
Ai = 93, Bi=64,
Q = 1, Di=4, C 2 = 47, D 2 = 128,
[0053] the Ve estimate has maximum error ε max = 3.45%.
[0054] FIG. 6 shows an example of the flowchart of the estimation algorithm when octant (0, π/4) is divided into three (S=3) sectors:
Θ) with borders (0, arct
θ 2 with borders (arctan tan (B2/A2)) and
θ 3 with borders (arctan (B 2 /A 2 ), π/4);
[0055] here Ai, Bi and A 2 , B 2 are the preset constant numbers.
[0056] and Ve estimate of V vector length is:
Ve = X + Ci*Y/D, for the Θ, sector,
Ve = X + C 2 *Y/D 2 for the θ 2 sector and
Ve = X + C 3 *Y/D 3 for the θ 3 sector;
[0057] here Ci, D ls C2, D 2 and C3, D3 are the preset constant numbers.
[0058] For example, if the following parameters are preset
A 2 = 81, B 2 =64,
C, = 7, D,=32,
C 2 = 5, D 2 =16,
C 3 = 49, D 3 =128,
[0059] the Ve estimate has maximum error ε max = 2.37%.
[0060] FIG. 7 shows an example of the flowchart of the estimation algorithm when octant (0, π/4) is divided into four (S=4) sectors:
Θ] with borders (0, arctan (B1/A1)),
θ 2 with borders (arctan (Bi/Aj), arctan (B 2 /A 2 )),
θ 3 with borders (arctan (B 2 /A 2 ), arctan (B 3 /A 3 )) and
θ 4 with borders (arctan (B 3 /A 3 ), π/4);
[0061] here Ai, Bi, A 2 , B 2 and A 3 , B 3 are the preset constant numbers, and Ve estimate of V vector length is:
Ve = X + C| *Y/D, for the Θ, sector,
Ve = X + C 2 *Y/D 2 for the θ 2 sector, Ve = X + C 3 *Y/D 3 for the θ 3 sector, and
Ve = X + C 4 *Y/D 4 for the G 4 sector;
[0062] here Ci , Di , C 2 , D2 , C3 , D3 and C 4 , D 4 are the preset constant numbers.
[0063] For example, if the following parameters are preset (the constants were chosen empirically):
Ai = 257, Bi=128,
A 2 = 95, B 2 =64,
A 3 = 19, B 3 =16,
C 2 = 35, D 2 =128,
C 3 = 43, D 3 =128,
C 4 = 25, D 4 =64,
[0064] the Ve estimate has maximum error ε ma x = 1.89%.
[0065] The higher the S value, the greater the potential accuracy of estimate of vector length.
[0066] All dividers are multiple of two in the above examples; therefore, the division operation can be performed with help of the shift operation.
[0067] A data analysis and collection logic 19 that receives spectrum data from the level detector output executes an analysis of data and sequentially stores them within one scanning period after which the spectrum data is updated, the spectral data may be used, e.g., for a displaying.
[0068] A sweep and data collection controller 20 sends a sweep signal to the frequency control input of LO to control the LO by stepwise changing of frequency in the range (0, fs/2), sequentially setting N » 1 of equidistant frequency values in = fs*(2«-l)/4N, where « = 1, 2, . , N; wherein each accumulator of the block of accumulators accumulates M samples at each frequency in: M (The floor function is also called the "greatest integer" or "entier" function, see, for example, http:**en.wikipedia.org/wiki/Floor_and ceiling_functions), where constant k is selected from the range (0.5, . . . , 5.0); wherein the spectrum analyzer forms 2N of spectrum points in the range (-fs/2, - fs/2) during each sweep period. The sweep controller synchronizes the spectrum analyzer sweep to the recirculating memory of a data analysis and storing logic.
[0069] Having thus described the invention, it should be apparent to those skilled in the art that certain advantages of the described apparatus have been achieved. it should also be appreciated that various modifications, adaptations, and alternative embodiments thereof may be made within the scope and spirit of the present invention. The invention is further defined by the following claims.