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Patent US4797923 - Super resolving partial wave analyzer-transceiver - Google PatentsSearch Images Maps Play YouTube News Gmail Drive More »Sign inAdvanced Patent SearchPatentsMethods and electronic apparatus for recovering relatively high-resolution information from a partial wave representation of an information signal. According to the invention, there is provided a method of operating on a given partial wave function, or of producing a partial wave representation of an...http://www.google.com/patents/US4797923?utm_source=gb-gplus-sharePatent US4797923 - Super resolving partial wave analyzer-transceiverAdvanced Patent SearchPublication numberUS4797923 APublication typeGrantApplication numberUS 06/802,980Publication dateJan 10, 1989Filing dateNov 29, 1985Priority dateNov 29, 1985Fee statusLapsedPublication number06802980, 802980, US 4797923 A, US 4797923A, US-A-4797923, US4797923 A, US4797923AInventorsWilliam L. ClarkeOriginal AssigneeClarke William LExport CitationBiBTeX, EndNote, RefManPatent Citations (3), Non-Patent Citations (6), Referenced by (27), Classifications (13), Legal Events (2) External Links: USPTO, USPTO Assignment, EspacenetSuper resolving partial wave analyzer-transceiverUS 4797923 AAbstract Methods and electronic apparatus for recovering relatively high-resolution information from a partial wave representation of an information signal. According to the invention, there is provided a method of operating on a given partial wave function, or of producing a partial wave representation of an information signal as part of the method. Portions of the partial wave representation of the information signal are synchronously selected as by signal sampling under control of an interpolated high frequency clock. An inverse partial wave representation of the selected portions of the transformed signal is then performed, and the inverse partial wave representation is linearly deconvolved to produce a high resolution equivalent of high-resolution information signal. The invention may include an analytic converter for removing the effects of any dispersion of the partial wave representation. While the concepts involved are universally adaptable to signal processing systems, preferred embodiments of the invention as applied to a modem network and to an interferometer are described.
I claim: 1. In a signal processing system having a system input, a signal transmission channel, and a system output, an improved apparatus for receiving a high resolution signal at the system input and recovering a corresponding high-resolution information signal at the system output, said apparatus comprising:means for producing a summation of wave functions representing said input signal; means for truncating said summation of wave functions to produce a corresponding summation of partial wave functions; means for transmitting the truncated summation of partial wave functions through the signal transmission channel; means for detecting the transmitted truncated summation of partial wave functions; means responsive to said detecting means for producing a partial wave spectrum from the detected summation of partial wave functions; and means for deconvolving said partial-wave spectrum to the values of said system input signal. 2. The apparatus as claimed in claim 1, wherein;said truncated summation of partial wave functions comprises real and imaginary parts; and only said real part is transmitted through the signal transmission channel of said system. 3. The apparatus as claimed in claim 1, wherein said summation of wave functions representing said input signal exists over a first interval, and said truncated summation of partial wave functions exists over a second interval, said second interval being M times shorter than said first interval, where M is the data compression ratio of the system.
Although adequate for many applications, this data rate capability is unacceptably low for applications such as digitized voice, facsimile, and other high density data transmission applications. For example, for a 1k about seven minutes at 2400 b/s. Almost all communications systems using analog channels would greatly benefit by the use of a modem able to transmit and receive at higher bit rates over existing channels. The super-resolving Z-transform signal processing system of the present invention satisfies this need.
SUMMARY OF THE INVENTION The present invention satisfies the needs set forth above in connection with the two examples to be described hereinafter in greater detail. Furthermore, the invention overcomes all of the disadvantages also described above with reference to the two implementation examples (modem and spectrometer).
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS The invention will now be described in more detail, first in rather general terms describing a trans-receive system application and an analyzer application, and then more specifically to the implementation of the invention in a modem network (trans-receive system) and then as applied to the processing of signal information developed in a Michelson interferometer spectrometer (analyzer).
FIG. 2A breaks the trans-receive configuration into smaller functional blocks than that of FIG. 1A. In FIG. 2A, the channel clock generator 9 is shown to have a reference frequency input 59 and a sync input 33. The reference frequency input 59 is preferably from a crystal controlled oscillator output and may be multiplied within clock generator 9 to any desirable multiple frequency of that crystal oscillator source (not shown). The sync input on line 33 is of a relatively lower frequency and causes the output from clock generator 9 to shift its frequency periodically. In this way, the information transmitted down transmission channel 5 will carry the sync information as occasional pulsed frequency shifts, and this sync information will be filtered out from the received signal and used to synchronize the detector with the wave packets at the receiving end. The sync signal also provides the timing to define frames of Z-transformed input (b.sub.k) resulting in the transmission of wave packets along the transmission channel 5.
System input is shown here as b.sub.k on line 3 representing a bit stream input to partial wave generator 1. Frames of the system input are transformed by partial wave generator 1 to wave packets and are transmitted along transmission channel 5 as sampled partial wave functions. The wave packets arrive at the receiving end at both sampler 35 and phase locked loop 11. The function of the phase locked loop 11 is to produce, from the transmitted wave packet information, a sync signal and a fast clock having a relationship to the transmitted information the same as did the sync and fast clock at the transmitting end of the transmission channel 5. The fast clock is used in sampler 35 which sends the sampled information to partial wave detector and deconvolver 39, 41, while the sync information from phase locked loop 11 is used by the partial wave detector and deconvolver 39, 41 to identify the separate wave packets for processing. Output 13 is then the bit stream reconstructed to match that of the system input on line 3.
The channel clock generator 9 is comprised of a frequency multiplying VCO (voltage controlled oscillator ) 8 having a reference frequency f.sub.r and a sync input from sync generator 33 as previously described. The output of VCO is a high speed clock herein referred to as a "fast clock". The fast clock is divided by a fixed number M in divider 29 the output of which is added to the information to be transmitted down the transmission channel 5 and will aid in developing the fast clock at the receiving end.
PREFERRED EMBODIMENT OF A SUPER-RESOLVING DIGITAL MODEM NETWORK The blocks shown in dashed lines in FIG. 6 correspond to the solid blocks in FIG. 5, previously described. The function and/or breakdown of each dashed block in FIG. 6 will now be described with reference to the series of drawings of FIGS. 7-12.
A binary bit stream of data is inputted to the system on line 3 and identified as the system input signal b.sub.k.
The Z-transformer 23 is a known functional block which receives the output from a sync generator 33 as a synchronizing signal. Accordingly, the output of Z-transformer 23 is a series of Z-transforms each transform being confined within a time frame t.sub.f (FIG. 7B) determined by the pulse recurrence frequency of the sync generator 33. Z-transformer 23 outputs a real and imaginary part, but only the real part is necessary to be transmitted, as will be explained later in the discussion of Hilbert transform pairs (FIG. 25A). The real part of the transformed input signal is multiplied by the truncating function e(t) from block 25 at the multiplier 61.
The VCO 8 is kept at a constant frequency by the supply of a highly accurate frequency f.sub.r from stabilized crystal controlled oscillator 59. In effect, VCO 8 is a frequency multiplying voltage controlled oscillator which outputs the fast clock necessary for high speed sampling in sampler 31, but is locked to the lower frequency (f.sub.r) highly stabilized frequency source. Being sensitive to voltage at its input for setting the frequency at its output; VCO 8 receives a voltage pulse from sync generator 33 at the sync rate, and this pulse is used by the system to cause the oscillator output from VCO 8 to temporarily shift its frequency for a short duration relative to the time period of the sync signal. This short burst of altered frequency from VCO 8 is used at the receiver to recreate the sync pulse timing necessary for the inverse Z-transformation at the partial wave detector 7. The divide-by M circuit 27 introduces a submultiple of the fast clock into the data stream from D/A converter 63 in order to multiplex a reference clock on the transmission channel 5. This reference clock will be used at the receiving end, in an interpolated form, as the sampling clock for extracting information from the sampled data at precisely the right frequency and phase. In the modem arrangement exemplified here, the channel clock is multiplexed with the channel data. In some applications, it may be advantageous to transmit the clock and data over different signal lines or channels, and such an arrangement is within the scope of the applied invention.
In operation, the phase locked loop 11 operates in the following manner. VCO 85 is designed to operate within a narrow range of the predetermined frequency of the fast clock used at the encoding (transmitting) end of the transmission channel 5 as determined by a reference frequency f.sub.rl oscillator 87. Its output is divided by M and sent to multiplier 93. Multiplier 93 is, in essence, a frequency comparator which compares the VCO 85 output divided by M with the extracted reference clock in the transmission channel 5. Depending upon whether the channel clock is lower or higher than the divide-by M output, a plus or minus voltage is generated by amplifer/low pass filter 95 which forces, via VCO 85, the divided-by M circuit 91 output to track the received channel clock extracted from line 5. When the channel clock frequency is shifted discontinuously at sync times, a pulse is developed at the output of filter 95 which serves to synchronize the inverse Z-transformer 39. The result of the phase lock loop operation just described not only produces the fast clock for extracting the data in sampler 35 in synchronous relationship to the transmitted data, but the sync signal produced on line 89 for phase locking VCO 85 is the same sync signal necessary for the inverse Z-transformer 39 to place the input data it sees in the center of its memory device. The inverse Z-transformer 39, like Z-transformer 23, is an off-the-shelf component and need not be described herein in detail.
The inverse Z-transformed signal from transformer 39 remains in the truncated form as it is routed to the linear deconvolver 41 which is designed to remove the truncation factor and restructure it to precisely the form b.sub.k which it has before processing by the partial wave generator 1 at the sending end. This is accomplished by selective multiplication of the inverse Z-transformed signal with deconvolution matrix 99. The zero crossing detector 98 operates in a conventional manner to produce the desired high resolution output b.sub.k.
The discrete Fourier transform of a real N/2-length sequence of pulses is shown in FIG. 7A. The pulses represent the real part of a complex N/2-length bit sequence b(f.sub.k), where a binary one is considered to be a pulse 101 of height +1, and a binary zero to be a pulse 103 of height -1. The count of bits for 0≦f≦f.sub.max /2 is the maximum number of bits per frame that can be transmitted within the available bandwidth f.sub.max of the transmission channel.
The transformed signal X(t) of FIG. 7B can be considered to be the Z-transform of the signal b(f.sub.k). ##EQU1## where Z=[e.sup.i2π/tf ].sup.t and φ.sub.k (t)=Z.sup.k.
The signal X(t) is then a sum of wave functions φ.sub.k (t)=[e.sup.i2π/tf ].sup.kt multiplied by a +1 or a -1 according to the value of the K.sup.th bit in the sequence. Because the sequence b(f.sub.k) is real and causal, i.e. b(f.sub.k)=0 for -f.sub.max /2≦f.sub.k &lt;0, the transformed signal's real and imaginary parts 109,115 form a Hilbert transform pair, and its real part 109 is purely even, and its imaginary part 115 purely odd. Accordingly, because the imaginary part 115 can be recovered from the real part at the receiver by a Hilbert transformer 90 degree phase shifter in analytic converter 37, only the real part 109 need be transmitted down the analog channel 5.
Similar arguments apply for a purely imaginary bit sequence, except that the real part of the transformed signal X(t) is purely odd, and the imaginary part purely even. By superposition, then, it is clear that using this form of Z-transform (e.g. Fourier transform) a total of N bits can be encoded into a wave packet tf in length, whose maximum wave frequency is f.sub.max /2. For example, assuming that N=256, and the maximum frequency which can be transmitted is 2500 Hz, then the period of the fundamental wave φ.sub.1 is (1/2500S) (N/2)=51.2 milliseconds. This can be seen to be the length of the time frame tf by noting that φ.sub.1 =[e.sup.i2π/tf ].sup.t =e.sup.i2π(1/tf)t, i.e. the fundamental wave has a freqeuncy 1/tf, or period tf.
The aforementioned wave functions φ.sub.k (t)=[e.sup.i2π/tf ].sup.kt are orthogonal over the interval -tf/2≦t≦tf/2, i.e., ##EQU2## where δ.sub.jk =1 if j=k, and
δ.sub.jk =0 elsewhere.
Because the wave functions φ.sub.k (t) are orthogonal, the bit sequence b.sub.k can be recovered easily at the receiver by using a known inverse discrete Fourier transform technique i.e. ##EQU4## where X(tn) is the sampled signal at the receiver and t.sub.n =nΔt, Δt=tf/N.
As shown, this decomposition of X(t) into orthogonal waves yields a bit rate about twice the maximum channel bandwidth. In the present invention, however, the N/2-length complex bit sequence is encoded into N non-orthogonal waves (partial waves), thereby achieving much higher bit rates. This is done by truncating the wave functions φ.sub.k (t).
With the present invention, one is able to encode and decode N bits in a time frame of only t.sub.x =tf/M, where M is the data compression ratio.
According to FIG. 9, we now have ##EQU5## where Z=[e.sup.i2π/tf As before (cf FIG. 7B), the real N/2-length bit sequence b.sub.k is Z-transformed and truncated into the signal X(t).sub.trunc (FIG. 10A) whose real part is shown. By similar arguments, we see that N bits can now be encoded into a time frame t.sub.x =tf/M which is M times shorter, and we are able to transmit data at M times the rate of the aforementioned Fourier transform modem. This requires, of course, the fast clock from VCO 8 of FIG. 6 to be M times faster than the transmitted reference channel clock.
At the receiver, however, the situation is more complicated because of the non-orthogonal nature of the functions. In this case the inverse Z-transform ##EQU6## where t.sub.n =nΔt, Δt=t.sub.x /N=tf/N low-resolution partial-wave spectrum whose elements are linear combinations of the M adjacent elements. This is due to the non-orthogonal nature of the partial wave functions ψ.sub.k (t). The situation resulting from the truncation of the wave functions ψ.sub.k (t) is shown in FIGS. 10A and 10B, where b(f)=.sup.b Hi(f) is the Fourier transform of X(t), and b(f)=b.sub.Lo (f) is the Fourier transform of the X(t).sub.trunc, and the latter is equivalent to convolving the high-resolution pulses b.sub.Hi (f) comprising the input bit sequence with the sync function (sin kf)/kf, resulting in the low resolution partial wave spectrum b.sub.Lo (f) (See FIG. 10B).
The basic problem may be stated: ##EQU7## When b.sub.Hi (F) is band-limited, i.e. has "a priori" known bounded support (see Prost and Goutte, supra) equation becomes a first kind Fredholm integral equation, and we must solve for b.sub.Hi (F), and R(f-F) is a non-causal impulse response function which has no inverse (i.e. it is a sync function or Gaussian).
The problem equation (1) can be solved exactly for the no-noise case by the method of Ville and Gerchberg (see Prost and Goutte, supra), and the discussion under the subtitle "N-Step Deconvolution Algorithm for Non-Causal Impulse Response Function" in the description of the Spectrometer embodiment of the invention, infra. This method involves a series of successive approximations and can be guaranteed to converge by a kernel-splitting technique. Repeated interations lead to an infinite series which can be summed, and result in a non-iterative solution. It can be proven that for the case of no noise, this series of approximations will always converge to the exact values of the function b.sub.HI (F) i.e. will retrieve exactly the high resolution bit spectrum from the transform of the truncated signal X.sup.(t) trunc .sup.when:
(a) N data points are taken (will converge in N steps), and
(b) R(f)&lt;2, always true for passive convolution.
PRACTICAL CONSIDERATIONS The signal X(t).sub.trunc contains the encoded N/2-length complex bit sequence, and is well-defined over the interval -t.sub.x /2≦t≦t.sub.x /2. However, discontinuities can exist at the frame boundaries (see FIG. 11A). For example, for M=1, t.sub.x =tf, and successive bit sequences b.sup.(i).sub.k whose second element b.sup.(i).sub.1 contains ones and zeros, the discontinuities can be seen clearly. The problem is solved by multiplying X(t).sub.trunc by an envelope function e(t) which is zero-valued at the frame boundaries, creating a wavepacket: ##EQU8## where: e(t)=0 at the frame boundaries,
S.sup.(i) (t) is the i.sup.th wave packet, and
b.sup.(i).sub.k is the i.sup.th bit sequence.
Another practical problem arises due to the effects of electronic dispersion in the analog channel, i.e. the effects of frequency-dependent phase shift factors. Because the bit information carried by each partial wave is determined at the receive by detecting a 180 degree phase shift (b.sub.k =1 or -1), any dispersion will reduce the effective deconvolver signal/noise ratio. Dispersion of the wave packets is eliminated by a unique arrangement referred to herein as the analytic signal converter (see FIG. 6), which is a digital signal processor for removing dispersion. The dispersed, sampled partial wave packet S(t) is first separated into even and odd parts (FIG. 6, Blocks 69 and 71, respectively). Then the odd part is 90 degree phase shifted by a Hilbert transformer 75 and added to the even part in adder 77, forming a real output R. The even part is likewise Hilbert transformed in block 73 and added to the odd part in adder 79, forming an imaginary output I. These real and imaginary outputs constitute an analytic signal, i.e. its Z -transform is causal. A real N/2-length bit sequence encoded into a partial-wave packet, and sent through a dispersive channel, will be converted by this means into an analytic signal whose deconvolved inverse Z-transform is real and causal and is precisely the received bit sequence b.sub.k. The wave packet need only be approximately centered in the inverse Z-transformer's memory, easily done by a simple analog circuit known to the skilled worker.
SPREAD SPECTRUM TECHNIQUES An alternate envelope function E(t) may also be employed to increase effective system jamming margin by spreading possible non-linear distortion over the maximum possible bandwidth, that is decreasing system error rate due to analog channel non-linearities. If E(t) is presumed to be a simple Gaussian function whose value is comparable to system noise level at the frame boundaries (of half-width HW), then its Fourier transforms will also be a Gaussian function of half-width 1/HW (see FIG. 11C). Therefore, the impulse response function R(f) in equation (1) is a Gaussian.
As shown by the above, the effects of dispersion are removed by the analytic signal converter. Using a spread function for the envelope function (FIG. 11C), the degradation of system performance due to non-linear distortion and channel noise can be minimized. For example, assuming the number of sample points/frame N=256, and a compression ratio M=16, and using a Gaussian envelope function e(t), the half-width R(f) of equation (1) is about f.sub.max /√N=f.sub.max /16.
e(t)=g(t)[e.sup.i2&#960;/N
where g(t.sub.n) is a Gaussian, and [e.sup.i2π/N is a chirp function of max frequency equal to channel bandwidth, then the bandwidth of the response function R(f) is equal to the full channel bandwidth HW distortion is spread over the entire channel bandwidth, so that any degradation of the deconvolver signal/noise ratio due to distortion will be minimized.
SUPER-RESOLVING Z-TRANSFORM SPECTROMETER EMBODIMENT The concepts of the present invention can be applied to systems other than a transmit-receive system, and a particularly natural application is found in the optical field where an interferogram is developed directly from an optical instrument, and no channel transmission line is involved. with reference to FIG. 12, a super-resolving Fourier transform spectrometer will now be described incorporating the advantages and concepts of the present invention.
ν is wavenumber in cm.sup.-1,
A practical Michelson interferometer includes a dispersive beamsplitter. Real-world beamsplitters made by coating KBr with germanium introduces an optical dispersion term, δ.sub.d, defined as optical phase retardation versus wavenumber. Equation (2) then becomes ##EQU10##
I(&#948;)=I(&#948;).sub.e +I(&#948;).sub.o, e&#8801;even, o&#8801;odd (4)
RESOLUTION OF THE FOURIER TRANSFORM SPECTROMETER In order to determine wavenumber resolution of a Fourier transform spectrometer, the usual approach is to assume a finite decomposition of the interferogram into orthogonal functions, i.e. (for the case of no dispersion): ##EQU11##
&#966;max=.+-.&#960; radians at &#948;=.+-.&#948;max/2.
For k=2, φmax=.+-.2.pi. radians, etc. The functions ψ.sub.k =cos [(2π/δ)δk] are orthogonal over the interval
where φ.sub.k =(2π/δmax)δ.sub.k for all values of δ over the range -δmax/2 to δmax/2. Therefore, ##EQU12## where δ.sub.ij is the discrete delta function, and
&#968;.sub.k =cos [(2&#960;/&#948;max)&#948;k].             (7)
Taking the Fourier transform of I(δ), for ψ.sub.k =e.sup.i2πδk, and defining Δν=1/δmax ##EQU13##
Therefore, S(ν) is a sum of sync functions of halfwidth=1/δmax, weighted by the S.sub.k 's and displaced by kΔν.
Accordingly, when digital resolution Δ.sub.d =νmax/N (where N is the number of data points, and equals optical resolution 1/δ.sub.max) we then sample at the frequency points where all the sync functions have zero-crossings except for the desired frequency component. Therefore, the functions are orthogonal. As δ.sub.max decreases, the spacing between zero-crossings increases so that the optical resolution is less than digital resolution, so the eigenfunctions become non-orthogonal. However, as long as they are linearly independent, optical resolution can be obtained comparable to the digital resolution by the present invention.
PRACTICAL CONSIDERATIONS Fourier transform spectrometers (FTS's) have intrinsic advantages and disadvantages.
(d) Disadvantages. The major disadvantage in FTS lies in the requirement that we must move the mirror so far (for Δν=cm.sup.-1 we have δmax=1/1 cm.sup.-1 =1 cm). In order to maintain coherence and a high signal/noise ratio, less than 2 arcseconds of alignment must be maintained as the mirror is moved over 0.5 to 1 cm. Complex electromechanical servo-mechanisms, as well as air bearings are required. These are delicate, expensive, hard to keep from tilting in use and in accurate alignment, and they introduce uncertainties into the spectral data. Other disadvantages include the complexity of the signal processing electronics, though increasing availability of inexpensive integrated circuit are bringing this cost down. If an entirely solid-state optical head were possible, as with the present invention, the optics could be easily replicated, leading to further cost savings.
SUPER-RESOLVING FOURIER TRANSFORM SPECTROMETER Recognizing these and other disadvantages of moving-mirror type Fourier transform spectrometers, the analyzer examples embodiment of the present invention concerns an improved Fourier spectrometer which requires the mirror to the moved only a very short distance, yet still retaining resolution and signal/ noise performance comparable to existing (Michelson) Fourier spectrometers.
THE SUPER-RESOLVING FTS OF FIG. 12 The basic goal here, then, is to contrive to calculate a high resolution spectrum but only require the mirror to move a small amount. This can be done and still keep the mirror flat very easily by an off-the-shelf piezoelectic translation stage 45. Super-resolution in electronic imaging systems has been studied extensively over the past ten years (see Prost and Guatto, supra), and much has been learned in this field. The superresolving FTS according to the present invention was conceived in full acknowledgement of the contribution and technical advance in the art by these procedures.
BASIC TECHNIQUE Referring to FIGS. 6 through 18, it can be seen that is we truncate the interferogram 189 at the dotted lines 194, 196, we are effectively simply decomposing the interferogram into a set of non-orthogonal functions, similar to the modem discussion of FIGS. 8 and 9. Truncating the interferogram is equivalent to moving the mirror only a small amount, i.e., taking a short scan. It will be shown that great advantage is attained by a significantly reduced mirror movement.
Truncating the interferogram 189 in this manner is equivalent to convolving the high resolution spectrum with a sync function δν�1/δ.sub.trunc wide, thereby smoothing the high resolution spectrum and resulting in the low resolution spectrum computed by simply transforming the truncated interferogram (see FIG. 19B(c)).
When S.sub.Hi (ν) is band-limited, i.e., has "a priori" known bounded support T, the equation becomes a first kind Fredholm integram equation, i.e., we must solve for S.sub.Hi (ν), and R(ν) is a non-casual impulse response function (also called a sync function and the instrument response function) which has no inverse. In formula (10), S.sub.Lo (ν) is effectively the convolution of S.sub.Hi (ν) and R(ν).
Equation (10) can be solved exactly for the no-noise case by the method of Ville and Gerchberg (see Prost and Goutte, supra), as will be more fully discussed lated in this description. This method involves a series of successive approximations and can be guaranteed to converge by a kernel-splitting technique, repeated iterations leading to an infinite series which can be summed, and results in a non-iterative solution. It can be proven that for the case of no noise, this series of approximations will always converge to the exact values of the function S.sub.Hi (ν), i.e., the solution will retrieve exactly the high resolution spectrum from the transform of the truncated interferogram when:
(b) R(ν)&lt;2, always true for passive convolution.
CHIRP-Z TRANSFORM We can consider that the sampled, truncated interferogram is the Z-transform of the low resolution spectrum. A chirp Z-transform is used to map the N points between 0→δtrunc/2 to the N points between 0→νmax/2, i.e.,
S.sub.Lo (&#957;.sub.n)=W.sup.n2/2 [(I(&#948;.sub.k)W.sup.k2/2)*W.sup.-k2/2 ]
W=[e.sup.-i2π/NM ]
I.sub.k =ΣS.sub.Lo (.sup.ν.sub.n)[e.sup.i2π/NM ].sup.nk
z=[e.sup.i2π/NM ].sup.k, and M=δmax/δ.sub.trunc is the truncation ratio.
Use of the Z-transform insures that the N statistically independent interferogram points are transformed into the N independent low resolution spectrum points, i.e., that S.sub.Lo (ν.sub.n) is oversampled. This is necessary to insure that S.sub.Lo (ν.sub.n) can properly be deconvolved in the presence of noise.
OPERATIONAL DESCRIPTION OF THE SUPER-RESOLVING SPECTROMETER OF FIG. 12 With the above as background, a detailed discussion of the separate major blocks of FIG. 12 will now be given
OPTICAL RETARDATION (a) As aforementioned, the present invention involves an improved version of a Michelson type Fourier transform spectrometer, and could employ any means of effecting the required optical retardation, including moving a mirror (FIG. 13) or a moving wedge (FIG. 14).
In the present invention, maximum phase retardation δ.sub.trunc is a factor of M smaller, so the required sample intervals Δδ are M times smaller, i.e., M clock pulses must be interpolated between the usual laser clocks (δmax/N=M.sub.trunc /N). This is accomplished by means of an optical phase interpolater as shown isolated from the rest of the apparatus in FIG. 21.
converts the purely even part into a purely odd part and converts the purely odd part into a purely even part using 90 or Hilbert transformers 169, 171 (HT), and
even part=[I(&#948;.sub.n)+I(&#948;.sub.N-n)]/2
odd part=[I(&#948;.sub.n)-I(&#948;.sub.N-n)]/2
where I(δ.sub.n)=I.sub.n are interferogram samples.
The even part 237 is phase shifted by 90 and added to the odd part 239 by adder 175, so forming the imaginary part 179 of the analytic signal. Similarly, the odd part 239 is phase shifted by 90 adder 173, so forming the real part 177 of the analytic signal.
&#968;.sub.k =[e.sup.i2&#960;/&#948;max ].sup.&#948;k
Alternatively, we may consider the detection process to be equivalent to computing a Z-transform, ##EQU16## where Z=[e.sup.i2π/N ].sup.k.
In the present invention, because only a truncated interferogram is generated, each of the eigen-functions of interest is also truncated, as can be seen by FIG. 18. Considering each of these truncated eigen-functions to be "partial waves", the present invention uses the process of partial wave coherent detection to compute an independent measurement of each partial wave. This can be thought of as a Z-transform ##EQU17## where Z=[e.sup.i2π/NM ].sup.k, and M is trunction ratio=δmax/δtrunc. S.sub.Lo (ν.sub.k) is the low resolution partial wave spectrum.
Each partial wave is linearly independent of all others, but not orthogonal, and each detected partial wave coefficient is dependent on about M others. Referring to FIG. 19B(c), the partial wave spectrum S.sub.Lo (ν) must be deconvolved in order to arrive at the high resolution spectrum S.sub.Hi (ν), FIG. 19B(a). That is, the linear dependencies between coefficients of the partial wave spectrum must be removed by the process of deconvolution.
Inputted to the chirp-z transformer 55 are a non-orthogonal set of basis functions (partial waves) which can be mathematically defined by the set (e.sup.i2π/NM).sup.nk where N is the desired digital resolution of the spectrometer (ν.sub.max /N)=Δν, M is the interpolation (truncation ratio) where n is any real number and k is a constant. The chirp-z transform (of the sampled interferogram) using this basis (set) is essentially just the outputs of a coherent detector which measures the strength of each partial wave of the truncated interferogram. These outputs are linearly dependent on each other because of the non-orthogonal nature of the basis functions. However, the chirp-z transform gives N statistically independent points each of which constitutes a linear combination of a spectral point and its adjacent spectral points.
The aforementioned linear deconvolution method takes time proportional to N.sup.2 to compute a high resolution spectrum. However, since the function to be deconvolved is a sync function (of the form sin KX/X), it can be factored into causal and anti-causal parts, both of which are shift-invariant. Therefore, the necessary deconvolution can be accomplished by a technique called "fast convolution" which takes time only proportional to N log.sub.2 N. This constitutes the preferred embodiment as will be seen in connection with the description of FIG. 24.
According to FIG. 12, the real output 181 of the Chirp-z transform 55 is a discrete real vector S.sub.Lo (ν.sub.K). In order to remove the aforementioned linear dependencies inherent in the partial-wave spectrum S.sub.Lo (ν.sub.K), the linear deconvolver section 57 operates on the spectrum S.sub.Lo (ν.sub.K) via a deconvolution matrix 185, by means of matrix-vector multiplication 187. This operation results in the high resolution spectrum S.sub.Hi (ν.sub.K)=S(ν.sub.K), i.e.,
S.sub.Hi (&#957;.sub.K)=LS.sub.Lo (&#957;.sub.K)
OPERATIONAL DESCRIPTION OF PREFERRED EMBODIMENT FIG. 24 The preferred embodiment of FIG. 24 is essentially identical to the embodiment of FIG. 12 except for the deconvolution means 57. The preferred embodiment of FIG. 12 employs a method called fast convolution in order to effect the deconvolution process in a time proportional to Nlog.sub.2 N, infra.
L=L.sup.+ L.sup.-
The diagonal element Toeplitz of these two matrices in the frequency domain correspond to vectors in the time (interferogram) domain.
The outputs from the analytic signal converter adders 173 and 175 are multiplied by the product of the Fourier transforms of the diagonal elements of L.sup.+ and L.sup.- (deconvolution vectors 223) and then connected to the Chirp-z transform section 55. Therefore, by the convolution theorem the output of the Chirp-z transform results in the deconvolved spectrum S.sub.Hi (ν.sub.K)=S(ν.sub.K).
The Chirp-z transform is accomplished by multiplying the complex product of the deconvolution vectors 223 and the outputs 173, 175 from the analytic signal converter 53 by the Chirp signal generator 57, then performing the FFT 225. The outputs of FFT 225 are then multiplied by the Chirp signal 229 by complex multiplier 227, and then connected to an inverse FFT section 231, whose output is multiplied by Chirp signal 235, resulting in the Chirp-z transformed, deconvolved analytic interferogram 21, i.e., the high resolution optical magnitude spectrum S.sub.Hi (ν.sub.K)=S(ν.sub.K).
The advantage of FIG. 24 over FIG. 12 may be seen by setting N=4096, so the fast FTS will enjoy a factor of N.sup.2 /Nlog.sub.2 N≅341 speed advantage.
N-STEP DECONVOLUTION ALGORITHM FOR NON-CAUSAL IMPULSE RESPONSE FUNCTION The N-steps method requires solving the integral equation (changing notation slightly) ##EQU18##
i.sup.n (&#957;)=0(&#957;)-[r(&#957;)-&#948;(&#957;)]* w(&#957;)i.sup.n-1 (&#957;)
i.sup.0 (&#957;)=0(&#957;)
i.sup.n (&#957;)=i.sub.0 (&#957;)-g(&#957;) * w(&#957;)i.sup.n-1 (&#957;)
i.sup.0 (&#957;)=i.sub.0 (&#957;)
g(&#957;)=d*.sup.-1 (&#957;) * k(&#957;)
d*.sup.-1 (ν) * d(ν)=δ(ν) and
i.sub.0 (ν)=d*.sup.-1 (ν) * 0(ν).
Therefore, the split kernel becomes ##EQU21## By substitution, ##EQU22## and the pattern becomes clear, i.e., ##EQU23## by forcing the diagonal elements to zero, since g.sub.ii =r(0) / d.sub.ii -1. Taking d.sub.ii =r(0), Then g.sub.ii =0, and
______________________________________G is lower triangular with diagonal zeroG.sup.2 is lower triangular with diagonal and sub-diagonal zero..G.sup.N all diagonals are zero______________________________________
Now define operator ##EQU24## and I=L I.sub.0
SOLUTION FOR NON-CAUSAL r(ν) A non-causal response consists of an anti-causal and causal part as seen in FIGS. 30 and 31. the Laplace transform of r(ν)→R(s), and by spectral factorization,
R(s)=R.sup.+ (s) R.sup.- (s)
R.sup.+ (s) is analytic in right half plane
R.sup.- (s) is analytic in left half plane
L[R.sup.+ (s)]=r.sup.+ (ν) causal part
L[R.sup.- (s)]=r.sup.- (ν) anti-causal part
so r(ν)=r.sup.+ (ν) * r.sup.- (ν) denotes inverse Laplance transform going over to the discrete case, then Toeplitz matrix
R=r.sub.ij
r.sub.ij =r[(i-j)]Δi=1 to N, and j=1 to N,
so R=R.sup.+ R.sup.- I
R.sup.+ lower triangle
R.sup.- upper triangle
&#966;=R.sup.+ R.sup.- I
R.sub.+ =&#966;
R.sup.- I=X
X=L(R.sup.+)&#966;
I=L(R.sup.-)X
So it is seen that L(R.sup.+) and L(R.sup.-) are composed of N submatrices apiece, equivalent to algorithm converging in N steps. Therefore 2N points must be sampled, requiring a double-sided transform, and for an analytic interferogram signal, yields a real, positive and causal restored spectrum, S.sub.Hi (ν.sub.n).
FAST SUPER-RESOLVING FTS Since the spectrum restoration is accomplished by applying matrix equation I.sub.0 =L I and ##EQU25## it can be done by faste* convolution in the interferogram domain. This allows us to accomplish all signal processing in approximately 4N log.sub.2 N time
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