Sep 9, 1995 - longing to Cohen's class is that due to their quadratic. (non-linear) nature ... tary signal dependent TFR introduced In[l]. The el- ementary TFR is ...
IEJCE TRANS. FUNDAMENTALS. VOL 0 8-A. NO . 9 SEPTEMBER 1995
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I PAPER
Special Section on Information Theory and Its Applications
Signal Dependent Time-Frequency and Time-Scale Signal Representat ions Designed Using the Radon Transform Branko RIST lct and Boua lem BOASH ASHt , Nonmembers
Time-frequency reprcsentations (TFRs) have been de\'elo~d a, lOols for analysis of non-,Ialiunary signals. Signal do:po:ndo:nt TFRs aro: known to po:rform well for a rnu of the AF: (b) ko:rnel de:;ig.ned using Ih.: ek menwry algorilh m; (c) resulting TFR.
for the automatic kernel design. A typical example of the Ro (J./!) function correspondlllg to a signal consisting of two cit irp signa Is wi th sl ightly di fferent freq uency rates, is given in Fig. I (a). The two peaks in Fig . I (a) correspond to the two signal components . The kernel is designed In the polar coordinates uSlllg the onedimensional function Ro (~J ). as:
(4)
g(±" ¢)
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Kernel Design
In this paper, lIlstead of using the RT in the timefrequency plane . we concentrate upon Its use in the ambiguity domain (0, T). The main benefit In lhls approach is that essentially two-dimensional problem s, such as the kernel design problem . can be solved in a single dimension. Recall thm lhe aim of lhe kernel design is to filter out the interference terms, without a!fecting the resolution of the auto-terms. Hence, the pass-band / stop-b;.md characteristic of the kernel y(O,r) should be close to unity in the rcgions of the (0 , T) plane where the signal auto-terms lie, and zero elsewhere. It is known that the interference terms are always located away from the origin of (8 , T) plane. while the auto-terms arc radially distributed in (e, T) and always traverse its origin [4J. Hence it was proposed [ I J to use the RT transform at distance 5 = 0 of the modulus of the AF,
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Ro(. » b n ,( ,p) ';; b
(6)
where r und lb are the polar coordinates and /1 IS a threshold which controls the amount or intcrlcrcnces suppression by the kernel. In practice. however. in order to reduce the sidelobe effects. the kernel edges are tapered. T he radial distance can be also taken into consideration ill Ihe kernel design, as described in [I J- The designed kernel and lls corresponding T F R for a twocomponent chirp signal are shown in Fig. I (b) and (c) respectively. 3.
Additional Requirements and the Customised Des ign
The TFR p{t, f) in (3 ) which correspond s to the kernel designed as described abo\'e is: real, because the kernel is real and g(O, T) = y ( - 0, -T ); time-shift inI'firianl, because y (B, T) is time'lIldependent: frequellcy shift invariant, beo.:ause g(O, T) is independent of frc-
quency. In addition , p(I,f) preserves the total signal energy, since .9 (0 ,0) = 1 and most Imponantly, p(t,1) suppresses the interference terms, since y(8, T) acts as a
IEICE TRANS . FUNDA"I ENl'ALS. VOL E78-A . .\'0. 9 SEPTEMBER I 'N;
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two-dimensional low-pass filter. In certain applications, ho\\'e\'er, the properties stated above may not be sufficient. An extensive list of desirablc properlies usually reqUired from a TFR IS given in [4]. The idea or customised signal -dependclll kernel design IS to incor porate additional requirements for a TFR, Into options which mayor may not be spcclfied, depcnding on the specific application. 3.1
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Proper Marginals
In a vicw of deSire to interpret a TFR as a distribution of signal energy over time and frcquency, it is usually requlrcd Ihat the TFR satisfies the time and frequency marginal distributions. Any T FR will have propcr marginals if its corrcspondlng kernel g((}, T) equals unity along the T and 8 axis [4 ], i.e. (VO)
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Fig. 2 Signal dependent TFR wilh propn margin
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to detect the prey regardless of its relati ve speed. bats emit Dopp!cr tolerant pulses_ A typical example of a Doppler toleram pulse is a hyperbolic FM signal T he duration of the O;1 t signal under anal ysis is 2.5 msec. and the sampling period IS 7 pscc. Figure 7 shows several TFRs of this signal. The spectrogram (window length 250psec). Ihe windowed WVD (window length 1. 7 msec) and the ChOl . Williams d istri buti on [ 14] (wi ndow-Iength 2;)() psec) are shown In Fig. 7 (a). (b) and (c ) respecti vely. The signal dependent TFR obtained hy the elementary algorithm and by the sequential a lgorithm (window length 890psec) arc displayed in Fig. 7 (d) and respectively. Each TFD is obtai ned after e:.;tensive parameter adjustments aimed at maximIsing their performance. By inspection of the TFOs shown 111 Fig_ 7 it ean be observed that the con ve ntional methods (such as the spectrogram, WVD. or the Choi · Wi1JJallls distribution) are inferior 10 the proposed signal-de pendell! TFRs. However. observe that the elementary algorithm tends 10 iinearise the time.frequency contems o f the len-hand side signal component. Hence. the TFR in Fig. 7 (e) IS preferred, though its resolution IS somewhat degraded compared to Fig. 7 (d).
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Kernel Design fo r T ime-Scale A nalysis
One of Ihe joint ("bi line;n") time-scale reprcsentalmns (TSRs) is the Marino\'ich-Altes (or Q) distribution [ 15].
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[ 16J
which represents a seale-inva nant counterpart o f the WYO. The Q distribution is related to Ihe WVO as fO[lows [ 16]:
(e) .'-ig. 7 TFR s of a bat signal: (a) spectrog ram: (b ) windowed WVD; (e) Choi ·Williams distribution: (d) sign~1 dependc nt TFR (block); (e) signal·dependent TFR (sequential)_
RISTlC and B(),\SHASH: SIGNAL DEf'fNf)ENT TI",IE -FREQUENCY AND TIME·SCALE REPRESeNTATIONS
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Usmg MoyaJ's formula for the WVD [-lJ. it was shown that the G LRT in (16) can be expressed using the RT of the WV D of the received signal I"n:
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Fig. 8
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[I ] II Ristie and B. BO
