ence of artifacts is eliminated. Fig.7 demonstrates the
time-frequency representation obtained by the algo-
rithm based on the expression (28) in the case, where
the number of analysis frequencies are varied accord-
ing to the sampling density. The chirp can be tracked
without any presence of artifacts.
7 CONCLUSION
The processing of non-stationary signal using level-
crossing sampling approach has been investigated.
On the one hand, such a sampling strategy provides
several interesting properties - signal to quantization
noise ratio does not depend on the number of quan-
tization bits, local sampling density reﬂects the in-
stantaneous bandwidth of signal, etc. On the other
hand, the captured samples are placed non-uniformly
and that requires rethinking of the processing method-
ology. The classical approaches of time-frequency
analysis have been discussed. Time-frequency repre-
sentations have been obtained using general forms of
them, which are suitable also for processing of non-
uniformly sampled signals. The simulation shows
that the main drawback of STFT is the appearance of
spurious components, while wavelet transform gives
low spectral resolution at high frequencies and low
temporal resolution at low frequencies.
Several enhancements have been proposed, which
are based on the idea of minimizing the error be-
tween the original signal and that reconstructed by
the Fourier series, not only at sampling time instants,
but also between them with the same accuracy. The
problem lies in the fact that the original signal values
are known only at sampling instants. One solution is
based on the consideration, that the continuous time
signal is constructed by interpolation of known sig-
nal samples. The expressions for zero-order and ﬁrst
order polynomial interpolation as well as for band-
limited interpolation with sinc functions have been
established. The other approach is to interpolate the
error samples in the same manner.
Simulation results show the improvement of TFRs
if enhanced algorithms are used instead of classical
ones. Additional beneﬁts can be gained if the band-
width of analysis is varied along the time axis accord-
ing to changes in local sampling density: the artifacts
are removed, the complexity of calculations can be
decreased. The common drawback of STFT based
methods is restrictions on the resolution. Extension of
the windows w(t) length improves the frequency res-
olution but at the same time degrades the temporal se-
lectivity. To overcome this rule, the signal-dependent
transformation described in (Greitans, 2005) can be
used. Due to the limited size of the paper, this method
is not discussed above, however the TFR obtained by
Figure 8: TFR of test-signal if signal-dependent transfor-
mation is used.
signal-dependent algorithm is shown in the Fig.8 for
the illustration. The increased resolution is achieved
by adapting the transformation functions to the local
spectral characteristics of the signal. As it is being
done in an iterative way, the mathematical complex-
ity is higher than for STFT based algorithms.
The proposed approach of processing non-
stationary signals using level-crossing sampling is
attractive for clock-less designs, which are now re-
ceiving increasing interest. Their advantages can play
a signiﬁcant role in future electronics’ development.
REFERENCES
Akay, M., editor (1998). Time frequency and wavelets in
biomedical signal processing. IEEE Press.
Baraniuk, R. G. and Jones, D. L. (1993). A signal-
dependent time-frequency representation: Optimal
kernel design. IEEE Trans. Signal Proc., 41(4):1589–
1602.
Chui, C. K. (1992). Wavelet Analysis and its Applications.
Academic Press, Boston, MA.
Cohen, L. (1995). Time-frequency analysis. Prentice-Hall.
E. Allier, G. Sicard, L. F. and Renaudin, M. (2003). A new
class of asynchronous a/d converters based on time
quantization. In Proc. of International Symposium on
Asynchronous Circuits and Systems ASYNC’03, pages
196–205, Vancouver, Canada.
Ellis, P. H. (1959). Extension of phase plane analysis to
quantized systems. IRE Transactions on Automatic
Control, AC(4):43–59.
Gabor, D. (1946). Theory of communication. Journal of
the IEE, 93(3):429–457.
Greitans, M. (2005). Spectral analysis based on signal de-
pendent transformation. In Proc. of the International
Workshop on Spectral methods and multirate signal
processing, pages 179–184, Riga, Latvia.
Hauck, S. (1995). Asynchronous design methodologies: An
overview. Proc. of the IEEE, 83(1):69–93.
Mark, J. W. and Todd, T. D. (1981). A nonuniform sam-
pling approach to data compression. IEEE Trans. on
Comm., 29(1):24–32.
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