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Next: A Simple Example With Up: CHIRPLET TRANSFORM SUBSPACES Previous: CHIRPLET TRANSFORM SUBSPACES

The Frequency-Frequency (FF) Plane

We begin by discussing the CF plane, and then present an argument for re-parameterizing this plane in terms of two frequency indices, leading to what we will be calling ``frequency-frequency'' (FF) analysis.

Consider a two-dimensional slice through the five-dimensional CCT parameter space that we defined in (11):

S_c,f_c = C_0,f_c,0,c,0 g(t) \: \: s(t)

 

where s(t) is an arbitrary time series, and the two dimensions of the transform space are the slope of the frequency rise, c, and the center frequency tex2html_wrap_inline1395. This transform is knownmannVI91 as the ``bowtie (tex2html_wrap_inline1563) subspace'' since the CF plane of a chirp is a sharp peak surrouned by faint bowtie-shaped contours (Fig. 6). Computing the CF plane of a signal, s(t), is equivalent to correlating the signal with a family of chirps that are parameterized by chirprate, c and center frequency, tex2html_wrap_inline1395. Calculating the CF plane from a signal that contains pure tones results in peaks on the tex2html_wrap_inline1571 axis. Downchirps in the signal result in peaks to the left of this line, and upchirps result in peaks to the right.

  
Figure: FIGURE GOES SOMEWHERE IN THIS GENERAL VICINITY

For a discrete functiongif, we would have periodicity in the CF plane, and the Nyquist boundary is diamond (tex2html_wrap_inline1573) shaped. The Nyquist limit dictates that the chirps with the highest (lowest) c values begin with a fractional frequency of -1/2 (+1/2) and end with a frequency of +1/2 (-1/2). These chirps will both lie on the tex2html_wrap_inline1585 axis of the CF plane. Consider a chirp that begins with a frequency 1/4 and ends with a frequency of 3/4. It has the same chirprate: c = 3/4-1/4 = 1/2, but it will violate the Nyquist limit because part of the chirp exceeds the fractional frequency of 1/2, and will therefore give rise to aliasing.

Ideally we would like this transform to have nice rectangular boundaries for convenient viewing on a video display, so we overcome the Nyquist problem by tilting the parameter space tex2html_wrap_inline1595. The new chirplets are then given by:

C_0, (f_end+f_beg)/2, 0, (f_end-f_beg)/2, 0 g(t)
= g(t) e^j 2 (f_end-f_beg2t + f_end+f_beg2)t

where g denotes the mother chirplet. The change of coordinates from the CF plane to the FF plane is given by tex2html_wrap_inline1599 and tex2html_wrap_inline1601. When the analysis interval (window) is of finite duration, tex2html_wrap_inline1603 may be taken to be the instantaneous frequency of the chirp at the beginning of the analysis interval (time window) and tex2html_wrap_inline1605 the instantaneous frequency at the end of this interval. Since the new parameterization involves two frequency coordinates, we will refer to the resulting parameter space as the ``frequency-frequency'' (FF) plane. Figure 7

  
Figure: FIGURE GOES SOMEWHERE IN THIS GENERAL VICINITY

shows the FF plane computed from a harmonic oscillation.

The value of the function defined on the FF plane, evaluated at the origin gives a measure of how strong the chirp component from 0 to 0 (the DC component) is. The value at coordinates (0,1/2), for example, gives the strength of the component of a chirp going from a frequency of 0 to 1/2. Values of the FF plane in the upper left half (above and to the left of the diagonal tex2html_wrap_inline1607) correspond to upchirps; those to the lower right correspond to downchirps. The values of the FF plane along the diagonal line, tex2html_wrap_inline1607, define the Fourier transform of the original time-domain signal; the windowed version of the signal may be entirely re-constructed from only the diagonal of the complex-valued FF plane.


next up previous
Next: A Simple Example With Up: CHIRPLET TRANSFORM SUBSPACES Previous: CHIRPLET TRANSFORM SUBSPACES

Steve Mann
Thu Jan 8 19:50:27 EST 1998