Noiseless data

There are many reasons why deconvolution algorithms produce unsatisfactory results. In the deconvolution of actual spectral data, the presence of noise is usually the limiting factor. For the purpose of examining the deconvolution process, we begin with noiseless data, which, of course, can be realized only in a simulation process. When other aspects of deconvolution, such as errors in the system response function or errors in base-line removal, are examined, noiseless data are used. The presence of noise together with base-line or system transfer function errors will, of course, produce less valuable results. 

Fig. 1 Deconvolution of simulated noiseless data using the Jansson weighting scheme. Trace (a) is the original spectrum o x trace (b) the convolved spectrum i x). Traces (c) and (d) are the power and phase spectra of o(x), traces (e) and (f) the power and phase spectra of i(x), traces (g) and (h) the power and phase spectra of the error spectrum E(jc). Traces (i)-(m) are the deconvolution result, the power and phase spectra of the deconvolution result, and the power and phase spectra of the error spectrum, respectively, after 10 iterations with r(jjjax = 1.0. Traces (n)-(r) are the same results after 20 additional iterations with r ax= 2.0. Traces (s)-(w) are the same results after 20 additional iterations with r(3.5. Traces (x)-(bb) are the same results after 20 additional iterations with r( Jax= 5.0.

Figure 2a. Profiles of the power-spectra of the true brightness distribution, the noiseless blurred image, and the actual data (noisy and blurred image). Clearly the noise dominates after frequency 80 frequels.

For a noiseless signal consisting of a sum of decaying sinusoids described in Eq. (101), each measured (complex) signal value x can be written as a weighted sum of the M preceding data points as... 

Figure 4.1 Time and frequency domain data in signal processing in the noiseless case using the fast Fourier transform (FFT) and fast Pad6 transform (FPT). Top panel (i) the input FID (to avoid clutter, only the real part of the time signal is shown). Middle panel (ii) absorption total shape spectrum (FFT). Bottom panel (iii) absorption component (lower curves FPT) and total (upper curve FPT) shape spectra. Panels (ii) and (iii) are generated using both the real and imaginary parts of the FID.

Fig. 2. Reconstructed images between 30-80 keV of (a) flight data during the Cygnus X-1 observation, and (b) simulated data of a noiseless source at the Cygnus X-1 position and flux. Images are significance plots with contours at Icr levels (dashed contours are at la and 2a).

Namely, an encoder converts the bit stream from the source into another bit stream applied to the medium, with the conversion rule being called a code. (In the context of data compression, the use of a code has nothing to do with encryption it is not the intention of the encoder to hide the information (though the codes we shall discuss can be used for that purpose).) Correspondingly, a decoder converts the stored bit stream (xi, X2,...) into that presented to the sink. Ideally, of course, the total processing is such that the sink bit stream (vi, V2,...) is identical to that of the source. If the encoder is such that, for an appropriate decoder, this is indeed the case for every possible source sequence, then the code is called lossless or noiseless otherwise, it is lossy. 

Samples displaying more eomplex decay dynamics are far more challenging. Here the photons must be distributed over more than 256 temporal channels in order to ensure that the lifetime components are adequately represented." For example, resolving with 25% accuracy the components of a noiseless double exponential decay with T2/ti = 6 and equal amplitudes A =A2 requires 900 photons and corresponds to a 15 minute long data acquisition with a 100 kHz excitation source. In the presence of noise and for a different ratio of amplitudes, these numbers can be considerably higher. 

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