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Deconvolution Techniques
Introduction
Deconvolution is usually defined as a mathematical operation which is the inverse of a convolution operation, i.e. the operation of solving for f (t) where y(t) = f (t) ∗ g(t) given y(t) and g(t) or only y(t) ; the latter is more complicated and is usually referred to as blind deconvolution [20]. Deconvolution is a very powerful signal processing technique and is widely used in many applications. In seismic processing, y(t) represents a series of acoustic reflections of an explosively generated signal due to rock inhomogeneities and g(t) models the earth medium [21]. In image processing, which is crucial to astronomy, medical images, etc, the same problem arises for two-dimensional signals in which the two-dimensional version of f (t) represents an image, while the two- dimensional version of g(t) models a distortion due to a poor lens, for example [22]. The deconvolution problem is mathematically classified as an ill-posed problem [23]. By Hadamard’s definition, a problem is well posed if a unique solution exists and the solution depends continuously on the data; otherwise, it is ill posed. Ill-posedness occurs in problems where the data or equations are inexact, i.e., they are only approximates. Consequently, the information represented by the data, or the equations are incomplete. Thus, there may not exist unique solutions to the deconvolution problem.
For this class of problems, more than one approximate solution may be achieved, some of which are “acceptable” estimates to the exact solution, and some of these acceptable solutions are just “better” estimates than others [23]. In selecting the “acceptable” estimate for an ill-posed problem, subjective judgment must be made [24]. This subjective judgment acts as a substitute to the problem’s missing information.
Frequency Domain Techniques
Inverse filtering
In the frequency domain, the most straight forward deconvolution technique is known as inverse filtering [25]. Time domain convolution transforms into multiplication in the frequency domain, and using X ( jω) , H ( jω) and Y ( jω) to denote the frequency domain forms of x(t) , h(t) and y(t) respectively, assuming no noise for now, Equation (3-1) Ideally, with the exact and complete knowledge of X ( jω) and Y ( jω) , deconvolution can be exactly performed to compute H ( jω) . By taking the inverse Fourier transform of H ( jω) , we can get the perfect estimate of the discrete channel impulse response h(t) . Inverse filtering is a very simple algorithm to implement. However, in practice, the inverse filtering is highly unstable and inaccurate. When Y ( jω) is noisy, at frequencies where X ( jω) is small the estimate of H ( jω) will be unreliable and even undefined where X ( jω) = 0 . Therefore, H ( jω) at these frequencies can be zeroed out to minimize the impact of noise and make the algorithm stable.
The continuous impulse response hc (t) is simply the inverse Fourier transform of H ( jω) . This method effectively bandlimits the estimated signal H ( jω) . Thus hc (t) will be the convolution of the true h(t) and a sinc function. By taking the amplitude and time delay of the peak of hc (t) by inspection or according to some threshold rules (local maximum greater than 0.2 of the global maximum and/or keeping some number of most significant paths) [25], an estimate of discrete channel impulse response h(t) can be derived.
Care must be taken when choosing the threshold for H ( jω) . hc (t) will have a narrower mainlobe but bigger sidelobes when the threshold is bigger and a wider mainlobe but smaller sidelobes when the threshold is smaller. The narrow mainlobe is desirable for pulse resolution while small sidelobes are desirable for false pulse rejection (large sidelobes may be mistaken as multipaths). Depending on the signal-to-noise-ratio (SNR) of the received signal, we need to choose this threshold carefully.
Generally speaking, the lower the SNR, the smaller the threshold should be. In practice, it would be beneficial to use the transmitted pulse, synthetic channel impulse response and estimated noise level to run some simple simulations to determine the appropriate threshold. To review the deconvolution accuracy and compare different thresholds, the reconstructed received signal y‘ (t) (derived from corrupted signal y(t) with different SNR values), and the simulated received signal (before adding Gaussian noise) ynoisefree (t) = x(t) ⊗ h(t) , are correlated. Table 3-1 gives an example of the correlation with different SNR values and thresholds when using a synthetic transmitted signal of a Gaussian pulse.
As seen from Table 3-1, the optimum threshold for various SNR is in the range of 15-25 dB, and the high correlation shows that this modified frequency division deconvolution works reasonably well because the neglected parts of H ( jω) comprise relatively little energy [25]. An example of the transmitted Gaussian pulse, synthetic channel impulse response, received signal with SNR of 20dB, deconvolved channel impulse response hc (t) using threshold of 25dB and the discrete impulse response h(t) are given in Figure 3-1, Figure 3-2, Figure 3-3, and Figure 3-4 respectively.
This iterative process is somewhat intuitive and ideally, is ended whenever hi (t ) ∗ x (t ) = y (t) , which necessarily yields hi (t) = h(t) .
The technique is very time consuming due to convolution performed at each iteration. Reference [26] proposed a Van-Cittert deconvolution procedure in the equivalent frequency domain. By turning the convolution into multiplication at each iteration, computation is reduced dramatically. Viewing this technique in the frequency domain is more helpful because of the Wiener-type filtering nature of this technique [26]. The conditions for the convergence of the Van-Cittert technique can be obtained in the frequency domain.
The same synthetic data used in inverse filtering section are used here to demonstrate the Van-Cittert deconvolution technique and the Bennia-Riad optimization criterion. The transmitted signal, synthetic channel impulse response and the received signal with SNR of 20dB are shown in Figure 3-1, Figure 3-2 and Figure 3-3 respectively.
The computed transfer function of the system under test is shown in Figure 3-5. In the figure, three cases are shown: the case of 2000 iterations, 20000 iterations, and the case of no filter (direct inverse filtering which is the same as the case of an infinite number of iterations). It can be seen from the figure that increasing the number of iterations increases the bandwidth of the filter. This results in a simultaneous increase in information content as well as the noise content.
A more direct view of the filtering nature of the Van-Cittert technique is shown in Figure 3-6. The figure plots the magnitude of the filter response B( f ) in Equation (3-8). The figure demonstrates the adaptive nature of this filter with its pass-band coinciding with the information frequency band and its stop-band affecting the noise interval. The number of iterations clearly affects the filter’s bandwidth. The larger the number of iterations, the wider the filter’s bandwidth is, although this does not necessarily improve the accuracy of the deconvolution result. We need to trade off between the fidelity of the signal and the noise content.
1 Introduction
1.1 Ultra-Wideband communications background
1.2 Motivation
1.3 Thesis Organization
2 Channel Measurement and Channel Modeling
2.1 Channel Sounding Techniques
2.2 Deterministic Channel Modeling vs. Statistical Channel Modeling
2.3 Large Scale Channel Modeling
2.4 Small Scale Channel Modeling
3 Deconvolution Techniques
3.1 Introduction
3.2 Frequency Domain Techniques
3.3 Time domain technique: CLEAN algorithm
4 Performance of CLEAN
4.1 Delay/Amplitude Estimate Performance
4.2 Other super-resolution techniques
5 Impact of Deconvolution on CLEAN Regenerated Channel Model
5.1 Saleh-Valenzuela Model
5.2 Comparison of Channel Statistics
6 Impact of Discrete Assumption on UWB Channel Modeling
6.1 Frequency distortion of materials
6.2 Distortion due to reflection
6.3 Antenna induced distortion
6.4 Diffraction induced distortion
7 Conclusions and Future Work
Appendix
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The Applicability of the Tap-Delay Line Channel Model to Ultra Wideband
