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Signal reconstruction from noisy multichannel samples

We consider the signal reconstruction problem under the case of the signals sampled in the multichannel way and with the presence of noise. Observing that if the samples are inexact, the rigorous enforcement of multichannel interpolation is inappropriate. Thus the reasonable smoothing and regularized corrections are indispensable. In this paper, we propose several alternative methods for the signal reconstruction from the noisy multichannel samples under different smoothing and regularization principles. We compare these signal reconstruction methods theoretically and experimentally in the various situations. To demonstrate the effectiveness of the proposed methods, the probability interpretation and the error analysis for these methods are provided. Additionally, the numerical simulations as well as some guidelines to use the methods are also presented.

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1 Introduction

The main specialty of the multichannel sampling [1, 2] is that the samples are taken from multiple transformed versions of the function. The transformation can be the derivative, the Hilbert transform, or more general liner time invariant system [3]. The classical multichannel sampling theorem [1]

is only available for the bandlimited functions in the sense of Fourier transform and it has been generalized for the bandlimited functions in the sense of fractional Fourier transform (FrFT)

[4], linear canonical transform (LCT) [5, 6] and offset LCT [7]. In a real application, only finitely many samples, albeit with large amount, are given in a bounded region [8]. That is, the underlying signal is time-limited. Thus, reconstruction by the sampling formulas for the bandlimited functions is inappropriate because the bandlimited functions cannot be time-limited by the uncertainty principle [9]. A time-limited function can be viewed as a period of a periodic function. Certain studies have been given to the sampling theorems for the periodic bandlimited functions [10, 11]. Moreover, the multichannel sampling approach has been extended to the time-limited functions [12].

Let be the unit circle and denote by , the totality of functions such that

Let , , and define

for . It was shown in [12] that there exist , , , such that

(1.1)

satisfies the following interpolation consistency:

(1.2)

Here, is a filtered function with the input and the impulse response , and , , , are determined by , , , . The continuous function is called a multichannel interpolation (MCI) for . The MCI reveals that one can reconstruct a time-limited function by using multiple types of samples simultaneously. If is periodic bandlimited, it can be perfectly recovered by (1.1).

It is noted that to find a function satisfying the interpolation consistency (1.2) is to solve a system of equations. And the matrix involved in this inverse problem may have a large condition number if the sample sets , have a high degree of relevance. In spite of this, in [8, 12], the authors showed that the large scale () inverse problem could be converted to a simple inversion problem of small matrices () by partitioning the frequency band into small pieces. Moreover, the closed-form of the MCI formula as well as the FFT-based implementation algorithm (see Algorithm 1) were provided.

The MCI guarantees that a signal can be well reconstructed from its clean multichannel samples, little has been said about the case where the samples are noisy. It is of great significance to examine the errors that arise in the signal reconstruction by (1.1) in the presence of noise. In this paper, we consider the reconstruction problem under the situation that a signal is sampled in a multichannel way and the samples are corrupted by the additive noise, i.e., we will use the noisy samples

(1.3)

to reconstruct . Here, is an i.i.d. noise process with , .

The interpolation of noisy data introduces the undesirable error in the reconstructed signal. There is a need to estimate the error of the MCI for the observations defined by (

1.3). An accurate error estimate of the MCI in the presence of noise helps to design suitable reconstruction formulas from noisy multichannel samples. Note that the MCI applies to various kinds of sampling schemes, thus the error analysis can also be used to analyze what kinds of sampling schemes have a good performance in signal reconstruction in the noisy environment. In the current paper, we provide an error estimate for the MCI from noisy multichannel samples, and express the error as a function of the sampling rate as well as the parameters associated with sampling schemes. In addition, we will show how sampling rate and sampling schemes affect the reconstruction error caused by noise.

Based on the error estimate of the MCI in the noisy environment, we will provide a class of signal reconstruction methods by introducing some reasonable smoothing and regularized corrections to the MCI such that the reconstructed signal could be robust to noise. In other words, the reconstruction should not be affected much by small changes in the data. Besides, we need to make sure that the reconstructed signal will be convergent to the original signal as the sampling rate tends to infinity.

If is a periodic bandlimited signal, only the error caused by noise needs to be considered. Otherwise, the aliasing error should be taken into account as well. It is noted that the smoothing and regularization operations will restrain high frequency in general. It follows that to reduce the noise error by the methods based on smoothing or regularization may increase the aliasing error. Thus it is necessary to make a trade-off between the noise error and the aliasing error such that the reconstructed signal can be convergent to in the non-bandlimited case as the sampling rate tends to infinity.

The objective of this paper is to study the aforementioned problems that arise in the signal reconstruction from noisy multichannel data. The main contributions are summarized as follows.

  1. The error estimate of the signal reconstruction by the MCI from noisy samples is given.

  2. We propose four methods, i.e., post-filtering, pre-filtering, regularization and regularization, to reduce the error caused by noise in the multichannel reconstruction. The parameters of post-filtering and pre-filtering are optimal in the sense of the expectation of mean square error (EMSE).

  3. The convergence property of post-filtering is verified theoretically and experimentally. The numerical simulations as well as some guidelines to use the proposed signal reconstruction methods are also provided.

The rest of the paper is organized as follows. Section 2 briefly reviews the multichannel interpolation (MCI) and its FFT-based fast algorithm. The error estimate for the MCI of noisy samples is provided. In Section 3, the techniques of post-filtering, pre-filtering and regularized approximation are applied to reconstruct from its noisy multichannel samples. The comparative experiments for the different methods are conducted in Section 4. Finally, conclusion and discussion are drawn at the end of the paper.

2 Error analysis of the MCI from noisy samples

2.1 The MCI and its fast implementation algorithm

We begin by reviewing the MCI in more detail. Let , and , we denote by the totality of the periodic bandlimited functions (trigonometric polynomials) with the following form:

The bandwidth of is defined by the cardinality of , denoted by . The set can be expressed as , where

We use the Fourier coefficients of to define the matrix

Suppose that is invertible for every and denote its inverse matrix as

Then the interpolating function in (1.1) is given by

where

It was shown in [12] that if is not bandlimited, the aliasing error of the MCI is given by

Moreover, the MCI can be implemented by a FFT-based algorithm (see Algorithm 1) and the well-known FFT interpolation [13] is a special case of the MCI.

Input : 
1. The multichannel samples , where

is a vector

    consisting of samples of ;
2. The location of lower bound for the frequency band: ;
3. The number of function values of :
Output : 
1. The vector consisting of function values of .
1 Multiply -th row of by , obtain ;
2 Take FFT of (for each column), obtain ;
3 Compute , where ;
4 Flatten w.r.t. column, obtain ( length vector) ;
5

Zero padding: add

zeros at the end of , obtain .;
6 Compute IFFT for , obtain ;
Multiply -th element of by , obtain .
Algorithm 1 FFT-based algorithm for MCI with complexity of .

2.2 The error estimate for the MCI of noisy samples

Given the noisy data (1.3), we define

If , then

Suppose that

are independent random variables with the same normal distribution

, it is easy to verify that , . Let

From Hölder inequality, we have that

It follows that

Therefore the variance of mean square error is bounded and is not larger than twice the square of the expectation.

In order to show the mean square error of MCI caused by noise more clearly, we consider three concrete sampling schemes, namely, the reconstruction problem of from (1) the samples of (single-channel); (2) the samples of and (two-channel); (3) the samples of and (two-channel). For simplicity, we abbreviate the MCI of the above types of samples as F1, FH2 and FD2 respectively and denote by the total number of samples. For F1, we have that , . It easy to see that

For FH2, we have that , . Since

It is clear that

It follows that

For FD2, by direct computations, we have that

It follows that

To study FH2 and FD2, we assume that is an even number and . It should be noted that for F1 because it is a single-channel interpolation. In contrast, for FH2 and FD2 as they are two-channel interpolations. Thus, to compare the performance of the three interpolation methods under the same total number of samples , one needs to keep in mind that has different values for F1 and FH2.

Figure 1: Left 1: for F1 (blue), FH2 (yellow) and FD2 (orange). Left 2: the average of the mean square error of experiments for F1 (blue), FH2 (yellow) and FD2 (orange). Left 3: the variance of the mean square error of experiments for F1 (blue), FH2 (yellow) and FD2 (orange). The horizontal axis in Left 1-3 represents the total number of samples, namely . Right 1-3 are the reconstructed results (red lines) for F1, FH2 and FD2 respectively in the noisy environment with ; the blue line is the original test function .

Having introduced the Fourier coefficients of the interpolation functions for F1, FH2 and FD2, we have that

Besides the theoretical error estimate, the experiments are conducted to compare the reconstructed results by F1, FH2 and FD2. Let

(2.1)
(2.2)

If , is the Dirichlet kernel of order . We use as the test function. Obviously, it is bandlimited with the bandwidth . The theoretical errors, the experimental errors and the reconstructed results are shown in Figure 1 and some conclusions can be drawn as follows.

  1. FD2 performs better than F1 in terms of noise immunity and FH2 has the worst performance.

  2. As the total number of samples increases, the expectation of mean square error (EMSE) would not decrease if there is no additional correction made in the multichannel reconstruction.

  3. The variance of mean square error (VMSE) is bounded and it decreases as increases.

Remark 2.1

In the second row first column of Figure 1, we see that the errors of F1, FH2 and FD2 become significantly large when . This is because the test function has the bandwidth , the reconstruction error is caused not only by noise but also by aliasing.

3 Multichannel reconstruction from noisy samples

The MCI cannot work well if one observes noisy data because does not converge to in the sense of the expectation of mean square error (EMSE). To alleviate this problem, some smoothing corrections are required. If is bandlimited and the number of samples is larger than the bandwidth, we only need to consider the error caused by noise. Suppose that and , where is the total number of samples. In this section, the techniques of post-filtering, pre-filtering, regularized approximation are applied to reconstruct from noisy samples.

3.1 Post-filtering

In [14], the ideal low-pass post-filtering is applied to the Shannon sampling formula and the error of signal reconstruction is also evaluated. Different from the previous work, we first derive the EMSE of the reconstruction by MCI and post-filtering. Then the filter is obtained by solving the optimization problem that minimizes the EMSE.

3.1.1 Formulation of post-filtering

A natural smoothing approach for the reconstructed signal is to convolute with a function . Let

Note that

provided that . It follows that

Since is an i.i.d. noise process with , then

Denote the Fourier coefficient of by , it follows that

Remark 3.1

Since the functions considered here are square integrable, the interchange of expectation and integral is permissible by the dominated convergence theorem. There are some similar cases happening elsewhere in the paper, we will omit the explanations.

Let and

(3.1)

Since and the equality holds only if , it follows that

where . Thus, if

then for every . To minimize , we rewrite it as follows:

where

Differentiating with respect to and solving , we obtain the optimal solution for minimizing the expectation of mean square error. That is,

(3.2)

3.1.2 Estimation of spectral density

The formula (3.2) gives the optimal values for the parameters of post-filtering, to minimize the difference (EMSE) between the filtered and the original (clean) signal . The key problem is that the square of absolute value of , namely the spectral density of , is unknown in typical cases. Thus we have to estimate the value of from the noisy multichannel samples.

There are various techniques for spectral density estimation. The representative methods are periodogram, Welch’s method, autoregressive model and moving-average model, etc. Here, we provide an unbiased estimation for

by using the uncorrelatedness of signal and noise.

Let

then

To estimate , we need to introduce the vector and , where

(3.3)

Here, is the -th order DFT matrix

(3.4)

with and is a diagonal matrix

(3.5)

Note that , and is an i.i.d. noise process, it follows that

Let be a by matrix and the entry in the -th row and -th column of is

where and . By direct computations, we have that

If is bandlimited, it can be verified that the diagonal element of is equal to (by a similar method for proving Lemma 1 in [12]). It follows that the diagonal element of

(3.6)

is an unbiased estimation for .

To validate the effectiveness of the above method for estimating spectral density, the noisy multichannel samples are applied to estimate by the formula (3.6) experimentally. We will perform a series of experiments under different quantities and types of samples. Let

(3.7)

be the test function, where . The mean square error (MSE) for estimating the spectral density of is defined by

where is the -th diagonal element of . To show the performance of the estimation more accurately, each experiment will be repeated times and the corresponding average MSE is an approximation of the expectation of MSE.

Average MSE
Table 1: The experimental results of multichannel based method for spectral density estimation. The first row displays the total number of samples used in each experiment. The second and third rows display the number of samples of and used in each experiment respectively. The error of spectral density estimation is given in the last row.

The experimental results are presented in Table 1. The second column indicates that if we use samples of to estimate spectral density, the expectation of MSE is approximately equal to . It can be seen that the expectation of MSE for spectral density estimation varies in inverse proportion to the total number of samples. In other words, the experimentally obtained MSE, i.e. , tends to as the total number of samples goes to infinity and if the same total number of samples are used to estimate spectral density, the fluctuations of MSE caused by different sampling schemes are not significant. Besides, it is noted that the traditional single-channel based method for spectral density estimation can not utilize the multichannel information to improve the accuracy. By contrast, the proposed multichannel based method fuses the different types of samples, thereby extending the scope of application and enhancing the precision, as seen from the column four and six of Table 1.

3.2 Pre-filtering

If is bandlimited, it can be expressed as