A Statistically Matched Perfect Reconstruction Filter Bank for Stationary Processes Nitin Nayar, Trishabh Chadda and Shiv Dutt Joshi Abstract— In this paper we propose an approach for finding a Perfect Reconstruction (PR) filter bank, which is optimally matched to a given stationary random sequence with strictly positive power spectral density (psd). The filter bank is obtained from the second order statistics of the random sequence. We present conditions so that the outputs of the filter bank are uncorrelated. We also present a method for estimating the filter bank.
guarantee the global optimum value. Gupta et al. [9], [10] have designed a whitening biorthogonal PR filter assuming 1/f β statistics of the random process. They have derived the wavelet analysis filter by minimizing the mean square error between the sample and its value estimated from its neighbourhood - an idea similar to sharpening in image processing operations.
I. INTRODUCTION
In this paper, we propose a novel scheme for obtaining a matched filter bank for a given stationary random sequence so that the outputs of the decomposition filters are uncorrelated. The filter bank is obtained from the autocorrelation function of the process and is computationally efficient. It is also valid over the entire time domain and not just a limited interval like in the Karhunen Loeve expansion. The only assumption however is that the autocorrelation function is positive over [−π, π].
With the growth of signal processing applications, the need for better techniques to represent signals by uncorrelated coefficients is important. Given a PR filter bank, a question naturally arises - what is the filter bank which completely removes the correlation between input values to give uncorrelated filter outputs. A statistically matched filter bank is the answer to the question. In mathematical terms, given any zero mean stationary random sequence x(n) with autocorrelation function R x (k), the goal is to find a PR filter bank whose outputs are uncorrelated. Considerable work can be found in literature in the field of designing filters based on signal characteristics. The Karhunen Loeve expansion gives a scheme for obtaining matched basis functions which yield uncorrelated expansion coefficients. However, the practical utility of the scheme is limited by two factors. First, the basis computation requires solution to an eigenvalue equation which is computationally expensive. And second, the expansion is only valid over a time interval of finite length. More recently, Tsatsanis and Giannakis [7] proposed a scheme to obtain an M-band PR filter bank which gives a minimum mean-square error low-resolution approximation of the input signal. Further, it also results in uncorrelated low resolution principal components that are automatically ordered by energy. Lu and Antoniou [8] have given a scheme for obtaining an M-band biorthogonal filter bank that is signal-adapted. Although the use of a first-order PR constraint improved the computational efficiency in their scheme but still the solution is approximate and does not
The paper is organised as follows: Section II gives a brief review of PR filter banks. The conditions on the filter bank to give uncorrelated outputs are presented in Section III. In Section IV we propose a method for estimating matched filter banks. Simulation results are presented in Section V. Throughout the paper, small letters would denote time domain quantities whereas the corresponding capital letters ¯ is defined would denote their Fourier transform. The filter h ¯ as h(n) = h(−n). The symbol , is used to denote inner product, ∗ will denote convolution. II. P ERFECT R ECONSTRUCTION FILTER BANKS A PR filter bank is shown in Fig. 1. The decomposition filters h and g are used to filter input a 0 . The filtered signal is then down-sampled by 2 to get outputs a 1 and d1 , ie. ¯ a1 (n) = (a0 ∗ h)(2n) and d1 (n) = (a0 ∗ g¯)(2n). At the reconstruction end, the signals a 1 and d1 are up-sampled by ˜ and g˜. Vetterli 2 and then filtered by reconstruction filters h [4] gave the following condition for a biorthogonal filter to
Nitin Nayar is with the Department of Electrical Engineering, Indian Institute of Technology, Delhi, Huaz Khas, New Delhi -110016, India Phone: 0091-9891-493434
[email protected] Trishabh Chadda is with the Department of Electrical Engineering, Indian Institute of Technology, Delhi, Huaz Khas, New Delhi -110016, India
[email protected] Shiv Dutt Joshi is with Faculty of Electrical Engineering, Indian Institute of Technology, Delhi, Huaz Khas, New Delhi -110016, India
[email protected]
Fig. 1.
PR Filter Bank
ensure perfect reconstruction.
Similarly (6b) is obtained by expanding condition (5b). Expanding (5c) we get
Theorem 2.1: The filter bank performs an reconstruction for any input signal if and only if
exact
˜ ˜ H ∗ (ω + π)H(ω) + G∗ (ω + π)G(ω) =0 ∗ ∗ ˜ ˜ H (ω)H(ω) + G (ω)G(ω) = 2
(1a) (1b)
The above theorem proves that the reconstruction filters are defined completely by the decomposition filters. This can be written as ∗ ˜ (ω) 2 H G(ω + π) (2) ˜ ∗ (ω) = Δ(ω) −H(ω + π) G where Δ(ω) = H(ω)G(ω + π) − H(ω + π)G(ω) where Δ(ω) = 0 ∀ ω ∈ [−π, π]. In particular, if we impose the condition that the decomposition filters, h and g and the reconstruction ˜ and g˜ are equal, i.e. h = h ˜ and g = g˜, the filter filters, h banks are known as conjugate mirror filters (CMF). The following properties follow from Theorem 2.1. 2
2
2
2
|H(ω)| + |H(ω + π)| = 2 |G(ω)| + |G(ω + π)| = 2
(3a) (3b)
E{a1 (n)d1 (m)} π 1 ˆ X (ω)H(ω)G∗ (ω)ejω2(m−n) dω = 0 = R 2π −π For decomposition filters to give uncorrelated outputs, should satisfy the following conditions. π 1 ˆ X (ω)|H(ω)|2 ejω2(m−n) dω = δ(m − n) R 2π −π π 1 ˆ X (ω)|G(ω)|2 ejω2(m−n) dω = δ(m − n) R 2π −π π 1 ˆ X (ω)H(ω)G∗ (ω)ejω2(m−n) dω = 0 R 2π −π
they
(6a) (6b) (6c)
ˆ X (ω)|H(ω)|2 , Discussion: Let us assume that F0 (ω) = R ˆ X (ω)|G(ω)|2 and F2 (ω) = R ˆ X (ω)H(ω)G∗ (ω). F1 (ω) = R Then the function f 0 (n), f1 (n) and f2 (n) have the following properties f0 (2n) = δ(n)
(7a)
f1 (2n) = δ(n) f2 (2n) = 0
(7b) (7c)
The following lemma ([1], pp. 263) would be useful in later sections. Lemma: If g and h are CMFs then
Hence to solve the problem of finding the matched filter bank we have to find functions satisfying the above properties.
H ∗ (ω)G(ω) + H ∗ (ω + π)G(ω + π) = 0
We are given a stationary discrete random process {x(n)} with autocorrelation function R x (n − m).
∀ ω ∈ [−π, π]
For a detailed discussion on PR filter banks, see [1], [2], [4]. III. C ONDITIONS FOR UNCORRELATED OUTPUTS In this section we define conditions on decomposition filters h and g so that the output samples are uncorrelated. Let {x(n)} be a given random sequence with autocorrelation ˆ X (ω). We shall also Rx (k). The psd of {x(n)} is given by R assume that ˆ X (ω) > 0 ∀ ω ∈ [−π, π] R
(4)
The output samples a 1 and d1 (Fig. 1) are uncorrelated. Hence we get the following properties. E{a1 (n)a1 (m)} = δ(m − n) E{d1 (n)d1 (m)} = δ(m − n)
(5a) (5b)
E{a1 (n)d1 (m)} = 0
(5c)
Expanding (5a) we get, E{a1 (n)a1 (m)} x(j1 − 2n)h(j1 ) x(j2 − 2m)h(j2 ) =E j
=
1
j1
j2
Rx (j1 − j2 + 2m − 2n)h(j2 )h(j1 )
j2
(using stationarity) π 1 ˆ X (ω)|H(ω)|2 ejω2(m−n) dω = δ(m − n) = R 2π −π
IV. T HE P ROPOSED A LGORITHM
Theorem 4.1: Let us choose any conjugate filter bank hcf , gcf . Let us define our function f 0 , f1 and f2 as f0 (n) = hcf (k), hcf (k − n) f1 (n) = gcf (k), gcf (k − n)
(8a) (8b)
f2 (n) = hcf (k), gcf (k − n)
(8c)
Then the functions thus defined will fulfill the requirements, defined in (7a), (7b) and (7c). Proof: f0 (2n) = hcf (k), hcf (k − 2n) π 1 = |Hcf (ω)|2 ejω(2n) dω 2π −π π 1 |Hcf (ω)|2 + |Hcf (ω + π)|2 ej(ω)(2n) dω = 2π 0 = δ(n) (using (3)) Thus (7a) holds and similarly (7b) holds. Now we prove (7c) f2 (2n) = hcf (k), gcf (k − 2n) π 1 Hcf (ω)G∗cf (ω)ejω(2n) dω = 2π −π π 1 Hcf (ω + π)G∗cf (ω + π)+ = 2π 0 Hcf (ω)G∗cf (ω) ej(ω)(2n) dω = 0 (using Lemma)
A. Proposed scheme for design of signal matched filter bank Now we present the algorithm to find the matched filter bank. The following results would be useful before we present the algorithm. 1) Theoretical foundation: We shall define the matched filters. Theorem 4.2: The decomposition filters defined as
Similarly we obtain (11b). (9a)
Gcf (ω) Gde (ω) = ˆ X (ω) R
(9b)
yield uncorrelated outputs when {x(n)} is given as input (Fig. 1). Proof: For the filter bank to give uncorrelated outputs it should satisfy (6). We shall check each of the conditions (6a), (6b) and (6c). π 1 ˆ X (ω)|Hde (ω)|2 ejω2(m−n) dω R 2π −π π 1 = |Hcf (ω)|2 ejω2(m−n) dω = δ(m − n) 2π −π Hence condition (6a) and similarly condition (6b) is satisfied. π 1 ˆ X (ω)Hde (ω)G∗de (ω)ejω2(m−n) dω = 0 R 2π −π π 1 = Hcf (ω)G∗cf (ω)ejω2(m−n) dω = 0 2π −π Condition (6c) is satisfied. Now we shall design the reconstruction filter bank for PR. Let hcf and gcf be any CMFs. Then the filter bank ˜ and g˜ form a PR filter bank, they satisfy defined by h, g, h (2). Hence we get 2Gcf (ω + π) Δcf (ω) −2H cf (ω + π) G∗cf (ω) = Δcf (ω)
(10a)
We can choose any CMF to define our matched filter bank. It is important to note that the phase response of the filters designed is same as that of the chosen CMFs. 2) Algorithm: (i) Calculate the psd of the random sequence {x(n)}. (ii) Choose suitable CMFs hcf and gcf . (iii) Estimate the matched decomposition filters h de and gde using (9). (iv) Estimate the matched reconstruction filters h re and gre using (11). V. S IMULATION R ESULTS In this section we will use our algorithm to estimate a PR Filter bank matched to random process x(n) with psd shown in Fig. 2. We use the Daubechies wavelet (vanishing moments = 6) CMFs hD6 and gD6 . The matched decomposition filter (hde and gde ) and the reconstruction filters (h re and gre ) are estimated using (9) and (11) respectively and are given in Table I. The autocorrelation and cross-correlation of filter outputs were calculated for both, the Daubechies CMFs and the matched filter bank. These have been shown in Fig. 3 and Fig. 4 respectively. It is clear from the figures that the autocorrelation of the Daubechies filter outputs is highly spread as compared to that of the matched filter, which is close to a delta function. The filter order is determined by the system itself, unlike other common filter design techniques. Power spectral density
2.5
2
1.5 PSD
Hcf (ω) Hde (ω) = ˆ X (ω) R
∗ Hcf (ω) =
hde and gde as
∗ 2Gde (ω + π) Hre (ω) = Hde (ω)Gde (ω + π) − Hde (ω + π)Gde (ω) ˆ X (ω) ∗ 2Gcf (ω + π) R = using (9) Δcf (ω) ˆ X (ω) using (10) = Hcf (ω) R
1
(10b)
where Δcf (ω) = Hcf (ω)Gcf (ω + π) − Hcf (ω + π)Gcf (ω) Theorem 4.3: The reconstruction filters h re and gre corresponding to the decomposition filters h de and gde for PR are given by ˆ X (ω) Hre (ω) = Hcf (ω) R (11a) ˆ X (ω) (11b) Gre (ω) = Gcf (ω) R Proof: Using (2) we get the reconstruction filters matched to
0.5
0 −3
−2
−1
0
1
2
3
Radians
Fig. 2.
ˆ x (ω) Power Spectral Density R
VI. C ONCLUSION In this paper, we have obtained conditions for the decomposition filters to give uncorrelated outputs. We have also proposed an algorithm to find such decomposition filters and the corresponding reconstruction filters. The simulation results validate the theory proposed.
correlation : scaling filterDaubechies6
correlation : wavelet filterDaubechies6
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Correlation
Cross−Correlation : CMFDaubechies6
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(a) Daub-6 Scaling filter
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(b) Daub-6 Wavelet filter Fig. 3.
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(c) Cross Correlation Daub-6
Correlation of output samples for Daubechies CMF correlation : matched wavelet filter
correlation : matched wavelet filter
Cross−Correlation : Matched PR Filter Bank
1
1
0.35
0.8
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Correlation
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(a) Matched filter h de Fig. 4.
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(c) Cross Correlation Matched Filter
Correlation of output samples for Matched Filter Bank TABLE I M ATCHED PR FILTER BANK
hde
gde
hre gre
0.0107 -0.0091 -0.0102 0.0020 0.0125 -0.0079 -0.0142 -0.0023 0.0137 0.0131 -0.0050 -0.0183 -0.0086 0.0137 0.0203 -0.0228 -0.0181 0.0119 0.0303 0.0099 -0.0278 -0.0337 0.0068 0.0464 0.0287 -0.0356 -0.0656 -0.0057 0.0852 0.0772 -0.0673 -0.1800 0.0104 0.6331 1.4374 1.6742 0.8671 -0.1571 -0.3883 -0.1004 0.0978 0.1229 0.0322 -0.0714 0.0693 0.0099 0.0585 0.0303 -0.0267 -0.0419 -0.0059 0.0309 0.0257 -0.0089 -0.0281 0.0115 0.0167 0.0215 0.0001 -0.0190 -0.0133 0.0078 0.0174 0.0052 -0.0122 -0.0133 0.0018 0.0135 0.0079 -0.0069 -0.0121 0.0155 -0.0109 0.0063 -0.0087 0.0018 0.0083 -0.0073 0.0110 -0.0207 0.0165 -0.0110 0.0154 -0.0072 -0.0075 0.0071 -0.0129 0.0293 -0.0263 0.0198 -0.0286 0.0186 0.0048 -0.0056 0.0164 -0.0487 0.0494 -0.0426 0.0655 -0.0512 0.0046 -0.0012 -0.0355 0.1376 -0.1689 0.2243 -0.3941 0.4705 -0.6877 1.0130 -0.6294 0.0671 -0.2158 0.2781 -0.0299 0.0412 -0.0370 -0.0694 0.0407 -0.0335 0.0783 -0.0477 0.0183 -0.0332 0.0118 0.0203 -0.0123 0.0164 -0.0367 0.0261 -0.0140 0.0217 -0.0112 -0.0070 0.0044 -0.0077 0.0215 -0.0174 0.0108 -0.0164 0.0108 -0.0090 -0.0116 0.0024 0.0182 0.0124 -0.0170 -0.0357 0.0014 0.0727 0.0252 -0.0903 0.1371 0.4542 0.1249 -0.2844 -0.0193 0.1932 -0.0289 -0.1000 0.0173 0.0398 0.0065 -0.0121 -0.0145 -0.0013 0.0102 0.0080 0.0081 0.0115 -0.0198 0.0169 -0.0133 0.0178 -0.0031 -0.0210 0.0354 -0.1154 0.2300 -0.0553 -0.5006 0.7631 -0.2115 -0.4264 0.3358 0.0466 -0.0914 -0.0259 0.0170 0.0347 -0.0241 0.0066 -0.0156 0.0122
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[7] M. K. Tsatsanis and G. B. Giannakis, ”Principal component filter banks for optimal multiresolution analysis”, IEEE Trans. Signal Processing, vol.43, no. 8, pp. 1766 1777, Aug. 1995 [8] W.-S. Lu and A. Antoniou, ”Design of signal-adapted biorthogonal filter banks”, IEEE Trans. Circuits Syst. I, Funam. Theory Appl., vol. 48, no. 1, pp. 90102, Jan. 2001. [9] Anubha Gupta, Shiv Dutt Joshi and Surendra Prasad, ”A new method for estimating wavelet with desired features froma given signal”, Signal Processing, Vol. 85, Issue 1, pp. 147-161, Jan. 2005. [10] Anubha Gupta, Shiv Dutt Joshi and Surendra Prasad, ”A New Approach for Estimation of Statistically Matched Wavelet”, IEEE Trans. Signal Processing, vol.53, no. 5, pp. 1778-1793, May 2005.
