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Multirate Filtering For Digital Signal Processing and Its Applications

1 Suverna Sengar and 2 Partha Pratim Bhattacharya

Department of Electronics and Communication Engineering Faculty of Engineering and Technology

Mody Institute of Technology & Science (Deemed University) Lakshmangarh, Dist. Sikar, Rajasthan,

Pin – 332311, India 1 [email protected], 2 [email protected]

ABSTRACT

Multirate Filtering techniques are used when conventional method becomes extremely costly and this technique is widely used in both sampling rate conversion system and in constructing filters with equal input and output rates. The basic concepts and building blocks in multirate digital signal processing are discussed here which includes down-sampler, up-sampler and analysis/synthesis representation. Applications of multirate digital filters in DS/CDMA code acquisition, Kalman filtering for optimal signal reconstruction from noisy subband system and lossy compression approach to transmultiplexed images are also reviewed. Keywords: Multirate filters, Decimation, Interpolation, Multistage system and Analysis/Synthesis filter.

1. INTRODUCTION

A multirate filter is a digital filter that changes the sampling rate of the input signal into another desired one. These filters are of essential importance in communications, image processing, digital audio, and multimedia. Unlike the single–rate system, the sample spacing in the multirate system can vary from point to point [1]-[2]. This often result in more efficient processing of signals because the sampling rates at various internal points can be kept as small as possible, but this also results in the introduction of a new type of error, i.e., aliasing.

1.1 TIME-DOMAIN REPRESENTATION OF DOWN-SAMPLING AND UP-SAMPLING

Two discrete signals with different sampling rates can be used to convey the same information. For example, a band limited continuous signal xc(t) might be represented by two different discrete signals {x[n]} and {y[n]} obtained by the uniform sampling of the original signal xc(t) with two different sampling frequencies Ft and Ft’.

x[n]=xc(nT) and y[n]=xc(nT') (1)

where T= 1/Ft and T’=1/ Ft

’ are the corresponding sampling intervals. When the sampling frequencies Ft and Ft’ are chosen in such a way that each of them exceeds at least two times the highest frequency in the spectrum of xc(t), the original signal xc(t) can be

reconstructed from either {x[n]} or {y[n]}. Hence, the two signals operating at two different sampling rates are carrying the same information. By using the discrete-time operations, signal {x[n]} can be converted to {y[n]}, or vice versa, with minimal signal distortions. The two basic operations in sampling rate alteration process are the down-sampler and up-sampler. These two operators can perform the sampling rate alteration: a down-sampler used for decreasing the sampling rate, and an up-sampler used for increasing the sampling rate.

1.2 DOWN-SAMPLING OPERATION The down-sampling operation with a down-sampling factor M, where M is a positive integer, is implemented by discharging M–1 consecutive samples and retaining every Mth sample. Applying the down-sampling operation to the discrete signal {x[n]}, produces the down-sampled signal

y[m]= x[ mM] (2). The down-sampling can be imagined as a two-step operation. In the first step, the original signal {x[n]} is multiplied with the sampling function {sM[n]} defined by,

sM[n]= (3).

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Multiplying the sequence {x[n]} by the sampling function {sM[n]} results in the intermediate signal {ys[m]},

ys[m]=x[n]sM[n]= (4).

This operation is called a discrete sampling. In the second step, the zero valued samples in {ys[m]} are omitted resulting in the down-sampled sequence {y[m]},

y[m]=ys[mM]=x[ mM] (5).

The down-sampling operation is sometimes called the signal compression, and the down-sampler is also known as a compressor. A block diagram representing the down-sampling operation is shown in Figure 1 [3].

Figure 1: Block diagram representation of a down-sampler

1.3 UP-SAMPLING OPERATION The up-sampling by an integer factor L is performed by inserting L-1 zeros between two consecutive samples. Applying the up-sampling operation to the discrete signal {x[n]}, produces the up-sampled signal {y[m]} is defined as,

y[m]= (6).

A block diagram presentation of equation-6 is given in Figure 2 [3].

Figure 2: Converting the sampling rate with an up-

sampler

The up-sampling is sometimes called the sequence expansion, and the term expander is sometimes used for the device.

1.4 DECIMATION AND INTERPOLATION The process of sampling rate decrease is called decimation, and the process of sampling rate increase is

called interpolation. Two devices, the down-sampler and the up-sampler, are elements that change the sampling rate of the signal. The drawback of the down-sampling is the aliasing effect, where as the up-sampling produces the unwanted spectra in the frequency band. Decimation has used to avoid the effects of aliasing, which occurs when the highest frequency in the spectrum of a down-sampled signal ωH exceeds the value π/M. In interpolation, the L-1 images caused by inserting L-1 zeros between the samples should be removed. 1.4.1 DECIMATION Decimation requires that aliasing should be prevented. So before, down sampling with the factor of M, the original signal has to be band limited to π/M. This means that the factor-of-M decimation has to be implemented in two steps:-

i) Band limiting of the original signal to π/M ii) Down-sampling by the factor-of-M.

Figure 3 [4] shows the block diagram of a decimator implemented as a cascade of the decimation Filter H(z), also called the anti-aliasing filter, and the factor-of-M down-sampler. The performance of a decimator is mainly determined by the filter characteristics.

Figure 3: Block diagram representation of

decimator

1.4.2 INTERPOLATION Interpolation requires the removal of the images. This means that the factor-of-L interpolation has to be implemented in two steps:

i) Up-sampling of the original signal by inserting L-1 zero-valued samples between two consecutive samples.

ii) Removal of the L-1 images from the spectrum of the up-sampled signal.

Figure 4 [4] shows the block diagram of an interpolator implemented as a cascade of a factor-of-L up-sampler and a lowpass filter, frequently called the anti-imaging filter. The cut-off frequency of the filter is π/L. The anti-imaging (interpolation [5]) filter H (z) is used to remove images from the spectrum of the up-sampled signal. Removal of images from the spectrum of the signal causes the interpolation of the sample values in time domain. The zero-valued samples in the up-sampled signal {xu[m]} are filled with the interpolated values.

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Figure 4: Block diagram representation of an interpolator

2. APPLICATIONS OF MULTIRATE FILTERS

A multirate filter is used to change the sampling rate of the input signal into another desired one and multirate filtering [6] can be used in various fields, like it is used in multirate adaptive filtering for DS/CDMA code acquisition, Multirate kalman filtering for optimal signal reconstruction from noisy subband system, multirate filters use in lossy compression approach to transmultiplexed image and many more. Different applications of multirate filters are described in the subsequent sections. 2.1 MULTIRATE ADAPTIVE FILTERING FOR DS/CDMA CODE ACQUISITION

In direct sequence code division multiple access (DS/CDMA) system [7], to recover the transmitted information, the received signal should first despreaded using a locally generated pseudo noise (PN) code sequence. The code acquisition performs initial code timing alignment between the locally generated and received signals. Code acquisition method is usually conducted using a correlator to serially search the code phase, which is related to the channel propagation delay. This approach performs well for the adaptive filtering approaches [8] have high acquisition-based capacity and drawback of these schemes is high computational complexity. Due to down-sampling operations, the computational complexity of the adaptive filters can be effectively reduced.

Code synchronization is a very essential and

important part of any spread spectrum (SS) system in order to remove the spreading effect induced by the transmitter and to exploit the processing gain of the spread signal. The receiver must be able to estimate the delay offset between the spreading code in the received signal and the locally generated replica of the code before data demodulation is started. Code synchronization is usually developed over two steps, namely acquisition and tracking [9]-[10].In this application of multirate processing, not only the computational complexity, but also the mean acquisition time can be effectively reduced. Here the basic idea of the scheme used in multirate adaptive filtering for DS/CDMA code acquisition is discussed.

2.1.1 SCHEME USED FOR MULTIRATE FILTER SYSTEM MODEL

There are K active users and each user is

specified with a pseudo-noise (PN) sequence with length L, where L is the processing gain. Consider an additive white Gaussian noise (AWGN) channel. The user-k transmitted signal is given by Xk(n)= (7) where dk(j) {1, -1} is the j-th BPSK(binary phase shifting keying) symbol of user- k, Ck(l) is the l-th chip of user-k’s PN sequence, and p(n) is the rectangular pulse having a unit amplitude and a chip-duration, Tc . Now, assume that user-1 is the desired user. The received signal can be written as r(n) = = (8)

where w(n) is AWGN (additive white Gaussian noise) and i(n) is the summation of interference and noise. Let the mean of i(n) be zero and the variance be . Assume that , k = 1, ... , K are integer multiples of Tc, and uniformly distributed over {0, 1. . . L - l}, The carrier synchronization is established before code acquisition and no data are transmitted during acquisition (i.e. d1(j) = 1). The main objective is to estimate the delay of the desired user, from the composite received signal, {r(n)}. Here proposed to use an adaptive filter (AF) to estimate the delay of the desired user from the tap-weight vector of the filter. The block diagram of the AF based DS/SS code acquisition system is shown in Figure 5 [8].

Figure 5: Block diagram for the adaptive acquisition

system

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THE SCHEME Figure 6 [11] shows the Structure of the proposed scheme, which consists of two adaptive filters. The input of the first filter (wc(.)) is the down-sampled locally generated PN sequence; the down-sampling factor is D. The reference signal is the down-sampled received signal. Here, the lowpass filter is an averaging filter with length D. Let where is an integer, and is a fractional delay, -D/2 < . To determine the fractional delay, then need the second filter (wf(.)). The filter appearing in front of wf(.) is named the delay tuning filter (DTF); its output is just a delay version of the input. If copy the delay identified by the first filter to the DTF, the second filter can identify the fractional delay. Feedback the peak position of wf(.) to the filter in front of wc(.), which is namely the phase tuning filter (PTF). The fractional delay will vanish. The convergent weights of the first filter only have a non-zero weight. In noisy environment, this will greatly reduce acquisition error. The cross adaptation of these two filters will effectively determine the code phase.

Figure 6: The Proposed Scheme

2.1.2 PERFORMANCE ANALYSIS Here, first consider the computational complexity. Now, only consider the required multiplications per chip duration. The second filter has 2D-1 taps and each tap needs two multiplications for LMS weight-updating. The first filter has Mp taps and its tap-weights updating rate is D times slower than the second filter. Thus, its computational complexity per chip duration, Cp, is given by

where N is the number of chips for tap-weight converging. Then consider the probability of acquisition error. An acquisition error may occur due to the first filter, the second filter, or both. The probability of acquisition error is then

(10)

where and are the correct taps of the first and the second filters, respectively; and denote the probabilities of correct acquisition of the first and the second filters, respectively. Finally, consider the mean acquisition time. The propose scheme gives an estimate

after N chips elapses. Figure 7 [11] shows the state transition diagram, whose transfer function is

where TP is the penalty time, z is a delay operator and Pe is the probability of acquisition error.

Figure 7: The state transition diagram of the

proposed acquisition system

2.1.3 EXISTING SIMULATION RESULTS

The simulation results are demonstrating the effectiveness of the proposed scheme. Here, K = 20 (SINR = -13dB), L = 128, D = 4, Mp = 32, and Tp = (chips) are considered. Both filters use a same step size p. For a 32-tap conventional adaptive acquisition method, it needs 64 multiplications per chip, but the proposed scheme only requires 30 multiplications per chip. Figure 8 shows the theoretical and simulated acquisition error probabilities Pe in various step sizes.

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Figure 8: Theoretical and simulated probabilities of

acquisition error P, versus step sizes p for the proposed scheme

The probability Pe was evaluated after N chips

using samples. For P = 2 samples were observed. Now, N is selected to be four times of the time constant of the first filter . These results are substituted into equation (13) for theoretical and simulated mean acquisition time calculation. The results are shown in Figure 9.

As shown in Figure 8, as step size increases the

probability of error also increases and from Figure 9 it can be seen that as step size increases the mean acquisition time decreases. So, not only the computational complexity is low, but also the mean acquisition time is low.

Figure 9: Theoretical and simulated mean acquisition

time versus step sizes of the proposed scheme

2.2 MULTIRATE KALMAN FILTERING FOR OPTIMAL SIGNAL RECONSTRUCTION FROM NOISY SUBBAND SYSTEM

The multirate signal processing is to decompose the original signal into complementary frequency bands and then process them separately in each subband. In this application use decimation and interpolation filters or analysis/synthesis filter banks that allow perfect reconstruction [12]. The perfect reconstruction (PR) filter bank systems are based on the assumption that the subband components are free of noise. While, in practical systems, the subband components are always contaminated by noises due to the effect of quantization [13], round-off [14] and other distortion, therefore, the perfect reconstruction is no longer possible. For improving the applicability of filter bank systems use the multirate kalman synthesis filter [15] in place of conventional synthesis filters to achieve optimal reconstruction of the input signal in noisy filter bank systems Figure 11 [16].

The input signal embedded in the state vector, the multichannel representation of subband signals is combined with the statistical model of input signal to derive the multirate state-space model for the filter bank systems and the subband noises are assumed to be additive ones. The multirate Kalman filter can be constructed to provide the minimum variance estimation of the input signal based on observations of noisy subband components. Here going to discuss first the basics idea of multirate kalman synthesis filtering and to explain the state-space model and also mention the multirate state-space model for 2-D kalman filtering.

Figure10: Noisy M-band filter bank system equipped

with a multirate Kalman synthesis filter

2.2.1 FILTER BANK SYSTEMS AND PROBLEM FORMULATION a. FILTER BANK SYSTEMS

Let {Hk (Z), Gk(Z): k = 0,1,... , M - 1} denotes the M band filter bank systems. The bank of filters {Hk (Z)} constitutes the analysis filters. Each filter output is

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down-sampled and transmitted to the receiver. Where they are up-sampled and fed into the bank of synthesis filters {Gk (Z)} for signal reconstruction Figure 11 [16].

Figure 11: M-band filter bank system with clean

subband paths

b. MULTICHANNEL REPRESENTATION OF SUBBAND SIGNALS

Let { hi , gi ; i=0,1} denote the impulse response of 2-band analysis/synthesis filter bank, the equivalent multichannel representation of subband signals . y(n) = (14)

where f(n) = = , y(n) =

and H(k) is the multichannel impulse response matrix of the form

H(k) = (15)

The above are the so-called polyphase components of respectively. If additive noisy corruption are included in the subband components, the received subband signal r(n) can be expressed as follows: r(n) = y(n)+ v(n) (16)

v(n) = in (16) is the additive-noise disturbance vector. 2.2.2 MULTIRATE STATE-SPACE MODEL FOR 1-D SIGNAL a. THE BASIC SIGNAL MODEL

Let p = , where are chosen such that .

By substituting k = k-λ, we can rewrite equation (14) as a causal form

y(n)= (17).

Now, can write the state vector as

x(n)= , then

X(n+1)=A x(n)+B (18).

Here further denote

and , the subband output is

y(n) = C x(n)+D (19).

Considering the effect of noisy disturbance vector v(n), now formulate the problem as the form of state-space model X(n+1)=A x(n)+B x(-1)=x-1 r(n)=C x(n)+D + v(n) (20). b. THE STATISTICAL MODEL

The state space model describing the statistical characteristic of the input signal, which has the form

z’(k+1)=

f(k) = H z’(k) (21)

where z’(k) and u(k) are the state vector and driving source, respectively. Let z(n) = z’[2(n+λ)], the equivalent block generation model is given below z(n+1)= f(n+λ)=

f[2(n+λ)]= H z(n) (22)

c. THE MULTIRATE STATE-SPACE MODEL

The multirate state space model will be given below w(n+1) =

r(n)= (23) where w(n) = , is the state vector of the system model. Similarly, the multirate state-space model for 2-D signal [15] can also be written. 2.2.3 NUMERICAL RESULTS

Simulation is carried out to show the feasibility and effectiveness of the proposed 2-D Kalman filtering for optimal 2-D signal reconstruction from noisy

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subband systems. Here compared the performance of the proposed 2-D multirate Kalman filter versus the conventional perfect reconstruction (PR) filters for 2-D signal reconstruction under different noise levels. The simulation was implemented with Matlab. Now adopted a set of 2-band quadrature mirror filter (QMF) PR filter bank {h (n)}of length 8 in the simulation [17]. The 2x2 subband decomposition of test image is implemented through two 1-D separate analysis filter bank { i=0,1. with

Two quantitative measures, SNRim,n and SNRr ,

are input noise level and reconstruction performance, respectively. The test image Hillside is of size 160 160

8 (Figure 12 (a) [15]). The image is converted to zero mean prior to 2 x 2 subband decomposition and its autocorrelation function. White additive noise at different SNR levels was added to all the four subband images. Figure 13 [15] demonstrates the reconstruction performance comparison with both the proposed multirate Kalman filtering and the conventional PR synthesis filters under different SNR level. It is observed that the improvement in reconstruction SNR with the proposed 2-D multirate Kalman filtering is considerable.

Figure 12: (a) Original image Hillside of size 160 160;

(b) 2 2 subband decomposition of Hillside; (c) noise-corrupted subband image at SNRi = 0 dB; (d) reconstructed image with conventional PR filter banks at SNRi = 0 dB; (e) reconstructed image with 2-D Kalman filtering at SNRi = 0 dB; (f) Reconstructed image with conventional PR filter banks at SNRi = 8 dB; (g) reconstructed image with 2-D Kalman filtering at SNRi = 8 dB

Figure 13: Reconstruction SNR versus additive noise SNR level

Figure 12(e) [15] show that, even in the

extremely low SNR, case, the main structure of the original image is still distinguishable on the reconstructed image with 2-D multirate Kalman filtering.

2.3 LOSSY COMPRESSION APPROACH TO TRANSMULTIPLEXED IMAGES

Multimedia content is more and more popular in many different types of telecommunications. That’s why use a new and efficient method for sending several images through a single transmission line is needed. The transmultiplexer is a structure that combines up-sampled and filtered signals for the transmission over a single transmission line. Transmultiplexing [18] is easy to apply because it needs only simple digital processing: up-sampling, filtering and summing. All these operations used in transmultiplexer are linear and time-invariant. Figure 14 [20] shows the structure of 4-channel (M=4) transmultiplexer.

Figure 14: Scheme of 4-channel image transmultiplexer

The main problem in transmultiplexers is preventing image distortion caused by the change of amplitude and phase as well as image leakage from one channel into another. The main motivation of this application, lossy compression is to achieve a perfect image reconstruction in the receiver by using appropriate filters [19].

In compression algorithm the number of bits needed to represent the signal (image) or its spectrum is

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minimized. The fundamental concept of compression is to split up the frequency band of a signal (image) and the quantized each subband using a bit rate accurately matched to compromise between the two opposite criteria: minimize distortions and maximize the compression rate.

Over the past several years, the wavelet methods have gained widespread acceptance in signal processing and in the signal compression. Multirate processing is related to signal transformation using wavelets. Wavelet packets are a way to analyze a signal using base functions which are well localized both in time and in frequency. The properties of wavelet transform make it useful in compression. The main advantage of the wavelet transform is used in lossy compression which reduces the signal (image) spectrum by eliminating redundant information. Lossy compression is generally used where a loss of a certain amount of information will not be detected by the users. 2.3.1 IMAGE TRANSMULTIPLEXING

Figure 14 [20] shows the structure of the four-channel (M=4) image transmultiplexer. The input images are up-sampled and filtered vertically and summed to obtain two combined images. These combined images are then up-sampled and filtered horizontally and summed to obtain the final version of combined image [19]. In presented system the combined image consists of four times more pixels than each input image. At the receiver end, the signal is relayed first two channels of the detransmultiplexing, where the signals are filtered and down-sampled horizontally. Then these signals are relayed to four channels where images are filtered and down-sampled vertically to recover the original image. 2.3.2COMPRESSION

Wavelet packet algorithm generates a set of orthogonal sub-images that are derived from a single combined image. The wavelet spectra are produced by cascading filtering and down-sampling operations in a tree-structure. Wavelet packets were introduced for splitting images into its frequency components so that to compress the image by non-uniform quantization.

The 2-D wavelet packet transform (WPT) can be viewed as a decomposition system in Figure 15 [20] for three levels. The basis data are the coefficients of wavelet series of the original image. The next level results of one step of the 2-D wavelet discrete transform (WDT). Subsequent levels are constructed by recursively applying the 2-D wavelet transform to the low and high frequency sub-bands of the previous wavelet transform. The higher level component has two times narrower frequency bands when compare with the subsequent lower frequency components. Multirate processing involves the application of filtering and down-sampling. The main objective is the design of lowpass and

highpass filters which gives useful transformations and allow recovering the transmitted images.

Figure 15: 2-D wavelet packet transforms structure

2.3.3 EXAMPLE

Four test images (boats, F-16, Lena and baboon) with 512 512 pixels resolution in 256 grayscale levels were selected for the analysis. The combined image has 1024 1024 pixel resolution. Its luminance values are integer numbers from -1088 to 1884. The combining filters and the separation filters were designed by algorithm presented in [21].

Linearity property of the transmultiplexer system enables to split the combined image into its frequency components. The periodicity of spectrum of the combined image is shown in Figure 16 [20]. The sampling densities in the frequency domain for all levels are the same but the amounts of samples are different. The spectrum of the up-sampled signal consists of the original spectrum and its components.

The three level 2-D wavelet packet decomposition was provided by the discrete Meyer wavelets. The absolute values of coefficients of wavelet packet decomposition are shown in Figure 17 [20]. The main part of energy (96.6%) of the signal is localized in last four sub-bands of the network AAA (2.5%), HAA (13.3%), VAA (10.4%) and DAA (70.4%). The compression with factor 16 was obtained by omitting information from other bands. The reconstructed output images are shown in Figure 18 [20]. The large part of the signal energy was sent and the compression was not very

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high, some distortions are visible. In the presented example data from bands with the energy higher than 0.2% of the whole energy was transmitted.

Another possible solution is the preliminary compression of the input images, e.g. using JPEG algorithm [22], and Figure of JPEG for lossy compression in transmultipexer system [20] and applying the 1-D transmultiplexer after converting of the bit streams into integer 1-D signal. The compression with factor 16 using the JPEG algorithm on the each of input images allows to gain the better quality of images than in case of the compression of the combined image.

Figure 16: Amplitude [dB] of spectrum of combined image

Figure 17: 2-D wavelet packet transform coefficients

Figure 18: Comparison of input and output images

3. CONCLUSION The fundamentals of multirate filters and its

applications in various fields like multirate adaptive filtering for DS/CDMA code acquisition, Kalman filtering for optimal signal reconstruction from noisy sub-band system and lossy compression approach to transmultiplexed images are discussed in this paper. In DS/CDMA code acquisition, it is proposed that a low complexity is achieved using fast acquisition adaptive filtering using the multirate signal processing technique. Unlike the conventional methods, the proposed scheme does not divide the uncertainty region into cells and search the code phase cell by cell. Instead, it uses a two-step procedure for searching. Firstly, it searches the code phase in the whole region with a lower resolution. Secondly, it searches the code phase in a small region with a high resolution. Due to this multirate processing and searching strategy the computational complexity and mean acquisition time can be effectively reduced. Multirate Kalman filtering combines the multichannel representation of sub-band signal with the statistical model of input signal to derive the multirate state-space model for noisy filter bank systems. The signal reconstruction can be formulated as the optimal state estimation with multirate Kalman filtering. Lastly lossy compression approach to transmultiplexed images uses the transmultiplexer for sending the several images over a single transmission line. The transmultiplexer is a structure that combines suitably up-sampled and filtered signals for the transmission by a single channel that uses different algorithms to reduce the high frequency components which causes major error in output images of the transmultiplexer.

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