A Windowing Approach for Designing Critically Sampled Nearly Perfect-Reconstruction Cosine-Modulated Transmultiplexers and Filter Banks

Pilar Martin 1, Fernando Cruz-Roldán 1and Tapio Saramäki2

1 Departamento Teoría de la Señal y Comunicaciones, Universidad de Alcalá, Alcalá de Henares, Madrid, Spain, e-mail: p.martin@uah.es and fernando.cruz@uah.es

2 Institute of Signal Processing, Tampere University of Technology, P. O. Box 553, FIN-33101 Tampere, Finland, e-mail: saram@vip.fi or ts@cs.tut.fi

Abstract

A very fast technique for designing nearly perfect-reconstruction (NPR) critically sampled cosine-modulated M-channel transmultiplexer (TMUX) and filter bank (FB) systems is proposed. This technique is based on using the windowing technique for designing the prototype filter so that its 3-dB cutoff frequency is located at ω = π / (2M). The motivation for this is the observation that if the prototype fil-ter cascaded with itself is a 2Mth filter, then the resulting TMUX (in the case of an ideal channel being a pure delay) or FB system has approximately a perfect-reconstruction (PR) property. This implies that for this cascade the 6-dB cutoff point should be approximately located at ω = π / (2M), and, correspondingly, the 3-dB cutoff frequency of the proto-type filter is located at this angular frequency.

The main advantage of the proposed design scheme is that it significantly lowers the computational complexity when compared with other existing techniques proposed for designing critically sampled cosine-modulated TMUX or FB systems. The PR property is not achieved, but for the result-ing TMUX (FB) systems, the intersymbol interference and crosstalk errors (the amplitude and alias errors) are small. Both of these errors are small enough in practical systems. Several examples are included illustrating these desired properties.

1. Introduction Critically sampled uniform M-channel transmultiplexer (TMUX) systems, as depicted in Fig 1, have been widely studied in the literature [1]−[4] due to their numerous com-munications applications [5]−[7]. As shown in this figure, the role of the synthesis bank is to combine M sub-signals to be transferred through the channel with the transfer function C(z). The role of the analysis filter banks is to separate the output signals of the transmission channel so that the lx̂ [n]’s for l = 0, 1, …, M−1 are approximately delayed versions of the original lx [n]’s.

Traditionally, the design of critically sampled (TMUX) systems is based on the design of critically uniform filter bank (FB) systems as shown in Fig.2. This is due to the fact that in the cases of the ideal channel for a TMUX system and when ignoring the effect of the processing unit for a FB sys-tem, similar properties are achieved for both systems [8].

The most efficient way for designing and implementing these systems is to first start with a linear-phase finite im-pulse response (FIR) prototype transfer function given by

Figure 1. M-channel critically sampled transmultiplexer.

Figure 2. M-channel critically sampled filter bank.

( ) [ ] [ ] [ ] NnnpnNpznpzP nN

n , ,1 ,0for ,

0L==−= −

=∑ (1)

and then to apply a proper cosine modulation scheme. One alternative is to generate the synthesis and analysis transfer functions, respectively, as follows:

( ) [ ] ( ) [ ] nN

nrr

nN

nrr znhzHznfzF −

=

−

=∑∑ ==

00 and (2a)

for 1 0 −≤≤ Mr , where

[ ] [ ] ( ) ( )

−−

−+=

41

2212cos2 ππ r

rNn

MrnMpnf (2b)

and

[ ] [ ] ( ) ( ) . 4

122

12cos2

−+

−+=

ππ rr

NnM

rnpnh (2c)

The main difference of the above equation compare to that given in [1] is that an additional constant M is included in Eq. (2b). This is due to the fact when interpolating by a factor of M the transfer function has to be multiplied by the same factor. This has two benefits. First, it the maximum amplitude value of the prototype filter is approximately equal to unity, then the same is true for those of the analysis and

synthesis filters. Second, the constant of the value 1/M ap-pearing in the input-output transfer functions of the systems of Figs. 1 and 2 can be omitted, as will be done throughout this paper.

For the TMUX system of Fig. 1, the z-transform of the lth-output signal is expressible on terms of the z-transforms of the input signals, denoted by Xk(z) for the kth input signal, as

),()()(ˆ 1

0zXzTzX k

M

klkl ∑

−

== (3a)

where

( ) ( ) ( )mMk

mMmMM

mllk WzFWzCWzHzT 111

1

0)( ∑

−

== (3b)

for k = 0, 1, …, M−1. Here, Tlk(z) is the transfer function be-tween the lth output and the kth input in Fig. 1.

When designing the prototype filter transfer function P(z), it usually assumed that the channel filter is ideal, that is, C(z) ≡ 1. In this case, Eq. (3b) deduces to

( ) ( )mMk

mMM

mllk WzFWzHzT 11

1

0)( ∑

−

== . (4)

The perfect reconstruction (PR) property implies that the output and input sequences are related through [2]

[ ] [ ]Knxnx ll −=ˆ for l = 0, 1, …, M−1, (5a)where

MNK = . (5b)This property is achieved by designing the prototype filter so that the following two conditions are met: 1) K

ll zzT −≡)( for l = 0, 1, …, M−1. 2) 0)( ≡zTkl for k ≠ l, k = 0, 1, …, M−1, and l = 0, 1,

…, M−1. For the FB system of Fig. 2, in turn, the relation between

the output signal x̂ [n] and the input signal x [n] is expressi-ble in the z domain as

( )∑−

=

−+=1

1

20 )()()()(

M

l

Mljl zeXzTzXzTzY π , (6a)

where

∑−

==

1

00 )()()(

M

kkk zHzFzT (6b)

is called the distortion transfer function and determines the distortion caused by the overall system for the unaliased component X(z) of the input signal and

( )∑−

=

−=1

0

2)()(M

k

Mljkkl zeHzFzT π (6c)

for l = 1, 2, …, M−1 are called the alias transfer functions and determine how well the aliased components X(ze−j2π l

/ M) of

the input signal are attenuated. In this case, the PR property implies that in the system of

Fig. 1, when omitting the processing unit, the output and in-put signals are related through [1], [2], [4], [9]

[ ] [ ]Nnxnx −=ˆ . (7)

This property is achieved by designing the prototype filter so that the following two conditions are met: 1) NzzT −≡)(0 . 2) 0)( ≡zTl for l = 1, 2, …, M−1.

For PR systems, the criteria for the prototype filter are very strict. Fortunately, in many practical applications, the PR property can be slightly relaxed, resulting in nearly per-fect reconstruction (NPR) systems. In these systems, the above-mentioned PR conditions should be met only ap-proximately. When practically constructing the overall sys-tems of Figs. 1 and 2, the main goal is that they are generated such that errors caused by the NPR property are hardly no-ticeable by a human being. In the case of audio or speech signals (images), our ears (eyes) serve as the final “referees”.

During the last two decades, numerous methods for de-signing the prototype filter for both PR and NPR systems have been introduced. Also, many structures have been pro-posed. See, e.g., [1]−[17] as well as the references in [1], [2], [4], and [9].

The main purpose of this paper is to further study the properties of NPR systems resulting when using the very fast technique originally proposed in [12] for designing prototype filters for TMUX systems. This technique is based on apply-ing the windowing technique for generating the prototype fil-ter such that its 3-dB cutoff frequency is located at

( )M2πω = . First, the applicability of this technique for generating FB systems is also considered. Second, the prop-erties of the resulting systems are studied more carefully by means of several comparisons, examples, and quantities measuring the differences between the resulting systems and the PR systems.

This paper is organized as follows. Section 2 proposes the basic approach, based on using windowing technique, for designing the prototype filter for both cosine-modulated TMUX and FB systems. In Section 3, several quantities are considered for measuring various errors cased by the NPR property. In Section 4, these quantities are used for evaluat-ing the performances of the systems resulting when applying the proposed design scheme. In addition, an example is in-cluded illustrating the performance of a resulting TMUX sys-tem in a practical application. Finally, concluding remarks are drawn in Section 5.

2. Proposed Design Scheme for the Prototype Filter This section reviews the efficient synthesis method proposed in [12] for designing prototype filters for NPR TMUX and FB systems. 2.1 Start-Up Idea for the Proposed Design Scheme

The key idea behind this technique is based on the follow-ing two observations made in [10]. First, if the prototype transfer function is designed to have a narrow transition bandwidth and high stopband attenuation, then the resulting TMUX and FB systems have good NPR properties. Second, most importantly, the prototype filter generating a PR TMUX or FB system is characterized by the property that if it is cascaded with itself, then the resulting filter is 2Mth band linear-phase FIR filter. According to the discussion in [18], this fact implies first that the passband and stopband edges of amplitude response of this cascade as given by [cf. Eq. (1) ]

( ) [ ]2

0

2 ωω jn

N

n

j enpeP −

=∑= (8)

are approximately related through ( ) ( )Mp 21 πρω −= (9a)

and ( ) ( )Ms 21 πρω += , (9b)

where ρ is a positive number. The second implication is that for any value ω = ω0, the following condition is valid:

( )( )( ) ( )( ) .11

2

1

21 00 =+ ∑∑=

+

=

+−M

k

MkjM

k

Mkj ePeP ωπωπ (10)

When selecting ω = ω0 = π/(2M) this implies that

( )( ) ( ) ( )( ) .21

1

22122 ∑=

+−=M

k

MkjMj ePeP ππ (11)

Hence, if the following properties are satisfied: 1) The frequency points ω = (2k+1)/(2M) for k = 1, 2, …, M in

the stopband region of the prototype filter, that is, in the region [ωs, π] with ωs given by Eq. (9c). This implies that ρ ≤ 2.

2) The amplitude response of the prototype filter cascaded with itself, as given by Eq. (8), provides enough attenua-tion at these points.

then the amplitude value of the prototype filter is very close to the value of 21 at ω = π/(2M).

2.2 Proposed Design Scheme Based on this fact, given N, the order of the prototype filter and M, the number of channels, the proposed design scheme based on windowing can be carried out as follows: Step 1: Select a proper window function w[n] being nonzero

for 0 ≤ n ≤ N and satisfying w[N−n] = w[n]. Step 2: Generate the impulse-response of the prototype

transfer function P(z), as given by Eq. (1), as [ ] [ ] [ ] , , ,1 ,0for Nnnwnpnp id L=⋅= (12a)

where [ ] ( )[ ] ( )[ ]πω 22sin NnNnnp cid −−= (12b)

are the impulse-response values of the ideal filter with cutoff frequency ωc. Step 3: Determine ωc so that the amplitude response of the

prototype achieves the value of 21 at ω = π/(2M). The value of ωc can be found by using any line search al-

gorithm, thereby making the overall synthesis extremely fast.

3. Performance Measures The NPR TMUX and FB systems of Figs. 1 and 2 resulting when using prototype filters designing the simple technique of the previous section do not suffer from any phase distor-tion. This is valid when it is assumed that the channel is ideal in Fig. 1 and the effects of the processing unit are ignored in Fig. 2. However, there exit various distortions between the input(s) and output(s) due to the NPR property. This section considers proper quantities for measuring these distortions.

3.1 Quality Measures for FB Systems For the FB system of Fig. 2, there exist two main distortions. The first one is the amplitude distortion between the output and input signals. This distortion can be conveniently meas-ured by determining the maximum and minimum values the amplitude response corresponding to T0(z), as given by Eq. (6b), as follows:

( ) 201 11 δδ ω −≤≤+ jeT . (13)

As the actual distortion measure the following quantity:

21 δδ +=ampE . (14)is used later. As a measure of the total alias distortion, the following quantity is used:

[ ]( ){ }ω

πω

jaliasalias eTE

,0max∈

= , (15a)

where

( ) ( )∑−

==

1

1

2M

k

jk

jalias eTeT ωω (15b)

is the total aliasing distortion and the amplitude responses in the summation of the above equation correspond to the trans-fer functions given by Eq. (6c).

For many techniques proposed for optimizing the proto-type filter, its stopband edge ωs is pre-specified according to Eq. (9b). For the proposed design scheme this is not true. Therefore, ωs for the resulting prototype filter is measured and the roll-off factor ρ, determined according to Eq. (9b), is used as the third quality measure. 3.2 Quality Measures for TMUX Systems For the PR TMUX system of Fig. 1 with the ideal channel, the lth output sequence is a delayed version of the the lth in-put sequence according to Eqs. (5a) and (5b). For the NPR system, there exist the following main errors. First, the kth input signals for k ≠ l has effects on the lth output. This is called inter-channel interference (ICI) and can be measured in the frequency domain by the quantity

[ ]( )

= ∑

−

≠=∈−≤≤

1

,0

2

,010maxmax

M

lkk

jlk

Mlici eTE ω

πω. (16)

Second, the delay for data samples passing through the single channel is not exactly K, as given by Eq. (5b), resulting in the fact that the transmitted sample does not occur at the out-put at the right time instant. In addition, also the other trans-mitted samples have an effect on the output sample at this time instant. This error is called intersymbol interference (ISI) and can be measured conveniently in the time domain by the following error quantity:

[ ] [ ]( )

−−= ∑−≤≤ n

llMl

isi KnntE 2

10max δ (17)

Here, tll[n] is the impulse response between the lth output and input sequences and δ[n] is the unit sample.

Also the following error quantities are used:

[ ] [ ]

−=

−≤≤nxnxMaxerror ll

nMlˆmaxlog20max 10

10 (18)

and

[ ]( ) [ ] [ ]( )

−= ∑∑

−≤≤ nll

nl

MlnxnxnxSNR 22

1010

ˆlog10max . (19)

4. Simulation Results This section studies the effects of N, the order of the proto-type filter, M, the number of channels, and the selection of the window on the resulting NPR systems. It is also shown, by means of an example, that the proposed technique results in slightly better TMUX and FB systems than those achiev-able using a similar technique introduced in [16].

Various prototype filters have been designed for several values of N and L using three windows, namely, the Black-man, Kaiser, and Saramäki windows [19]. It has been ex-perimentally observed that NPR TMUX and FB systems with very good properties are achieved by selecting the adjustable parameter of the Kaiser and Saramäki windows such that the minimum stopband attenuation of the resulting prototype fil-ter is approximately 100 dB. Various quality measures, as in-troduced in Section 3, are given in Tables I, II, and III for various 4-channel, 8-channel, and 16-channel FB and TMUX systems, respectively.

A very similar technique for designing prototype filters has been proposed by Lin and Vaidyanathan in [16]. In order to compare the proposed synthesis scheme with this tech-nique, consider the example included in [16]. In this exam-

ple, M = 32 channels are used, the order of the prototype fil-ter is N = 466, the adjustable parameter of the Kaiser window is selected to give approximately a 100-dB attenuation for the prototype filter. Figures 3 and 4 compare the perform-ances the TMUX and FB systems resulting when using the proposed technique and the Lin-Vaidyathan technique. It can be seen from Fig. 3 that the resulting TMUX systems are practically the same. However, as seen from Fig. 4, the pro-posed technique slightly reduces the maximum amplitude deviation from unity for the unaliased component (0.0018 compared to 0.0020) in the resulting FB systems.

In order to compare the efficiency of these two TMUX systems in a practical application, an electrocardiogram sig-nal has been used as the input for the first channel, whereas there are no input signals to the remaining channels. It has been assumed that the power (energy per sample) of this sig-nal is equal to –15.8721 dB, the transmission channel has an AWGN impairment with noise power equal to –19.9933 dB, no channel equalization has been performed. These condi-tions are similar to those used in VDSL environments. The solid and dashed lines in Figs. 5 show the first input and out-put sequences, respectively, for the proposed and Lin-Vaidyanathan techniques. More detailed quality measures for these two TMUX systems are given in Table IV, showing that the proposed technique gives slightly better results.

Table I. 4-channel transmultiplexers and filter banks designed by the proposed method

N K=N/2M Window ρ Eamp Ealias (dB) EICI (dB) EISI (dB) 64 8 Blackman 0.8356 0.0016 −97.0281 −97.0499 −66.7800 64 8 Kaiser 0.9549 0.0037 −106.0701 −108.1598 −56.0070 64 8 Saramäki 0.9522 0.0036 −102.4349 −102.4349 −56.3366 96 12 Blackman 0.5545 0.0019 −103.8314 −110.0339 −67.2522 96 12 Kaiser 0.6334 0.0033 −104.3283 −104.3283 −59.4184 96 12 Saramäki 0,6319 0.0035 −102.6998 −102.6998 −60.1737

128 16 Blackman 0.4150 0.0019 −109.1875 −119.1823 −68.4681 128 16 Kaiser 0.4739 0.0036 −106.8078 −106.8078 −61.7293 128 16 Saramäki 0.4727 0.0035 −105.8325 −105.8325 −60.1783

Table II. 8-channel transmultiplexers and filter banks designed by the proposed method

N K=N/2M Window ρ Eamp Ealias (dB) EICI (dB) EISI (dB) 64 4 Blackman 1.6692 0.0054 −48.5692 −90.8108 −50.7769 64 4 Kaiser 1.9033 0.0118 −41.6208 −92.2248 −44.0331 64 4 Saramäki 1.8982 0.0112 −42.0985 −90.9870 −44.5892 96 6 Blackman 1.1090 0.0017 −88.5438 −88.5438 −65.0391 96 6 Kaiser 1.2667 0.0035 −87.7372 −105.5739 −55.8483 96 6 Saramäki 1.2634 0.0037 −87.8525 −102.8346 .55.0165 128 8 Blackman 0.8300 0.0015 −96.9505 −97.0023 −66.7478 128 8 Kaiser 0.9476 0.0033 −100.3057 −100.3057 −57.5924 128 8 Saramäki 0.9452 0.0038 −98.6888 −98.6888 −55.9667

Table III. 16-channel transmultiplexers and filter banks designed by the proposed method

N K=N/2M Window ρ Eamp Ealias (dB) EICI (dB) EISI (dB) 64 2 Blackman 2.9661 0.1064 −22.5434 −26.9199 −23.7833 64 2 Kaiser 3.1973 0.1132 −22.0594 −25.0879 −23.3184 64 2 Saramäki 3.2185 0.1170 −21.9656 −24.9693 −23.1546 96 3 Blackman 2.1873 0.0380 −31.5911 −55.3548 −32.9065 96 3 Kaiser 2.4724 0.0537 −28.4338 −43.1023 −29.7515 96 3 Saramäki 2.4677 0.0528 −28.5715 −43.4864 −29.8907 128 4 Blackman 1.6580 0.0016 −48.5799 −90.8795 −50.8803 128 4 Kaiser 1.8888 0.0034 −42.1487 −90.4121 −44.6405 128 4 Saramäki 1.8846 0.0036 −42.3440 −89.7817 −44.8614

(a)

(b)

(c)

(d)

Figure 3. Various responses for 32-channel-TMUX systems designed using the proposed technique (solid line) and the Lin-Vaidyanathan technique (dashed line). (a) Amplitude responses for the prototype filters. (b) Synthesis filter bank. (c) Total aliasing distortions as given by Eq. (14b). (d) Amplitude responses between the first input and output sequences.

(a)

(b)

Figure 4. Amplitude responses |T0(ejω)| for the unaliased component for 32-channel FB systems. (a) Proposed technique. (b) Lin-Vaidyanathan technique. Because of the periodicity of |T0(ejω)| is π/32, only a part of the response is shown.

(a)

(b)

Figure 5: The earliest and retarded lines show first input and output sequence of TMUX system of Fig. 1 in an example practical application, respectively. For more details, see the text. (a) Proposed technique. (b) Lin-Vaidyanathan technique.

Table IV. Quality measures of two techniques for generating prototype filters for an example practical application. For

more details, see the text.

M Method Maxerror (dB) SNR (dB) 32 Proposed -69.7457 31.2470 32 Lin-Vaidyanathan -69.4564 30.9577

5. Conclusions This paper proposed an extremely fast technique for de-

signing prototype filters for nearly perfect-reconstructing (NPR) cosine-modulated maximally decimated transmulti-plexers and multirate filter banks. The technique is based on the windowing technique for finding the prototype filter such that its 3-dB cutoff frequency is located at ω = π/(2M). Simulations indicated that good candidate windows result-ing in very good NPR systems are the Blackman window as well as the Kaiser and Saramäki windows, where the ad-justable parameter is selected to give approximately a 100-dB stopband attenuation for the prototype filter.

Among the above-mentioned three windows, the Blackman window resulted for some reason in transmulti-plexers and filter banks with the properties closest to the PR systems. Therefore, the future is devoted to finding a window among the existing windows in the literature or even to generating a new window for this purpose so that the errors caused by the nearly perfect reconstruction prop-erty will be reduced even further.

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[9] H. S. Malvar, Signal Processing with Lapped Transforms. Norwood, MA: Artec House, 1992.

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[12] F. Cruz-Roldán, P. Amo-López, S. Maldonado-Bascón, and S. S. Lawson, "An efficient and simple method for designing pro-totype filters for cosine-modulated pseudo-QMF banks," IEEE Signal Processing Letters, Vol. 9, No. 1, pp. 29-31, January 2002.

[13] T. Saramäki, “A generalized class of cosine-modulated filter banks,” in Proc. Workshop on Transforms and Filter Banks, Tampere, Finland, Feb. 1998, pp.336-365.

[14] T. Saramäki and R. Bregović, “An efficient approach for de-signing nearly perfect-reconstruction cosine-modulated and modified DFT filter banks,” in Proc. of IEEE Int. Conf. Acoust. Speech, Signal Processing, Salt Lake City, Utah, May 2001 vol. VI, pp. 3617-3620.

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[16] Y.-P. Lin and P. P. Vaidyanathan, “A Kaiser window approach for the design of prototype filters of cosine modulated filter banks,” IEEE Signal Proc. Letters, vol. 5, pp. 132-134, June 1998.

[17] D. Creusere and S. K. Mitra, “A simple method for designing high-quality prototype filters for M-band pseudo-QMF banks", IEEE Trans. Signal Processing, vol. 43, No 4, April 1995, pp. 1005-1007.

[18] T. Saramäki and Y. Neuvo, “ A class of FIR (Nth-band) Ny-quist filters with zero intersymbol interference," IEEE Trans. Circuits Syst., vol. CAS-34, pp. 1182-1190, Oct. 1987.

[19] T. Saramäki, “Finite impulse response filter design”, Chapter 4 in Handbook for Digital Signal Processing , edited by S. K. Mi-tra and J. F. Kaiser. John Willey & Sons, NY, 1993.