結合類比前端特性之訊號整形對稱式FIR濾波器設計

行政院國家科學委員會專題研究計畫 成果報告

結合類比前端特性之訊號整形對稱式 FIR 濾波器設計 研究成果報告(精簡版)

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執 行 期 間 ： 98 年 08 月 01 日至 99 年 07 月 31 日 執 行 單 位 ： 國立臺灣科技大學電機工程系

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計畫參與人員： 碩士班研究生-兼任助理人員：盧欣伯 碩士班研究生-兼任助理人員：吳華祥 碩士班研究生-兼任助理人員：蔡佩容 碩士班研究生-兼任助理人員：柯宜欣 碩士班研究生-兼任助理人員：周亞忻 博士班研究生-兼任助理人員：夏偉鈞 博士班研究生-兼任助理人員：何永祥

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中 華 民 國 99 年 08 月 16 日

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□期中進度報告 結合類比前端特性之訊號整形對稱式 FIR 濾波器設計

計畫類別：5 個別型計畫 □ 整合型計畫 計畫編號：NSC 98－2221－E －011 －085－

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中 華 民 國 九十九 年 八 月 十七 日

The Design of Hybrid Symmetrical-FIR/Analog Pulse-Shaping Filters

NSC 98-2221-E-011-085

Chia-Yu Yao (e-mail: [email protected]) Department of Electrical Engineering

National Taiwan University of Science and Technology Taipei, Taiwan

Abstract— In this paper, we propose a method for designing linear-phase (LP) square-root (SR) FIR filters incorporating with the effects of analog parts in a communication system.

In the proposed method, the nonlinear phase distortion caused by the analog filters is compensated first by inserting an FIR filter that is the time reversal of the discrete-time equivalence of the analog parts. Next, the proposed method employs a constrained optimization algorithm to design the coefficients of the desired FIR filter. Unlike the conventional design method for LP FIR filters, the proposed method relaxes the passband equiripple constraint imposed by the Remez exchange algorithm.

The proposed method takes the filter’s stopband attenuation, the system’s tolerable ISI, and the opening of the eye pattern into account simultaneously. Design examples show that using these constraints can result in a more robust eye opening and thus a more robust ISI performance in the presence of receiver clock jitter.

Index Terms— square-root filter, Nyquist filter, linear-phase FIR filter, inter-symbol interference, eye pattern

I. INTRODUCTION

A

[6], then getting the matched SR transmitter and receiver filters by performing a spectral factorization on the Nyquist filter polynomial. In this way, the SR filters usually have asymmetrical coefficients.

LetN denote the filter length of the SR filters. Fig. 1 shows the block diagram of a band-limited digital communication system. For convenience, let ht[n] and hr[n] = ht[−n], n =

−(N −1)/2, −(N −3)/2, . . . , (N −1)/2, represent transmitter and receiver FIR filter coefficients, respectively. Notably, we allow indices to be . . . , −3/2, −1/2, 1/2, 3/2, . . . when N is even. Sm(s) and An(s) represent the transfer functions of the analog smoothing filter and the analog anti-aliasing filter, respectively. Having the analog parts in the system, we are facing a hybrid FIR-filter/analog-filter pulse-shaping problem.

The designing ofht[n] and hr[n] must consider to compensate the effects of the analog parts.

In [7], Coleman and Lytle design the optimal autocorrelation sequence, h[n] = ht[n] ∗ hr[n], according to the magnitude

Fig. 1. Block diagram of a band-limited digital communication system.

responses ofSm(s) and the digital-to-analog converter (DAC) at the transmitter, andAn(s) at the receiver. Their FIR filters, ht[n] and hr[n], operate at 2 samples per baud. The problem of such a low oversampling ratio is described as follows: The mainlobe of the spectrum of the FIR filter will extend to the transition band of the analog filters. However, the transition band of an analog filter is sensitive to the variation of the analog component values caused by temperature, aging, etc..

This will degrade the performance of the design in [7]. Another problem of the design of [7] is that they only consider the magnitude response of an analog filter. However, the phase response of an analog filter is generally not linear. This nonlinear phase distortion will disperse the signal waveform in time domain such that compensating only the magnitude distortion can not resolve the dispersion completely. This situation is even worse at low over-sampling ratio that the design of [7] employs.

A better way to cope with the component value variations of analog parts is to employ higher oversampling ratio [8], [9]. A design with high oversampling ratio allows the mainlobe of the spectrum of the FIR filter to completely locate in the (relatively flat) passbands of the analog filters. Since the passband of an analog filter is relatively flat, its sensitivity to component value variations is relatively small compared with the sensitivity of the transition band. Therefore, in this paper, we study the cases with oversampling ratio T > 2. In [8], all examples employ oversampling ratioT = 4. However, [8] does not consider the zero-order hold feature of a DAC. Also, [8] implicitly assumes symmetry between the smoothing filter and the antialiasing filter. Hence, the formulation in [8] for compensating analog parts is not complete.

On the other hand, it is well known that there are several advantages of an FIR filter possessing symmetric coefficients.

For example, the number of multipliers is about halved and the phase response becomes linear. These advantages make designing symmetric SR filters ht[n] and hr[n] attractive.

(In this paper, we consider symmetric matched filter pair, so ht[n] = hr[n].)

Without compensating the effects of analog parts, in the case of T = 2, [10] developed a strategy incorporating with the Remez exchange algorithm (e.g., McClellan et al.’s program [11]) to design linear-phase (LP) FIR transmitter and receiver filters. Since the Remez exchange algorithm is used, a passband equiripple constraint is imposed in the method of [10]. However, the passband equiripple constraint can be relaxed for SR filters such that the filter length can be slightly shortened, while the ISI performance and stopband attenuation are still remained [12]. The other problems of the method in [10] are: First, the oversampling ratioT = 2. Thus, the overall performance is sensitive to analog component value variation in a communication system. Second, the effects of analog parts are ignored. Thus, if we put the FIR filters designed by the method of [10] in the Fig. 1 system, the performance will inevitably degraded.

Another work about LP FIR SR filters was recently given in [13]. In [13], Farhang-Boroujeny uses a ‘soft penalty function’ to help people design the LP FIR SR coefficients.

His soft penalty function is not directly related to the practical specifications of a pulse-shaping FIR filters. Even though in the design iterations the value of the penalty function decreases step by step, before the design is complete and the performance indices are checked, we cannot know if the design meets the specifications or not. Also, the method given in [13]

does not compensate the effects of the analog parts. Therefore, the same disadvantage as the method of [10] remains.

In [12], [14], a recognition of the “tolerable ISI” is pro- posed. The idea is that zero ISI may not always be necessary in practice. The tolerable ISI should depend on the operating point (BER vs SNR) of a band-limited digital communication system. As long as the ISI is sufficiently small (not necessarily zero), the BER performance of a communication system can be acceptable at the operating point. A small but non-zero ISI can give us an increased degree of freedom in designing the SR filters. Thus, the tolerable ISI can be taken as an input parameter for designing SR filters.

In practical implementations of band-limited communica- tion systems, since there exists interference and noise in the channel, the recovered receiver clock signal must suffer from some timing jitter. Thus, the synchronization between the transmitter and the receiver cannot be perfect. This timing jitter will deteriorate the BER performance if the opening of the eye pattern is not sufficiently large. Therefore, [15], [16] proposed several improved Nyquist pulses that possess wider eye openings than the raised-cosine pulse has. These improved Nyquist pulses are continuous-time pulses and non- causal. Therefore, directly applying improved Nyquist pulses to practical applications is impossible. We need to convert the improved Nyquist pulses into FIR form by sampling and trun- cating them. Truncating leads to nonzero stopband spectrum

and that may violate the frequency domain specifications of the design. On the other hand, what we obtain from sampling and truncating the improved Nyquist pulses are discrete-time Nyquist filter approximations. The existence of SR solutions from spectral factorization of the approximated Nyquist pulses is not guaranteed. If the SR solutions exist, they are generally asymmetric.

Therefore, to cope with the designing of symmetric SR pulse-shaping filter that compensates the effects of analog parts with hybrid eye-opening/magnitude-response/tolerable- ISI constraints, we propose a new design method. The pro- posed method does not employ the Remez exchange algo- rithm. Hence, the passband equiripple constraint is relaxed.

A measurement for the eye-opening robustness is developed such that we can take the eye opening into account in problem formulation. The proposed method considers three kinds of constraints: the frequency domain constraint, the tolerable ISI constraint, and the eye-opening constraint.

The last thing we need to mention is that the nonlinear phase response of analog parts has to be compensated. We construct an equivalent FIR channel model for an analog channel that includs the DAC, the analog filters, and the analog-to-digital converter (ADC). By cascading the time reversal of the FIR channel model in the system, the nonlinear phase response caused by the analog filters is approximately compensated.

However, this will introduce an extra magnitude distortion to the signal. Fortunately, the extra magnitude distortion can be taken care of by properly formulating the design equations of the SR pulse-shaping filters.

The paper is organized in the following manner: In Sec- tion II, we present a FIR channel model and the way of compensating the nonlinear phase response of analog parts.

In Section III, the basic time-domain and frequency-domain responses are formulated. In Section IV, the formulation of ISI is reviewed and a measurement of the eye-opening robustness is introduced. In Section V, the formulation of the proposed design method is presented. Numerical design examples are given in Section VI. Finally, a summary of conclusions is given in Section VII.

II. THEANALYSIS OF ANFIR CHANNELMODEL

Assume that the channel in Fig. 1 is a wide-band linear- phase channel such that the channel model shown in Fig. 2 can be used for analysis. Leta[m] denote the m-th transmitted data sample. The DAC is modeled as a zero-order hold block in Fig. 2 whereT0represents the sampling period. The ADC is simply modeled as a sampler that takes samples at multiple instants nT0+ ǫ where ns are non-negative integers and ǫ denotes the timing offset between the transmitter and the receiver. We assume that 0 ≤ ǫ < T0 for convenience. Let hsm(t) and han(t) be the impulse responses of Sm(s) and An(s), respectively.

Denote

hana(t) = hsm(t) ∗ han(t)

where ∗ is the continuous-time convolution operation. Since

Fig. 2. A baseband channel model.

hana(t) is causal, r(t) can be derived as r(t) =

∞

X

m=0

a[m]

Z t−mT0

max(0,t−(m+1)T0)

hana(τ ) dτ.

Denote r[n] = r(nT0+ ǫ) and let ˆh^t_t²₁ =

Z t2

max(0,t1)

hana(τ ) dτ, max(0, t1) ≤ t2, then

r[n] =

∞

X

m=0

a[m]ˆh^(n−m)T_(n−m−1)T^0+ǫ_0+ǫ. (1)

Sincehana(τ ) is causal, ˆh^(n−m)T_(n−m−1)T^0+ǫ

0+ǫ= 0 when (n−m)T0+ ǫ ≤ 0, which implies that only those terms with m ≤ n in (1) are nonzero. Therefore, (1) can be rewritten as

r[n] =

n−1

X

m=0

a[m]ˆh^(n−m)T_(n−m−1)T^0+ǫ

0+ǫ+ a[n]ˆh^ǫ₀ (2) Next, denote

h^ǫ_Π[n] =

( ˆh^ǫ₀, n = 0

ˆh^nT_(n−1)T^0+ǫ_0+ǫ, n = 1, 2, 3, . . . . Thus, (2) can be written as

r[n] = a[n] ∗ h^ǫ_Π[n]

where∗ represents the discrete-time convolution operation.

Hence, the channel in Fig. 2 can be modeled as a discrete- time convolution of the transmitted data samples a[n], n = 0, 1, 2, . . . , and the channel’s equivalent discrete-time impulse response h^ǫ_Π[n]. Notably, hana(t) is essentially time limited in practice [8], so is h^ǫ_Π[n]. Therefore, the channel can be modeled as an FIR channel in practice.

In general, since the sampling rate 1/T0 and the analog filtersSm(s) and An(s) are known a priori, we can use them to construct an M -tap FIR channel equivalence ˜h⁰_Π[n], n = 0, 1, . . . , M − 1, by truncating the first M samples of h⁰_Π[n].

Next, let

hins[n] = ˜h⁰_Π[M − 1 − n]. (3) Thus, ˜h⁰_Π[n] ∗ hins[n] will possess a linear phase response.

This implies that the nonlinear phase distortion introduced by Sm(s) and An(s) can be approximately compensated by inserting hins[n] in the transmitter or receiver.

It is noted that the values ofh^ǫ_Π[n] depend on ǫ. Therefore, different timing offset ǫ leads to different h^ǫ_Π[n]. The phase equalization FIR filter hins[n] does not correct the sampling timing offset, it just equalize the phase response of the system.

In order to obtain an optimum sampling timing, a clock recovery subsystem is required, which is beyond the scope of this paper.

III. THEFORMULATIONS OFBASICTIME-DOMAIN AND

FREQUENCY-DOMAINRESPONSES

By inserting hins[n] in the system, the previous section has shown that the phase response of the analog parts can be approximately linearized. Next, denote

hD[n] = ˜h⁰_Π[n] ∗ hins[n].

The length ofhD[n] is 2M − 1. For convenience, we assume thathD[n] spans across n = −M +1, . . . , −1, 0, 1, . . . , M −1.

Recall that N denotes the length of the SR filter. Since we discuss symmetrical SR filters andht[n] = hr[n], h[n] = ht[n] ∗ hr[n] = ht[n] ∗ ht[n]. Then for |n| ≤ N − 1

h[n] =



























N −1 2

X

k=1

ht[k]

ht[n + k] + ht[n − k]

+ ht[0]ht[n], N ∈ odd

N −1 2

X

k=1/2

ht[k]

ht[n + k] + ht[n − k]

, N ∈ even (4) whereht[l] = 0 when |l| > (N − 1)/2.

We can further employ the technique introduced in [13] to express (4) in the matrix form. Let

h_t=

 ( ht[0] ht[1] · · · ht[^{N −1}₂ ] )^T, N ∈ odd ( ht[¹₂] ht[³₂] · · · ht[^{N −1}₂ ] )^T, N ∈ even

(5) and let S_n,|n| ≤ N − 1, be an N × N matrix whose kth-row, lth-column element is given by [13]

(Sn)k,l=

 1, k − l = n

0, otherwise. (6)

Denote Il and Jl as the l × l identity matrix and the l × l antidiagonal matrix, respectively. Let

E=















0 J_{N −1}

2

I_{N +1}

2

!

, N ∈ odd J_N

2

I_N

2

!

, N ∈ even

(7)

Then (4) can be rewritten as, for|n| ≤ N − 1, h[n] = htTE^TS_nEh_t. Denote ˆS_n= E^TS_nE, then

h[n] = h^tTˆS_nh_t. (8) The complete impulse response of the system, including the equivalent FIR channel model of the analog parts and the phase-equalizing FIR filterhins[n], can be written as

hc[n] =

N −1

X

m=1

h[m]

hD[n + m] + hD[n − m]

+ h[0]hD[n]

=

N −1

X

m=1

h_t^TˆS_mh_t

hD[n + m] + hD[n − m]

+h^tTˆS₀h_thD[n] (9)

where|n| ≤ N + M − 2 and hD[l] = 0 when |l| > M − 1. It is noted that bothh[n] and hD[n] are symmetrical, so is hc[n].

It is understood [5] that hc[0] = 1/T for Nyquist filters.

Therefore,

N −1

X

m=1

2h^tTˆS_mh_thD[m] + h^tTSˆ₀h_thD[0] = 1

T (10)

Next, we will examine the frequency domain constraints.

The zero-phase frequency response of hc[n] at ω = π/T should be equal to1/2 [10]; i.e.,

N −1ˆ

X

n=1

"_{N −1} X

m=1

h_t^TˆS_mh_t

hD[n + m] + hD[n − m]

+htTˆS₀h_thD[n]

# cosnπ

T + 1 T =1

2. (11)

The magnitude response of the symmetrical SR filter can be expressed in the following manner. Denote A(ω) = (1 2 cos ω 2 cos 2ω · · · 2 cos^{N −1}₂ ω) when N is odd and A(ω) = 2(cos^ω₂ cos^3ω₂ · · · cos^{N −1}₂ ω) when N is even.

Then the zero-phase frequency response of the SR filter can be expressed as

Ht(ω) = A(ω)ht (12)

As mentioned in Section I, we do not need to constrain the passband-ripple size for designing SR filters. What matters in frequency domain is the frequency response of hc[n] at ω = π/T and the stopband attenuation (SBATT) of Ht(ω).

The stopband edgeωsof an SR filter is specified by the over- sampling factorT and the roll-off factor α as

ωs=(1 + α)π

T .

IV. THEFORMULATION OFISIANDEYE-OPENING

ROBUSTNESS

In this section, we express the ISI and the eye-opening robustness in terms of the SR filter coefficients. Let

N = N + M − 1.ˆ The formula for calculating the ISI [17] is:

ISI=

⌊^{N −1ˆ}

T ⌋

X

j=1

h^c[±jT ]  

|hc[0]| . (13)

Since hc[n] is symmetrical, substituting hc[0] = 1/T and (9) into (13), we obtain

ISI 2T =

⌊^{N −1ˆ}

T ⌋

X

j=1

N −1

X

m=1

h_t^TˆS_mh_t

hD[jT + m] + hD[jT − m]

+ht

TSˆ₀h_thD[jT ]     

(14) LetγISIdenote the tolerable ISI. To derive a constraint about the ISI, we introduce⌊( ˆN −1)/T ⌋ non-negative slack variables

0 2 4 6 8 10 12

−2

−1 0 1 2

(a)

0 2 4 6 8 10 12

−1.5

−1

−0.5 0 0.5 1 1.5

(b)

Fig. 3. Two discrete-time eye patterns with T= 6 result from two different Nyquist filters possessing the same ISI.

(aj, j = 1, 2, . . . , ⌊( ˆN − 1)/T ⌋) to relax the absolute values in (14). For j = 1, 2, . . . , ⌊( ˆN − 1)/T ⌋, let

± (_{N −1}

X

m=1

h_t^TSˆ_mh_t

hD[jT + m] + hD[jT − m]

+h^tTSˆ₀h_thD[jT ] )

≤ aj (15)

The constraint about ISI can now be written as

⌊^{N −1}_T ⌋

X

j=1

aj≤ γISI

2T . (16)

Next, let us develop a measurement for the eye-opening robustness of an eye pattern. Consider the two normalized discrete-time eye patterns shown in Fig. 3, where the over- sampling ratioT is six. Hence, the time indices 0, 6, and 12 correspond to perfectly synchronous sampling instants. These two eye patterns result from two different Nyquist filters that possess the same ISI at a multiple ofT . One can see that the eye shown in Fig. 3(b) opens wider than that shown in Fig.

3(a).

Since system performance in the presence of timing jitter is of great interest, the time indices one step away from the perfectly synchronous sampling instants are used to measure the eye-opening robustness (as is indicated in Fig. 3). The opening of eye depends on how much interference from hc[kT + 1], k 6= 0, intrudes on hc[1]. Since hc[−kT − 1] = hc[kT + 1] for all k, it is equivalent that the interference intruding on hc[1] comes from the samples of hc[kT ± 1], k > 0, as is depicted in Fig. 4. Notably, since hc[1] > 0 in practice, we do not need to worry about the sign of hc[1] in the following formulations. In a similar manner of defining ISI and considering the worst case scenario, we define the

Fig. 4. The sources of interference to hc[1].

measure of eye-opening robustness as

EYE= hc[1]

⌊^Nˆ

T⌋

X

j=1

h^c[jT − 1]

  +

⌈^{N −1ˆ}

T ⌉−1

X

j=1

h^c[jT + 1]

. (17)

A larger EYE value corresponds to a better eye opening. We note that if T = 2, both the numerator and the denominator of (17) contain the term hc[1]. In this case, even the maxi- mum value of EYE is still poor. Therefore, the eye-opening technique presented in this paper applies only for cases where T > 2, which coincides with our argument stated in Section I.

It is also noted that the upper index of the second summation in the denominator of (17) is ⌈( ˆN − 1)/T ⌉ − 1 in order that the summation will not contain the term hc[ ˆN ] if ˆN − 1 is divisible by T .

In a manner similar to the ISI case, we express EYE in terms of ht. We introduce⌊ ˆN /T ⌋ + ⌈( ˆN − 1)/T ⌉ − 1 slack varables,bj,j = 1, 2, . . . , ⌊ ˆN/T ⌋, and cj,j = 1, 2, . . . , ⌈( ˆN − 1)/T ⌉ − 1, and write

± (_{N −1}

X

m=1

h_t^TSˆ_mh_t

hD[jT − 1 + m]

+hD[jT − 1 − m]

+ htTˆS₀h_thD[jT − 1]

)

≤ bj, (18)

± (_{N −1}

X

m=1

h_t^TSˆ_mh_t

hD[jT + 1 + m]

+hD[jT + 1 − m]

+ htTˆS₀h_thD[jT + 1]

)

≤ cj, (19)

and

hc[1] =

N −1

X

m=1

htTSˆ_mht

hD[m + 1] + hD[m − 1]

+htTˆS₀h_thD[1].

Similar to the ISI case, we denoteγEYEas the specification of the eye opening. Then, the eye opening constraint can be

expressed as

⌊^Nˆ

T⌋

X

j=1

bj+

⌈^{N −1ˆ}

T ⌉−1

X

j=1

cj≤

N −1

X

m=1

h_t^TSˆ_mh_t

hD[m + 1] + hD[m − 1]

+ htTSˆ₀h_thD[1]

γEYE

(20) It is noted that in the above formulations, we keep all equations and inequlities in the first-order or the second-order form of ht. On the other hand, the formulation of [13] is in the fourth-order form of ht.

V. THEDESIGNPROBLEMFORMULATION

Having Expressedhc[0], the frequency responses, ISI, and EYE, in terms of of ht, in Eqs. (10)–(20), we can now formulate our design problem. Letηmax denote the specified gain limit in stopband and η denote the achieved maximum stopband gain (in linear scale) of the FIR SR filter. The design problem becomes:

minimizeη (21)

subject to

±A(ω)h^t≤ η, ω ∈ [ωs, π]

N −1ˆ

X

n=1

cosnπ T

"_{N −1} X

m=1

h_t^TˆS_mh_t

hD[n + m]

+hD[n − m]

+ h^tTˆS₀h_thD[n]

#

= 1 2− 1

T

N −1

X

m=1

2htTSˆ_mh_thD[m] + htTˆS₀h_thD[0] = 1 T

⌊^{N −1ˆ}_T ⌋

X

j=1

aj≤ γISI

2T

± (_{N −1}

X

m=1

h_t^TSˆ_mh_t

hD[jT + m]

+hD[jT − m]

+ htTSˆ₀h_thD[jT ] )

≤ aj, j = 1, 2, . . . , ⌊( ˆN − 1)/T ⌋

⌊^N_T^ˆ⌋

X

j=1

bj+

⌈^{N −1ˆ}_T ⌉−1

X

j=1

cj≤

N −1

X

m=1

h_t^TˆS_mh_t

hD[m + 1] + hD[m − 1]

+ htTˆS₀h_thD[1]

γEYE

± (_{N −1}

X

m=1

h_t^TSˆ_mh_t

hD[jT − 1 + m] + hD[jT − 1 − m]

+htTSˆ₀h_thD[jT − 1]

)

≤ bj, j = 1, 2, . . . ⌊ ˆN /T ⌋

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−60

−50

−40

−30

−20

−10 0 10

Normalized Frequency (cycle/sample)

Normalized Magnitude Response

ht(n) = ht(24 − n), n = 13, 14, . . . , 24 ht(0) = 1.9923082 × 10⁻³ ht(7) = 3.9731293 × 10⁻² ht(1) = −1.2248102 × 10⁻³ ht(8) = 2.3698283 × 10⁻² ht(2) = −5.9807801 × 10⁻³ ht(9) = −8.9814112 × 10⁻² ht(3) = 6.4517182 × 10⁻³ ht(10) = −2.8427998 × 10⁻² ht(4) = 1.0883640 × 10⁻² ht(11) = 3.1262843 × 10⁻¹ ht(5) = −1.7035807 × 10⁻² ht(12) = 5.2999323 × 10⁻¹ ht(6) = −1.7624002 × 10⁻²

Fig. 5. Normalized magnitude response and coefficients of the 25-tap SR filter designed in Example 1.

± (_{N −1}

X

m=1

h_t^TˆS_mh_t

hD[jT + 1 + m] + hD[jT + 1 − m]

+htTˆS₀h_thD[jT + 1]

)

≤ cj, j = 1, 2, . . . , ⌈( ˆN − 1)/T ⌉ − 1

whereγISIandγEYErepresent the specifications of the toler- able ISI and eye-opening, respectively. The unknowns of the above optimization problem areη, ht,aj, j = 1, 2, . . . , ⌊( ˆN − 1)/T ⌋, bj, j = 1, 2, . . . , ⌊( ˆN )/T ⌋, and cj,j = 1, 2, . . . , ⌈( ˆN − 1)/T ⌉ − 1. It is noted that the above optimization problem is a ‘second-order’ one, unlike the optimization problem given in [13], which is a fourth-order one.

The problem can be solved by the fmincon (constrained optimization) package in the Matlab optimization toolbox.

However, since the problem is not convex, the initial choice of ht is critical for fmincon to find a solution in a reasonable period of time. Through numerical experiments, we found that either the square-root-raised-cosine pulse, or the square-root pulses of the pulses given in [15], [16] are good initial choices for ht.

VI. DESIGNEXAMPLES

Example 1: In this example, we redesign the matched LP SR filters with nonzero ISI reported in [10]. The filter’s specifications are: the roll-off factorα = 0.3, the filter length is 25, and the over-sampling factor T = 2. Since T = 2, the eye-opening constraint is not applied in this example.

Using the proposed formulation described in the previous section (except the eye-opening constraint) and the fmincon

TABLE I

COEFFICIENTS OF THE24-TAPSR FILTERDESIGNED INEXAMPLE1.

ht(n) = ht(23 − n), n = 12, 13, . . . , 23 ht(0) = 1.6651682 × 10⁻³ ht(6) = 8.4301280 × 10⁻³ ht(1) = −4.2276138 × 10⁻³ ht(7) = 4.9360383 × 10⁻² ht(2) = −1.8865956 × 10⁻³ ht(8) = −3.2383526 × 10⁻² ht(3) = 1.1603980 × 10⁻² ht(9) = −9.9621419 × 10⁻² ht(4) = −7.7589064 × 10⁻⁵ ht(10) = 1.2252438 × 10⁻¹ ht(5) = −2.5405473 × 10⁻² ht(11) = 4.6979021 × 10⁻¹

package of Matlab, we obtain a symmetric SR filter whose magnitude response and coefficients are shown in Fig. 5.

The SBATT and ISI resulting from the coefficients are 52.39 dB and −53.05 dB, respectively. They are superior to the 48.84 dB SBATT and the −50.87 dB ISI reported in [10]. As mentioned in Section III, the passband ripple is not constrained in the formulation. The peak passband ripple of the resulting magnitude response is6.25 × 10⁻³ dB, which is worse than the2.22×10⁻³dB passband ripple reported in [10]. However, the peak passband ripple is so small that it can be ignored in most applications.

To achieve comparable SBATT and ISI with the SR filter given in [10], we redesign an SR filter with 24 taps using the proposed method. The resulting coefficients are summarized in Table I. The SBATT and ISI obtained are49.56 dB and −50.91 dB, respectively. On the other hand, the peak passband ripple is raised to9.08 × 10⁻³ dB.

From the two new design results, we conclude that slightly sacrificing the passband ripple helps the proposed method yield better SBATT and ISI performance or shorter filter length. Since the passband ripple is not important in SR filter design, the proposed formulation (21), in company with the fmincon package of Matlab, provides a suitable systematic design method for LP SR filters.

In conventional FIR filters, symmetric and asymmetric filters possess approximately the same length if the frequency domain specifications are the same. However, the symmetric SR filter is longer than its asymmetric counterpart because, for SR filters, we have both frequency domain specifications and time domain constraints. Thus, we need a sufficient degree of freedom to achieve the design goal. For an asymmetric SR filter with N taps, we have N degrees of freedom, but for a symmetric SR filter with N coefficients, the degree of freedom is only ⌈N/2⌉. Hence, when new constraints such as ISI and eye-opening robustness are imposed on the design problem, the symmetric SR filter requires more taps than the asymmetric SR filter requires.

In [12], [14], we have already shown that employing the tolerable ISI can help us to reduce the length of SR filters. In the next example, we also consider the eye-opening constraint and we design symmetric SR filters having reasonable lengths to endure receiver clock jitter.

Example 2: In [12], [14], we showed that the BER perfor- mance of a BPSK communication system was not deteriorated under−23 dB ISI when SNR = 10 dB. In this example, four SR filters are designed for the BPSK application. The system specifications are: SNR = 10 dB, the minimum stopband