Remezアルゴリズムに基づくShort Time DFTのデシメーションフィルタの最適化

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愛知工業大学研究報告第

29

号平成

6

年

2

3

7 On t

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e

O

p

t

i

m

i

z

a

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i

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o

f

Decimation F

i

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e

Short Time DFT

Based on Remez Algorithm

Remez

アルゴリズムに基づく

ShortTime DFT

の

デシメーションフィルタの最適化

岸政七

t

，

Masahichi KISHI

，

前島利行

f

T

o

s

h

i

y

u

k

i

MAESHIMA

ABSTRACT Decimαtionfilters ofthe ST gDFT Hilbert transformerαre examined through frequency responseωith employing Remez algorithm. Optimiz，αtion is performedαlternαte般

か

onthe frequencyαndtime domωn ωith restricting filter co司fficients.This optimized re

-sponse ofthe Hilbert transformer also defines the optimized weighting functionfor the dec -Lmαtion filters. In otherωords， the optimized Hilbert trar時former‘responsebαsed0πthe

Remezα19orithm deduces the optimizing weighting function of the significαnt decimαtion filter in the ST DFT.

1. INTRODUCTION

The instantaneous spectrum in so important concept that the previously reported short time DFT(8T DFT) Hilbert transformer is realized to be almost free from any error both in phase shifting and amplitude owing to employing shifting the phase on the phase plane[IJ. This

concept is provided with 8T DFT to make many applications being feasible in such radio communication systems as high speed MO-DEM， highly efficient CODEC， and distortion free filters in addition to the Hilbert trans・

former. The significant functions in the 8T DFT are mainly characterized with the decimation filters which a playa important role during analyzing input signals[2J •

In this paper， we discuss about optimization of the decimation filters used in the general・

ized short time DFT(8T gDFT) Hilbert trans・

T

愛知工業大学情報通信工学科(豊田市)

formers with employing Remez Algorithm [3J •

Optimization is performed alternatively on the frequency and time domains with adjusting time domain response

，

i.e. unit sample re幽

sponse. The unit sample response optimized in frequency domain is transformed via inverse 8T gDFT (8T gIFT) to define the coefficients of the decimation filters[4J • 80 long as this optimized Hilbert trans・ former is realized ideally

，

the optimized unit sample response of the 8T gDFT Hilbert transformer also suggests the optimum weighting function for the decimation filter in a 8T DFT

，

which is also universally adopted to filter bank systems. As discussed details in following session

，

the ideal Hilbert transformer is realized with em -ploying the relation G(ω)=一jsignωG(ω)， here G(ω)is Hilbert transform of the Fourier transform of the arbitrary function g(t)・The

8T gDFT guar釘lteesthe solver for avoiding

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2

3

8

愛知工業大学研究報告，第29号B，平成6年， Vo1.29-B， Ma.r.1994 Therefore， the optimized ST gDFT Hilbert transformer consequently gives the newly proposing optimization algorithm for the sig -nificant decimation filters in the analysis of the instantaneous spectrum. 2. PRINCIPLE OF THE ST gDFT The previously reported ST DFT Hilbert transformer is provided with causality based on restricting theOthand ~ thsub -channels being null[1]. Nullification on these sub-channel reduces the bandwidth over subjective domain as shown in fig .1 (b ) . Fortunately， the ST gDFT is able to release Imag 軍 - 宜 Real (a) Ideal Hilbert Transformer Imag l11

I

1 c

t 〉Real -1l (b) ST OFT Hilbert Transformer Imag Real (c) ST 90FT Hilbert Transformer

Fig.l Comparison of the pαssbandαmong ideal(<α)， ST DFT(b)αnd ST gDFT Hilbert transformers(c). this restriction from causality. As shown in fig.1(c)

，

the ST gDFT is able to adjust its channel allocation to avoid zero cross in theOth sub -channel in order to coincide with that of ideal Hilbert transformer shown in fig.1(a). That is

，

the instantaneous spectrum仇(n)is given by ST gDFT asfollows.

。

k(η) =

L

h

(

η-r)x(r)WNー(知的r ( 11 ) r=-o。 0，5kくN where h( * ) is叩 aprioridecimation filter;x(r)

is a sampled data at time η 巧~k+f)rare ST gDFT operators，

W N -(k+f)r = e-j存(k+f)r

， 0

，5fく1 ₍₂₎

here

，

fis newly introduced p町 ameterto

ad-just channel allocation of the existing ST DFT. Where the parameter f is叫 obe

t

，the channel allocation of the ST gDFT is moved up by half of sub幽channelwidth to coincideOth

lower fringe with zero frequency. Attentions must be paid on this channel allocation to get ideal Hilbert transformer based on the phase place relation

a

(

ω)=一jsignωG(ω)， and on that bandwidth of realized Hilbert trans -former will become to equal to that of ideal one as shown in fig.2 because there exist no sub司

channel eliminations in the ST gDFT Hilbert transformer. In fig.2， solid curve shows amplitude fre -quency response of the ST gDFT Hilbert trans・

2 .

0 一

一

8TgOFT 凶可コ @ 万

三 0

.

0

a

E

《

-

2 .

0 .

O π/2 _z Normalized Angular Frequency， radian Fig.2 Comparison ofαmplitude frequency response be -tween ST gDFT and ST DFT Hilbert tran司formers(2m= 8，N=8).

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2

3

9

Let's consider what effect will be introduced byw巴ightingthe truncated Nyquist by Kaiser

function as follows.

On the Optimization of Decimation Filters in the Short Time DFT Based on Remez Algorithm

former and dott自d curve shows that of the

ST DFT one. Wideness of the ST gDFT Hilbert transform巴ris shown clearly in the

figure if both window length 2m = 8 and frame

length N = 8. h(n) = n(n)k(η) ， (5)

where， 3. DECIMATION FILTERS IN THE ST DFT

ん

(

s

J

1 -

(n

'

J

v

)

2 )

ANDSTgDFT

here Io(* ) is the modified Othorder Bessel of the first kind，βis an arbitrary positive real number to adjust the width and energy ofOth

the mainlobe.

Solid curve in fig.3 shows the amplitude fre -quency response of the ST gDFT Hilbert trans -former adopted with Kaiser weighted Nyquist of eq.5， here frame length N = 8， and frame number 2m = 8 ， and

s

=6.3. Dotted curve in fig.3 simultan巴ouslyshows th色amplitudefre

-quency response of the ST gDFT Hilbert trans -former in which the d巴cimationfilter h(.) is

adopted with Nyquist merely truncated by 2m frame number. It is clearly shown that the amplitude error over subjective domain (0，π) is remarkably improved by employing Kaiser weighting function given by eq.6.

Figure 4 shows that the maximum amplitude error on the subjective domain of the ST gDFT Hilbert transformer adopted Kaiser weighted

(6) - m Ns，η三;mN k(n) = 3.1. Definition of the Decimation Filters Necessary condition for the decimation filt日r defined by h(.)in eq.l is d巴ducedfrom spec出向 cation as an ideallow -pass filter， π π 一一一〈川〈一一 -_N 一山一N 寸上ハ u r a E ' l t P 4 t i -色、 ρ U N (3)

Inverse Fourier transform for N(ej_ω)_g_i_v色S

impulse response of the decimation filtern(n)

so called Nyquist， else. ( ' - j ，ト句 _sin(nπIN) n(η)= 市 r~rrH(eJω)

e

J

w

切ω一一三n;y一(4) 1 v " rιL /1 V As shown clearly in eq4.， infinite frame number Nyquist behaves as an ideal decima-tion filter with fatal victim of paying infinite processing amount during convolution. When the Nyquist function is truncated by finite length， amplitude peak values both of main and side lobes becomes to be greater， and the sharpness of cut -off becomes to be vague to show Gibbs's phenomenon on the frequency

1.0 0:0 0.0

宅

1.5 」 O l.. L !.u <D tココ c.. E <( E コ E H 国 2 domain. 2.0 III ℃ 、田万コ H ニ

a

E

︿ 10.0 8 Fig.4 MαXUγLUmαmplitude error0/the ST gDFT Hilbert trans/ormer with K

，

ωser -weighted Nyquist vs. β(2m=8， N=8). 5.0 Fig.3 Compαrison0/αmplitude /requency response be tωeeenIGαiser -weightedαndηon -weighted ST gDFT Hilbert trαn司formers(2m=8， N=8).

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where E(ω) is eva1uated over both passband and eliminating band of the desired filter W(ω) as a w色ightingfunction， and H(ejW) is

frequency response of the optimization target. Optimizing response in the Chebycheff mean -ings is estimated by the maximum absolut自

value of E(ω) at M

+

2 p芭ak frequency

{ω小 i= 0， 1， 2，…， M + 1. These frequencies

should locate in the regions 0:::;ωSωIpand ωeSω :::;7[，here，ωIpis cut -off frequency on the

passband and ωe is cut -off frequency on the eliminating band. Where the values of the magnitude at these fr巴quenciesare given by

unique value by δ

M

+

20rder simultaneous equations are de-rived from eq.9，

M ar.1994

Following to Rabiner's discussion， the ap-proximation error function is defined as fo1 -lows.

E

(

ω)

=W(

ω)[Hd(eJω) -H(eJω₎_J

₍

₉

₎

Vo1.29-B，平成6年，第29号B，

Nyquist is monotonously improved as the pa-rameter

s

goes large， and is improved to be less than 0.01dB if

s

is greater than 6.3.

Contradictorily， the bandwidth of the trans -former is slightly shrinked from increasing the valu巴of

s

as shown in fig .5.If

s

is set to

be 10， the bandwidth is shrinked by 4.5 point by percent. Kaiser weighted Nyquist is conse-quently recognized as a suitable decimation filter in the meanings both of minimum ampli司

tude error and maximum bandwidth under re -striction condition of

s

=6.3 as indicated in figs.4 and 5.

愛知工業大学研究報告，

2

4

0

4. OPTIMIZATION OF THE DECIMATION

Hilbert of the ST gDFT

FILTERS

4.1 Optimization Transform町

The frequency response of the ST gDFT Hilbert transformer H(ejω) of length 2M + 1 is glven as

M

H(eJW) =

エ

h(η)e -juJn，

九=-M

There exists M

+

2unknowns forh(n) and δ in these simultaneous equations. Rabiner suggested thatδis effici巴ntto solve more than

solving about h(η) in concerning with peaks. In general， these peaks are given by searching around the points of dividing passband and eliminating band. Uu) W 町恥州

ω

川

ω

叫

ω

(

i

い

l

何

←

ト

(

伊

川

作

)

ト

一

→

州附刷叶(刊仰h0

的

十

)

ト

一

b

か

五

会

劫

制

h

(

山O町) =一(ト一1り)'δ i= 0， 1， 2， …， M+1. (7)

The causality is well known given by merely adding delay by Msamples. H(ejW) is also modified from the symmetry ofh(n) as fo1 -lows. M =m N (8) M H(eJω) = h(O)

+

L

2h(n) cos(ωη)

-

0 .

2

1.0 X 7[ E

'

"

可コ悶』 Z H てコ亘百 E 田 D (/) ul 国 ι 0.9X 7[ 0.0

0 .

2

0

，

0

国司、 L O ﹄ L 凶申万三一目立

E

︽ π

/

2

71: Normalized Angular Frequency， radian Fig.6 Amplitude response of the optimized ST gDFT Hilbert traT同former，2m=8， N=8 and δ=1.0233巳 10.0 3 Fig.5 Bandwidth of the ST gDFT Hilbert transformer with Kaiser -weighted 1ゆqULstvsβ(2m=8， N=8).

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On the Optimization of Decimation Filters in the Short Time DFT Based on Remez Algorithm 241

It is clearly shown in fig.6 that the amplitude response of the optimized ST gDFT Hilbert transformer is featured with equalripple with-in the subjective domain， where frame number 2m = 8 ， frame lengthN = 8， and δin eq.lO is set to be O.ldB， i.e.δ=1.0233. Figure 7 shows the relationship between the amplitude error δand passband width of the optimized ST gDFT Hilbert tr百 lsform邑rs. In similar to

Kaiser weighted Nyquist， the passband width of the optimized transformer based on Remez algorithm is slightly shrinked from improving amplitude error.

4.2. Extraction of the Optimized Weighting Function

As reported previously， the unit sample re -sponse of the ST gDFT ig(凡)is given as follow

r

U

.

2

/

H

_，

_h(n)_，

ig(n)=~Nsin( πη/N) for odd n (11)

for even n

It is adequate that decimation function h(η) described by

h

(

η)=η(η)ω(η) ， (1~

凡(η)is infinite Nyquist function and ω(n)is

such a weighting function as Kaiser， Black-man， or the desired optimized weighting func -tion. Substituting eq.12 into the unit sample response of eq.ll， it gives， 0.95x7c E 伺百咽

.

..c 官判主てコ E 句 . ロ UU3 3 <tJ ι 0.85XZ 0.10 ₀_.₀₅ ₀_.₀₁ Amplitude Error. dB

Fig.7 Passbαnd width vs.αmplitude error pf the opti -mized ST gDFT Hilbert tr，αnsformer. n 川山﹁的 m国一︽

一川

一円 Mll 川 2

一π

何一

-m

ω

一仇 2 一 m n ( 13)

As well known， the term 2/π凡onthe right

band of eq.13 meIlns the unit sample respons巴

of the infinite Hilbert transformer. Once the unit sample response of the optimized ST gDFT Hilbert transformer， the optimized dec-imation function in the Chebycheff meanings is given by eq.13. That is， the optimized deci -mation filter， which is significant in the in -stantaneous spectrum analysis both in the ex-isting ST DFT and in the ST gDFT， is defined by multiply the optimized response by the re -ciprocal numberππ/2. Figure 8 shows the op嶋

timized weighting function by solid curve over 8 frame durations. Kaiser weighting function is also shown in the figure by dotted by curve in comparison with the optimized one

5. CONCLUSION

The optimization of the weighting function was discussed through Remez algorithm with em-phasis on reducing amplitude error both of passband and eliminating band in the mean-ings of Chebycheff. These optimized weight-ing functions ensure that such concept of the instantaneous spectrum as ST DFT， ST gDFT and etc. are put on the stage of developing 由刀コ一日

a

E

︿ _一_一_回 _O_p_t_i_m_i_z_e_d 四"..K aiser O -4N・3N -2N -N 0 N 2N 3N 4N Fig.8 Amplitude response of the optimized weighted function， 2m=8

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2

4

2

愛知工業大学研究報告，第29号B，平成6年 Vo1.29-B，Mar.1994

Hilbert transformer， CODEC and MODEM op -timized in the Chebycheff meanings.

6. REFERENCES

[1] M.Kishi，“The Properties and Configura

-tion of the Short Time DFT Hilbert Trans-formers" ， IEEE ICASSP'89， Glasgow， Scot

-land， Proc. Vo1.2 No.D4.10，PP.1019-1022， May 1989. [2] M.Kishi， H.Koga，“On the Optimization of the Prototype Filter used in the Short Time DFT Hilbert Transformers" ， IEEE VTC'91， St. Louis， Missouri， Proc. PP .161-165， May 1992.

[3] L.R.Rabiner釘ldB.Gold，“Theory and

Applications of Digital Signal Processing" ， Prentice -Hall

，

1975・

i

[4] M.Kishi， T.Ishiguro， Y.Kozaki，“.A，pplica

-tion of the Generalized Short Time DFT to the Hilbert Transformer and Its Characteristics" ， IEEE VTC'93， Secaucus， New Jersey， May 1993.