Design of Two-Dimensional Notch Filter Using Bandpass Filter and Fractional Delay Filter

Chien-Cheng Tseng Depart. of Computer and Communication Engineering National Kaohsiung First University of Sci. and Tech.

Kaohsiung, Taiwan tcc@nkfust.edu.tw

Su-Ling Lee Depart. of Computer Sci. and Information Engi.

Chang-Jung Christian University Tainan, Taiwan

lilee@mail.cjcu.edu.tw

Abstract—In this paper, the design of two-dimensional (2D) notch filter using bandpass filter and fractional delay filter is presented. First, the design of 2D notch filter is decomposed into the designs of 2D parallel-line filter and straight-line filter. Then, parallel-line filter is designed by bandpass filter and straight-line filter is designed by fractional delay filter. Next, the FIR fractional delay filters are used to implement the designed notch filter. Finally, several numerical examples are demonstrated to show the effectiveness of the proposed design method.

I. INTRODUCTION In many signal processing applications, there is a need for a notch filter which is characterized by a unit gain at all frequencies except at the sinusoidal frequencies where their gain is zero. These applications include communication, control, biomedical engineering and image processing etc. In one-dimensional (1D) case, one typical example is to cancel 50 or 60 Hz power line interference in the recording of electrocardiograms [1]. In two-dimensional (2D) case, two examples are to eliminate a 2D sinusoidal interference pattern superimposed on an image [2] and to reduce blocking artifact from DCT coded image [3]. Thus, it is interesting to design notch filter to remove the sinusoidal interferences corrupted on a desired signal. On the other hand, fractional delay has become an important element in several applications like modeling of music instruments, time adjustment in digital receivers, speech coding and comb filter design etc. [4]-[8]. An excellent survey of the fractional delay filter design is presented in the tutorial paper [4]. When the fractional delay is fixed, we are dealing with the so-called the fixed fractional delay (FFD) design. The typical design methods include the window method [4], Lagrange interpolation method [4], allpass filter method [5] and discrete Fourier transform method [6]. If the fractional delay is adjustable, it

is called the variable fractional delay (VFD) design. So far, several methods have been proposed to solve this design problem such as decoupling approach [7], and second order cone programming [8] etc. The purpose of this paper is to establish the relation between 2D notch filter and the fractional delay filter such that 2D notch filter can be designed by using well-documented design methods of fractional delay filter.

II. PROBLEM STATEMENT The ideal frequency response of 2D notch filter is given by

⎩⎨⎧ ±=

=others

D NN

1),(),(0

),( 212121

ωωωωωω (1)

where ),( 21 NN ωω is the prescribed notch frequency. The problem is how to design 2D filter ),( 21 zzH to approximate ),( 21 ωωD as well as possible. So far, the design of 2D notch filter has been classified into two categories. One is the FIR filter design, the other is IIR filter design. In the FIR case, a typical method is the singular value decomposition approach [9]. In the IIR case, two design approaches are simple algebraic method [10] and outer product expansion [11]. Because IIR filter is more computational efficient than the FIR filter, 2D IIR notch filter design is studied in this paper. In the simple algebraic method [10], the design of 2D notch filter ),( 21 zzH is decomposed into the designs of 2D parallel-line filter ),( 21 zzH p and 2D straight-line

filter ),( 21 zzH s . The transfer function of 2D notch filter in [10] is given by

),(),(1),( 212121 zzHzzHzzH sp−= (2) where the parallel-line filter is chosen as

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

+−−= −−

−−

222

121

22

1212

21 11

21),(

zazazzaazzH p

(3)

978-1-4673-5762-3/13/$31.00 ©2013 IEEE 89

with the coefficients

( )2

21 tan1

)cos(2BW

Na+

= ω , ( )( )2

22 tan1

tan1BW

BW

a+−= (4)

and BW is a small positive number. Moreover, in [10], the analog inductance-resistance network and bilinear transformation are used to design straight-line filter ),( 21 zzH s below:

1

21

11

21

1

12

11

12

11

21

21212121

1

),(

−−−−−−+−+−++

−−−−

++++++=

zzzzzzzz

zzH

RLLR

RLLR

RLLR

RLLR

s

(5)

where the parameters

)tan(

1

21 1 N

L ω= , )tan(

1

22 2 N

L ω−= (6)

and R is a small positive number. In order to ensure the bounded-input/bounded output (BIBO) stability of the designed 2D notch filter, the parameters need to satisfy the constraints 01 >L and 02 >L . This implies that notch frequency needs to satisfy the following condition:

01 >Nω and 02 <Nω (7) The above constraint limits the applicability of notch filter, so we will use the fractional delay filter to design straight-line filter ),( 21 zzH s in this paper such that this constraint can be removed. The details are described in next section.

III. PROPOSED DESIGN METHOD

Given the notch frequency ),( 21 NN ωω , the proposed design method will be divided into the following three cases to discuss: Case 1 is |||| 21 NN ωω > , case 2 is

|||| 21 NN ωω < , and case 3 is |||| 21 NN ωω = . Now, the details of every case are described below: A. Design Method in Case 1 In this case, the design is based on the following fact: Fact 1: If |||| 21 NN ωω > ,

N

N

1

2ωωα −= , horizontal

parallel-line filter and straight-line filter have the ideal frequency response below:

⎩⎨⎧ ±=

=others

eeH Njjp 0

1),( 2221

ωωωω (8a)

⎩⎨⎧ =+

=others

eeH jjs 0

01),( 1221

αωωωω (8b)

then it can be shown that the frequency response of filter ),( 21 zzH in Eq.(2) is given by

),(),( 2121 ωωωω DeeH jj = (9)

That is, ),( 21 zzH is an ideal 2D notch filter. Proof: After some algebra, it can be shown that the following equality is valid:

⎩⎨⎧ ±=

=others

eeHeeH NNjjs

jjp 0

),(),(1),(),( 21212121

ωωωωωωωω (10)

Thus, it yields the result

),(),(),(1),(

21

212121

ωω

ωωωωωω

DeeHeeHeeH jj

sjj

pjj

=

−= (11)

The proof is completed. Because |||| 21 NN ωω > and

N

N

1

2ωωα −= hold, it yields

1|| <α (12) That is, parameter α is a fractional number in the interval

)1,1(− . Now, the remaining problem is how to design horizontal parallel-line filter and straight-line filter to fit the ideal response in Eq.(8) as well as possible. The design method is described below: The horizontal parallel-line filter can be easily designed by choosing ),( 21 zzH p as

2

|)(),( 121 zzBp zHzzH == (13)

where )(1 zH B is a one-dimensional (1D) bandpass filter whose transfer function is

221

21

1 111)( −−

−−

++++−=

zrrazzazzH B (14)

with )cos(2 0ω−=a (15)

The 0ω is the center frequency and )1( r−π is the 3-dB bandwidth of bandpass filter )(1 zH B . Hence, we only require to choose N20 ωω = and to let r be as small as possible, then ),( 21 zzH p is the desired horizontal parallel-line filter. Moreover, the straight-line filter can be designed by choosing ),( 21 zzH s as

21

|)(),( 221 zzzBs zHzzH α== (16)

where )(2 zH B is another 1D bandpass filter whose transfer function is

1

1

2 111)( −

−

−−−=

rzzzHB (17)

Substitute Eq.(17) into Eq.(16), we have

121

121

21 11

1),( −−

−−

−−

−=zrzzz

zzH s α

α

(18)

Hence, we only require to make r be as small as possible, then ),( 21 zzH s is the desired straight-line filter. Combining equations (2), (13), (14) and (18), the designed 2D notch filter is given by

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

−⎟⎟⎠

⎞⎜⎜⎝

⎛

++++

−−= −−

−−

−−

−−

121

121

22

212

22

12

21 11

11

111),(

zrzzz

zrrazzaz

zzH α

α

(19)

where )cos(2 2 Na ω−= . This is a closed-form design, so it is easy to use. Now, one example is used to demonstrate

90

the effectiveness of the above design method. The parameters are chosen as πω 5.01 −=N , πω 2.02 =N , and 999.0=r , so the delay α is 4.0 . Fig.1(a)-(c) show the magnitude responses |),(| 21 ωω jj

p eeH ,

|),(| 21 ωω jjs eeH and |),(| 21 ωω jj eeH . Clearly, the specifications

of three filters are all fitted well. However, the details of notch are underneath the unit gain plane, so the loss

|),(|1 21 ωω jj eeH− is shown in Fig.1(d). Obviously, the design error is samll. B. Design Method in Case 2 In this case, the design is based on the following fact: Fact 2: If |||| 21 NN ωω < ,

N

N

2

1ωωβ −= , vertical parallel-

line filter and straight-line filter have the ideal frequency response below:

⎩⎨⎧ ±=

=others

eeH Njjp 0

1),( 1121

ωωωω (20a)

⎩⎨⎧ =+

=others

eeH jjs 0

01),( 2121

βωωωω (20b)

then it can be shown that ),(),( 2121 ωωωω DeeH jj = is

valid. That is, ),( 21 zzH is an ideal 2D notch filter. The proof of Fact 2 is similar to one of Fact 1, so it is omitted. Because |||| 21 NN ωω < and

N

N

2

1ωωβ −= hold, it

yields 1|| <β (21)

That is, parameter β is a fractional number in the interval )1,1(− . Now, the remaining problem is how to design

vertical parallel-line filter and straight-line filter to fit the to fit the ideal response in Eq.(20) as well as possible. These designs are similar to those in case 1, so the results are reported below: The vertical parallel-line filter can be designed by choosing ),( 21 zzH p as

1

121

1

21

11

121 111|)(),(

1 −−

−−

= ++++−==

zrrazzazzHzzH zzBp (22)

where )cos(2 1Na ω−= . Moreover, the straight-line filter can be also designed by fractional delay filter. Its result is given by

β

β

β −−

−−

= −−−==

21

1

21

1221 1

11|)(),(21 zrz

zzzHzzHzzzBs

(23)

Substituting equations (22) and (23) into Eq.(2), we get the designed notch filter below:

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

−⎟⎟⎠

⎞⎜⎜⎝

⎛

++++

−−= −−

−−

−−

−−

β

β

21

1

21

12

121

1

21

11

21 11

11

111),(

zrzzz

zrrazzaz

zzH (24)

where )cos(2 1Na ω−= . Now, one example is used to demonstrate the effectiveness of the above design method. The parameters are chosen as πω 2.01 =N ,

πω 4.02 −=N , and 999.0=r , so the delay β is 5.0 . Fig.2(a)-(d) show the magnitude responses |),(| 21 ωω jj

p eeH ,

|),(| 21 ωω jjs eeH , |),(| 21 ωω jj eeH and the loss |),(|1 21 ωω jj eeH− .

Clearly, the specifications of three filters are all fitted well. C. Design Method in Case 3 In the case of |||| 21 NN ωω = , the designed results can be summarized as the following fact: Fact 3: If NN 21 ωω = , the designed 2D notch filter is

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

−⎟⎟⎠

⎞⎜⎜⎝

⎛

++++

−−= −

−

−−

−−

121

121

22

212

22

12

21 11

11

111),(

zrzzz

zrrazzaz

zzH (25)

And, if NN 21 ωω −= , the designed 2D notch filter is

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−⎟⎟

⎠

⎞⎜⎜⎝

⎛++++−−= −−

−−

−−

−−

12

11

12

11

22

212

22

12

21 111

1111),(

zrzzz

zrrazzazzzH (26)

In the above, )cos(2 2 Na ω−= . The proof of Fact 3 is similar to one of Fact 1 if 1±=α is chosen, so it is omitted. Note that the fractional delay filter is not necessary to be used in this case.

IV. DISCUSSION For the transfer functions of the designed 2D notch filters in Eqs.(19)(24), there is a fractional delay element. The remaining problem is how to implement the fractional delay. Without losing generality, only the implementation of fractional delay α−

1z in Eq.(19) is discussed below. The denominator and numerator of ),( 21 zzH s in Eq.(18) are

multiplied by an integer delay Nz −1 to yield:

12

)(11

12

)(11

21 1),( −+−−

−+−−

−−−=

zrzzzzzzzH NN

NN

s α

α

(27)

where N is a prescribed integer. In [12], the fractional delay filter is approximated by the FIR filter below:

∑=

−+− =≈L

n

nN znpzPz0

)( )()(α (28)

To let notch filter have exactly zero gain at notch frequency, the following constraint needs to be satisfied by fractional delay filter: )(11 )( αωω +−= Njj NN eeP (29)

Therefore, the filter coefficients can be determined by solving the following constrained optimization problem:

Minimize ωλπ

λπαωω deePJ Njj

2)()(∫−

+−−=

Subject to )(11 )( αωω +−= Njj NN eeP (30) This problem has been solved in [12]. Finally, substituting Eq.(28) into (27), we have

1211

1211

21 )()(

1),( −−

−−

−−

−=zzrPzzzPz

zzH N

N

s (31)

91

Thus, the fractional delay has been replaced by an FIR filter, so it can be implemented. Now, one example is used to demonstrate the effectiveness of the above design method. The parameters are chosen as πω 4.01 −=N ,

πω 2.02 =N , 999.0=r , 8.0=λ 5=N and 10=L , so the delay α is 5.0 . Fig.3(a)(b) show the

magnitude response and group delay response of the fractional delay filter )( zP . And, Fig.3(c)(d) show the magnitude response |),(| 21 ωω jj eeH and the loss

|),(|1 21 ωω jj eeH− after fractional delay is replaced by FIR filter. Clearly, the specification of notch filters is fitted well.

V. CONCLUSIONS In this paper, 2D notch filter has been designed by using

bandpass filter and fractional delay filter. Several numerical examples have been also demonstrated to show the effectiveness of the proposed design method. However, only two dimensional notch filter design is studied here, so it is interesting to extend this decomposition method to design three dimensional notch filter in the future.

REFERENCES [1] S.C. Pei and C.C. Tseng, "Elimination of AC interference in

electrocardiogram using IIR notch filter with transient suppression" IEEE Trans. on Biomedical Engineering, pp.1128-1132, Nov. 1995.

[2] R.C. Gonzalez and R.E. woods, Digital Image Processing, 2nd Edition, Prentice-Hall, 2002.

[3] V.K. Srivastava and G.C. Ray, "Design of 2D-multiple notch filter and its application in reducing blocking artifact from DCT coded image," Proc. of the 22nd Annual EMBS International Conference, pp.2829-2833, July 2000.

[4] T. I. Laakso, V. Valimaki, M. Karjalainen and U.K. Laine, "Splitting the unit delay: tool for fractional delay filter design," IEEE Signal Processing Magazine, pp.30-60, Jan. 1996.

[5] C.C. Tseng, "Design of 1-D and 2-D variable fractional delay allpass filters using weighted least-squares method," IEEE Trans. on Circuits and Systems-I: Fundamental Theory and Applications, vol.49, pp.1413-1422, Oct. 2002.

[6] C.C. Tseng and S.L. Lee, "Design of fractional delay filter using discrete Fourier transform interpolation method," Signal Processing, vol.90, pp.1313-1322, April, 2010.

[7] T.B. Deng, "Decoupling minimax design of low-complexity variable fractional-delay FIR digital filters," IEEE Trans. on Circuits and Systems-I: Regular Papers, vol.58, pp.2398-2408, Oct. 2011.

[8] T.B. Deng, "Minimax design of low-complexity even-order variable fractional-delay filters using second-order cone programming," IEEE Trans. on Circuits and Systems-II: Express Briefs, vol.58, pp.692-696, Oct. 2011.

[9] S.C. Pei, W.S. Lu and C.C. Tseng, "Two-dimensional FIR notch filter design using singular value decomposition," IEEE Trans. on Circuits and Systems-I, pp.290-294, Mar. 1998.

[10] S.C. Pei and C.C. Tseng, "Two dimensional IIR digital notch filter design," IEEE Trans. on Circuits and Systems-II, pp.227-231, Mar. 1994.

[11] S.C. Pei, W.S. Lu and C.C. Tseng, "Analytical two-dimensional IIR notch filter design using outer product expansion," IEEE Trans. on Circuits and Systems-II, pp.765-768, Sept. 1997.

[12] S.C. Pei and C.C. Tseng, "A comb filter design using fractional sample delay," IEEE Trans. on Circuits and Systems-II: Analog and Digital Signal Processing, vol.45, pp.649-653, June 1998.

(a) (b)

(c) (d) Fig.1 The designed 2D notch filter in case 1. (a) |),(| 21 ωω jj

p eeH . (b)

|),(| 21 ωω jjs eeH . (c) |),(| 21 ωω jj eeH . (d) |),(|1 21 ωω jj eeH− .

(a) (b)

(c) (d) Fig.2 The designed 2D notch filter in case 2. (a) |),(| 21 ωω jj

p eeH . (b)

|),(| 21 ωω jjs eeH . (c) |),(| 21 ωω jj eeH . (d) |),(|1 21 ωω jj eeH− .

0 0.2 0.4 0.6 0.8 10

0.5

1

normalized frequency ω (xπ)

mag

nitu

de

0 0.2 0.4 0.6 0.8 14.5

5

5.5

6

6.5

normalized frequency ω (xπ)

grou

p de

lay

(a) (b)

(c) (d) Fig.3 (a) Magnitude response of )( zP . (b) Group delay response of

)( zP . (c) |),(| 21 ωω jj eeH . (d) |),(|1 21 ωω jj eeH− .

92