The Design of Compensation Filter Based on Digital Receiver

Bing Li, Ya Qiu, Bao Ma School of Mathematical and Computer Science, Xihua University, 610037,China

libinglyl@yahoo.cn Abstract--In order to solve the problem which the pass band

tolerance of the CIC filter is large and the stop band is small

in the digital receiver, the paper uses the ISOP filters to

compensate it after the analysis about the transfer function

of the traditional CIC filter and its spectral characteristics.

Finally the simulation result shows that the compensating

filter could improve the inherent pass band tolerance of CIC

filter effectively.

Keywords-compensation filter; digital receiver; CIC filter;

transfer function

I. INTRODUCTION

It’s hard to real-time processing the received information which is converted by A/D converter in digital receiver because of computational complexity. Therefore, in the case of real-time processing, DSP is the "bottleneck" of system. A feasible solution is the use of digital down-conversion technology and multi-rate signal processing, turning high-speed data streams into low-speed data streams which can be real-time processing in DSP [1].

Multi-rate is defined as two or more of the sampling rate in a system. In order to save computation and storage capacity, the multi-rate signal processing technology would be used in the digital receiver system; .i.e. the system handles the base-band signal with a lower sampling rate and deals with modulated signal with higher sampling rate. The core of multi-rate digital signal processing is the conversion of sampling rate and filter banks.

Decimation filter in the digital receiver is implemented through three stages. As which has no multiplication, CIC filter is used as the head of the low-pass filter and decimation. HBF filter is used as a second stage of the low-pass filter and decimation, since almost half of which coefficient is zero and which multiplication is reduced by

half. FIR filter is used as the last of the shaping filter to improve the performance of the decimation system.

II. CIC FILTER

A. The Integrator-comb (CIC) Filter CIC filter, which has higher operational efficiency, is

not required for the multiplication of general FIR filter, and is mostly used as decimation-filter at the first stage of down-conversion. The impulse response function is:

others

Dnh

,010 , 1

(1)

In Eq.1，D is the order of CIC filter (D is actually decimation factor)[2].The CIC filter transfer function is ：

)()()1(1

111)()( 21

D11

D1D

0zHzHz

zzzznhzH

n

n

(2)

In Eq.2， 11 1

1)(

z

zH

)1()(2DzzH

The implementation diagram is shown in Fig.1.

z 1 z D

)(nx)(1 ZH )(2 ZH

)(my

（a）

z 1

z 1

)(nx )(my

（b） Figure 1. The implementation of CIC filter

Fig.1 shows that CIC filter consists of two parts,

)(1 zH is an integrator, it would be realized for an accumulator; )(2 zH is a comb filter. CIC filter is the

cascade of the integrator and comb filter, so it is also known as the cascaded integrator comb filter [3].

2011 Fourth International Conference on Intelligent Computation Technology and Automation

978-0-7695-4353-6/11 $26.00 © 2011 IEEE

DOI 10.1109/ICICTA.2011.475

767

2011 Fourth International Conference on Intelligent Computation Technology and Automation

978-0-7695-4353-6/11 $26.00 © 2011 IEEE

DOI 10.1109/ICICTA.2011.475

753

B. The Characteristics of CIC Decimation Filter The frequency response of CIC filter is[4]:

)2

()2

()2/sin()2/sin()( 1

aa

j SDSDDeH (3)

In Eq.3, x

xxSa

)sin()( is the sampling function,

and 1)0( aS .So the magnitude of CIC filter is D

where = 0, that is in the amplitude-frequency characteristic curve of the CIC, the range of (0 ~ 2 / D) is as the main lobe of the CIC filter, and the other range is the side-lobe. As the frequency increasing, the side-lobe level would be decreasing, where the first side-lobe level is:

)23sin(

1

)23sin(

)2

3sin()( 25.1

DD

eHD

j

（4）

When D>> 1, the first side-lobe level A1 is approximately 3/2D , its difference with the main lobe

level (D) is: dBADa s 46.13

23lg20

1lg20

/D2 /D22 /D23

)(H e j

1

Figure 2. CIC filter frequency characteristics

As Eq.4 and Fig.2 shown, the side-lobe level of the single-stage CIC filter is relatively large, only lower than the main lobe 13.46 dB, this also means that stop-band attenuation is poor, so it is generally that one-level CIC filter is difficult to meet the practical requirements. In order to meet the practical demands, we could cascade multi-level CIC filter to solve [5]. For example, when using Q-level CIC filter to achieve, the frequency response is:

Qj

QNeH

)2/sin()2/sin()(

In this case, the suppression of the side-lobe is

sQ

S Qaa , if Q = 5, then 67.3asdB, the actual needs

would be met basically. At the same time, the selection of D should be to meet

the requirements of the band pass tolerance, it means that

selecting the bandwidth factor, we should consider that the rate could not fall too much when 1 . If the tolerance is s , we could obtain:

)2

sin(

)2

sin(lg20

)()(lg20

1

10

D

D

eHeH

j

j

s

（5）

Set Db 2

1 into (5) yields:

)sin()sin(

lg20

bD D

b

s

While 1bD , D

bD

b )sin( , )sin(lg20 b

bs .If the

Q-level CIC filter is used, the tolerance in band is :

sQs Q ,it means that the Q-level CIC filter's tolerance

in band is Q as much as single-stage one. It can be seen that although multi-stage cascade can increase stop-band attenuation and reduce the aliasing effects, but the tolerance in band would be increased. Therefore, the number of the cascaded CIC filter is limited, generally limited to 5 orders. Compensation filter can be used to reduce the excessive pass band tolerance is, compensate.

III. ISOP COMPENSATION FILTER

Second-order polynomial interpolation filter compensation is also known as ISOP filter [6]. Compensation filter ISOP will be placed at the back of the CIC, which can reduce the computation of the compensator greatly. ISOP filter’s system transfer function )(zP is:

)1(2

1)( 2 II zczc

zP

（6）

Eq.6, the positive integer I is insert-factor, c is real number. The frequency response is:

The frequency response of P(z) is monotonically

increasing in the interval of I/,0 , and that cycle

is I/2 , “I” is the interpolation rate. Obviously, it just enough compensates for monotonically decreasing pass-band frequency amplitude of the CIC. In order to achieve this compensation effect, ISOP filter’s monotonically increasing bandwidth I

,0 should

be consistent with CIC pass band bandwidthpf2 , namely:

Icos22

1)(P

cc

e j

768754

)2(12

pfpI If

In the transfer function of Compensation filter, the power of z should be a multiple of decimation ratio (D). Suppose I = Dk, k is a positive integer, then the minimum of the amplitude response would be appeared at a multiple of f = 1 / (Dk).In this case, the position of each minimum value should consistent with the position of CIC filter zero-response, so that after ISOP filtering, the anti-aliasing features of CIC decimation filter could be preserved. When I = Dk, the deformation of the above inequality would be:

pDfk 2

11 (7)

Therefore the transfer function of the pass-band roll-off compensation filter is designed as:

)1(2

1)( 2DKDK

zczc

zP (8)

As the frequency response function of ISOP filter known, as long as (k, c) is determined ,its frequency response function should also be determined. For each k which satisfies Eq.7, minimum :

1)()( jj ePeH (9)

In Eq.9, )H(e j is the frequency response of CIC filter

and )P(e j is the frequency response of ISOP filter. How

to select k and c? Set )H(e j as the frequency response of

the compensated CIC, and then be sure the value of k which satisfies Eq.7. In an effective range, continuously changes the value of C in a certain step. For each value of C, maximum error ’ is obtained after pass band compensation. And then compare to find the value of C when ’ is smallest, we can approximate the value that

is c of the best ISOP filter. Where,pf 20 , is the

setting compensation precision.

V. SIMULATIONS

According to the above-mentioned principle, a design example is given. As the characteristics of CIC filter stop-band and transition-band are not very good, we usually use five-level Cascade CIC to increase the attenuations of the transition-band and the stop-band. For

this design, when B = 4kHz, MHzf s 6.1 , D = 32,

08.0D/

sfB

b, Q = 5, so the tolerance in band of the

designed CIC filter would be as following.

4582.0)sin(

lg20

b

bQQS

The spectrum simulated with Matlab is shown in

Fig.3.It can be seen from Fig.3 that the pass band tolerance of the CIC filter could be up to 0.4582 which is greater than the system requirement’s 0.2dB.So it need to be compensated. For the previously designed CIC, its frequency response is:

5)sin()16sin(

2)(

jeH

0025.06

3

106.1104

sff

pf

So the range of k is from 1 to 7 as Eq.7, and then according to Eq.9 to optimize the design. When parameters (k, c) equal to (4, -8.3), it could obtain optimum compensation. Then an optimal compensation ISOP filter is realized.

Figure 3. The frequency response of CIC (5 order)

Figure 4. The magnitude response of the ISOP filter

769755

Figure 5. The compensation effect of the ISOP

It can be seen from Fig.4 that the frequency response of ISOP will increase up to 0.45dB when the frequency is 4 KHz. It is just the reverse of the previously designed CIC’s the pass band roll-off rate. If compensation filter and the CIC are cascaded, the compensation filter could eliminate the pass band attenuation of CIC,Fig 5 shows the compensation effect. In the Fig.5, the bottom curve means the pass band spectrum of CIC filter, the top curve means the response spectrum of the compensation filter, the middle curve means the response spectrum of the overall cascaded system .It can be seen from Fig 5 that the excessive CIC’s pass band roll-off has been greatly eliminated.

VI. CONCLUSIONS

Multi-rate signal processing is key technology in the digital receiver, so the design of the digital filter what is good or bad will directly impact on the effect of digital down conversion and real-time processing capabilities, and thus affect the reception of the entire system. The results of the simulation prove the method of the second-order polynomial interpolation can effectively compensate for the inherent pass band attenuation of the CIC filter, and make the frequency response curve of the pass band flat.

REFERENCES

[1] Jaeyoung Kwak, Kwyro Lee．Design of dividable

inter-leaver for parallel decoding in codes [J].

Electronics Letters，2002，38(22):1362-1364．

[2] Stephen.G, Stewart.R.W. High-speed sharpening of

decimating CIC filter. Electronics

Letters,2004,40(21):1383-1384.

[3] Jovanovic-Dolecek, G.Mitra.S.K. Efficient

sharpening of CIC decimation filter.ICASSP,IEEE

International Conference on Acoustics, Speech and

Signal Processing- Proceedings,2003,6:385-388.

[4] Li Zhi Jian, Zeng DaZhi, LongTeng, Analysis of

phase truncation spurs and Signal Processing of the

DDS.2009,25(11):1706-1709.

[5] Jou Shyh-Jye, Jheng Kai-Yuan,Chen Hsiao-Yun, Wu

An-Yeu. Multiplierless multirate

decimator/interpolator module generator.

Proceedings of IEEE Asia-Pacific Conference on

Advanced System Integrated Circuits,2004,pp58-61.

[6] Oh Hyuk J, Kim Sunbin,Choi Ginkyu, Lee Yong H.

On the use of interpolated second-order polynomials

for efficient filter design in programmable

down-conversion, IEEE Journal on Selected Areas in

Communications,1999,17(4):551-560

770756