noise improvement for different bp filter...

Noise improvement for different BP filter designs

Z. Šverko*, N. Stojković*, S. Vlahinić* and I. Markovinović* * University of Rijeka – Faculty of Engineering/Department of automatics and electronics, Rijeka, Croatia

[email protected], [email protected], [email protected], [email protected]

Abstract - In this paper an analysis of noise through four

different approaches to filter design is performed. The

voltage noise spectral density and root mean square (RMS)

value of noise voltage for the simply designed (SD) active

RC, noise optimized (NO) active RC, operational

transconductance amplifiers (OTA-C) and switched

capacitor (SC) filter design are calculated for fourth-order

Chebyshev band-pass filter, with central frequency fc=4kHz,

bandwidth 1kHz and pass band ripple αmax=-0.1dB. For the

achievement realization of all four designs a cascade of two

second order sections are used. The analyses is performed

using Matlab and SPICE (LTspice) programming tools.

Keywords - band-pass filter; noise; SC filter

I. INTRODUCTION

There are different ways to achieve the desired transfer function of the filter. The most commonly used realization is with the operational amplifiers (OA) which gives a stable (solid) voltage point output and also low impedance output. On the other hand, if high impedance output is desired, then realization with general transconductance amplifier (OTA-C) sections will be used. Such design gives us a constant current source on output.

Likewise, one of the realization of the same transfer function can be with switched capacitors (SC) where resistors are replaced with switched capacitors [1, 2]. This realization is good for design in monolithic integrated technology, where the filter is completely implemented in chip using thick and thin film technology.

Except satisfaction of elementary specifications on amplitude and phase filter characteristics, there exist additional parameters of filter quality. One of them is noise. The goal is to reduce noise as much as possible, because with superposition of noise on observed signal the information that the signal transmits can be covered. Through the sections of the paper will be demonstrating how to design filters on above suggested approaches with the desire to reduce noise in filter.

II. REALIZATION

The normed transfer functions of the observed Chebyshev band-pass filter are given by:

)(2)(1)( sHsHsH a

7066.02456.02

4525.0)(1

ss

ssH b

415.13475.02

4525.0)(2

ss

ssH c

15931.02207.235931.04

22048.0)(

ssss

ssH d

The transfer functions of fourth-order Chebyshev band-pass filter sections denormed on a frequency of 4kHz are given by:

)(2)(1)( sHsHsH a

8105469.73101764.42

3103542.5)(1

ss

ssH b

8102869.53104956.32

3103969.6)(2

ss

ssH c

171099.312108462.4

29102979.13310672.74

27104251.3)(

s

sss

ssH d

A. Continuous-time active-RC filters

In Fig. 1 the Single Amplifier Biquad (SAB) topology of the band-pass filter is given.

Realization is performed as a cascade of two second order sections. The circuit transfer function is given as:

2,1 )()( i si

HsH a

R111

R121

R21

R31

R41

C21

C11

-

+

R112

R122

R22

R32

R42

C22

C12

-

+

Uin

Uout

Figure 1. Fourth-order OA band-pass filter

126 MIPRO 2019/MEET

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iCiC

iGiG

iGiC

iGiG

iC

iG

iC

iGss

s

iC

iG

iG

iG

siH

21

21

31

41

2

2

1

22

1

11

3

41

)(

b

Using given filter parameters, element values can be calculated by comparing equations (2a) to (2c) with (3a) and (3b) with assumptions C1i=C2i and R1i=R2i. Values taken over from [3] are given in Table 1.

Fig. 2 shows the frequency responses of the filter. The simulated values of the frequency response are given by simulating circuit shown in Fig.1 using element values from Table 1. Unlike the former, frequency response of calculated values is given by the filter transfer function (2a) to (2c). In the zoomed inset of Fig. 2 there can be seen a minimal difference between these two characteristics (simulated-dashed, calculated-full).

B. OTA-C filter

Realization is performed as a cascade of two second order general transconductance amplifier sections. This

circuit is shown in Fig. 3. The circuit transfer function is given as:

2,1 )()( i si

HsH a

iCiC

imgimg

iC

imgss

s

iC

img

siH

21

21

2

32

2

4

)( b

Comparing the equations (4a, 4b) with (2a-2c) element values of OTA-C filter were calculated and shown in Table 2. Frequency responses are shown in Fig. 4.

4000-100

-50

0

20

Ma

gn

itu

de

[d

B]

Frequency [Hz]

400 4000 40000-200

-100

0

100

200

Ph

ase

[d

eg

]

Frequency [Hz]

4000

-0.1

0

Frequency [Hz]

Figure 2. Frequency response of the Fourth-order continuous-time

active-RC (SD) band-pass filter

TABLE II. ELEMENT VALUES OF OTA-C FILTER

1st section 2nd section

gm1i 0.27472 mS 0.22993 mS

gm2i 0.27472 mS 0.22993 mS

gm3i 0.041764 mS 0.034956 mS

gm4i 0.053542 mS 0.063969 mS

C1i 100 nF 100 nF

C2i 100 nF 100 nF

-

+

+

-

-

+

+

-

C11

C21

gm11

gm21

gm31

gm41Uin

-

+

+

-

-

+

+

-

C12

C22

gm12

gm22

gm32

gm42

Uout

Figure 3. Fourth-order OTA-C band-pass filter

4000-100

-50

0

20

Ma

gn

itu

de

[d

B]

Frequency [Hz]

400 4000 40000-200

-100

0

100

200

Ph

ase

[d

eg

]

Frequency [Hz]

4000

-0.1

0

Frequency [Hz]

Figure 4. Frequency response of the Fourth-order OTA-C BP filter

TABLE I. ELEMENT VALUES OF SIMPLY DESIGNED (SD) AND

NOISE OPTIMIZED (NO) CONTINUOUS ACTIVE-RC FILTERS [3]

SD NO

1st section 2nd section 1st section 2nd section

R11i 53.1915 kΩ 44.5208 kΩ 21.3238 kΩ 21.3238 kΩ

R12i 3.9075 kΩ 4.81995 kΩ 1.5299 kΩ 1.8538 kΩ

R2i 3.6401 kΩ 4.3491 kΩ 9.2824 kΩ 11.0902 kΩ

R3i 7.3529 kΩ 7.35294 kΩ 0.9866 kΩ 0.9866 kΩ

R4i 3.9789 kΩ 3.9789 kΩ 3.9789 kΩ 3.9789 kΩ

C1i 10 nF 10 nF 10 nF 10 nF

C2i 10 nF 10 nF 10 nF 10 nF

ki 1 1 2.55 2.55

MIPRO 2019/MEET 127

C. Discrete time SC filter

The discrete time SC filter based on Sallen-Key (SAK) topology is shown in Fig. 5.

The forward Euler transformation (s=>(z-1)/Ts) was applied for calculation of the discrete time transfer function. Where Ts (fs=4900fc) is a sampling time and which has been calculated as Ts=1/fs .

)(2)(1)( zHzHzH a

9997.09997.12

)1(4108023.5)(1

zz

zzH b

9996.09996.12

)1(4108023.5)(2

zz

zzH c

9992.09977.3

29977.539992.34

1227103667.3

)(

z

zzz

zz

zH d

The transfer function of discrete time SC filter based on Sallen-Key topology is given by:

2,143

22

)1(1)(

ii

azi

azi

a

zi

a

iGzH a

i

Ci

Ci

G5

/4

1 b

iR

Ci

Ci

Ci

a1211

c

RfiC

iRC

iC

RfiC

iRC

iC

RfiC

iC

iC

iRC

iRC

iC

iRC

iRC

iC

iRC

iC

iC

iRC

iC

iRC

iC

iC

iC

iC

ia

2221

21212

211221

2

2

11212

2

12

d

RfiC

iC

iC

iG

RfiC

iRC

iC

RfiC

iC

iC

iRC

iRC

iC

iRC

iC

iC

iRC

iC

iRC

iC

iC

iC

iC

ia

21

2221

212221

2

2

11212

2

12

3

e

Rf

Ci

Ci

Ci

Gi

Ci

Ci

a212

2

14 f

Comparing the equations (5a-5c) with (6a-6f) element values were calculated and presented in Table 3. From Fig. 6 and Fig 2. it can be seen that the Z-transformation hasn't got undesired effects on filter specifications.

TABLE III. ELEMENT VALUES OF SC FILTER

1st section 2nd section

CR1i 37.9 pF 53.7 pF

CR2i 2.82 pF 2.52 pF

C1i 120 nF 120 nF

C2i 3.3 nF 3.3 nF

CRfi 125.5 pF 313.5 pF

C4i 2.15 nF 1.53 nF

C5i 2.55 nF 5.1 nF

CR11 C11

C21

CR21

+

-

CRf1

CR51

CR41

CR12 C12

C22

CR22

+

-

CRf2

CR52

CR42

UoutUin

Figure 5. Fourth-order SC band-pass filter (SAK topology)

4000-100

-50

0

20

Ma

gn

itu

de

[d

B]

Frequency [Hz]

400 4000 40000-200

-100

0

100

200

Ph

ase

[d

eg

]

Frequency [Hz]

4000

-0.1

0

Ma

gn

itu

de

[d

B]

Frequency [Hz]

Figure 6. Frequency response of the Fourth-order SC band-pass filter

128 MIPRO 2019/MEET

III. NOISE

A. Noise in filter with OA

Fig 7. shows an equivalent circuit of the active-RC

filter for the calculation of the voltage noise spectral

density. Parallel to the resistors have been added current

noise sources: i/RkTniI 4 , whilst to the voltage

noise sources En = 2.5 nV/√Hz for LT1007 have been

added at the input of the operational amplifiers. The input

current noise source of operational amplifiers were

neglected, because its values are much smaller than

voltage noise sources, for instance In=0.4 pA/√Hz .

The voltage noise spectral density is calculated by the given equation [4]:

n

lnEj

lVT

knI

m

jkI

Tn

V2

),

(

2

)(,

2)

,(

2

)(,

)(2

In this equation TV,l and TI,k represent the transfer functions of the voltage noise sources En,i and the current noise sources In,i, respectively. Fig. 8 (a) shows the total voltage noise spectral density while Fig. 8 (b, c) shows noise components for both sections. The differences between simulated and calculated values can be considered as a result of using theoretical values based on thermal noise sources in calculations and real elements noise models using SPICE. The RMS value has been calculated by the equation:

2

1

)(2)2

(

dn

Vefn

E

In the desire to reduce voltage noise spectral density of filter we used the optimization algorithm given in [3]. All element values are taken from [3], recalculated and tested in Matlab and SPICE in order to give us smaller noise results. The total voltage noise spectral density of NO filter design is shown in Fig. 9 (a). Noise components for both sections are given through the Fig. 9 (b, c).

Comparing the noise RMS values of filters, we can conclude that optimized filter gave smaller noise RMS value.

B. Noise in OTA-C filter

The equivalent circuit for the calculation of the

voltage noise spectral density of the observed OTA-C

filter is shown in Fig. 10.

Voltage noise spectral density (7) is calculated. The

total voltage noise spectral density components are given

in Fig. 11 (a). The differences between simulated and

calculated values can be considered as a result of using

theoretical values based on thermal noise sources in

calculations using MATLAB and real elements noise

models using SPICE.

(a)

400 4000 400000

100

200

300

Frequency [Hz]

Nois

e s

pectr

al d

ensity

[nV

/Hz1

/2]

In11,1

In12,1

In2,1

In3,1

In4,1

En1,1

(b)

400 4000 400000

100

200

300

400

Frequency [Hz]

Nois

e s

pectr

al d

ensity

[nV

/Hz1

/2]

In11,2

In12,2

In2,2

In3,2

In4,2

En1,2

(c)

Figure 8. Noise voltage spectral density of Fourth-order SD band-pass

filter and its components. (a) Total noise voltage spectral density: simulated (dashed) and calculated (full). Noise RMS value is

En=54.556µV. (b) Section 1 noise voltage spectral density components.

Noise RMS value is En=34.805 µV. (c) Section 2 noise voltage spectral density components. Noise RMS value is En=42.011 µV.

R111

R121

R21

R31

R41

C21

C11

-

+

R112

R122

R22

R32

R42

C22

C12

-

+

Uin

Uout

+

+

In111

In21

In121

In31

In41

In42

In112

In22

In122

In32

En11

En12

Figure 7. Equivalent electrical noise circuit of the Fourth-order SD and

NO band-pass filter

MIPRO 2019/MEET 129

Fig. 11 (b, c) shows noise components for both

sections.

LT1228 was used as a transconductance amplifier in

this design. The input voltage noise source of this

-

+

+

-

-

+

+

-

C11

C21

gm11

gm21

gm31

gm41

-

+

+

-

-

+

+

-

C12

C22

gm12

gm22

gm32

gm42Uin

Uout

+

En11 +

En21

+

En41

+

En31

+

En12 +

En22

+

En42

+

En32

Figure 10. Equivalent electrical noise circuit of the Fourth-order OTA-C band-pass filter

(a)

400 4000 400000

40

80

120

Frequency [Hz]

Nois

e s

pectr

al d

ensity

[nV

/Hz1

/2]

In11,1

In12,1

In2,1

In3,1

In4,1

En1,1

(b)

400 4000 400000

50

100

150

Frequency [Hz]

Nois

e s

pectr

al d

ensity

[nV

/Hz1

/2]

In11,2

In12,2

In2,2

In3,2

In4,2

En1,2

(c)

Figure 9. Noise voltage spectral density of Fourth-order NO band-pass

filter and its components. (a) Total noise voltage spectral density:

simulated (dashed) and calculated (full). Noise RMS value is

En=20.412µV. (b) Section 1 noise voltage spectral density components. Noise RMS value is En=12.422 µV. (c) Section 2 noise voltage spectral

density components. Noise RMS value is En=16.197 µV.

(a)

400 4000 400000

10

20

Frequency [Hz]

Nois

e s

pectr

al d

ensity

[nV

/Hz1

/2]

En1,1

En2,1

En3,1

En4,1

(b)

400 4000 400000

10

20

Frequency [Hz]

Nois

e s

pectr

al d

ensity

[nV

/Hz1

/2]

En1,2

En2,2

En3,2

En4,2

(c)

Figure 11. Noise voltage spectral density of Fourth-order OTA-C band-

pass filter and its components. (a) Total noise voltage spectral density: simulated (dashed) and calculated (full). Noise RMS value is

En=2.4384µV. (b) Section 1 noise voltage spectral density components.


130 MIPRO 2019/MEET

transconductance amplifier is En=6 nV/√Hz and the

input current noise source is In=1.4 pA/√Hz. The current

noise source is much smaller and because of that it was

neglected in calculation procedure.

C. Noise in SC filter

The fourth-order SC band-pass filter is shown in Fig.

5. The equivalent circuit for the calculation of voltage

noise spectral density is based on [5]. Fig. 12 shows total

voltage noise spectral density and its components.

D. Comparison

If the RMS voltage noise value is measured between

0.1fc to 10fc as the indicator of noise reduction,

comparison of noise in all presented designs is given in

Table 4. Fig. 13 shows voltage noise spectral density for

all filter designs which are discussed in this paper.

IV. CONCLUSION

In this paper, noise improvement using different filter

designs is analyzed. Analyses of four different designs

are done. As noise measure RMS voltage noise value is

calculated between 400 Hz and 40 kHz. Noise analysis is

done for SD and for NO design filters based on

operational amplifiers. Noise improvement for NO is 2.5

times. Noise analyses for OTA-C and SC designs are also

done. They gave improvement of more than 20 times for

OTA-C design in comparison with SD and 10 times for

SC design in comparison with SD. In the end, considering

RMS voltage noise as a numeric indicator, it is shown

that the OTA-C filter design gives the best noise

reduction.

REFERENCES

[1] H. A. de Andrade Serra, “Design of Switched-Capacitor Filters using Low Gain Amplifiers”, PhD thesis, Faculdade de Ciencias e Technologia, Universidade Nova de Lisboa, 2012.

[2] I. Volarić, N. Stojković and S. Vlahinić, "Noise improvement using SC filters," 2015 38th International Convention on Information and Communication Technology, Electronics and Microelectronics (MIPRO), Opatija, 2015, pp. 121-126.

[3] N. Stojković, E. Kamenar, M. Šverko, “Optimized Second- and Fourth- Order LP and BP Filters”, Automatika, no 52, vol. 2, pp. 152-168, 2011.

[4] W. K. Chen, The Circuits and Filters Handbook, 2nd ed. CRC Press, 2003.

[5] J. H. Fisher, “Noise sources and calculation techniques for switched capacitor filters”, IEEE Journal of Solid-State Circuits, vol. 17., pp. 742-752, August 1982.

TABLE IV. NOISE RMS VALUE COMPARISON

SD NO OTA-C SC

En 54.556 µV 20.412 µV 2.4384 µV 5.3954 µV

(a)

(b)

(c)

Figure 12. Noise voltage spectral density of Fourth-order discrete time

SC band-pass filter and its components. (a) Total noise voltage spectral density: simulated (dashed) and calculated (full). Noise RMS value is

En=5.3954 µV. (b) Section 1 noise voltage spectral density components.


Figure 13. Noise voltage spectral density for: SD (dashed), NO

(dashed-dot), OTA-C (full), SC (dot-dot)

MIPRO 2019/MEET 131

noise improvement for different bp filter...

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