CHAPTER -2

NOTCH FILTER DESIGN TECHNIQUES

electronic systems. These techniques are rapidly developing day by day due to

tremendous technological developments in high speed computers, integrated

circuit fabrication and field programmable arrays (FPGA). With these, digital

signal processing has now become more reliable and speed processing is almost

near infinity [1-3]. DSP techniques find applications in variety of areas such as in

speech processing, data transmission on telephone channels, image processing,

instrumentation, bio-medical engineering, seismology, oil exploration, detection

of nuclear explosion etc. There are many commonly used DSP operations such as

Differentiation, Integration, Hilbert transformation and Filtering. Among these,

filtering is the operation that is invariably used in most of the applications. There

are various types of filters such as High pass (HP), Low pass (LP), Band stop

(BS), Band pass (BP), and Notch filters. Digital notch filters are of two types, IIR

and FIR. In this research work, we have focused our attention on the design

techniques of application based digital FIR notch filters.

2.1 NOTCH FILTER CHARACTERISTICS AND APPLICATIONS

Detection, estimation and filtering of narrow band signals in the presence

of noise represent some of the important applications of signal processing

techniques. In most of these applications it is desired to remove the narrow band

signal while leaving the broad band energy unchanged. This can be achieved by

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passing the signals through a notch filter where the notches are centered on the

narrow band signals.

2.1.1 Notch Filter Characteristics

A notch filter is essentially a band stop filter with a very narrow stop band

and two pass bands. The amplitude response, H1(ω), of a typical notch filter

(designated as NF1) shown in Figure 2.1 is characterized by a notch frequency ωd

(radians) and -3 dB rejection band width (RBW). For an ideal notch filter, this

RBW should be zero, the pass band magnitude should be unity and the attenuation

at the notch frequency should be infinite. We may alternatively have the

amplitude response H2(ω) of a notch filter (designated as NF2) as shown in Figure

2.2. H2(ω) has 180 degrees phase shift at the notch frequency ωd i.e. it has

opposite signs in the two pass bands. However, the magnitude response | H2(ω) |

is of the same type as that shown in Figure 2.1. In this thesis, we propose the

design methodologies for both these type of filters. Before considering the design

aspects of the notch filters, it is relevant to highlight some of the important

applications of the notch filters.

2.1.2 Applications of Notch Filters

Some of the applications of the notch filters are as follows:

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 Notch filter characteristics of NF1 type

H1(

(0dB) 1 RBW
- 3dB

Normalized
amplitude
response

(-∞ dB) 0
d (radians)

Frequency in radians

Figure 2.1 The normalized amplitude response H1(ω) of the notch filter NF1

14
 Notch filter characteristics of NF2 type

H2(|H2(|
1

Normalised
amplitude
response H (
 2

 
0
d

(radians)

H2(

-1 



Figure 2.2 The normalized amplitude response H2 (ω) and | H2 (ω) | of the notch filter NF2

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(i) In the area of communications, control, instrumentation and bio medical

engineering, notch filters are generally used to eliminate 50/60 Hz power

line interference.

(ii) Spatial notch filters are used in transmitting antenna arrays, which are

omni-directional except for one null. This feature is used when it is

desired to null out one listener in a known direction.

(iii) Notch filters are used to remove spectral lines from a broad band

spectrum, so as to eliminate noise due to electromagnetic interference

(EMI), as encountered in radars and direction finders [4].

(iv) Switching type of AC & DC motor drives, converters and inverters cause

sinusoidal disturbances at certain harmonics of the line frequency. This

can be a problem when measuring the velocity of an elevator using an

analog tacho generator [14]. This is so because, the primary velocity

signal now contains an additive ripple component consisting of line

frequency and its low order (2nd, 3rd and 4th order) harmonics. Use of notch

filter eliminates such unwanted signals and enables accurate

measurements.

(v) Besides the line frequency disturbances, mechanical resonance also cause

narrow band interference in many industrial and automatic control

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 applications. A servomotor is commonly coupled to its load by inherently

resilient drive shafts, and the combination possesses selective resonance

frequencies. The overall system may not operate smoothly if the resonance

frequency signals are not eliminated. Such interference is easily removed

by using notch filters [12,13].

(vi) In practical electrocardiogram (ECG) measurements, the primary signal is

often contaminated by strong disturbances, which must be removed before

the signal is registered for further analysis. The varying ECG contact

potentials and breathing artefacts (below 0.5 Hz) cause unwanted base line

drift [15]. For stress ECG recordings, this drift may some times make the

recording impossible. Such unwanted signals are effectively removed by

using appropriate linear phase notch filters. Elimination of dc component

of an ECG is another use of notch filters [16]. Notch filters are also used

to suppress the secondary artefacts that arise due to electrical interference,

as well as cross coupling from frontal EEG while analyzing electro-

oculogram [17].

(vii) Adaptive and broad null width notch filters find application in electronic

counter countermeasure (ECCM) systems. An important measure for

reducing the effects of either electronic countermeasure (ECM) or mutual

interference is to avoid saturating, or overloading of the receiver, with

17
 large interfering signals. This can be achieved with proper tuning of radar,

associated with adaptive notch filters to evade narrow band spot jamming.

(viii) A filter with broad null width can be used to suppress interference

frequencies that may arise due to accidental spikes around 50 Hz in anti

spike circuits as a protection measure.

The aforementioned list of application of notch filters is only indicative and not

complete. Now, we shall discuss the techniques commonly used to design the

notch filters.

2.2 DIGITAL NOTCH FILTER DESIGN TECHNIQUES

Digital notch filters are of two types.[1-3, 5-9].

 Infinite Impulse Response(IIR) filters

 Finite Impulse Response(FIR) filters

Now the salient features of these techniques with special emphasis on FIR digital

notch filters are discussed.

2.2.1 IIR Design Techniques

IIR filters are preferred in situations where linearity of phase response is

not that important. These filters require much lower order compared to the FIR

18
ones for the same set of magnitude response characteristics. This implies

simplified structure with fewer multipliers and adders. The commonly used IIR

filter design methods require transforming the given specification to an equivalent

analog filter. Through the use of known analog filter design techniques, analog

notch filter is designed first and then it is converted back to the digital domain

through inverse transformation. This approach has the advantage that the standard

results of analog filter design can be conveniently used. Based upon this

approach, one can design Butterworth, Chebyshev (i.e.equiripple) or least-squares

error (LSE) design or the design based on filter parameter optimization

techniques. The most commonly used transformation is the bilinear

2  1  z 1 
transformation connecting‘s’ and ‘z-1’ by the relation: s    . This is a
T  1  z 1 

conformal transformation, and hence the most popular.

As an alternative, one can design digital IIR filters directly in the z-

domain without reference to the analog domain. There are three methods for

designing the IIR filters directly.

(i) Pad/e approximation method.

(ii) Least-Squares design in time domain.

(iii) Least-Squares design in frequency domain.

Among the above, we discuss the first method as this approach has influence on

our approach.

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 Suppose that the desired impulse response hd(n) is specified for n ≥ 0. The

filter to be designed has the system function

b z
k 0
k
1
(2.1a)
H ( z)  N
1   ak z k
k 0


(2.1b)
  h( k ) z  k
k 0

where h(k) is the unit sample response. The filter has L = M + N + 1 parameters,

namely, the coefficients {ak} and {bk}, which can be selected to minimize some

error criterion. Suppose that we minimize the sum of the squared errors

U
   [h
n0
d ( n )  h ( n )] 2
(2.2)

with respect to the filter parameters {ak} and {bk}, where U is some preselected

upper limit in the summation. In general, h(n) is a nonlinear function of the filter

parameters and, hence, the minimization of ε involves the solution of a set of

nonlinear equations. However, if we select the upper limit as U = L – 1, it is

possible to match h(n) perfectly to the desired response hd(n) for 0 ≤ n ≤ M + N.

This can be achieved in the following manner.

The difference equation for the desired filter is

y(n) = - a1 y(n-1) – a2 y(n-2) - . . . – aN y(n-N)

Suppose that the input to the filter is a unit sample [i.e. x(n) = δ(n)]. Then the

response of the filter is y(n) = h(n) and hence (2.3) becomes:

20
 h(n) = - a1 h(n-1) – a2 h(n-2) - . . . – aN h(n-N)
(2.4)
+ b0 δ(n) – b1 δ(n-1) + . . . + bM δ(n-M)

Since δ(n-k) = 0 except for n = k, (2.4) reduces to

h(n) = - a1 h(n-1) – a2 h(n-2) - . . . – aN h(n-N) + bn , 0≤n≤M (2.5)

For n > M, since {bk} are all zero, (2.4) becomes

h(n) = - a1 h(n-1) – a2 h(n-2) - . . . – aN h(n-N), 1≤n≤N (2.6)

The set of ( M + N + 1 ) linear equations in (2.5) and (2.6) can be used to solve

for the filter parameters {ak} and {bk}. We set h(n) = hd(n) for 0 ≤ n ≤ M + N, and

use the linear equations in (2.6) to solve for the filter parameters {ak}, 1 ≤ k ≤ N.

Then we use these values of {ak} in (2.5) and solve for the parameter {bk}, 0 ≤ k

≤ M. Thus we obtain a perfect match between h(n) and the desired response hd(n)

for the first L = M + N + 1 values of the impulse response. This design technique

is usually called the Pad /e approximation procedure.

Some modified IIR notch filter designs have also been reported in the

literature. Hirano et al. [18] have realized IIR notch filter function by applying

bilinear transformation on second order analog transfer function. The design

requires only two multipliers and offers independent variation of notch frequency

(ωd) and the 3 dB rejection bandwidth (BW). Laakso et al. [13] have proposed

21
first and second order IIR notch filters with zeros strictly on the unit circle and

poles close to the zeros to ensure a narrow notch width. The first order notch filter

is useful in eliminating zero frequency (i.e. dc component) only. The second order

notch filter given by

1  2 cos  0 z 1  z 2 (2.7)
H 0 ( z)  K 0
1  2r cos  0 z 1  r 2 z 2

can be designed for an arbitrary notch frequency ω0. In (2.7) ω0 is the normalized

frequency in radians, r is the radius of the complex conjugate pole pair located at

the frequency ω0 and K0 is a scaling factor such that the maximum gain of the

filter equals unity. In this design, the rejection bandwidth (RBW) can be

controlled through ‘r’. RBW decreases as ‘r’ goes closer to unit circle [13].

However, the quantization error increases when ∆ = 1-| r | is made small (since the

variance of the quantization error is proportional to 1 / ∆2 [1].

Much research has also been carried out in the area of adaptive notch

filtering. In certain applications of signal processing, where it is desired to

eliminate unknown or time varying narrow band or sine wave components from

the observed time series, it becomes necessary to use adaptive notch filters

(ANF). ANFs also find use in retrieval of sinusoids in noise and in tracking and

enhancing time varying narrow band / sinusoidal signals in wide band noise [53-

56]. Adaptive notch filter designs have been proposed by Thompson [54], Rao

and Kung [55], Friedlander and Smith [40], and Nehorai [41], amongst others.

22
The computational efficiency, stability, convergence and numerical robustness of

these methods depend upon the algorithm used for adaptation.

Juan E.Cousseau, Stefan Werner and Pedro Dario Donate have introduced

a new family of IIR adaptive notch filters [56] that forms multiple notches using a

second-order factorization of an all-pass transfer function. They have proposed

two new adaptive filtering algorithms that can achieve fast convergence at low

computational cost. This all-pass based notch realization introduces a different

compromise between bias and signal to noise ratio (SNR) when compared with

realizations previously reported in the literature. It achieves lower bias than other

approaches at low SNR. This property is particularly attractive for the estimation

and tracking of multiple sinusoids.

One of the major problems in IIR filters is that these designs have non

linear phase response and, therefore, introduce phase distortion in general. It is

possible to reduce phase distortion by cascading an all-pass phase equalizer.

However, in this case, the advantage of lower order IIR filter is lost as cascading

an all-pass filter results in an increased order of the over all filter which may some

times be comparable to that of an equivalent FIR filter. Moreover, the IIR filters

tend to be unstable (due to quantization) when the pole is very near to the unit

circle. We, now examine some of the design techniques used for FIR notch filters.

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2.2.2 FIR Design Techniques

There are essentially three types of standard design techniques for linear

phase FIR filters.

 Frequency Sampling method

 Window Method

 Optimal Filter Design Method

The frequency sampling method is often not used for notch filter design because,

the desired frequency response changes rapidly across the notch point leading to

large interpolation error. The window method has the advantages that it is easy to

use and closed form expressions are available for the window coefficients.

Several windows have been reported in the literature, such as Hamming, Hann,

Blackman, Bartlett, Papoulis, Lanczos, Tukey, Kaiser and Dolph-Chebyshev [1,

19] etc.

Vaidyanathan and Nguyen [20] introduced FIR eigen filters which are

optimal in the least squares sense. Here, the objective function is defined only as

the sum of pass band and stop band errors; the error of approximation in the

transition band is not included. One of the advantages of eigen filters over other

FIR filters is that, they can be designed to incorporate a wide variety of time

domain constraints such as the step response and Nyquist constraint in addition to

the usual frequency domain characteristics. This method has also been extended

to include flatness constraints in the pass band.

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 The filters designed by using Kaiser window and eigen filter approaches

are such that the ripple size grows as we move closer to the band edge. Because of

this, the filter performance exceeds (i.e. better than the specification) for most

frequencies, except around ωp and ωs [10], where ωp and ωs are respectively the

pass band and stop band edges. The filter is thus, over designed and it is possible

to reduce the filter length ‘N’ to meet the given set of specifications. This is

achieved by the mini-max design method where the error is uniformly distributed

in the pass band and stop band. Using Alternation Theorem and Remez –

exchange algorithm, Parks and McClellan [21,22] have proposed versatile

algorithm [23] for designing various types of FIR filters including differentiators

and Hilbert transformers.

Out of all the FIR designs, Parks and McClellan iterative design yields the

best results, although, it too has some inherent limitations. Equiripple designs

only consider the specified pass bands and stop bands but the transition bands are

not considered in the numerical solution. In fact, the transition regions are

considered as ‘don’t care’ regions in the design procedure. As a result, the

numerical solution may fail, especially in the transition region. For the optimum

design, the filter length is determined by the narrow transition band. If the

transition band is too wide, the algorithm will fail in the transition region resulting

in overshoot of the frequency response [9].

25
 Besides the standard techniques, a number of other methods are proposed

for the design of notch filters in the literature. Among them, a few are discussed

here. Tian-Hu yu et al. [24] has proposed two methods for designing the notch

filters by exploiting the aforementioned design techniques. In one of the methods,

a notch filter H(ω) is derived from a low pass filter HLP(ω) by using the relation

(2.8)
(2.8)
H(ω) = 2HLP(ω) - 1

when HLP(ω) has the passband 0 to ωd. This transformation provides a change of

phase by 180 degrees at the notch frequency ωd. The frequency response H(ω)

may further be sharpened by using the amplitude change function (ACF) [25] . An

alternative method in [24] is based on complementing a narrow pass band (tone)

filter B(ω) to obtain the desired notch filter by using

H(ω) = 1 - B(ω) (2.9)

Obviously, a narrow band filter B(ω) will have a large filter order. A number of

techniques are, however, available in the literature [26-28] for reducing the

number of multiplications.

Byrnes [29] considered spatial notch filters, and derived a polynomial

whose magnitude on the unit circle is close to a constant in all the directions

except one. Such polynomials are used to design transmitting antenna arrays,

which are omni-directional except for one null. The designed filter, however, is a

26
nonlinear phase FIR filter. Also, in this design the weights are complex in nature

and, therefore, the number of multiplications and additions required for

computation increases. This in turn also increases the memory requirements.

Another method for designing FIR notch filter was proposed by M.H.Er

[30] where the symmetry constraint for the coefficients h(n) was relaxed and

therefore the design yields nonlinear phase FIR filters. Two procedures have been

proposed in [30] for varying the null width. In the first approach, the mean

squared error between the desired unity response and the response of the filter

over frequency band of interest is minimized subject to a null constraint and its

zero derivative constraints at the frequency of interest. The null width can be

increased in discrete steps by setting more derivatives to zero at the notch

frequency. In the second approach, a null power constraint over a frequency band

of interest is introduced. This approach is found to be more effective in

controlling the null width as compared to the derivative constraint methodology.

Both of these approaches adopt optimization techniques, which have been

efficiently solved in [30]. The limitation of such a design, however, is that it

yields nonlinear phase, and does not provide closed form mathematical formula

for computation of design weights. Every time the notch frequency ωd changes,

the optimization procedure has to be re run to obtain the design weights.

Novel analytical designs for maximally flat (MF) and equiripple (ER) FIR

notch filters have been proposed in [31-33, 57], based on Zolotarev polynomials

27
and Jacobs’s elliptical functions. In [31] P. Zahradnik and M. Vlcek have

presented a fast analytical tuning procedure for FIR notch filters. This analytical

tuning procedure is based on the differential equation of the transformed

Chebyshev polynomial. Proposed tuning procedure adjusts a single frequency of

the frequency response to the desired value while preserving the nature of the

filter.

The same authors have presented a novel analytical design method [32]

for highly selective digital optimal equiripple FIR comb filter. So designed comb

filters are optimal in the Chebyshiv sense. In this method the number of notch

bands, the width of the notch bands and the attenuation in the pass bands can be

independently specified. Next with these specifications they have evolved the

degree formula and differential equation for generating polynomial of the comb

filter. Based on the differential equation, they have described an algebraic

recursive procedure for the evolution of impulse response of the filter. This

procedure is very robust but highly involved. By using this, highly selective

equiripple FIR comb filters with thousands of coefficients can be designed.

P. Zahradnik and M. Vlcek have suggested recursive algorithms [33] for

computation of impulse response of notch filters, using many results proposed by

them in [35]. These algorithms also, require complex mathematical

manipulations. These notch filter designs are optimal and very useful ones though

the calculation of the impulse response h(n) requires computations of a number of

28
parameters. In this method the recursive algorithms are derived which lead to the

impulse response coefficients of the notch filter.

Recently in 2009, another design method proposed by P. Zahradnik et al.

[57] is an extension to the analytical design of digital FIR comb filter [32]. This

design runs from the filter specification through the degree formula to evaluate

the impulse response coefficients, by an extremely efficient recursive algorithm.

The proposed design method outperforms the standard procedures in terms of

speed and robustness. Here also, the design approach requires highly involved

mathematics.

To summarize, the aforementioned design approaches for FIR notch filters

have the following drawbacks:

(1) The window technique is a non optimal design [9].

(2) The eigen filter approach results in excessive ripples near the band edges

[20].

(3) Parks-McClellan method [22-24] as well as Er’s method [30] are iterative

and give no explicit or recursive formulas for computing the design weights.

(4) The classical method of Byrnes [29] uses complex weights and requires

excessive multiplications.

(5) The adaptive notch filters are algorithm dependent.

29
(6) Algorithms proposed by Zahradnik and Vlcek [31-33, 57] require highly

involved mathematical manipulations. The calculation of the impulse

response h(n) requires computation of a number of parameters.

We, in this thesis, have proposed alternative design approaches for the

design of application specific FIR notch filters. These approaches are free from

most of the above listed undesired features.

30