I am trying to design a digital notch filter that stops the power line frequency of 50 Hz. My approach to design the filter was first to realize the circuit using passive components inductor and capacitor (the output is taken from the L and C component which has resonant frequency of 50Hz). Then, I determined the transfer function of the system in 's' domain and converted it to equivalent transfer function in 'z' domain. Then, I determined the equivalent difference equation from the transfer function in 'z' domain. But, when I implemented that difference equation in coded form, I was unable to get the desired response. I am new to digital filter implementation, so I wanted to ask whether I missed on some crucial consideration ? And, I also want to know more about the implementation technique of digital filter.
Here is the image of the notch filter that I implemented using passive components
There is an immensely simpler way of implementing a phase accurate notch filter: N-path filter. It can be designed as a notch with ridiculously high Q factor.
The approach is indirect and as you have found the results uncertain. Indeed the notch frequency can be controlled by an appropriate constant attached to the bilinear transformation but then you cannot control the bandwidth to be to the required value.
I produced a solution to the design in 1969 in the publication
Digital notch filters
Constantinides, A.G. Electronics Letters Volume: 5 , Issue: 9
Digital Object Identifier: 10.1049/el:19690150
Publication Year: 1969 , Page(s): 198 - 199
Subsequent to this there were several papers relying on this approach by other authors
You can download this paper from Researchgate.
Article Digital notch filters
I forgot to add a simple MATLAB code as below
clear all;close all
theta0=.3*pi;rho=.94; % Selectable: theta0=notch frequency, rho=controls BW
b=[1 -2*rho*cos(theta0) rho^2];a=b(end:-1:1);
num=(b+a)/2;den=a;
fvtool(num,den);title('Notch Filter response')
Rojan, see http://doc.utwente.nl/86403/ for new implementations of such filters in RF. They also come as "commutating filters" and this principle is very old. You can easily adopt this approach in a digital world and 50Hz offending noise as well - your signal is already sampled.
To go from "s" to "Z" you use the bilinear transformation for which you need to calculate the transition frequency between analog and digital domain.
The easiest way is to work directly in Z like Constantinides (see his matlab code).
If you are working in low frequencies (i.e. EEG or ECG signals), the simplest approach is to delay the signal with a half of period (10 ms) and then add the two streams.
Here are some further comments and a short Matlab code to help
The object is this: Given a discrete time signal x(n) that contains a frequency (normalised) theta0 to process x(n) in such a way as to produce a discrete time signal y(n) that does not contain the sinusoidal component of frequency theta0.
This is not a very good definition because it does not say to what extent is the rest of the spectrum of x(n) allowed to be modified by the required processing. However it does have a simple answer and this is
y(n)=x(n) – 2*cos(theta0)*x(n-1)+x(n-2)
The equation represents a system a transfer function H(z) with
input x(n), output y(n) where
H(z)=1-2*cos(theta0)*z^(-1)+z^(-2);
This is a very crude notch filter.
However if the disturbance of the rest of the spectrum of x(n) is to be reduced then a proper notch filter needs to be designed which has both a numerator and a denominator. The numerator essentially will be the same as above and the denominator in effect corrects the distortions of H(z) on the rest of the spectrum.
A very simple and obvious way is to create a denominator is to shift the zeros slightly off (and inside the unit circle) as 1-2*r*cos(theta0)*z^(-1)+r^2*z^(-2), where 0
In defence of N-path filters, here is a nice document that describes using them to notch out vuvuzele sounds from audio in sports coverage: http://www.cypress.com/?docID=34215
Try this code
% Notch Filter 50 Hz
%
% theta = 2*pi*50/fs (50 hertz and fs = Sampling Frequency)
% we define now a 2nd order digital filter*
%
jj = sqrt(-1);
len = length(data); % length of data vector
vecteur_freq = fe/len*(0:1:len-1); % frequency vector
theta = 2*pi*50/fe; % 50 Hz
module_pole = 0.99;
zero = [exp(jj*theta) exp(-jj*theta)];
pole = module_pole*zero;
% Polar representation
% Num and Denum coeffs
B = poly(zero);
A = poly(pole);
zplane(B,A);
% thetas definition
V_theta = linspace(0,2*pi,length(vecteur_freq));
% tranfert function
figure(2);
freqz(B,A,V_theta);
The cascade connection of a lowpass (LP) and a highpass (HP) filter will produce a bandstop behaviour, and if a notch filter is considered a bandstop (BS) filter then the task is achieved.
However, a notch filter essentially produces a notch in the frequency domain in which there a transmission zero at the notch frequency and the bandwidth of the notch defined at the 3dB points on either side of the notch frequency is controllable as design parameter.
In the cascade connection of an LP and an HP filter we have to make sure that the respective cutoff (or corner) frequencies must be so arranged to produce a “notch “ type behaviour (ie the LP cutoff frequency is less than the HP cutoff frequency) and that the transition bands of the two filters interact to produce the notch. Moreover, the transition frequencies (the points along the frequency axis at which the respective stopbands start) dictate the bandwidth of the notch. The attenuation at the notch so achieved, may not necessarily be high. A transmission zero is produced in the cascade only when both the LP and the HP happen to have a coincident transmission zero at the notch frequency. This means that either one has to design the constituent LP and HP filters to have a zero at the point, or just accept the fact that there will be a residual signal at the notch frequency of the cascade.
Another significant issue is that a cascade of two filters which have lengths L1 and L2 produces a system that has an impulse response of L1+L2-1. Thus is “rapid response” applications such an arrangement is not tractable in that significant latency is inherent.
The design most ably demonstrated by Alain Loussert in his Matlab code would be the first option to avoid the pitfalls outlined above
Hi, this article maybe helps you. It's about design and implementation of high Q notch filters.
Chapter High Q Notch Filter Analysis
I think that our new filter designer + signal analyser tool may help you here. Please have a look at: http://www.advsolned.com/asn_filter_designer.html and have a look at the filter script examples. There's an interactive example for notch filtering.
There is another way to design Notch filter, would you like to have a look at the paper "DESIGN OF DIGITAL FILTERS USING PASCAL’S TRIANGLE".You can download this paper from Researchgate.
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