Anti-aliasing Filter

Table of Contents

Introduction Antialiasing example Anti-aliasing Filter Model Configure Input Plot Sampled Signal Plot amplitude response

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details <https://www.gnu.org/licenses/>.

This code was developed by Miodrag Bolic for the book PERVASIVE CARDIAC AND RESPIRATORY MONITORING DEVICES: https://github.com/Health-Devices/CARDIAC-RESPIRATORY-MONITORING

% Changing the path from main_folder to a particular chapter
main_path=fileparts(which('Main_Content.mlx'));
if ~isempty(main_path)
    %addpath(append(main_path,'/Chapter2'))
    cd (append(main_path,'/Chapter3/ADC'))
    addpath(append(main_path,'/Service'))
end
SAVE_FLAG=0; % saving the figures in a file

Introduction

This notebook provides introduction to a model for an anti-aliasing filter.

Aliasing occurs when high and low frequency samples are indistinguishable thus resulting in improper conversion of the input signal. According to the Nyquist criterion, the sampling rate should be at least twice the maximum frequency component of the signal. Aliasing usually exists when Nyquist criterion is violated and should be prevented through the use of low-pass (anti-aliasing) filters. These low pass filters are added before the sampler and the ADC.

The goal is to model an anti-alias filter that attenuates the higher frequencies (higher than the Nyquist frequency) which would prevent the aliasing components from being sampled.

Antialiasing example

clear_all_but('SAVE_FLAG')

fs=12;

fsine=10;

Ts=1/fs;

t=0:0.001:1; %time;

y=sin(2*pi*fsine*t);

n=0:fs;

ys=sin(2*pi*fsine*Ts*n);

figure

plot(t,y)

hold on

falias=-mod(fs,fsine); % modify this line for different sampling frequencies than 12Hx

plot(t, sin(2*pi*falias*t))

plot(Ts*n, ys, 'o', 'MarkerFaceColor', 'r')

legend (["Signal without aliasing", "Signal with aliasing"])

title('Sampled signal below the Nyquist rate')

xlabel('Time (s)')

ylabel('Voltage (v)')

ylim([-1.1, 1.4])

annonation_save('',"Fig3.20.jpg", SAVE_FLAG);

Anti-aliasing Filter Model

The following simple model was created to model an analog anti-aliasing filter. Since it is before the sampler and the ADC this is the analog world and the anti-aliasing filter is an analog filter. An input voltage is confiured here and and the results will be plotted as a sampled signal with and without antialiasing filter.

Configure Input

The input voltage is configured here as a sin function. The reshape function is included as it can be helpful for preprocessing the data for subsequent computations or analyzing the data.

model_name = 'elec_a2d_sd_anti_alias_graph';
open_system(model_name);
T=0.001; 
Vin(:,1)=0:T:60; %time;
t=Vin(:,1); Vin(:,2)=1.5+sin(2*pi*0.25*t)+0.25*sin(2*pi*60*t);
sim(model_name)
simOut = sim('elec_a2d_sd_anti_alias_graph', 'CaptureErrors', 'on');
temp=simOut.vOut.signals.values(:,1,:);
alias=reshape(temp,4801,1);
temp=simOut.vOut.signals.values(:,2,:);
no_alias=reshape(temp,4801,1);

Plot Sampled Signal

The time domain signal is plotted here.

figure

plot(simOut.vOut.time,alias)

hold on

plot(simOut.vOut.time,no_alias)

xlim([0,4])

legend (["Vo1", "Vo"])

title('Sampled signal with and without antialiasing filter')

xlabel('Time (s)')

ylabel('Voltage (v)')

annonation_save('a)',"Fig3.22a.jpg", SAVE_FLAG);

Plot amplitude response

fs=80;
L=length(no_alias)-1;
n = 2^nextpow2(L);

Compute the two-sided spectrum P2. Then compute the single-sided spectrum P1 based on P2 and the even-valued signal length L.

Y = fft(alias,n);
P2a = abs(Y/n);
P1a = P2a(1:n/2+1);
P1a(2:end-1) = 2*P1a(2:end-1);
Y = fft(no_alias,n);
P2n = abs(Y/n);
P1n = P2n(1:n/2+1);
P1n(2:end-1) = 2*P1n(2:end-1);

Define the frequency domain f and plot the single-sided amplitude spectrum P1. The amplitudes are not exactly at 0.7 and 1, as expected, because of the added noise. On average, longer signals produce better frequency approximations.

f = fs*(0:(n/2))/n;

figure

plot(f,mag2db(P1a))

hold on

plot(f,mag2db(P1n))

title('Amplitude Spectrum')

xlabel('f (Hz)')

ylabel('|V(f)| (dB)')

ylim([-90 , -10])

xlim([17.5, 22.5])

legend (["Vo1", "Vo"])

annonation_save('b)',"Fig3.22b.jpg", SAVE_FLAG);