One Dimensional Low pass , High Pass and band pass filtering

Consider a one dimensional signal in time domain.

For instance, generate cosine waves of different amplitudes and different frequencies and combine them to form a complicated signal.

Consider four cosines waves with frequencies 23, 18, 10 and 5 and amplitudes 1.2, 0.8, 1.1 and 2 respectively and combine them to generate a signal.

Eg.

Cosine wave with frequency 23 and amplitude 1.2.

 

Figure.1

Cosine wave with frequency 18 and amplitude 0.8

 

Figure.2

Cosine wave with frequency 10 and amplitude 1.1

 

Figure.3

Cosine wave with frequency 5 and amplitude 2.

Figure.4

MATLAB code:

%To generate a signal

%Input function A cos(2pift)

freq =[23 18 10 5];

Amp = [1.2 0.8 1.1 2];

fs = 100;

ts = 1/fs;

t = 0:ts:5;

xt1 = 0;

for Ind = 1:numel(freq)

Freq = freq(Ind);

A = Amp(Ind);

xt1 = xt1+A*cos(2*pi*Freq*t);

end

figure,plot(t,xt1);

 

Figure.5

Let’s make a little understanding of this signal in frequency domain.

i.e in Fourier domain.

Fourier transform is one of the various mathematical transformations known which is used to transform signals from time domain to frequency domain.
The main advantage of this transformation is it makes life easier for many problems when we deal a signal in frequency domain rather than time domain.

After fourier transform, the signal will look like the below figure.

MATLAB code:

f1 = linspace(-fs/2,fs/2,numel(xt1));

xf = fftshift(fft(xt1));

str = 'Frequency domain';

figure,plot(f1,abs(xf)),title(str);

 

Figure.6

EXPLANATION:

In the figure.6, there are 4 different colors to denote four different signals.

Due to symmetric property, we see two peaks of the same signal.

fftshift is useful for visualizing the Fourier transform with the zero-frequency component in the middle of the spectrum.

The yellow color peaks denote the frequency with 5

The blue color peaks denote the frequency with 10

The green color peaks denote the frequency with 18

And the red color denotes the frequency with 23.

From this, we can understand that the low frequency components are close to the centre of the plot and the high frequency components are away from the centre.

Now, lets try to separate each of the original 4 signals from the combined ones.

We will keep only one signal at a time and remove/filter the rest.

MATLAB code:

       Kernel_BP = abs(f1) < 8;

   BPF = Kernel_BP .* xf;

  

   figure,plot(f1,abs(BPF));

Figure.7

Figure.8

A kernel is used to obtain the essential frequency component alone.

This can be done by making the values one for the frequency range less than 8 and rest zero.

We have obtained the lowest frequency component and the kernel we designed is for low pass filtering. The value which we use to find the limit is the cut off frequency. Here 8 is the cut of frequency.

Now , let’s go back to time domain.

MATLAB code:

BP_t = ifft(ifftshift(BPF));

plot(t,real(BP_t));

Compare the figure.4 and figure.9, both have same frequency right!!

Figure.9

Now let’s try to design the kernel for filtering high frequency component.

High Pass filter:MATLAB code :

    

   Kernel_BP = abs(f1) > 20;

   BPF = Kernel_BP .* xf;

  

   figure,plot(f1,abs(BPF));

 

Figure.10

   

Figure.11

  BP_t = ifft(ifftshift(BPF));

  plot(t,real(BP_t));

Figure.12

We have obtained the same signal as in figure.1

We will try to filter the signal of frequency 18.

This is band pass filter.

Band Pass Filter:
MATLAB code:

   Kernel_BP = abs(f1) > 15 & abs(f1) < 20;

   BPF = Kernel_BP .* xf;

  

   figure,plot(f1,abs(BPF));

         

Figure.13

   BP_t = ifft(ifftshift(BPF));

   plot(t,real(BP_t));

 

Figure.14

Figure.15

This same type of filtering can be applied on noisy signals to remove undesired signals.

MATLAB code:

Noise = 2 * randn(size(t));

xt1 = xt1 + Noise;

Using this signal as input, we can apply low, high and band pass to remove noise and obtain the original signal(to some extent).

Wait for my next post on two dimensional low and high pass filtering. i.e on images in space domain.

Optical Illusion – Circle

Let us define a circle and assign different color to each one of the four quadrants.

MATLAB CODE:

%Radius

rad=400;

[X,Y]=meshgrid(-rad:rad);

nh = X.^2+Y.^2 <= rad.^2;

color_cir= uint8([size(nh,1) size(nh,2) 3]);

nh =uint8(nh);

%Color Blue

color_cir(1:rad,1:rad,1) = nh(1:rad,1:rad)*10;

color_cir(1:rad,1:rad,2) = nh(1:rad,1:rad)*100;

color_cir(1:rad,1:rad,3) = nh(1:rad,1:rad)*180;

%color pink

color_cir(rad:rad*2,1:rad,1) = nh(rad:rad*2,1:rad)*180;

color_cir(rad:rad*2,1:rad,2) = nh(rad:rad*2,1:rad)*10;

color_cir(rad:rad*2,1:rad,3) = nh(rad:rad*2,1:rad)*100;

%color Green

color_cir(rad:rad*2,rad:rad*2,1) = nh(rad:rad*2,rad:rad*2)*100;

color_cir(rad:rad*2,rad:rad*2,2) = nh(rad:rad*2,rad:rad*2)*180;

color_cir(rad:rad*2,rad:rad*2,3) = nh(rad:rad*2,rad:rad*2)*10;

%Color Red

color_cir(1:rad,rad:rad*2,1) = nh(1:rad,rad:rad*2)*180;

color_cir(1:rad,rad:rad*2,2) = nh(1:rad,rad:rad*2)*10;

color_cir(1:rad,rad:rad*2,3) = nh(1:rad,rad:rad*2)*10;

imshow(color_cir);

Concentric Circles:

Colored:

MATLAB CODE:

clc

clear all

rad1 = 600;

[X,Y] = meshgrid(-rad1:rad1);

nh  =  X.^2+Y.^2  <=  rad1.^2;

color_cir =  uint8([size(nh,1) size(nh,2) 3]);

nh  = uint8(nh);

color_cir(1:size(nh,1),1:size(nh,2),1) = nh*10;

color_cir(1:size(nh,1),1:size(nh,2),2) = nh*1;

color_cir(1:size(nh,1),1:size(nh,2),3) = nh*2;

for rad = 550:-50:50

[X,Y] = meshgrid(-rad:rad);

diff = rad1-rad;

nh = X.^2+Y.^2  <= rad.^2;

nh  = uint8(nh);

Pos1 = diff:(rad+diff-1);

Pos2 = diff+rad:(rad*2)+diff-1;

%Color Red

color_cir(Pos1,Pos1,1) = nh(1:rad,1:rad)*100 + color_cir(Pos1,Pos1,1);

color_cir(Pos1,Pos1,2) = nh(1:rad,1:rad)*10 + color_cir(Pos1,Pos1,2);

color_cir(Pos1,Pos1,3) = nh(1:rad,1:rad)*10 + color_cir(Pos1,Pos1,3);

%Color Green

color_cir(Pos1,Pos2,1) = nh(1:rad,rad:rad*2-1)*10 + color_cir(Pos1,Pos2,1);

color_cir(Pos1,Pos2,2) = nh(1:rad,rad:rad*2-1)*100 + color_cir(Pos1,Pos2,2);

color_cir(Pos1,Pos2,3) = nh(1:rad,rad:rad*2-1)*10 + color_cir(Pos1,Pos2,3);

%Color Blue

color_cir(Pos2,Pos1,1) = nh(rad:rad*2-1,1:rad)*10 + color_cir(Pos2,Pos1,1);

color_cir(Pos2,Pos1,2) = nh(rad:rad*2-1,1:rad)*10 + color_cir(Pos2,Pos1,2);

color_cir(Pos2,Pos1,3) = nh(rad:rad*2-1,1:rad)*100 + color_cir(Pos2,Pos1,3);

%Color yellow

color_cir(Pos2,Pos2,1) = nh(rad:rad*2-1,rad:rad*2-1)*150 + color_cir(Pos2,Pos2,1);

color_cir(Pos2,Pos2,2) = nh(rad:rad*2-1,rad:rad*2-1)*100 + color_cir(Pos2,Pos2,2);

color_cir(Pos2,Pos2,3) = nh(rad:rad*2-1,rad:rad*2-1)*10 + color_cir(Pos2,Pos2,3);

end

figure,imshow(color_cir);

EXPLANATION:

1.      Draw a circle of radius 600.

2.      Fill the circle with black color.

3.      Draw another circle of radius 550 and fill the first quadrant will green, second quadrant with red, third quadrant with blue and the fourth quadrant with yellow color.

4.      Place the smaller circle (radius 550) on the bigger circle (radius 600).

5.      Draw another circle of radius 500 and repeat the same process till the radius of the smaller circle is 50.

Black and White:

MATLAB CODE:

clc

clear all

rad1 = 1250;

[X,Y] = meshgrid(-rad1:rad1);

nh = X.^2+Y.^2  <=  rad1.^2;

color_cir =  double([size(nh,1) size(nh,2) 3]);

nh  = double(nh);

color_cir(1:size(nh,1),1:size(nh,2),1) = nh*1;

color_cir(1:size(nh,1),1:size(nh,2),2) = nh*1;

color_cir(1:size(nh,1),1:size(nh,2),3) = nh*1;

c1 = 0;

c2 = 255;

for rad = 1200:-50:50

[X,Y] = meshgrid(-rad:rad);

diff = rad1-rad;

nh = X.^2+Y.^2 <= rad.^2;

nh  = double(nh);

Pos1 = diff:(rad+diff-1);

Pos2 = diff+rad:(rad*2)+diff-1;

%Second Quadrant

color_cir(Pos1,Pos1,1) = nh(1:rad,1:rad)*c1 + color_cir(Pos1,Pos1,1);

color_cir(Pos1,Pos1,2) = nh(1:rad,1:rad)*c1 + color_cir(Pos1,Pos1,2);

color_cir(Pos1,Pos1,3) = nh(1:rad,1:rad)*c1 + color_cir(Pos1,Pos1,3);

%First Quadrant

color_cir(Pos1,Pos2,1) = nh(1:rad,rad:rad*2-1)*c2 + color_cir(Pos1,Pos2,1);

color_cir(Pos1,Pos2,2) = nh(1:rad,rad:rad*2-1)*c2 + color_cir(Pos1,Pos2,2);

color_cir(Pos1,Pos2,3) = nh(1:rad,rad:rad*2-1)*c2 + color_cir(Pos1,Pos2,3);

%Third Quadrant

color_cir(Pos2,Pos1,1) = nh(rad:rad*2-1,1:rad)*c2 + color_cir(Pos2,Pos1,1);

color_cir(Pos2,Pos1,2) = nh(rad:rad*2-1,1:rad)*c2 + color_cir(Pos2,Pos1,2);

color_cir(Pos2,Pos1,3) = nh(rad:rad*2-1,1:rad)*c2 + color_cir(Pos2,Pos1,3);

%Fourth Quadrant

color_cir(Pos2,Pos2,1) = nh(rad:rad*2-1,rad:rad*2-1)*c1 + color_cir(Pos2,Pos2,1);

color_cir(Pos2,Pos2,2) = nh(rad:rad*2-1,rad:rad*2-1)*c1 + color_cir(Pos2,Pos2,2);

color_cir(Pos2,Pos2,3) = nh(rad:rad*2-1,rad:rad*2-1)*c1 + color_cir(Pos2,Pos2,3);

if(c2 > 1)

    c1 = 255;

    c2 = -255;

else

    c1 = -255;

    c2 = 255;

end

end

color_cir = uint8(color_cir);

figure,imshow(color_cir);

EXPLANATION:

1.      Draw a circle of radius 1250.

2.      Fill it with black color.

3.      Draw another circle with radius 1200.

4.      Fill second and fourth quadrant of the smaller circle with white color.

5.      Fill First and Third quadrant of the smaller circle with black color.

6.      Place the smaller circle on the bigger circle.

7.      Define another circle of radius 1150.

8.      Fill second and fourth quadrant of the smaller circle with black color.

9.      Fill First and Third quadrant of the smaller circle with white color.

10.  Now place this circle on the bigger one. Repeat this process with smaller circles and by interchanging black and white color on the quadrants for each circle.

Concentric circle - 2:

MATLAB CODE:

%Define the matrix

[x,y] = meshgrid(-600:600);

%Preallocate

ccircle = zeros(size(x));

for i = 500:-10:1

   

    Tmp = x.^2 + y.^2 <= i*i;

   

    %Find the gradient of the image

    [Gmag,Gdir] = imgradient(Tmp);

   

    ccircle = ccircle + Gmag;

end

figure,imshow(ccircle);