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Description

English: Impulse response

h_{1}(n)={0.8}^{n}\quad n=0,\ldots ,N-1

of a discrete-time low-pass filter.

Italiano: Risposta all'impulso

h_{1}(n)={0,8}^{n}\quad n=0,\ldots ,N-1

di un filtro passa-basso a tempo discreto.

Date

23 September 2017

Source

Own work

Author

Luca Ghio

Permission
(Reusing this file)

I, the copyright holder of this work, release this work into the public domain. This applies worldwide.
In some countries this may not be legally possible; if so:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

Category:Self-published work#Discrete-time%20low-pass%20filter%20impulse%20response%20(r=0.8).pdf Category:PD-self#Discrete-time%20low-pass%20filter%20impulse%20response%20(r=0.8).pdf

gnuplot source

click to expand

N=11;
n=[0:N-1];
x=ones(1,N);
r1=0.8;
r2=0.5;
r3=0.2;
h1=r1.^n;
h2=r2.^n;
h3=r3.^n;
figure
set (gca,'FontSize',14)
stem(n,x)
xlabel('n')
ylabel('x(n)')
axis([0 10 0 2])
grid on
figure
set (gca,'FontSize',14)
stem(n,h1)
xlabel('n')
ylabel('h_1(n)')
axis([0 10 0 1])
grid on
figure
set (gca,'FontSize',14)
stem(n,h2)
xlabel('n')
ylabel('h_2(n)')
axis([0 10 0 1])
grid on
figure
set (gca,'FontSize',14)
stem(n,h3)
xlabel('n')
ylabel('h_3(n)')
axis([0 10 0 1])
grid on

N=11;
n=[0:N-1];
x=ones(1,N);
r1=0.8;
r2=0.5;
r3=0.2;
k1=(1-r1)/(1-r1^N);
k2=(1-r2)/(1-r2^N);
k3=(1-r3)/(1-r3^N);
h1=k1*r1.^n;
h2=k2*r2.^n;
h3=k3*r3.^n;
%  figure
%  set (gca,'FontSize',14)
%  stem(n,x)
%  xlabel('n')
%  ylabel('x(n)')
%  axis([0 10 0 2])
%  grid on
%  figure
%  set (gca,'FontSize',14)
%  stem(n,h1)
%  xlabel('n')
%  ylabel('h_1(n)')
%  axis([0 10 0 1])
%  grid on
%  figure
%  set (gca,'FontSize',14)
%  stem(n,h2)
%  xlabel('n')
%  ylabel('h_2(n)')
%  axis([0 10 0 1])
%  grid on
%  figure
%  set (gca,'FontSize',14)
%  stem(n,h3)
%  xlabel('n')
%  ylabel('h_3(n)')
%  axis([0 10 0 1])
%  grid on

f=linspace(-0.5,0.5,1000);
H1=k1*(1-(r1*exp(-j*2*pi*f)).^N)./(1-r1*exp(-j*2*pi*f));
H2=k2*(1-(r2*exp(-j*2*pi*f)).^N)./(1-r2*exp(-j*2*pi*f));
H3=k3*(1-(r3*exp(-j*2*pi*f)).^N)./(1-r3*exp(-j*2*pi*f));
figure
set (gca,'FontSize',14)
plot(f,abs(H1))
hold on
plot(f,abs(H2),'r')
plot(f,abs(H3),'g')
xlabel('f')
ylabel('|H_i(e^{j\omega})|')
axis([-0.5 0.5 0 1.5])
grid on
legend('|H_1(e^{j\omega})|','|H_2(e^{j\omega})|', '|H_3(e^{j\omega})|')

N = 11;
X = (sin(pi*f*N) ./ sin(pi*f)) .* exp(-j*pi*f*(N-1));
figure
set (gca,'FontSize',14)
plot(f,abs(X))
xlabel('f')
ylabel('|X(e^{j\omega})|')
axis([-0.5 0.5 0 11])
grid on

Y1 = X .* H1;
Y2 = X .* H2;
Y3 = X .* H3;
figure
set (gca,'FontSize',14)
plot(f,abs(Y1))
hold on
plot(f,abs(Y2), 'r')
plot(f,abs(Y3), 'g')
xlabel('f')
ylabel('|Y_i(e^{j\omega})|')
axis([-0.5 0.5 0 11])
grid on
legend('|Y_1(e^{j\omega})|','|Y_2(e^{j\omega})|', '|Y_3(e^{j\omega})|')

N=11;
n=[0:N-1];
x=ones(1,N);
r1=0.8;
r2=0.5;
r3=0.2;
k1=(1-r1)/(1-r1^N);
k2=(1-r2)/(1-r2^N);
k3=(1-r3)/(1-r3^N);
h1=k1*r1.^n;
h2=k2*r2.^n;
h3=k3*r3.^n;
y1=conv(x,h1);
y2=conv(x,h2);
y3=conv(x,h3);
n=[0:length(y1)-1];
figure
set (gca,'FontSize',14)
stem(n,y1)
xlabel('n')
ylabel('y_1(n)')
axis([0 20 0 1.5])
grid on
figure
set (gca,'FontSize',14)
stem(n,y2)
xlabel('n')
ylabel('y_2(n)')
axis([0 20 0 1.5])
grid on
figure
set (gca,'FontSize',14)
stem(n,y3)
xlabel('n')
ylabel('y_3(n)')
axis([0 20 0 1.5])
grid on

Category:Images with Octave source code Category:Created with GNU Octave Category:Lowpass filters Category:Plots of discrete time

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File:Discrete-time low-pass filter impulse response (r=0.8).pdf