High pass filter lower amplitude of the filtered signal
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Inspired by a similar question, can someone please help me understand why amplitudes of the frequencies that are in the pass zone are also decreased?
For example, I have a composite signal consists of 1 Hz and 30 Hz. Amplitude of the 30 Hz signal is 0.5. If I design a filter with cutoff frequency of 15 Hz, output signal which is frequency of 30 Hz is two times lower amplitude.
I wonder why is that. Is it because of the filter's ripple?
clear
clc
close all
fs = 100; % sampling frequency
f1 = 1;
f2 = 30; % frequency of the signal
t = 5; % time duration
n = 0:1/fs:t; % sample period
x1 = sin(2*pi*f1*n); % single tone signal 1 Hz
x2 = 0.5*sin(2*pi*f2*n); % single tone signal 30 Hz
x = x1 + x2; % composite signal contains 1 Hz and 30 Hz
subplot(4,1,1)
plot(n,x)
grid on
title('Composite signal in time domain');
xlabel("time (s)")
ylabel("Signal value")
% Perform FFT and look at the amplitude spectrum
Y = fft(x);
L = length(x);
f = fs*(0:(L/2))/L;
P2 = abs(Y/L);
P1 = P2(1:L/2+1);
P1(2:end-1) = 2*P1(2:end-1);
subplot(4,1,2)
plot(f,P1)
title("Single-Sided Amplitude Spectrum of x(t)")
xlabel("f (Hz)")
ylabel("|P1(f)|")
% Now design high pass filter to pass all above 7 Hz
fc = 15; % cutoff freq 15 Hz
[b,a] = cheby1(6,5,fc/(fs/2),'high');
xfilt = filtfilt(b,a,x);
subplot(4,1,3)
plot(n,xfilt)
% Perform FFT and look at the amplitude spectrum of filtered signal
Y = fft(xfilt);
L = length(x);
f = fs*(0:(L/2))/L;
P2 = abs(Y/L);
P1 = P2(1:L/2+1);
P1(2:end-1) = 2*P1(2:end-1);
subplot(4,1,4)
plot(f,P1)
title("Single-Sided Amplitude Spectrum of x(t)")
xlabel("f (Hz)")
ylabel("|P1(f)|")
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Accepted Answer
Bora Eryilmaz
on 6 Dec 2022
Try instead a filter that is more smooth at the transition region and has less ripple at the left of the high-pass region:
fs = 100;
fc = 15;
[b,a] = butter(6,fc/(fs/2),'high');
freqz(b,a,512,fs)
a = gca;
a.YLim = [-25 0];
The Chebychev filter still has -2.81 dB attenuation at 30 Hz, so you would get an attenuation of
0.5*10^(-2.81/20)
fs = 100;
fc = 15;
[b,a] = cheby1(6,5,fc/(fs/2),'high');
freqz(b,a,512,fs)
a = gca;
a.YLim = [-5 0];
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