To smooth the noisy signal with integral

Hi all,

I am trying to learn the signal progressing with MATLAB. I have a nosiy signal with frequency of f=1/20 Hz, it takes 1 minute to collect one sample and so the signal along with noise as

f = 1/20;  
t=0:0.2:50; 
noise = rand(size(t));
y = sin (2*pi*f*t) + noise;

I need to smooth the signal. I read a book about signal process and it integrate the signal within certain time for the purpose. I assume I need to smooth the signal every 4 samples, is the following code correct to do the job?

Y = zeros(size(y)); % the signal after smooth
len = 4; % smooth every 4 samples
for n=1:length(y)
  if (n>len)
    Y(n) = trapz(y(n-len:n))/len;
  end
 end

I got the code by trail and error. The plots looks ok. But after I read the article for numerical integral, I find that we need to multiple the time difference Delta t=0.2 to the trapz to get the correct intergral, that is

for n=1:length(y)
  if (n>len)
    Y(n) = trapz(y(n-len:n))*0.2/len;
  end
 end

But to do so, the amplitude of the smoothed signal is pretty off, am I missing something here? Thanks.

Answer 1

[Star Strider]

I believe you have the correct approach.

I would do something like this:

f = 1/20;  
t=0:0.2:50; 
noise = randn(size(t));
y = sin (2*pi*f*t) + noise;

Y = zeros(size(y)); % the signal after smooth
len = 4; % smooth every 4 samples
for n=1:length(y)-len
    if (n>len)
        Y(n) = trapz(y(n:n+len))/len;
    end
end

figure
plot(t, y)
hold on
plot(t, Y, 'LineWidth',1.5)
hold off
grid

Note that I use randn here instead of rand. The randn function produces Gaussian-distributed (normally-distributed) random numbers with a mean of 0 and a standard deviation of 1. The rand function produces uniformly-distributed random numbers going from 0 to 1, with a mean of 0.5. This may be the source of the amplitude deviation you were seeing, since the mean would integrate, creating a positive slope instead of the 0 slope created by using randn (that approximates real white noise).

.