Signals and Systems - Electrical Engineering

8.3 Discrete-Time Systems 495

ThisM-order filter averagesMpast input values{y[n−k],k=0,...,M− 1 }and assigns this average
to the outputz[n]. The effect is to smooth out the input signal by attenuating the high-frequency
components of the signal due to the noise. The larger the value ofMthe better the results, but at the
expense of more complexity and a larger delay in the output signal (this is due to the linear-phase
frequency response of the filter, as we will see later).

We use a third-order and a fifteenth-order filter, implemented by our functionaveragergiven below.
The denoising is done by means of the following script.

%%%%%%%%%%%%%%%%%

% Linear filtering

%%%%%%%%%%%%%%%%%

N = 200; n = 0:N−1;

x = cos(pi∗n/16); % input signal

noise = 0.2∗randn(1, N); % noise

y = x + noise; % noisy signal

z = averager(3, y); % averaging linear filter with M = 3

z1 = averager(15, y); % averaging linear filter with M = 15

Our functionaveragerdefines the coefficients of the averaging filter and then uses the MATLAB func-
tionfilterto compute the filter response. The inputs offilterare the vectorb=( 1 /M)[1···1], the
coefficients connected with the input, the unit coefficient connected with the output, andxis a vector
with the entries the signal samples we wish to filter. The results of filtering using these two filters are
shown in Figure 8.12. As expected, the performance of the filter withM=15 is a lot better, but a
delay of 8 samples (or the integer larger thanM/2) is shown in the filter output.

function y = averager(M,x)

% Moving average of signal x

% M: order of averager

% x: input signal

%

b = (1/M)∗ones(1, M);

y = filter(b, 1, x);

Nonlinear Filtering
Is linear filtering always capable of getting rid of noise? The answer is: It depends on the type of noise.
In the previous example we showed that a high-order averaging filter, which is linear, performs well
for Gaussian noise. Let us now consider animpulsivenoise that is either zero or a certain value at
random. This is the type of noise occurring in communications whenever cracking sounds are heard
in the transmission, or the “salt-and-pepper” noise that appears in images.

It will be shown that even the 15th-order averager—that did well before—is not capable of denoising
the signal with impulsive noise. Amedian filterconsiders a certain number of samples (the example
shows the case of a 5th-order median filter), orders them according to their amplitudes, and chooses
the one in the middle value (i.e., the median) as the output of the filter. Such a filter is nonlinear as it
does not satisfy superposition. The following script is used to filter the noisy signal using a linear and