Moving maximum and LTI in MATLAB - matlab

Question

I've implement a moving maximum filter in matlab and now I have to determine if it's LTI or not. I've already prove it's linear but I can't prove if it's time ivnariant. To prove this I have to give as an input a signal x[n] and get the output X[n]. After that I'm giving the same signal but this time it have to delay it that is x[n-a] and the output must be X[n-a] the same as the X[n] but delayed. My problem is that I don't know what input to give. Also, the moving maximum is a LTI system or not?
Thanks in advance.

Answer 1

Max-Filter Using For-Loop and Testing System for Linear and Time-Invariance
Filter Construction:
To filter an array/matrix/signal a for-loop can be applied. Within this for-loop, portions of the signal that overlaps the window must be stored in this case called Window_Sample. To apply the maximum filter the max() of the Window_Sample must be taken upon each iteration of the loop. Upon each iteration of the loop the window moves. Here I push all the code that calculates the maximum filtered signal into a function called Max_Filter().
Time-Invariant or Not-Time Invariant:
With proper padding, I think the maximum filter is time-invariant/shift-invariant. Concatenating zeros to the start and ends of the signal is a good way to combat this. The amount of padding I used on each end is the size of the window, the requirement would be half the window but there's no harm in having a little bit of buffer. Unfortunately, without proper padding, this may appear to break down in MATLAB since the max filter will be introduced to points early. In this case without padding the ramp the max filter will take the 5th point as the max right away. Here you can apply a scaling factor to test if the system is linear. You'll see that the system is not linear since the output is not a scaled version of the input.
clf;
%**********************************************************%
%ADJUSTABLE PARAMETERS%
%**********************************************************%
Window_Size = 3; %Window size for filter%
Delay_Size = 5; %Delay for the second test signal%
Scaling_Factor = 1; %Scaling factor for the second test signal%
n = (0:10);
%Ramp function with padding%
Signal = n; %Function for test signal%
%**********************************************************%
Signal = [zeros(1,Window_Size) Signal zeros(1,Window_Size)];
Filtered_Result_1 = Max_Filter(Signal,Window_Size);
%Ramp function with additional delay%
Delayed_Signal = Scaling_Factor.*[zeros(1,Delay_Size) Signal];
Filtered_Result_2 = Max_Filter(Delayed_Signal,Window_Size);
plot(Filtered_Result_1);
hold on
plot(Filtered_Result_2);
xlabel("Samples [n]"); ylabel("Amplitude");
title("Moving Maximum Filtering Signal");
legend("Original Signal","Delayed Signal");
xticks([0: 1: length(Filtered_Result_2)]);
xlim([0 length(Filtered_Result_2)]);
grid;
function [Filtered_Signal] = Max_Filter(Signal,Filter_Size)
%Initilizing an array to store filtered signal%
Filtered_Signal = zeros(1,length(Signal));
%Filtering array using for-loop and acccesing chunks%
for Sample_Index = 1: length(Signal)
Start_Of_Window = Sample_Index - floor(Filter_Size/2);
End_Of_Window = Sample_Index + floor(Filter_Size/2);
if Start_Of_Window < 1
Start_Of_Window = 1;
end
if End_Of_Window > length(Signal)
End_Of_Window = length(Signal);
end
Window_Sample = Signal(Start_Of_Window:End_Of_Window);
Filtered_Signal(1,Sample_Index) = max(Window_Sample);
end
end
Alternatively Using Built-In Functions:
To test the results of the created filter it is a good idea to compare this to the output results of MATLAB's built-in function movmax(). Here the results appear to be the same as the for-loop implementation.
clf;
Window_Size = 3;
n = (0:10);
%Ramp function with padding%
Signal = n;
Signal = [zeros(1,Window_Size) Signal zeros(1,Window_Size)];
Filtered_Result_1 = movmax(Signal,Window_Size);
%Ramp function with additional delay%
Delay_Size = 5;
Delayed_Signal = [zeros(1,Delay_Size) Signal];
Filtered_Result_2 = movmax(Delayed_Signal,Window_Size);
plot(Filtered_Result_1);
hold on
plot(Filtered_Result_2);
xlabel("Samples [n]"); ylabel("Amplitude");
title("Moving Maximum Filtering Signal");
legend("Original Signal","Delayed Signal");
xticks([0: 1: length(Filtered_Result_2)]);
xlim([0 length(Filtered_Result_2)]);
grid;
Ran using MATLAB R2019b