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SignalsandsystemsChapter2LinearTime-InverariantSystems2.1Discrete-timeLTIsystem: theconvolutionsum discrete signals can be expressed by the delta function of different overlapping frames Thediscret-timeunitimpulseresponseandtheconvolutionsumrepresentationo

Signals and systems Chapter 2 Linear Time-Inverariant Systems 2.1 Discrete-time LTI system: the convolution sum discrete signals can be expressed by The delta function of different overlapping frames. the discret-time unit impulse response and the convolution sum representation o

< > Chapter 2

Linear Time-Inverariant Systems

2.1 Discrete-time LTI system: the convolution sum

Discrete signals can be expressed by delta functions with different overlapping amplitude values.

The discret-time unit impulse response and the convolution sum representation of LTI systems

The above example clearly parses the convolution process step by step.

For the convolution and part, see the article I wrote.《Why shoshould we use convolution?

For why x [k] * h [n-k]

Here we are studying the LTI system. h [n] is an LTI system. For k input x [k] at different times, the system's response is only offset,

X [0] The input h is h [0], and x [1] corresponds to h [n-1]... x [k] corresponds to h [n-k]

To deepen our understanding of the concept, let's look at the convolution and process of time-varying systems.

The input is X [n], and the response is h. Note that the input response of the time-varying system is different at different times, so here there are three different responses.

We regard the input as an impluse sequence. In this way, using the properties of the delta function, we can easily get the output ~

You need to know that the system inputs a series of impulse, so all the results (x [-1] h [-1],..., x [1] h [1]) is accumulated to obtain the output y [n],

This is why the y [n] convolution and formula are suffixed!

It is precisely because of the characteristics of time-varying systems that will lead to an interesting phenomenon. For input x [n] and response h [n],

During the calculation process, h [n] is directly reversed, and k units are offset. The original input signal is used for multiplication, and then the results of each unit are accumulated, the output is y [n] at the moment. Finally, the output of the system is somewhat misleading ", the reason for double quotation marks is that h [n] is an infinitely long step function, so the following infinity approaches 1/(1-alpha ).

In a computer, it is impossible to simulate infinite sequences... the input sequence is finite, so the output will be

(Length of x [n]) + (length of h [n])-1.

Why is it reduced by one? Think about it. If the output is at (length of x [n]) + (length of h [n]), there is no overlap between the two, and the result is 0. we will not consider this meaningless point here. so the output is only (length of x [n]) + (length of h [n])-1 point

Here is an example.

%code writer:EOF%code date:2014.10 .1%e-mail:jasonleaster@gmail.com%code file:demo_for_convolution%code purpose:%             A demo for convolution in LTI-systemclear allclose all% you could use this varible to define how many number of points in the input sequence.points = 10;% x is used as input points% h is used as responce sequnce.% %% input sequence one% x = exp(-[0: (points-1)]);% h = ones(1,points*10);%% Input sequence twoalpha = 2;x = [1 1 1 1 1];h = alpha.^([0:6]);length_x = size(x,2);length_h = size(h,2);figure(1);subplot(121);scatter(1:length_x,x,'r');title('x[n]');subplot(122);scatter(1:length_h,h,'g');title('h[n]');output = zeros(1,length_x+length_h -1);%% Kernel part of our convolution sum   :- )for current_point_n= 1:length_x + length_h        tmp = current_point_n;        while(tmp > 0)            if  current_point_n <= length_x  &&  (current_point_n - tmp + 1) <= length_h                     output(current_point_n) = output(current_point_n) + x(tmp)*h(current_point_n - tmp + 1);            end                        if current_point_n > length_x && current_point_n < (length_h + length_x -1)                                if tmp > length_x                    tmp = tmp -1;                    continue;                else                    if (current_point_n - tmp + 1) <= length_h                     output(current_point_n) = output(current_point_n) + x(tmp)*h(current_point_n - tmp + 1);                           end                end                            end                        tmp = tmp -1;        endendfigure(2);scatter(1:size(output,2),output,'filled');title('output[n]');

The above input can be adjusted at will, and the program is robust.

Properties of LTI systems

Exchange Law, combination law, distribution law

Demo of reversible descriptions:

To explore the cause and effect,

Exploration of stability:

Finally, we must realize that the differential and difference equations only describe the input and output relations of the continuous and discrete systems respectively. They are similar to the descriptions of the system input and output, and cannot be confused and compared. I made a random comparison before, so I was so worried.

The process of thinking is recorded here.

Http://blog.csdn.net/cinmyheart/article/details/39499967