## Discrete systems in series and parallel Discrete systems in series Let as say that we have two discrete systems and their impulse responses are h1(n) and h2(n). Then when these discrete systems are connected in series, then overall impulse response: Where: As you noticed there were changing made: This is nothing more than convolution of impulse responses of both discrete systems: Discrete systems in parallel If we have two discrete systems connected in parallel: As we see there is simple sum of output queues of each discrete system. So we can assume, that overall impulse response is as sum of systems connected in parallel:

## Impulse Response of discrete system Impulse signal can be represented as: d[n] = 1, if n=0 d[n] = 0, otherwise it can also be written like d=[1,0,0,0,â€¦] Impulse Response The impulse response h(n) is the response of filter L() at time n to unit impulse occurring at time 0. h(n)=L(d(n)) Lets see how discrete system can be described when impulse response is known We know that: In the linear system this can be written as follows: Because h(n-k)=L(d(n-k)) Then: What do we get? There is obvious, that linear system can be described by its impulse response. The last expression is called convolution. This is the heart of DSP Filtering. To write this sum in more convenient matter is assumed that: Matlab example Matlab example: % Plot an unit impulse signal n = -7:7; x = [0 0 0 0 0 0 0 1 2 3 0 0 0 0 0]; subplot(4,2,1); stem(n, x); limit=[min(n), max(n), 0, 5]; axis(limit); title(‘Input x[n]’); subplot(4,2,3); x0=0*x; x0(8)=x(8); stem(n, x0); axis(limit); h=text(0, x0(8), ‘x’); set(h, ‘horiz’, ‘center’, ‘vertical’, ‘bottom’); subplot(4,2,4); y0=0*x; index=find(x0); for i=index:length(n) y0(i)=x0(index)*exp(-(i-index)/2); end stem(n, y0); axis(limit); h=text(0, x0(8), ‘x*h[n-0]’); set(h, ‘vertical’, ‘bottom’); subplot(4,2,5); x1=0*x; x1(9)=x(9); stem(n, x1); axis(limit); h=text(1, x1(9), ‘x’); set(h, ‘horiz’, ‘center’, ‘vertical’, ‘bottom’); subplot(4,2,6);…

## What is a linear system? Discrete system is nothing more than algorithm, where input is transformed to output. The output is transformed by operator L() which describes discrete system. Lets see few most common operators of discrete systems. Delay This means that output queue is delayed by on sample. Multiplication This operator takes each sample of input queue and multiplies by constant a. Sum operator Takes two or more sample queues and adds them in the output. Assuming we can say, that the system is linear if input sum reaction is equal to sum of inputs reactions: The system has stable parameters if: y(n-k)=L(x(n-k)), this means that output delay should be the same as input. It is obvious that delay, multiply and sum operators are linear and has stable parameters. We will need then for further lessons.

## Understanding of discrete signals Discrete signals can be generated by software or obtained from real world through ADC. Discrete signals are sampled from analog signals. So you get samples in fixed time intervals. Discrete signal is as sequence of numbers. The element number n of sequence is marked as x(n). The most common number rows: Unit sample sequence d[n] = 1, if n=0 d[n] = 0, otherwise You can describe it in Matlab like % Plot an unit impulse signal n = -5:5; x = 0*n; index=find(n==0); x(index)=1; % plot stem(n, x); axis([-inf, inf, -0.2, 1.2]); xlabel(‘n’); ylabel(‘x’); title(‘Unit Impulse Signal delta[n]’); Unit Step Sequence u[n] = 1, if n>=0 u[n] = 0, otherwise You can describe it in Matlab like: % Plot an unit impulse signal n = -5:5; x = 0*n; index=find(n>=0); x(index)=1; % plot stem(n, x); axis([-inf, inf, -0.2, 1.2]); xlabel(‘n’); ylabel(‘x’); title(‘Unit Step Signal u[n]’); As you can see step is nothing more than set of impulses. And impulse can be expressed as d[n]=u(n)-u(n-1); Thus any sequence of numbers can be expressed asset of impulses like this: For example sin() sequence can be written like this: Matlab script would look like this: % Plot a sinusoidal signal n = 0:40;… 