Notes For Discrete-Time System Analysis [Chapter I. Fundamental Concepts]

Note for the course Discrete-Time System Analysis.

1.1 Introduction

1.2 Discrete-Time Signals

Discrete-time Variable

If the time variable $t$only takes on discrete values $t=t_n$for some range of integer values of$n$, then$t$ is called discrete-time variable.

Discrete-time Signal

If a continuous-time signal $x(t)$is a function of discrete-time variable $t_n$, then the signal $x(t_n)$ is a discrete-time signal, which is called the sampled version of the continuous-time function$x(t)$.

If we let $t_n=nT$, then obviously $T$is the sampling interval. When $T$is a constant, such a sampling process is called uniform sampling, instead, nonuniform sampling. Then we could write$x(t_n)$ in form of$x[n]$, in which square brackets is needed. And also,$x[n]=x(t_n)=x(t){|}_{t=nT}=x(nT)$ .

Some typical examples of discrete-time signals

Discrete-Time Unit-Step Function

$$ u[n] = \left{$$\begin{array}{c}1,n=0,1,2,\cdots 0,n=-1,-2,-3,\cdots \end{array}$$\right. $$

Discrete-Time Unit-Ramp Function

$$ r[n]=nu[n]=\left{$$\begin{array}{c}n,n=0,1,2,\cdots 0,n=-1,-2,-3,\cdots \end{array}$$\right. $$

Unit Pulse

$$ \delta[n] = \left{$$\begin{array}{c}1,n=00,n\ne 0\end{array}$$\right. $$

Periodic Discrete-Time Signals

For a discrete-time signal $x[n]$, if there exists a positive integer $r$which makes that $x[n+r]=x[n]$for all integers n, Then $x[n]$ is called a periodic discrete-time signal and the integer$r$ is period. Fundamental period is the smallest value for positive integer$r$.

For example, if we let $x[n]=A\mathrm{c}\mathrm{o}\mathrm{s}(\mathrm{\Omega }n+\theta )$, then the signal is periodic if $x[n+r]=A\mathrm{c}\mathrm{o}\mathrm{s}(\mathrm{\Omega }(n+r)+\theta )=A\mathrm{c}\mathrm{o}\mathrm{s}(\mathrm{\Omega }n+\theta )$ .

Cause $\mathrm{c}\mathrm{o}\mathrm{s}$is periodic, there is $A\mathrm{c}\mathrm{o}\mathrm{s}(\mathrm{\Omega }n+\theta )=A\mathrm{c}\mathrm{o}\mathrm{s}(\mathrm{\Omega }n+2\pi q+\theta )$ for all integers$q$ .

Obviously, $x[n]$is periodic when there exists an integer $r$which makes $\mathrm{\Omega }r=2\pi q$for some integers$q$, in equivalent, the discrete-time frequency$\mathrm{\Omega }={\displaystyle \frac{2\pi q}{r}}$ for some integers$q,r$.

Discrete-Time Complex Exponential Signals

$$x[n]=C{a}^n=|C||a{|}^n\mathrm{c}\mathrm{o}\mathrm{s}({\omega }_{0}n+\theta )+j|C||a{|}^n\mathrm{s}\mathrm{i}\mathrm{n}({\omega }_{0}n+\theta )$$

where $C=|C|e^{j\theta }$and$a=|a|e^{j{\omega }_0}$ .

Discrete-Time Rectangular Pulse

$$ p_{L}[n] = \left{$$\begin{array}{c}1,;n=-{\displaystyle \frac{L-1}{2}},\cdots ,-1,0,1,\cdots ,{\displaystyle \frac{L-1}{2}}0,;\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{s}\end{array}$$\right. $$

where $L$ is a positive odd integer.

Digital Signals

When a discrete-time signal $x[n]$satiesfies that its values are all belongs to a finite set$\text{Missing or unrecognized delimiter for left}$ , then the signal called a digital signal.

However, the sampled signals don’t have to be digital signals. For example, the sampled unit-ramp function values on a infinite set $\text{Missing or unrecognized delimiter for left}$.

Binary Signal is a digital signal whose values are all belongs in to set $\text{Missing or unrecognized delimiter for left}$.

Time-Shifted Signals

Giving a discrete-time signal $x[n]$and a positive integer$q$ , then

• $x[n-q]$is the $q$-step right shifts of$x[n]$
• $x[n+q]$is the $q$-step left shifts of$x[n]$

Discrete-Time Signals defined Interval by Interval

Discrete-Time Signals also may be defined Interval by Interval. For example, $$ x[n]=\left{$$\begin{array}{c}{x}_1[n],;n_1\le n<n_2{x}_2[n],;n_2\le n<n_3{x}_3[n],;n\ge n_3\end{array}$$\right. $$CausetheUnit-StepFunctionsatisfieswhen\$n\ge 0\$,\$u[n]=1\$,wecanuseittowrite\$x[n]\$insuchaform$$ x[n]=x_{1}[n]\cdot(u[n-n_{1}]-u[n-n_{2}]) +x_{2}[n]\cdot(u[n-n_{2}] - u[n - n_{3}]) +x_{3}[n]\cdot u[n - n_{3}] \ = u[n - n_{1}]\cdot x_{1}[n] +u[n - n_{2}]\cdot(x_{2}[n] - x_{1}[n]) +u[n - n_{3}]\cdot(x_{3}[n] - x_{2}[n]) $$

1.3 Discrete-Time Systems

Definition of Discrete-Time Systems and Analysis

The Discrete-Time System is a system which transforms discrete-time inputs to discrete-time outputs.

The Discrete-Time System Analysis is a process to solve the discrete-time output with discrete-time inputs and discrete-time system.

For example. Consider the differential equation ${\displaystyle \frac{dv(t)}{dt}}+av(t)=bx(t)$, now we resolve time into discrete interval forms of length$△$ , so the equation will become $$\frac{v(n△)-v((n-1)△)}{△}+av(n△)=bx(n△)$$ which equals to $$\frac{v[n]-v[n-1]}{△}+av[n]=bx[n]$$ and $$v[n]-\frac{1}{1+a△}v[n-1]=\frac{b△}{1+a△}x[n]$$

1.4 Basic Properties of Discrete-Time Systems

System with or without memory

Given a discrete-time system with input of $x[n]$and output with $y[n]$ , we call the system memoryless when$y[n]$ is only related to input at present time, otherwise we call it is the one with memory.

For example, $$y[n]=\sum _{k=-\mathrm{\infty }}^{n}x[k]$$ and $$y[n]=x[n]+x[n-1]$$ are systems with memory, and $$y[n]=(2x[n]-x[n{]}^2{)}^2$$ is an example of system in memoryless.

Causality

Given a discrete-time system with input of $x[n]$and output with $y[n]$ , we call the system causal when$y[n]$ is only related to input at present and past time, or we call it not causal.

For example, $$y[n]=\sum _{k=-\mathrm{\infty }}^{n}x[k]$$ is system in causality, but $$y[n]=x[n-1]+x[n+1]$$ not because $x[n+1]$ is input in future, and $$y[n]=x[-n]$$ is also not because when $n$is negative, there is$-n>n$ .

Time Invariance

To a discrete-time system with input of $x[n]$and output of $y[n]$ , we call the system time invariant when a time shifts in the input signal results identical time shifts in the output signal. This is also to say, output$y[n]$ is not explicity related on the varaible of time.

For example, $$y[n]=(n+1)x[n]$$ is not time invariant because $y[n]$has an explicit relationship with time variable$n$.

and, the system $$y[n]=x[2n]$$ is also not time invariant, because any time shift in input will be compressed by factor 2.

As an example of system which is time invariant, $$y[n]=10x[n]$$ which is obvious.

Linearity

A system is to be called a linear system when the input consists of a weighted sum of several signals, the output will also be a weighted sum of the responses of the system for each of those signals.

To make a proof of a system to be in linearity, we let $y_1[n]$is the response of the system to the input $x_1[n]$, and$y_2[n]$ is the response of the input$x_2[n]$ . The system is a linear system if and only if

• Addivity Property

  The response to $x_1[n]+x_2[n]$is$y_1[n]+y_2[n]$.

• Homogeneity Property

  The response to $a{x}_1[n]$is $a{y}_1[n]$, for$a$ is any complex constant.

It’s interesting to find that a system with a linear equation may not be a linear system.

For example, considering the system $y[n]=2x[n]+3$, it’s easy to find the system is not linear, because

For two inputs $x_1[n]$and$x_2[n]$, there are

$x_1[n]\to y_1[n]=2x_1[n]+3$

$x_2[n]\to y_2[n]=2x_2[n]+3$

However, the response to input $(x_1[n]+x_2[n])$ is

$y_3[n]=2(x_1[n]+x_2[n])+3\ne y_1[n]+y_2[n]$.

Notice that $y[n]=3$when $x[n]=0$, it reminds us that the system violates the “zero-in/zero-out” property and the zero-input response of the system is$y_0[n]=3$.