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

1.1 Introduction

1.2 Discrete-Time Signals

Discrete-time Variable

If the time variable ttonly takes on discrete values t=tnt=t_{n}for some range of integer values ofnn, thentt is called discrete-time variable.

Discrete-time Signal

If a continuous-time signal x(t)x(t)is a function of discrete-time variable tnt_{n}, then the signal x(tn)x(t_{n}) is a discrete-time signal, which is called the sampled version of the continuous-time functionx(t)x(t).

If we let tn=nTt_{n}=nT, then obviously TTis the sampling interval. When TTis a constant, such a sampling process is called uniform sampling, instead, nonuniform sampling. Then we could writex(tn)x(t_{n}) in form of$ x[n],inwhichsquarebracketsisneeded.Andalso,, 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]={1,n=0,1,2,0,n=1,2,3,u[n] = \left\{\begin{matrix} 1,n=0,1,2, \cdots \\ 0,n=-1,-2,-3,\cdots \end{matrix}\right.

Discrete-Time Unit-Ramp Function

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

Unit Pulse

δ[n]={1,n=00,n0\delta[n] = \left\{\begin{matrix} 1,n=0 \\ 0,n \neq 0 \end{matrix}\right.

Periodic Discrete-Time Signals

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

For example, if we let x[n]=Acos(Ωn+θ)x[n]=A\mathrm{cos}(\Omega n + \theta), then the signal is periodic if x[n+r]=Acos(Ω(n+r)+θ)=Acos(Ωn+θ)x[n+r]=A\mathrm{cos}(\Omega (n + r)+\theta) = A\mathrm{cos}(\Omega n+\theta) .

Cause cos\mathrm{cos}is periodic, there is Acos(Ωn+θ)=Acos(Ωn+2πq+θ)A\mathrm{cos}(\Omega n + \theta) = A\mathrm{cos}(\Omega n + 2 \pi q + \theta) for all integersqq .

Obviously, x[n]x[n]is periodic when there exists an integer rrwhich makes Ωr=2πq\Omega r = 2 \pi qfor some integersqq, in equivalent, the discrete-time frequencyΩ=2πqr\Omega = \dfrac{2 \pi q}{r} for some integersq,rq,r.

Discrete-Time Complex Exponential Signals

x[n]=Can=Cancos(ω0n+θ)+jCansin(ω0n+θ)x[n]=Ca^{n}=|C||a|^{n}\mathrm{cos}(\omega _{0}n+\theta)+j|C||a|^{n}\mathrm{sin}(\omega _{0}n + \theta)

where C=CejθC = |C|e^{j\theta}anda=aejω0a = |a|e^{j\omega _{0}} .

Discrete-Time Rectangular Pulse

pL[n]={1,  n=L12,,1,0,1,,L120,  othersp_{L}[n] = \left\{\begin{matrix} 1,\;n=-\dfrac{L-1}{2},\cdots,-1,0,1,\cdots,\dfrac{L-1}{2} \\ 0,\; \mathrm{others} \end{matrix}\right.

where LL is a positive odd integer.

Digital Signals

When a discrete-time signal x[n]x[n]satiesfies that its values are all belongs to a finite set{a1,a2,,an}\left\{ a_{1},a_{2},\cdots,a_{n} \}\right. , 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 {0,1,2,}\left\{ 0,1,2,\cdots \}\right..

Binary Signal is a digital signal whose values are all belongs in to set {0,1}\left\{ 0,1 \}\right..

Time-Shifted Signals

Giving a discrete-time signal x[n]x[n]and a positive integerqq , then

  • x[nq]x[n-q]is the qq-step right shifts ofx[n]x[n]
  • x[n+q]x[n + q]is the qq-step left shifts ofx[n]x[n]

Discrete-Time Signals defined Interval by Interval

Discrete-Time Signals also may be defined Interval by Interval. For example,

x[n]={x1[n],  n1n<n2x2[n],  n2n<n3x3[n],  nn3x[n]=\left\{\begin{matrix} x_{1}[n],\;n_{1}\leq n < n_{2} \\ x_{2}[n],\;n_{2} \leq n < n_{3} \\ x_{3}[n], \; n \geq n_{3} \end{matrix}\right.

Cause the Unit-Step Function satisfies when n0n \geq 0, u[n]=1u[n]=1 , we can use it to writex[n]x[n] in such a form

x[n]=x1[n](u[nn1]u[nn2])+x2[n](u[nn2]u[nn3])+x3[n]u[nn3]=u[nn1]x1[n]+u[nn2](x2[n]x1[n])+u[nn3](x3[n]x2[n])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 dv(t)dt+av(t)=bx(t)\dfrac{dv(t)}{dt}+av(t)=bx(t), now we resolve time into discrete interval forms of length\bigtriangleup , so the equation will become

v(n)v((n1))+av(n)=bx(n)\frac{v(n\bigtriangleup)-v((n-1)\bigtriangleup)}{\bigtriangleup}+av(n \bigtriangleup)=bx(n \bigtriangleup)

which equals to

v[n]v[n1]+av[n]=bx[n]\frac{v[n]-v[n-1]}{\bigtriangleup}+av[n]=bx[n]

and

v[n]11+av[n1]=b1+ax[n]v[n]-\frac{1}{1+a\bigtriangleup}v[n-1]=\frac{b\bigtriangleup}{1+a\bigtriangleup}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]x[n]and output with y[n]y[n] , we call the system memoryless wheny[n]y[n] is only related to input at present time, otherwise we call it is the one with memory.

For example,

y[n]=k=nx[k]y[n] = \sum_{k=- \infty}^{n}x[k]

and

y[n]=x[n]+x[n1]y[n] = x[n] + x[n - 1]

are systems with memory, and

y[n]=(2x[n]x[n]2)2y[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]x[n]and output with y[n]y[n] , we call the system causal wheny[n]y[n] is only related to input at present and past time, or we call it not causal.

For example,

y[n]=k=nx[k]y[n]=\sum_{k=-\infty}^{n}x[k]

is system in causality, but

y[n]=x[n1]+x[n+1]y[n]=x[n-1]+x[n+1]

not because x[n+1]x[n + 1] is input in future, and

y[n]=x[n]y[n]=x[-n]

is also not because when nnis negative, there isn>n-n > n .

Time Invariance

To a discrete-time system with input of x[n]x[n]and output of y[n]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, outputy[n]y[n] is not explicity related on the varaible of time.

For example,

y[n]=(n+1)x[n]y[n]=(n+1)x[n]

is not time invariant because y[n]y[n]has an explicit relationship with time variablenn.

and, the system

y[n]=x[2n]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]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 y1[n]y_{1}[n]is the response of the system to the input x1[n]x_{1}[n], andy2[n]y_{2}[n] is the response of the inputx2[n]x_{2}[n] . The system is a linear system if and only if

  • Addivity Property

    The response to x1[n]+x2[n]x_{1}[n]+x_{2}[n]isy1[n]+y2[n]y_{1}[n]+y_{2}[n].

  • Homogeneity Property

    The response to ax1[n]ax_{1}[n]is ay1[n]ay_{1}[n], foraa 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]+3y[n]=2x[n]+3, it’s easy to find the system is not linear, because

For two inputs x1[n]x_{1}[n]andx2[n]x_{2}[n], there are

x1[n]y1[n]=2x1[n]+3x_{1}[n] \rightarrow y_{1}[n]=2x_{1}[n]+3

x2[n]y2[n]=2x2[n]+3x_{2}[n] \rightarrow y_{2}[n]=2x_{2}[n]+3

However, the response to input (x1[n]+x2[n])(x_{1}[n]+x_{2}[n]) is

y3[n]=2(x1[n]+x2[n])+3y1[n]+y2[n]y_{3}[n]=2(x_{1}[n] + x_{2}[n])+3 \neq y_{1}[n]+y_{2}[n].

Notice that y[n]=3y[n]=3when x[n]=0x[n]=0, it reminds us that the system violates the “zero-in/zero-out” property and the zero-input response of the system isy0[n]=3y_{0}[n]=3.

  • 本文作者: Panelatta
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