We shall meet in place where there is no darkness.


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




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]


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.


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,


is system in causality, but


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


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,


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

and, the system


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,


which is obvious.


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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