Discrete Time Signals and Systems

Discrete Time Signals and Systems

Signal classification

• Periodic and non-periodic
• Odd and even signals
• energy signal and power signal

basic signal

• Impulse function
• unit step function
• Ramp function

Operation on signal

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Periodic and Aperiodic Discrete-Time Sinusoids
$$x(n)=Acos[2\pi f_{0}n]=x(n+N)=Acos[2\pi f_0(n+N)]=Acos[2\pi f_{0}n+2\pi f_{0}N]$$
$2\pi f_{0}N=2\pi k$ just $f_0=\frac{k}{N}$

n is integer: Periodic

n is not integer: Aperiodic

Periodic judgment of composite signals

1. Find N for each signal

   if $\frac{N_1}{N_2}=rationalnumber$ it is Periodic

2. Find the lowest common multiple($ LCM(N_1,N_2)$)

   Periodic is $ LCM(N_1,N_2)$

Odd and even signals

odd: $x_0(t)=\frac{1}{2}[x(t)?x(?t)]$

even: $x_e(t)=\frac{1}{2}[x(t)+x(?t)]$

$x(t)=x_0(t)+x_e(t)$

energy signal and power signal

energy: $E={\sum }_{n=?\infty }^{\infty }|x[n]{|}^2$

power:

Periodic: $P_{\infty }={\lim }_{N\to \infty }\frac{1}{2N+1}{\sum }_{n=?\infty }^{+\infty }|x[n]{|}^2$

Aperiodic: $P_x=\frac{1}{N}{\sum }_{n=0}^{N?1}|x[n]{|}^2$

energy signal:energy is finite,power is zero

power signal:energy is infinite,power is finite

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find the energy and power for

• Impulse function
• unit step function
• Ramp function

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1. Time Shifting(left is +;right is -)
2. Time-scale
3. Time Reversal

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System of discrete signal

• Linear systems and nonlinear systems

• Causal and Acausal Systems

• Time-varying and time-invariant systems

• static system and dynamic system

• Stable and unstable systems

• convolution

• convolution sum

• circular convolution

• Stability of linear time-invariant systems

Linear systems and nonlinear systems

• Linear systems satisfy uniformity and superposition
• A system that satisfies uniformity and superposition is a linear system

Must have：$x(n)=y(n)=0$

Four steps to solve problems:

1. x1 to y1 = F(x1)
2. x2 to y2 = F(x2)
3. y3 = ay1 + by2
4. F(ax1 +bx2) = y4

if y4 = y3 ,the system is linear

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Causal and non-causal Systems

casual system: The output depends only on present and past signals

Acausal Systems：The output depends on at least one future input

eg：Y(n) = x(-n) is non-casual (at n = -1)

even and odd is non-casual

ps: anti-casual system: output only upon “only future input” for all time

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Time-varying and time-invariant systems

Time-varying system(TVS):

y(n) = F[x(n)] = x(n)cos(2$w\pi$n)

y(n,k) = F[x(n-k)] = x(n-k)cos(2$w\pi$n)

y(n-k) $\ne $ y(n,k)

y(n) to y(n,k):only change n for x(n)

time-invariant systems :

y(n,k) = y(n-k)

eg:y(n-k) = sin (x(n-k))

y change n for all n

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static system and dynamic system

static system (memory-less system):

output only depends on now input for all time

dynamic system：

output depends on past and/or future inputs

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Stable and unstable systems

Stable system: BIBO

bounded input to bounded output

unstable systems:

bounded input to unbounded output

eg: $y(n)=y^2(n?1)+2\delta (n)$

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convolution

• time-invariant systems

$y(n)={\sum }_{k=?\infty }^{\infty }x(k)h(n?k)$

$y(n)=x(n)?h(n)$

1. time reversal
2. shifting of h(-k) to h(n-k)
3. multiply x(k)h(n-k)
4. Sum

• linear convolution
• circular convolution

matrix method:

x(n) = {1,2,3,4} h(n)={0,2,2,2}

length of x(n):4 samples;k

length of h(n):4 samples;m

length of y(n): = k + m - 1 = 4 + 4 - 1 = 7 samples

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circular convolution

x(n) = {2,1,2} h(n) = {1,2,3}

$y(n)={\sum }_{k=?\infty }^{\infty }x(k)h(n?k)$

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Stability of linear time-invariant systems

$S={\sum }_{k=?\infty }^{\infty }|h(n)|<\infty$

it is stability

eg: $h(n)=(0.8{)}^{n}u(n+2)$