Discrete Time Signals and Systems

2023-12-24 21:28:13

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 ) = A c o s [ 2 π f 0 n ] = x ( n + N ) = A c o s [ 2 π f 0 ( n + N ) ] = A c o s [ 2 π f 0 n + 2 π f 0 N ] 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] x(n)=Acos[2πf0?n]=x(n+N)=Acos[2πf0?(n+N)]=Acos[2πf0?n+2πf0?N]
2 π f 0 N = 2 π k 2\pi f_0 N =2 \pi k 2πf0?N=2πk just f 0 = k N f_0 = \frac{k}{N} f0?=Nk?

n is integer: Periodic

n is not integer: Aperiodic

Periodic judgment of composite signals

  1. Find N for each signal

    if N 1 N 2 = r a t i o n a l ? n u m b e r \frac{N_1}{N_2} = rational \ number N2?N1??=rational?number 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 ) = 1 2 [ x ( t ) ? x ( ? t ) ] x_0(t)=\frac{1}{2}[x(t)-x(-t)] x0?(t)=21?[x(t)?x(?t)]

even: x e ( t ) = 1 2 [ x ( t ) + x ( ? t ) ] x_e(t) = \frac{1}{2}[x(t)+x(-t)] xe?(t)=21?[x(t)+x(?t)]

x ( t ) = x 0 ( t ) + x e ( t ) x(t) = x_0(t) + x_e(t) x(t)=x0?(t)+xe?(t)

energy signal and power signal

energy: E = ∑ n = ? ∞ ∞ ∣ x [ n ] ∣ 2 E=\sum_{n=-\infty}^{\infty}|x[n]|^2 E=n=??x[n]2

power:

Periodic: P ∞ = lim ? N → ∞ 1 2 N + 1 ∑ n = ? ∞ + ∞ ∣ x [ n ] ∣ 2 P_\infty=\lim_{N\to\infty}\frac1{2N+1}\sum_{n=-\infty}^{+\infty}|x[n]|^2 P?=limN?2N+11?n=?+?x[n]2

Aperiodic: P x = 1 N ∑ n = 0 N ? 1 ∣ x [ n ] ∣ 2 P_x=\frac1{N}\sum_{n=0}^{N-1}|x[n]|^2 Px?=N1?n=0N?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

image-20231222222522112

Must have: x ( n ) = y ( n ) = 0 x(n) = y(n) = 0 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 π w \pi wπn)

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

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 δ ( n ) y(n)=y^2(n-1)+2\delta (n) y(n)=y2(n?1)+2δ(n)

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convolution

  • time-invariant systems

y ( n ) = ∑ k = ? ∞ ∞ x ( k ) h ( n ? k ) y(n)=\sum_{k=-\infty}^{\infty}x(k)h(n-k) y(n)=k=??x(k)h(n?k)

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

image-20231223180642723

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circular convolution

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

y ( n ) = ∑ k = ? ∞ ∞ x ( k ) h ( n ? k ) y(n)=\sum_{k=-\infty}^{\infty}x(k)h(n-k) y(n)=k=??x(k)h(n?k)

image-20231223181124148

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Stability of linear time-invariant systems

S = ∑ k = ? ∞ ∞ ∣ h ( n ) ∣ < ∞ S=\sum_{k=-\infty}^{\infty}|h(n)|<\infty S=k=??h(n)<

it is stability

eg: h ( n ) = ( 0.8 ) n u ( n + 2 ) h(n) = (0.8)^n u(n+2) h(n)=(0.8)nu(n+2)

文章来源:https://blog.csdn.net/u011146203/article/details/135186253
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