Discrete Time Signals and Systems

2023-12-26 00:28
文章标签 time signals systems discrete

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Discrete Time Signals and Systems

文章目录

  • Discrete Time Signals and Systems
    • Signal classification
    • basic signal
    • Operation on signal
    • System of discrete signal
      • Linear systems and nonlinear systems
      • Causal and non-causal Systems
      • Time-varying and time-invariant systems
      • static system and dynamic system
      • Stable and unstable systems
      • convolution
      • circular convolution
      • Stability of linear time-invariant systems
    • Fourier Series
      • Fourier Series basic concepts
      • Discrete time fouriew series (DTFS)
      • CTFS
    • Continuous time Fourier transform (CTFT)
      • Dirichlet's conditions
      • lecture(77)
      • lecture(78)
      • properties of CTFT
        • linearity
        • time shifting
        • time scaling
      • frequency shifting
      • time differenciation
      • frequency differenciation
      • convolution
      • Parseval's Theorem
    • Fourier transform of Periodic signal
    • Discrete time fouriew series (DTFT)
      • basic properties
      • DTFT for common signals
      • properties
        • linear
        • time shifting
        • frequency shifting
        • Frequency domain differential
    • Discrete Fourier Transform(DFT)
      • Calculation of DFT
        • Rotation matrix of inverse DFT transform(IDFT)
      • Properties of DFT
        • 1.linear
        • 2.circular convolution
        • 3.periodicity
        • 4.cyclic frequency shift
      • DTF calculates linear convolution and circular convolution
        • linear convilution
        • cyclical convolution
      • another way to calculate linear convolution
        • Overlap-Add
        • Overlap-Save
    • z-transformation
      • Basic concepts of z-transformation
      • region of convergence (ROC)
      • Properties of z-transformation
        • linear
        • time shifting
        • scaling
        • differential
        • convolution
        • initial value theorem
        • terminal value theorem
      • Inverse transformation of z-transform
        • long division method
        • partial fraction expansion method
        • Residue method
      • stability

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

 

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[2πf0n]=x(n+N)=Acos[2πf0(n+N)]=Acos[2πf0n+2πf0N]
2 π f 0 N = 2 π k 2\pi f_0 N =2 \pi k 2πf0N=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 N2N1=rational number it is Periodic

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

    Periodic is L C M ( N 1 , N 2 ) LCM(N_1,N_2) LCM(N1,N2)

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=limN2N+11n=+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=N1n=0N1x[n]2

energy signal:energy is finite,power is zero

power signal:energy is infinite,power is finite

 

find the energy and power for

  • Impulse function
  • unit step function
  • Ramp function

 

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

 

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

 

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

 

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

 

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

 

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(n1)+2δ(n)

 

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(nk)

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

 

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(nk)

image-20231223181124148

 

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)

 

Fourier Series

  • Fourier Series basic concepts
  • Discrete time fouriew series (DTFS)

Fourier Series basic concepts

Condition for existence of Fourier Series

  1. Finite Number of maxima and minima over the time period
  2. Finite number of discontinuities over the time period T
  3. Signal should be absolutely integrable over the range of time period T

period function:

f ( x ) = a 0 + ∑ n = 1 ∞ ( a n c o s ( n w 0 t ) + b n s i n ( n w 0 t ) f(x)=a_0+\sum_{n=1}^\infty(a_ncos(nw_0t)+b_n sin(nw_0t) f(x)=a0+n=1(ancos(nw0t)+bnsin(nw0t)

a 0 = 1 T 1 ∫ 0 T 1 f ( t ) d t a_{0}=\frac{1}{T_{1}}\int_{0}^{T_1}f(t)dt a0=T110T1f(t)dt

a n = 2 T 1 ∫ 0 T 1 f ( t ) cos ⁡ n w 1 t d t a_{n}=\frac{2}{T_{1}}\int_{0}^{T_{1}}f(t)\cos nw_1tdt an=T120T1f(t)cosnw1tdt

b n = 2 T 1 ∫ 0 T 1 f ( t ) sin ⁡ n w 1 t d t b_{n}=\frac{2}{T_{1}}\int_{0}^{T_{1}}f(t)\sin nw_1tdt bn=T120T1f(t)sinnw1tdt

eg:image-20231223183235229

 

Discrete time fouriew series (DTFS)

x n = ∑ k = 0 N − 1 c ( k ) e J k 2 π N n x_n=\sum_{k=0}^{N-1}c(k)e^{Jk\frac{2\pi }{N}n} xn=k=0N1c(k)eJkN2πn

c ( k ) = 1 N ∑ n = 0 N − 1 x ( n ) e − J k 2 π N n c(k)=\frac{1}{N}\sum_{n=0}^{N-1}x(n)e^{-Jk\frac{2\pi}{N}n} c(k)=N1n=0N1x(n)eJkN2πn

 

CTFS

x ( w ) = ∫ − ∞ + ∞ x ( t ) e − J w t d t x(w)=\int_{-\infty}^{+\infty}x(t)e^{-Jwt}dt x(w)=+x(t)eJwtdt

Inverse transformation: x ( t ) = 1 2 π ∫ − ∞ + ∞ x ( w ) e j w t d w x(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}x(w)e^{jwt}dw x(t)=2π1+x(w)ejwtdw

 

Continuous time Fourier transform (CTFT)

  • basic concept
  • fourier fransform
  • magnitude specturm
  • phase spectrum

properties of CTFT

  • linearity
  • time shifting
  • frequency shifting
  • time scaling
  • time differenciation
  • frequency differenciation
  • convolution

Parseval’s Theorem

  1. Any signal is built up addition of elementary signals which are at different frequencies.

  2. CTFT is used to transform the signal from time domain to frequency domain.

  3. With the help of CTFT,plot the amplitude and phase spectrum.

  4. CTFT can be efxpressed as:

    x ( w ) = ∫ − ∞ + ∞ x ( t ) e − J w t d t x(w)=\int_{-\infty}^{+\infty}x(t)e^{-Jwt}dt x(w)=+x(t)eJwtdt

    Inverse transformation: x ( t ) = 1 2 π ∫ − ∞ + ∞ x ( w ) e j w t d w x(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}x(w)e^{jwt}dw x(t)=2π1+x(w)ejwtdw

 

Dirichlet’s conditions

  1. f(t) should be absolutely integrable.
  2. f(t) should have a finite number of maxima and minima over any finite interval.
  3. f(t) should have finite number of discontinues over any finite interval

These conditions are sufficient but not necessary Condition.

If the conditions are not met, there is not necessarily no Fourier transform.

 

lecture(77)

  • fourier fransform
  • magnitude specturm
  • phase spectrum

 

lecture(78)

Inverse transformation for function

important

 

properties of CTFT

linearity

F [ f ( t ) ] = F ( ω ) , F [ g ( t ) ] = G ( ω ) \mathscr{F}[f(t)]=F(\omega),\quad\mathscr{F}[g(t)]=G(\omega) F[f(t)]=F(ω),F[g(t)]=G(ω)

F [ α f ( t ) + β g ( t ) ] = α F ( ω ) + β G ( ω ) F − 1 [ α F ( ω ) + β G ( ω ) ] = α f ( t ) + β g ( t ) \begin{aligned}\mathscr{F}[\alpha f(t)+\beta g(t)]&=\alpha F(\omega)+\beta G(\omega)\\\mathscr{F}^{-1}[\alpha F(\omega)+\beta G(\omega)]&=\alpha f(t)+\beta g(t)\end{aligned} F[αf(t)+βg(t)]F1[αF(ω)+βG(ω)]=αF(ω)+βG(ω)=αf(t)+βg(t)

time shifting

F [ f ( t ) ] = F ( ω ) \mathcal{F}[f(t)]=F(\omega) F[f(t)]=F(ω)

F [ f ( t − t 0 ) ] = e − j ω t 0 F ( ω ) F − 1 [ F ( ω − ω 0 ) ] = e j ω 0 t f ( t ) \begin{aligned}\mathscr{F}[f(t-t_0)]&=e^{-j\omega t_0}F(\omega)\\\mathscr{F}^{-1}[F(\omega-\omega_0)]&=e^{j\omega_0t}f(t)\end{aligned} F[f(tt0)]F1[F(ωω0)]=et0F(ω)=ejω0tf(t)

time scaling

F [ f ( t ) ] = F ( ω ) \mathcal{F}[f(t)]=F(\omega) F[f(t)]=F(ω)

F [ f ( a t ) ] = 1 ∣ a ∣ F ( ω a ) \mathscr{F}[f(at)]=\frac{1}{|a|}F\left(\frac{\omega}{a}\right) F[f(at)]=a1F(aω)

frequency shifting

F [ f ( t ) ] = F ( ω ) \mathcal{F}[f(t)]=F(\omega) F[f(t)]=F(ω)

F [ f ( t ) e j w 0 t ] = F ( ω − ω 0 ) F − 1 [ f ( t ) e − j w 0 t ] = F ( ω + ω 0 ) \begin{aligned}\mathscr{F}[f(t)e^{jw_0t}]&=F(\omega - \omega_0)\\\mathscr{F}^{-1}[f(t)e^{-jw_0t}]&=F(\omega + \omega_0)\end{aligned} F[f(t)ejw0t]F1[f(t)ejw0t]=F(ωω0)=F(ω+ω0)

time differenciation

F [ f ( t ) ] = F ( ω ) \mathcal{F}[f(t)]=F(\omega) F[f(t)]=F(ω)

F [ d n f ( t ) d t n ] = ( j ω ) n F ( ω ) F − 1 [ d n F ( ω ) d ω n ] = ( − j t ) n f ( t ) \begin{gathered} \mathscr{F}\left[\frac{d^{n}f(t)}{dt^{n}}\right]=(j\omega)^{n}F(\omega) \\ \mathscr{F}^{-1}\left[\frac{d^{n}F(\omega)}{d\omega^{n}}\right]=(-jt)^{n}f(t) \end{gathered} F[dtndnf(t)]=()nF(ω)F1[dωndnF(ω)]=(jt)nf(t)

frequency differenciation

F [ f ( t ) ] = F ( ω ) \mathcal{F}[f(t)]=F(\omega) F[f(t)]=F(ω)

F [ ( − j t ) n f ( t ) ] = d n F ( w ) d w n \mathscr{F} [(-jt)^{n}f(t)]= \frac{d^{n}F(w)}{dw^{n}} F[(jt)nf(t)]=dwndnF(w)

F [ t f ( t ) ] = j d F ( w ) d w \mathscr{F} [tf(t)]= j\frac{dF(w)}{dw} F[tf(t)]=jdwdF(w)

convolution

F [ f ( t ) ] = F ( ω ) , F [ g ( t ) ] = G ( ω ) \mathscr{F}[f(t)]=F(\omega),\quad\mathscr{F}[g(t)]=G(\omega) F[f(t)]=F(ω),F[g(t)]=G(ω)

F [ f 1 ( t ) ∗ f 2 ( t ) ] = F 1 ( ω ) ⋅ F 2 ( ω ) \mathscr{F}[f_1(t) *f_2(t)]=F_1(\omega)·F_2(\omega) F[f1(t)f2(t)]=F1(ω)F2(ω)

F [ f 1 ( t ) ⋅ f 2 ( t ) ] = 1 2 π F 1 ( ω ) ∗ F 2 ( ω ) \mathscr{F}[f_1(t) ·f_2(t)]=\frac{1}{2\pi}F_1(\omega)*F_2(\omega) F[f1(t)f2(t)]=2π1F1(ω)F2(ω)

 

Parseval’s Theorem

if x(t) CTFT to x ( w ) x(w) x(w) or x ( f ) x(f) x(f)

E = ∫ − ∞ ∞ ∣ x ( t ) ∣ 2 d t = ∫ − ∞ ∞ ∣ x ( f ) ∣ 2 d f E=\int_{-\infty}^{\infty}|x(t)|^{2}dt=\int_{-\infty}^{\infty}|x(f)|^{2}df E=x(t)2dt=x(f)2df

 

Fourier transform of Periodic signal

  • spectrum of e j ω 0 t e^{j\omega_0t } ejω0t and e − j ω 0 t e^{- j\omega_0t } ejω0t
  • spectrum of s i n w t sinwt sinwt and c o s w t coswt coswt
  • Spectrum of a general periodic signal

spectrum of e j ω 0 t e^{j\omega_0t } ejω0t and e − j ω 0 t e^{- j\omega_0t } ejω0t

F [ e j ω 0 t ] = 2 π δ ( ω − ω 0 ) F [ e − j ω 0 t ] = 2 π δ ( ω + ω 0 ) \begin{aligned}\mathscr{F}[e^{j\omega_0t}]&=2\pi\delta(\omega-\omega_0)\\\mathscr{F}[e^{-j\omega_0t}]&=2\pi\delta(\omega+\omega_0)\end{aligned} F[ejω0t]F[ejω0t]=2πδ(ωω0)=2πδ(ω+ω0)

 

spectrum of s i n w 0 t sinw_0t sinw0t and c o s w 0 t cosw_0t cosw0t

F [ cos ⁡ ( ω 0 t ) ] = F [ e j ω 0 t + e − j ω 0 t 2 ] = π [ δ ( ω − ω 0 ) + δ ( ω + ω 0 ) ] F [ sin ⁡ ( ω 0 t ) ] = F [ e j ω 0 t − e − j ω 0 t 2 j ] = j π [ δ ( ω + ω 0 ) − δ ( ω − ω 0 ) ] \begin{gathered} \mathscr{F}[\cos(\omega_{0}t)]=\mathscr{F}[\frac{e^{j\omega_{0}t}+e^{-j\omega_{0}t}}{2}]=\pi[\delta(\omega-\omega_{0})+\delta(\omega+\omega_{0})] \\ \mathscr{F}[\sin(\omega_{0}t)]=\mathscr{F}[\frac{e^{j\omega_{0}t}-e^{-j\omega_{0}t}}{2j}]=j\pi[\delta(\omega+\omega_{0})-\delta(\omega-\omega_{0})] \end{gathered} F[cos(ω0t)]=F[2ejω0t+ejω0t]=π[δ(ωω0)+δ(ω+ω0)]F[sin(ω0t)]=F[2jejω0tejω0t]=[δ(ω+ω0)δ(ωω0)]

 

Spectrum of a general periodic signal

f ( t ) = ∑ n = − ∞ ∞ F ( n w 0 ) e j n w 0 t f(t)=\sum_{n=-\infty}^{\infty}F(nw_0)e^{jnw_0 t} f(t)=n=F(nw0)ejnw0t

F ( w ) = F [ f ( t ) ] = 2 π ∑ n = − ∞ ∞ F ( n w 0 ) δ ( ω − n ω 0 ) F(w) = \mathscr{F}[f(t)] = 2\pi \sum_{n=-\infty}^{\infty}F(nw_0)\delta(\omega-n\omega_{0}) F(w)=F[f(t)]=2πn=F(nw0)δ(ωnω0)

F ( n w 1 ) = 1 T ∫ 0 T f ( t ) e − j n w 0 t d t F(nw_1) = \frac{1}{T}\int_{0}^{T} f(t)e^{-jnw_0t}dt F(nw1)=T10Tf(t)ejnw0tdt

 

Discrete time fouriew series (DTFT)

  • basic properties
  • DTFT for common signals

basic properties

DTFT: X ( e j ω ) = ∑ n = − ∞ + ∞ x [ n ] e − j ω n X\left(e^{j\omega}\right)=\sum_{n=-\infty}^{+\infty}x[n]e^{-j\omega n} X(e)=n=+x[n]ejωn

Inverse transform of DTFT: x [ n ] = 1 2 π ∫ 2 π X ( e j ω ) e j ω n d ω x\left[n\right]=\frac1{2\pi}\int_{2\pi}X(e^{j\omega})e^{j\omega n}d\omega x[n]=2π12πX(e)ejωndω

DTFT for common signals

x ( n ) = a n u ( n ) ; ∣ a ∣ < 1 x(n) = a^n u(n);|a|<1 x(n)=anu(n);a<1

DTFT [X(n)] = ∑ n = − ∞ ∞ x ( n ) e − j w n = ∑ n = 0 + ∞ a n e − j w n = ∑ n = 0 + ∞ ( a ⋅ e − j w ) n = 1 1 − a e − j w = 1 1 − a ( c o s w − j s i n w ) = 1 ( 1 − a c o s w ) + j a s i n w \begin{aligned}\text{DTFT [X(n)]}&=\sum_{n=-\infty}^{\infty}x({n})e^{-jwn}\\&=\sum_{n=0}^{+\infty}a^ne^{-jwn}\\&=\sum_{n=0}^{+\infty}(a\cdot e^{-jw})^n\\&=\frac1{1-ae^{-jw}} \\&=\frac{1}{1-a(cosw-jsinw)} \\&=\frac{1}{(1-acosw)+jasinw}\end{aligned} DTFT [X(n)]=n=x(n)ejwn=n=0+anejwn=n=0+(aejw)n=1aejw1=1a(coswjsinw)1=(1acosw)+jasinw1

magnitude:

∣ x ( e j w ) ∣ = 1 ( 1 − a c o s w ) 2 + ( a s i n w ) 2 = 1 1 + a 2 − 2 a c o s w |x(e^{jw})| = \frac{1}{\sqrt{(1-acosw)^2}+(asinw)^2 }=\frac{1}{\sqrt{1+a^2-2acosw} } x(ejw)=(1acosw)2 +(asinw)21=1+a22acosw 1

phase:

∠ x ( e j w ) = − a r c t a n a s i n w 1 − a c o s w \angle x(e^{jw}) = -arctan\frac{asinw}{1-acosw} x(ejw)=arctan1acoswasinw

eg:x(n) = {-4,3,-2,3,-4}

find:(1). x ( e j 0 ) x(e^{j0}) x(ej0) (2). x ( e j w ) x(e^{jw}) x(ejw) (3). ∫ π − π x ( e j w ) d w \int_{\pi }^{-\pi} x(e^ {jw} )dw ππx(ejw)dw

 

properties

linear

a x 1 [ n ] + b x 2 [ n ] ↔ a X 1 ( e j ω ) + b X 2 ( e j ω ) ax_1\left[n\right]+bx_2\left[n\right]\leftrightarrow aX_1(e^{j\omega})+bX_2(e^{j\omega}) ax1[n]+bx2[n]aX1(e)+bX2(e)

time shifting

x ( n ) ↔ x ( e j w ) x(n) \leftrightarrow x(e^{jw}) x(n)x(ejw)

x ( n − n 0 ) ↔ x ( e j w ) ⋅ e − j w n 0 x( n - n_0) \leftrightarrow x(e^{jw})·e^{-jwn_0} x(nn0)x(ejw)ejwn0

frequency shifting

x ( n ) ↔ x ( e j w ) x(n) \leftrightarrow x(e^{jw}) x(n)x(ejw)

x ( n ) ⋅ e − j w n 0 ↔ x ( e j ( w − w 0 ) ) x(n)·e^{-jwn_0} \leftrightarrow x(e^{j(w-w_0)}) x(n)ejwn0x(ej(ww0))

Frequency domain differential

x ( n ) ↔ x ( e j w ) x(n) \leftrightarrow x(e^{jw}) x(n)x(ejw)

n x ( n ) ↔ j d x ( e j w ) d w n x(n) \leftrightarrow j \frac{dx(e^{jw})}{dw} nx(n)jdwdx(ejw)

 

eg:

1. y ( n ) − A y ( n − 1 ) = x ( n ) ; ∣ A ∣ < 1 y(n) - Ay(n-1) = x(n);|A|<1 y(n)Ay(n1)=x(n);A<1

2. y ( n ) − 3 4 y ( n − 1 ) + 1 8 y ( n − 2 ) = 2 x ( n ) y(n) - \frac{3}{4}y(n-1)+\frac{1}{8}y(n-2)=2x(n) y(n)43y(n1)+81y(n2)=2x(n)

3. x ( e j w ) = e − j w x(e^{jw})=e^{-jw} x(ejw)=ejw for − π ≤ w ≤ π -\pi \le w \le \pi πwπ

4.find convolution x ( n ) = 1 2 n u ( n ) x(n) = \frac{1}{2}^n u(n) x(n)=21nu(n) and x ( n ) = 1 3 n u ( n ) x(n) = \frac{1}{3}^n u(n) x(n)=31nu(n)

 

Discrete Fourier Transform(DFT)

DFT is equally spaced sampling of DTFT

DFT: x ( n ) —— > x ( k ) x(n) ——> x(k) x(n)——>x(k)

x ( k ) = ∑ n = 0 N − 1 x ( n ) e − j 2 π k N n ; k = 0 , 1 , 2 , 3 , . . . , N − 1 x(k) = \sum_{n=0}^{N-1}x(n)e^{-j\frac{2\pi k}{N}n};k=0,1,2,3,...,N-1 x(k)=n=0N1x(n)ejN2πkn;k=0,1,2,3,...,N1

making W N = e − j 2 π N W_N = e^{-j\frac{2\pi}{N}} WN=ejN2π

x ( k ) = ∑ n = 0 N − 1 x ( n ) W N k n x(k) = \sum_{n=0}^{N-1}x(n)W_N^{kn} x(k)=n=0N1x(n)WNkn

 

Calculation of DFT

eg: x ( k ) = ∑ n = 0 4 − 1 x ( n ) e − j 2 π k 4 n x(k) = \sum_{n=0}^{4-1}x(n)e^{-j\frac{2\pi k}{4}n} x(k)=n=041x(n)ej42πkn

1. x ( k ) = ∑ n = 0 N − 1 x ( n ) e − j 2 π k N n ; k = 0 , 1 , 2 , 3 , . . . , N − 1 x(k) = \sum_{n=0}^{N-1}x(n)e^{-j\frac{2\pi k}{N}n};k=0,1,2,3,...,N-1 x(k)=n=0N1x(n)ejN2πkn;k=0,1,2,3,...,N1

2.matrix method

image-20231223221845915

image-20231224153828648
Rotation matrix of inverse DFT transform(IDFT)

image-20231224153845233

 

Properties of DFT

1.linear

X 1 ( k ) = D F T [ x 1 ( n ) ] X_1(k)=DFT[x_1(n)] X1(k)=DFT[x1(n)]

X 2 ( k ) = D F T [ x 2 ( n ) ] X_2(k)=DFT[x_2(n)] X2(k)=DFT[x2(n)]

D F T [ a x 1 ( n ) + b x 2 ( n ) ] = a X 1 ( k ) + b X 2 ( k ) DFT[ax_1(n)+bx_2(n)]=aX_1(k)+bX_2(k) DFT[ax1(n)+bx2(n)]=aX1(k)+bX2(k)

2.circular convolution

DFT: x [ n ] ⊛ h [ n ] ⇔ X [ k ] ⋅ H [ k ] x[n]\circledast h[n]\Leftrightarrow X[k]·H[k] x[n]h[n]X[k]H[k]

DFT: x [ n ] ⋅ h [ n ] ⇔ X [ k ] ⊛ H [ k ] x[n]· h[n]\Leftrightarrow X[k]\circledast H[k] x[n]h[n]X[k]H[k]

3.periodicity

D F T [ x ( n ) ] = x ( k ) DFT[x(n)] = x(k) DFT[x(n)]=x(k)

x ( n + N ) = x ( n ) x(n+N) = x(n) x(n+N)=x(n);for all n

x ( k + N ) = x ( k ) x(k+N) = x(k) x(k+N)=x(k);for all k

4.cyclic frequency shift

D F T [ x ( n ) e j 2 π N l n ] = x ( ( k − l ) ) N DFT[x(n)e^{j\frac{2\pi}{N}ln}]=x((k-l))_N DFT[x(n)ejN2πln]=x((kl))N

(( )) is cyclic frequency

 

DTF calculates linear convolution and circular convolution

linear convilution

[ y ( n ) = x ( n ) ∗ h ( n ) ] [y(n) = x(n)*h(n)] [y(n)=x(n)h(n)]

then the length of x(n) is L,and h(n) is M

the length of linear convolution: N = L + M -1

  1. filling the length of x(n) and h(n) to N (add 0)
  2. calculates DTF[x(n)] and DTF[h(n)]
  3. then$ y(k) = x(k)·h(k)$
  4. calculate I D F T [ Y ( k ) ] = y ( n ) IDFT[Y(k)] = y(n) IDFT[Y(k)]=y(n)

 

cyclical convolution

[ y ( n ) = x ( n ) ⊛ h ( n ) ] [y(n) = x(n)\circledast h(n)] [y(n)=x(n)h(n)]

then the length of x(n) is L,and h(n) is M(add 0)

the length of cyclical convolution is :N = max(L,M)

  1. filling the length of x(n) and h(n) to N
  2. calculates DTF[x(n)] and DTF[h(n)]
  3. then y ( k ) = x ( k ) ⋅ h ( k ) y(k) = x(k)·h(k) y(k)=x(k)h(k)
  4. calculate I D F T [ Y ( k ) ] = y ( n ) IDFT[Y(k)] = y(n) IDFT[Y(k)]=y(n)

 

another way to calculate linear convolution

  • Overlap-Add and Overlap-Save
Overlap-Add

then the length of x k ( n ) x_k(n) xk(n) ( x k ( n ) x_k(n) xk(n) is not x ( n ) x(n) x(n)) is L,and h(n) is M

the length of convolution is :N = L + M - 1 (the size of N is depend on you)

eg: x ( n ) = { 3 , − 1 , 0 , 1 , 3 , 2 , 0 , 1 , 2 , 1 } x(n) = \{ 3,-1,0,1,3,2,0,1,2,1\} x(n)={3,1,0,1,3,2,0,1,2,1} h ( n ) = { 1 , 1 , 1 } h(n) = \{ 1,1,1\} h(n)={1,1,1}

1.we take N = 6,then 6= L + 3 - 1;so L = 4.

x 1 ( n ) = { 0 , 0 , 3 , − 1 , 0 , 1 } x_1(n) = \{ 0,0,3,-1,0,1 \} x1(n)={0,0,3,1,0,1}

x 2 ( n ) = { 0 , 1 , 3 , 2 , 0 , 1 } x_2(n) = \{ 0,1,3,2 ,0,1\} x2(n)={0,1,3,2,0,1}

x 3 ( n ) = { 0 , 1 , 2 , 1 , 0 , 0 } x_3(n) = \{ 0,1,2,1,0,0 \} x3(n)={0,1,2,1,0,0}

h ( n ) = { 1 , 1 , 1 , 0 , 0 , 0 } h(n) = \{1,1,1,0,0,0\} h(n)={1,1,1,0,0,0}

2.calculate

x 1 ( n ) ⊛ h ( n ) = { 1 , 1 , 3 , 2 , 2 , 0 } x_1(n)\circledast h(n) =\{1,1,3,2,2,0\} x1(n)h(n)={1,1,3,2,2,0}

x 2 ( n ) ⊛ h ( n ) = { 1 , 2 , 4 , 6 , 5 , 3 } x_2(n)\circledast h(n)=\{1,2,4,6,5,3\} x2(n)h(n)={1,2,4,6,5,3}

x 3 ( n ) ⊛ h ( n ) = { 0 , 1 , 3 , 4 , 3 , 1 } x_3(n)\circledast h(n) =\{0,1,3,4,3,1\} x3(n)h(n)={0,1,3,4,3,1}

3.Remove first (M-1) points ,Concatenate all results

the result is { 3 , 2 , 2 , 0 , 4 , 6 , 5 , 3 , 3 , 4 , 3 , 1 } \{3,2,2,0,4,6,5,3,3,4,3,1\} {3,2,2,0,4,6,5,3,3,4,3,1}

 

Overlap-Save

then the length of x k ( n ) x_k(n) xk(n) ( x k ( n ) x_k(n) xk(n) is not x ( n ) x(n) x(n)) is L,and h(n) is M

the length of convolution is :N = L + M - 1 (the size of N is depend on you)

eg: x ( n ) = { 3 , − 1 , 0 , 1 , 3 , 2 , 0 , 1 , 2 , 1 } x(n) = \{ 3,-1,0,1,3,2,0,1,2,1\} x(n)={3,1,0,1,3,2,0,1,2,1} h ( n ) = { 1 , 1 , 1 } h(n) = \{ 1,1,1\} h(n)={1,1,1}

1.we take N = 5,then 5 = L + 3 - 1;so L = 3.

x 1 ( n ) = { 0 , 0 , 3 , − 1 , 0 } x_1(n) = \{ 0,0,3,-1,0 \} x1(n)={0,0,3,1,0}

x 2 ( n ) = { − 1 , 0 , 1 , 3 , 2 } x_2(n) = \{ -1,0,1,3,2 \} x2(n)={1,0,1,3,2}

x 3 ( n ) = { 3 , 2 , 0 , 1 , 2 } x_3(n) = \{ 3,2,0,1,2 \} x3(n)={3,2,0,1,2}

x 4 ( n ) = { 1 , 2 , 1 , 0 , 0 } x_4(n) = \{ 1,2,1,0,0 \} x4(n)={1,2,1,0,0}

h ( n ) = { 1 , 1 , 1 , 0 , 0 } h(n) = \{1,1,1,0,0\} h(n)={1,1,1,0,0}

2.calculate

x 1 ( n ) ⊛ h ( n ) = { − 1 , 0 , 3 , 2 , 2 } x_1(n)\circledast h(n) =\{-1,0,3,2,2\} x1(n)h(n)={1,0,3,2,2}

x 2 ( n ) ⊛ h ( n ) = { 4 , 1 , 0 , 4 , 6 } x_2(n)\circledast h(n)=\{4,1,0,4,6\} x2(n)h(n)={4,1,0,4,6}

x 3 ( n ) ⊛ h ( n ) = { 6 , 7 , 5 , 3 , 3 } x_3(n)\circledast h(n) =\{6,7,5,3,3\} x3(n)h(n)={6,7,5,3,3}

x 4 ( n ) ⊛ h ( n ) = { 1 , 3 , 4 , 3 , 1 } x_4(n)\circledast h(n) =\{1,3,4,3,1\} x4(n)h(n)={1,3,4,3,1}

3.Remove first (M-1) points ,Concatenate all results

the result is { 3 , 2 , 2 , 0 , 4 , 6 , 5 , 3 , 3 , 4 , 3 , 1 } \{3,2,2,0,4,6,5,3,3,4,3,1\} {3,2,2,0,4,6,5,3,3,4,3,1}

 

 

z-transformation

  • Basic concepts of z-transformation
  • ROC
  • Properties of z-transformation
  • Inverse transformation of z-transform
  • stable

Basic concepts of z-transformation

  • Bilateral Z Transform

  • Unilateral Z Transform

Bilateral Z Transform:

X ( Z ) = Z { x [ n ] } = ∑ n = − ∞ + ∞ x [ n ] Z − n , Z ∈ R x X(Z)=Z\left\{x[n]\right\}=\sum_{n=-\infty}^{+\infty}x\left[n\right]Z^{-n}\text{,}Z\in Rx X(Z)=Z{x[n]}=n=+x[n]Zn,ZRx

Unilateral Z Transform:

X ( Z ) = Z { x [ n ] } = ∑ n = 0 + ∞ x [ n ] Z − n , Z ∈ R x X(Z)=Z\left\{x[n]\right\}=\sum_{n=0}^{+\infty}x\left[n\right]Z^{-n},Z\in Rx X(Z)=Z{x[n]}=n=0+x[n]Zn,ZRx

 

region of convergence (ROC)

making z = r ⋅ e j w z = r·e^{jw} z=rejw

x ( z ) = ∑ n = − ∞ ∞ x ( n ) z − n = ∑ n = − ∞ ∞ x ( n ) r − n e − j w n x(z) = \sum_{n=-\infty}^{\infty}x(n)z^{-n} \\=\sum_{n=-\infty}^{\infty}x(n)r^{-n}e^{-jwn} x(z)=n=x(n)zn=n=x(n)rnejwn

if r = 1 r = 1 r=1,then x ( z ) = D T F T x(z) = DTFT x(z)=DTFT

eg: x ( n ) = { 1 , 2 , 3 , 4 , 0 , 1 } x(n) = \{1,2,3,4,0,1 \} x(n)={1,2,3,4,0,1}

x ( z ) = ∑ n = − ∞ ∞ x ( n ) z − n = ∑ n = 0 5 x ( n ) z − n = x ( 0 ) z − 0 + x ( 1 ) z − 1 + x ( 2 ) z − 2 + x ( 3 ) z − 3 + x ( 4 ) z − 4 + x ( 5 ) z − 5 = 1 + 2 z − 1 + 3 z − 2 + 4 z − 3 + z − 5 x(z) = \sum_{n=-\infty}^{\infty}x(n)z^{-n}=\sum_{n=0}^{5}x(n)z^{-n}=x(0)z^{-0}+x(1)z^{-1}+x(2)z^{-2}+x(3)z^{-3}+x(4)z^{-4}+x(5)z^{-5}\\=1+2z^{-1}+3z^{-2}+4z^{-3}+z^{-5} x(z)=n=x(n)zn=n=05x(n)zn=x(0)z0+x(1)z1+x(2)z2+x(3)z3+x(4)z4+x(5)z5=1+2z1+3z2+4z3+z5

ROC:exist entire z-plane except z = 0 z = 0 z=0

using plot:

eg: x ( n ) = { a n ; n ≥ 0 0 ; n < 0 x(n)=\left\{\begin{matrix} a^n\;;n\ \ge 0 \\ 0\;;n\ < 0 \end{matrix}\right. x(n)={an;n 00;n <0

x ( z ) = ∑ n = 0 + ∞ ( a z − 1 ) n = 1 1 − a z − 1 = z z − a x(z) = \sum_{n=0}^{+\infty}(az^{-1})^{n}\\=\frac{1}{1-az^{-1}}=\frac{z}{z-a} x(z)=n=0+(az1)n=1az11=zaz

needing that : ∣ a z − 1 ∣ < 1 |az^{-1}|<1 az1<1 just ROC: ∣ z ∣ > ∣ a ∣ |z|> |a| z>a

eg : x ( n ) = − a n u ( − n − 1 ) x(n) = -a^nu(-n-1) x(n)=anu(n1)

ROC: ∣ a − 1 ∣ < 1 |a^{-1}|<1 a1<1 is just ∣ z ∣ < ∣ a ∣ |z| < |a| z<a

 

Properties of z-transformation

  • linear
  • time shifting
  • scaling
  • differential
  • convolution
  • initial value theorem
  • terminal value theorem
linear

x 1 ( n ) ⟶ x 1 ( z ) ; R O C 1 x_1(n) \longrightarrow x_1(z);ROC_1 x1(n)x1(z);ROC1

x 2 ( n ) ⟶ x 2 ( z ) ; R O C 2 x_2(n) \longrightarrow x_2(z);ROC_2 x2(n)x2(z);ROC2

a x 1 ( n ) + b x 2 ( n ) ⟶ a x 1 ( z ) + b x 2 ( z ) ; R O C : [ R O C 1 ∩ R O C 2 ] ax_1(n) + bx_2(n) \longrightarrow ax_1(z)+bx_2(z);\\ROC:[ROC_1\cap ROC_2] ax1(n)+bx2(n)ax1(z)+bx2(z);ROC:[ROC1ROC2]

time shifting

x ( n ) ⟶ x ( z ) x(n) \longrightarrow x(z) x(n)x(z)

x ( n − n 0 ) ⟶ x ( z ) z − n 0 x(n - n_0) \longrightarrow x(z)z^{-n_0} x(nn0)x(z)zn0

scaling

x ( n ) ⟶ x ( z ) ; R O C : ∣ z ∣ > 1 x(n) \longrightarrow x(z);ROC: \;|z|>1 x(n)x(z);ROC:z>1

a n x ( n ) ⟶ x ( z a ) a^nx(n) \longrightarrow x(\frac{z}{a}) anx(n)x(az)

differential

x ( n ) ⟶ x ( z ) x(n) \longrightarrow x(z) x(n)x(z)

n x ( n ) ⟶ − z d x ( z ) d z nx(n) \longrightarrow -z \frac{dx(z)}{dz} nx(n)zdzdx(z)

convolution

x 1 ( n ) ∗ x 2 ( n ) ⟶ x 1 ( z ) ⋅ x 2 ( z ) x_1(n) * x_2(n) \longrightarrow x_1(z)·x_2(z) x1(n)x2(n)x1(z)x2(z)

initial value theorem

x ( 0 ) = lim ⁡ n → 0 x ( n ) = lim ⁡ z → + ∞ x ( z ) x(0) = \lim_{n \to 0}x(n) = \lim_{z \to +\infty}x(z) x(0)=limn0x(n)=limz+x(z)

terminal value theorem

x ( + ∞ ) = lim ⁡ n → ∞ x ( n ) = lim ⁡ z → 1 ( 1 − z − 1 ) x ( z ) x(+\infty) = \lim_{n \to \infty}x(n) = \lim_{z \to 1}(1-z^{-1})x(z) x(+)=limnx(n)=limz1(1z1)x(z)

 

Inverse transformation of z-transform

  • long division method
  • partial fraction expansion method
  • Residue method
long division method
image-20231224203722689
partial fraction expansion method

x ( z ) = z ( z − 1 2 ) ( z + 1 2 ) ( z + 1 4 ) x(z) = \frac{z(z-\frac{1}{2})}{(z+\frac{1}{2})(z+\frac{1}{4})} x(z)=(z+21)(z+41)z(z21)

x ( z ) z = z − 1 2 ( z + 1 2 ) ( z + 1 4 ) = 4 z + 1 2 − 3 z + 1 4 \frac{x(z)}{z} = \frac{z-\frac{1}{2}}{(z+\frac{1}{2})(z+\frac{1}{4})}=\frac{4}{z+\frac{1}{2}}-\frac{3}{z+\frac{1}{4}} zx(z)=(z+21)(z+41)z21=z+214z+413

x ( z ) = 4 z z + 1 2 − 3 z z + 1 4 x(z) = \frac{4z}{z+\frac{1}{2}} - \frac{3z}{z+\frac{1}{4}} x(z)=z+214zz+413z

x ( n ) = 4 ( − 1 2 ) n u ( n ) − 3 ( − 1 4 ) n u ( n ) x(n) = 4(-\frac{1}{2})^{n}u(n) -3 (-\frac{1}{4})^{n}u(n) x(n)=4(21)nu(n)3(41)nu(n)

Residue method

x ( z ) = 1 ( z − 2 ) ( z − 3 ) x(z) = \frac{1}{(z-2)(z-3)} x(z)=(z2)(z3)1

1. z n − 1 x ( z ) = z n − 1 ( z − 2 ) ( z − 3 ) z^{n-1}x(z) = \frac{z^{n-1}}{(z-2)(z-3)} zn1x(z)=(z2)(z3)zn1

2. R 1 = ( z − 2 ) z n − 1 ( z − 2 ) ( z − 3 ) ∣ z = 2 = − 2 n − 1 R_1 =(z-2)\frac{z^{n-1}}{(z-2)(z-3)}|_{z=2} = -2^{n-1} R1=(z2)(z2)(z3)zn1z=2=2n1

3. R 2 = ( z − 3 ) z n − 1 ( z − 2 ) ( z − 3 ) ∣ z = 3 = 3 n − 1 R_2 =(z-3)\frac{z^{n-1}}{(z-2)(z-3)}|_{z=3} = 3^{n-1} R2=(z3)(z2)(z3)zn1z=3=3n1

4. x ( n ) = − 2 n − 1 + 3 n − 1 x(n) = -2^{n-1} + 3^{n-1} x(n)=2n1+3n1

 

stability

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

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