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  • Binbaşı
    1816 Mesaj
    Table 1: Properties of the Continuous-Time Fourier Series
    x(t) =
    +∞Xk=−∞
    akejk!0t =
    +∞Xk=−∞
    akejk(2/T)t
    ak =
    1
    T ZT
    x(t)e−jk!0tdt =
    1
    T ZT
    x(t)e−jk(2/T)tdt
    Property Periodic Signal Fourier Series Coefficients
    x(t)
    y(t)  Periodic with period T and
    fundamental frequency !0 = 2/T
    ak
    bk
    Linearity Ax(t) + By(t) Aak + Bbk
    Time-Shifting x(t − t0) ake−jk!0t0 = ake−jk(2/T)t0
    Frequency-Shifting ejM!0t = ejM(2/T)tx(t) ak−M
    Conjugation x∗(t) a∗
    −k
    Time Reversal x(−t) a−k
    Time Scaling x( t), > 0 (periodic with period T/ ) ak
    Periodic Convolution ZT
    x( )y(t −  )d Takbk
    Multiplication x(t)y(t)
    +∞X l=−∞
    albk−l
    Differentiation
    dx(t)
    dt
    jk!0ak = jk
    2
    T
    ak
    Integration Z t
    −∞
    x(t)dt
    (finite-valued and
    periodic only if a0 = 0)  1
    jk!0ak =  1
    jk(2/T)ak
    Conjugate Symmetry
    for Real Signals
    x(t) real
    
    
    
    ak = a∗
    −k
    ℜe{ak} = ℜe{a−k}
    ℑm{ak} = −ℑm{a−k}
    |ak| = |a−k|
    <) ak = −<) a−k
    Real and Even Sig-
    nals
    x(t) real and even ak real and even
    Real and Odd Signals x(t) real and odd ak purely imaginary and odd
    Even-Odd Decompo-
    sition of Real Signals  xe(t) = Ev{x(t)} [x(t) real]
    xo(t) = Od{x(t)} [x(t) real]
    ℜe{ak}
    jℑm{ak}
    Parseval’s Relation for Periodic Signals
    1
    T ZT
    |x(t)|2dt =
    +∞Xk=−∞
    |ak|2
    Table 2: Properties of the Discrete-Time Fourier Series
    x[n] = X k=<N>
    akejk!0n = X k=<N>
    akejk(2/N)n
    ak =
    1
    N X n=<N>
    x[n]e−jk!0n =
    1
    N X n=<N>
    x[n]e−jk(2/N)n
    Property Periodic signal Fourier series coefficients
    x[n]
    y[n] Periodic with period N and fun-
    damental frequency !0 = 2/N
    ak
    bk  Periodic with
    period N
    Linearity Ax[n] + By[n] Aak + Bbk
    Time shift x[n − n0] ake−jk(2/N)n0
    Frequency Shift ejM(2/N)nx[n] ak−M
    Conjugation x∗[n] a∗
    −k
    Time Reversal x[−n] a−k
    Time Scaling x(m)[n] =  x[n/m] if n is a multiple of m
    0 if n is not a multiple of m
    1
    m
    ak viewed as
    periodic with
    period mN !
    (periodic with period mN)
    Periodic Convolution Xr=hNi
    x[r]y[n − r] Nakbk
    Multiplication x[n]y[n] Xl=hNi
    albk−l
    First Difference x[n] − x[n − 1] (1 − e−jk(2/N))ak
    Running Sum
    nX
    k=−∞
    x[k] finite-valued and
    periodic only if a0 = 0   1
    (1 − e−jk(2/N))ak
    Conjugate Symmetry
    for Real Signals
    x[n] real
    
    
    
    ak = a∗
    −k
    ℜe{ak} = ℜe{a−k}
    ℑm{ak} = −ℑm{a−k}
    |ak| = |a−k|
    <) ak = −<) a−k
    Real and Even Signals x[n] real and even ak real and even
    Real and Odd Signals x[n] real and odd ak purely imaginary and odd
    Even-Odd Decomposi-
    tion of Real Signals
    xe[n] = Ev{x[n]} [x[n] real]
    xo[n] = Od{x[n]} [x[n] real]
    ℜe{ak}
    jℑm{ak}
    Parseval’s Relation for Periodic Signals
    1
    N Xn=hNi
    |x[n]|2 = Xk=hNi
    |ak|2
    Table 3: Properties of the Continuous-Time Fourier Transform
    x(t) =
    1
    2 Z ∞
    −∞
    X(j!)ej!td!
    X(j!) = Z ∞
    −∞
    x(t)e−j!tdt
    Property Aperiodic Signal Fourier transform
    x(t) X(j!)
    y(t) Y (j!)
    Linearity ax(t) + by(t) aX(j!) + bY (j!)
    Time-shifting x(t − t0) e−j!t0X(j!)
    Frequency-shifting ej!0tx(t) X(j(! − !0))
    Conjugation x∗(t) X∗(−j!)
    Time-Reversal x(−t) X(−j!)
    Time- and Frequency-Scaling x(at)
    1
    |a|
    X j!
    a 
    Convolution x(t) ∗ y(t) X(j!)Y (j!)
    Multiplication x(t)y(t)
    1
    2
    X(j!) ∗ Y (j!)
    Differentiation in Time
    d
    dt
    x(t) j!X(j!)
    Integration Z t
    −∞
    x(t)dt
    1
    j!
    X(j!) + X(0)(!)
    Differentiation in Frequency tx(t) j
    d
    d!
    X(j!)
    Conjugate Symmetry for Real
    Signals
    x(t) real
    
    
    
    X(j!) = X∗(−j!)
    ℜe{X(j!)} = ℜe{X(−j!)}
    ℑm{X(j!)} = −ℑm{X(−j!)}
    |X(j!)| = |X(−j!)|
    <)X(j!) = −<)X(−j!)
    Symmetry for Real and Even
    Signals
    x(t) real and even X(j!) real and even
    Symmetry for Real and Odd
    Signals
    x(t) real and odd X(j!) purely imaginary and odd
    Even-Odd Decomposition for
    Real Signals
    xe(t) = Ev{x(t)} [x(t) real]
    xo(t) = Od{x(t)} [x(t) real]
    ℜe{X(j!)}
    jℑm{X(j!)}
    Parseval’s Relation for Aperiodic Signals
    Z +∞
    −∞
    |x(t)|2dt =
    1
    2 Z +∞
    −∞
    |X(j!)|2d!
    Table 4: Basic Continuous-Time Fourier Transform Pairs
    Fourier series coefficients
    Signal Fourier transform (if periodic)
    +∞X k=−∞
    akejk!0t 2
    +∞X k=−∞
    ak(! − k!0) ak
    ej!0t 2(! − !0)
    a1 = 1
    ak = 0, otherwise
    cos !0t [(! − !0) + (! + !0)]
    a1 = a−1 = 1
    2
    ak = 0, otherwise
    sin !0t
    
    j
    [(! − !0) − (! + !0)]
    a1 = −a−1 = 1
    2j
    ak = 0, otherwise
    x(t) = 1 2(!)
    a0 = 1, ak = 0, k 6= 0
    this is the Fourier series rep-
    resentation for any choice of
    T > 0 !
    Periodic square wave
    x(t) =  1, |t| < T1
    0, T1 < |t| ≤ T
    2
    and
    x(t + T) = x(t)
    +∞X k=−∞
    2 sin k!0T1
    k
    (! − k!0)
    !0T1
    
    sinc k!0T1
      =
    sin k!0T1
    k
    +∞Xn=−∞
    (t − nT)
    2
    T
    +∞X k=−∞
     ! −
    2k
    T  ak =
    1
    T
    for all k
    x(t) 1, |t| < T1
    0, |t| > T1
    2 sin !T1
    !

    sinWt
    t
    X(j!) =  1, |!| < W
    0, |!| > W

    (t) 1 —
    u(t)
    1
    j!
    + (!) —
    (t − t0) e−j!t0 —
    e−atu(t),ℜe{a} > 0
    1
    a + j!

    te−atu(t),ℜe{a} > 0
    1
    (a + j!)2 —
    tn−1
    (n−1)!e−atu(t),
    ℜe{a} > 0
    1
    (a + j!)n —
    Table 5: Properties of the Discrete-Time Fourier Transform
    x[n] =
    1
    2 Z2
    X(ej!)ej!nd!
    X(ej!) =
    +∞ X n=−∞
    x[n]e−j!n
    Property Aperiodic Signal Fourier transform
    x[n]
    y[n]
    X(ej!)
    Y (ej!)  Periodic with
    period 2
    Linearity ax[n] + by[n] aX(ej!) + bY (ej!)
    Time-Shifting x[n − n0] e−j!n0X(ej!)
    Frequency-Shifting ej!0nx[n] X(ej(!−!0))
    Conjugation x∗[n] X∗(e−j!)
    Time Reversal x[−n] X(e−j!)
    Time Expansions x(k)[n] = x[n/k], if n = multiple of k
    0, if n 6= multiple of k
    X(ejk!)
    Convolution x[n] ∗ y[n] X(ej!)Y (ej!)
    Multiplication x[n]y[n]
    1
    2 Z2
    X(ej)Y (ej(!−))d
    Differencing in Time x[n] − x[n − 1] (1 − e−j!)X(ej!)
    Accumulation
    nX
    k=−∞
    x[k]
    1
    1 − e−j!X(ej!)
    +X(ej0)
    +∞Xk=−∞
    (! − 2k)
    Differentiation in Frequency nx[n] j
    dX(ej!)
    d!
    Conjugate Symmetry for
    Real Signals
    x[n] real
    
    
    
    X(ej!) = X∗(e−j!)
    ℜe{X(ej!)} = ℜe{X(e−j!)}
    ℑm{X(ej!)} = −ℑm{X(e−j!)}
    |X(ej!)| = |X(e−j!)|
    <)X(ej!) = −<)X(e−j!)
    Symmetry for Real, Even
    Signals
    x[n] real and even X(ej!) real and even
    Symmetry for Real, Odd
    Signals
    x[n] real and odd
    X(ej!) purely
    imaginary and odd
    Even-odd Decomposition of
    Real Signals
    xe[n] = Ev{x[n]} [x[n] real]
    xo[n] = Od{x[n]} [x[n] real]
    ℜe{X(ej!)}
    jℑm{X(ej!)}
    Parseval’s Relation for Aperiodic Signals
    +∞ X n=−∞
    |x[n]|2 =
    1
    2 Z2
    |X(ej!)|2d!
    Table 6: Basic Discrete-Time Fourier Transform Pairs
    Fourier series coefficients
    Signal Fourier transform (if periodic)
    X k=hNi
    akejk(2/N)n 2
    +∞X k=−∞
    ak ! −
    2k
    N  ak
    ej!0n 2
    +∞X l=−∞
    (! − !0 − 2l)
    (a) !0 = 2m
    N
    ak =  1, k = m,m ± N,m ± 2N, . . .
    0, otherwise
    (b) !0
    2 irrational ⇒ The signal is aperiodic
    cos !0n 
    +∞X l=−∞
    {(! − !0 − 2l) + (! + !0 − 2l)}
    (a) !0 = 2m
    N
    ak =  1
    2 , k = ±m,±m ± N,±m ± 2N, . . .
    0, otherwise
    (b) !0
    2 irrational ⇒ The signal is aperiodic
    sin !0n
    
    j
    +∞X l=−∞
    {(! − !0 − 2l) − (! + !0 − 2l)}
    (a) !0 = 2r
    N
    ak = 

    1
    2j , k = r, r ± N, r ± 2N, . . .
    − 1
    2j , k = −r,−r ± N,−r ± 2N, . . .
    0, otherwise
    (b) !0
    2 irrational ⇒ The signal is aperiodic
    x[n] = 1 2
    +∞X l=−∞
    (! − 2l) ak =  1, k = 0,±N,±2N, . . .
    0, otherwise
    Periodic square wave
    x[n] =  1, |n| ≤ N1
    0, N1 < |n| ≤ N/2
    and
    x[n + N] = x[n]
    2
    +∞X k=−∞
    ak ! −
    2k
    N  ak = sin[(2k/N)(N1+1
    2 )]
    N sin[2k/2N] , k 6= 0,±N,±2N, . . .
    ak = 2N1+1
    N , k = 0,±N,±2N, . . .
    +∞X k=−∞
    [n − kN]
    2
    N
    +∞X k=−∞
     ! −
    2k
    N  ak =
    1
    N
    for all k
    anu[n], |a| < 1
    1
    1 − ae−j! —
    x[n] 1, |n| ≤ N1
    0, |n| > N1
    sin[!(N1 + 1
    2 )]
    sin(!/2)

    sinWn
    n = W
     sinc 􀀀Wn
    0 < W <   
    X(ej!) =  1, 0 ≤ |!| ≤ W
    0, W < |!| ≤ 
    X(ej!)periodic with period 2

    [n] 1 —
    u[n]
    1
    1 − e−j! +
    +∞X k=−∞
    (! − 2k) —
    [n − n0] e−j!n0 —
    (n + 1)anu[n], |a| < 1
    1
    (1 − ae−j!)2 —
    (n + r − 1)!
    n!(r − 1)!
    anu[n], |a| < 1
    1
    (1 − ae−j!)r —
    Table 7: Properties of the Laplace Transform
    Property Signal Transform ROC
    x(t) X(s) R
    x1(t) X1(s) R1
    x2(t) X2(s) R2
    Linearity ax1(t) + bx2(t) aX1(s) + bX2(s) At least R1 ∩ R2
    Time shifting x(t − t0) e−st0X(s) R
    Shifting in the s-Domain es0tx(t) X(s − s0) Shifted version of R [i.e., s is
    in the ROC if (s − s0) is in
    R]
    Time scaling x(at)
    1
    |a|
    X s
    a “Scaled” ROC (i.e., s is in
    the ROC if (s/a) is in R)
    Conjugation x∗(t) X∗(s∗) R
    Convolution x1(t) ∗ x2(t) X1(s)X2(s) At least R1 ∩ R2
    Differentiation in the Time Domain
    d
    dt
    x(t) sX(s) At least R
    Differentiation in the s-Domain −tx(t)
    d
    ds
    X(s) R
    Integration in the Time Domain Z t
    −∞
    x( )d( )
    1
    s
    X(s) At least R ∩ {ℜe{s} > 0}
    Initial- and Final Value Theorems
    If x(t) = 0 for t < 0 and x(t) contains no impulses or higher-order singularities at t = 0, then
    x(0+) = lims→∞ sX(s)
    If x(t) = 0 for t < 0 and x(t) has a finite limit as t → ∞, then
    limt→∞ x(t) = lims→0 sX(s)
    Table 8: Laplace Transforms of Elementary Functions
    Signal Transform ROC
    1. (t) 1 All s
    2. u(t)
    1
    s
    ℜe{s} > 0
    3. −u(−t)
    1
    s
    ℜe{s} < 0
    4.
    tn−1
    (n − 1)!
    u(t)
    1
    sn ℜe{s} > 0
    5. −
    tn−1
    (n − 1)!
    u(−t)
    1
    sn ℜe{s} < 0
    6. e− tu(t)
    1
    s +
    ℜe{s} > −ℜe{ }
    7. −e− tu(−t)
    1
    s +
    ℜe{s} < −ℜe{ }
    8.
    tn−1
    (n − 1)!
    e− tu(t)
    1
    (s + )n ℜe{s} > −ℜe{ }
    9. −
    tn−1
    (n − 1)!
    e− tu(−t)
    1
    (s + )n ℜe{s} < −ℜe{ }
    10. (t − T) e−sT All s
    11. [cos !0t]u(t)
    s
    s2 + !2
    0
    ℜe{s} > 0
    12. [sin !0t]u(t)
    !0
    s2 + !2
    0
    ℜe{s} > 0
    13. [e− t cos !0t]u(t)
    s +
    (s + )2 + !2
    0
    ℜe{s} > −ℜe{ }
    14. [e− t sin !0t]u(t)
    !0
    (s + )2 + !2
    0
    ℜe{s} > −ℜe{ }
    15. un(t) =
    dn(t)
    dtn sn All s
    16. u−n(t) = u(t) ∗ · · · ∗ u(t)
    | {z } n times
    1
    sn ℜe{s} > 0
    Table 9: Properties of the z-Transform
    Property Sequence Transform ROC
    x[n] X(z) R
    x1[n] X1(z) R1
    x2[n] X2(z) R2
    Linearity ax1[n] + bx2[n] aX1(z) + bX2(z) At least the intersection
    of R1 and R2
    Time shifting x[n − n0] z−n0X(z) R except for the
    possible addition or
    deletion of the origin
    Scaling in the ej!0nx[n] X(e−j!0z) R
    z-Domain zn
    0 x[n] X z
    z0 z0R
    anx[n] X(a−1z) Scaled version of R
    (i.e., |a|R = the
    set of points {|a|z}
    for z in R)
    Time reversal x[−n] X(z−1) Inverted R (i.e., R−1
    = the set of points
    z−1 where z is in R)
    Time expansion x(k)[n] =  x[r], n = rk
    0, n 6= rk
    X(zk) R1/k
    for some integer r (i.e., the set of points z1/k
    where z is in R)
    Conjugation x∗[n] X∗(z∗) R
    Convolution x1[n] ∗ x2[n] X1(z)X2(z) At least the intersection
    of R1 and R2
    First difference x[n] − x[n − 1] (1 − z−1)X(z) At least the
    intersection of R and |z| > 0
    Accumulation Pn
    k=−∞ x[k] 1
    1−z−1X(z) At least the
    intersection of R and |z| > 1
    Differentiation nx[n] −z dX(z)
    dz R
    in the z-Domain
    Initial Value Theorem
    If x[n] = 0 for n < 0, then
    x[0] = limz→∞ X(z)
    Table 10: Some Common z-Transform Pairs
    Signal Transform ROC
    1. [n] 1 All z
    2. u[n] 1
    1−z−1 |z| > 1
    3. u[−n − 1] 1
    1−z−1 |z| < 1
    4. [n − m] z−m All z except
    0 (if m > 0) or
    ∞ (if m < 0)
    5. nu[n] 1
    1− z−1 |z| > | |
    6. − nu[−n − 1] 1
    1− z−1 |z| < | |
    7. n nu[n] z−1
    (1− z−1)2 |z| > | |
    8. −n nu[−n − 1] z−1
    (1− z−1)2 |z| < | |
    9. [cos !0n]u[n] 1−[cos !0]z−1
    1−[2 cos !0]z−1+z−2 |z| > 1
    10. [sin !0n]u[n] [sin !0]z−1
    1−[2 cos !0]z−1+z−2 |z| > 1
    11. [rn cos !0n]u[n] 1−[r cos !0]z−1
    1−[2r cos !0]z−1+r2z−2 |z| > r
    12. [rn sin !0n]u[n] [r sin !0]z−1
    1−[2r cos !0]z−1+r2z−2 |z| > r



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    _____________________________

    Adam gibi adam Machete Sağlam!




  • Yarbay
    2475 Mesaj
    20. satırda hata var
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  • Binbaşı
    1816 Mesaj
    quote:

    Orjinalden alıntı: YrkndrY

    20. satırda hata var



    olabilir daha kontrol etmemiştim
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    Adam gibi adam Machete Sağlam!
  • Yarbay
    2369 Mesaj
    Son satırda da "cos" var.
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    _____________________________
    Ben Türküm!
  • Binbaşı
    1365 Mesaj
    bu ne
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    _____________________________
  • Binbaşı
    1320 Mesaj
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    _____________________________
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