Table 1: Properties of the Continuous-Time Fourier Series x(t

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Table 1: Properties of the Continuous-Time Fourier Series +∞ X

x(t) =

jkω0 t

ak e

k=−∞

1 ak = T Property

Periodic Convolution

−jkω0 t

x(t)e

T

ak ejk(2π/T )t

k=−∞

1 dt = T

Z

x(t)e−jk(2π/T )t dt

T

Periodic Signal

x(t) y(t) Linearity Time-Shifting Frequency-Shifting Conjugation Time Reversal Time Scaling

Z

=

+∞ X



Fourier Series Coefficients

Periodic with period T and fundamental frequency ω0 = 2π/T

Ax(t) + By(t) x(t − t0 ) ejM ω0 t = ejM (2π/T )t x(t) x∗ (t) x(−t) x(αt), α > 0 (periodic with period T /α) Z x(τ )y(t − τ )dτ

ak bk Aak + Bbk ak e−jkω0 t0 = ak e−jk(2π/T )t0 ak−M a∗−k a−k ak T ak b k

T

Multiplication

+∞ X

x(t)y(t)

al bk−l

l=−∞

Differentiation

dx(t) Z dt t

Integration

x(t)dt

−∞

(finite-valued and periodic only if a0 = 0)

Conjugate Symmetry for Real Signals

x(t) real

Real and Even Signals

x(t) real and even

Real and Odd Signals

x(t) real and odd

Even-Odd Decomposition of Real Signals



2π jkω0 ak = jk ak T    1 1 ak = ak jk(2π/T )  jkω0 ∗ ak = a−k     ℜe{ak } = ℜe{a−k } ℑm{ak } = −ℑm{a−k }   |ak | = |a−k |    < ) ak = −< ) a−k ak real and even ak purely imaginary and odd

xe (t) = Ev{x(t)} [x(t) real] xo (t) = Od{x(t)} [x(t) real] Parseval’s Relation for Periodic Signals Z +∞ X 1 |ak |2 |x(t)|2 dt = T T k=−∞

ℜe{ak } jℑm{ak }

Table 2: Properties of the Discrete-Time Fourier Series X X x[n] = ak ejkω0 n = ak ejk(2π/N )n k=

1 N

ak =

Property

Time Scaling

Periodic Convolution

x[n]e−jkω0 n =

n=

1 N

X

x[n]e−jk(2π/N )n

n=

Periodic signal

x[n] y[n] Linearity Time shift Frequency Shift Conjugation Time Reversal

X

k=



Fourier series coefficients

Periodic with period N and fundamental frequency ω0 = 2π/N

Ax[n] + By[n] x[n − n0 ] ejM (2π/N )n x[n] x∗ [n] x[−n]  x[n/m] if n is a multiple of m x(m) [n] = 0 if n is not a multiple of m (periodic with period mN ) X x[r]y[n − r]



Periodic with period N

Aak + Bbk ak e−jk(2π/N )n0 ak−M a∗−k a−k viewed as 1 ak periodic with m period mN

!

N ak bk

r=hN i

Multiplication

ak bk

X

x[n]y[n]

al bk−l

l=hN i

First Difference Running Sum

x[n] − x[n − 1] n  X finite-valued and x[k] k=−∞

periodic only if a0 = 0

Conjugate Symmetry for Real Signals

x[n] real

Real and Even Signals

x[n] real and even

Real and Odd Signals

x[n] real and odd

Even-Odd Decomposition of Real Signals



(1 − e−jk(2π/N ) )ak   1 ak (1 − e−jk(2π/N ) )  ak = a∗−k     ℜe{ak } = ℜe{a−k } ℑm{ak } = −ℑm{a−k }   |a  k | = |a−k |   < ) ak = −< ) a−k ak real and even ak purely imaginary and odd

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 X 1 X |x[n]|2 = |ak |2 N n=hN i

k=hN i

Table 3: Properties of the Continuous-Time Fourier Transform Z ∞ 1 X(jω)ejωt dω x(t) = 2π −∞ Z ∞ X(jω) = x(t)e−jωt dt −∞

Property

Aperiodic Signal

Fourier transform

x(t) y(t)

X(jω) Y (jω)

Linearity Time-shifting Frequency-shifting Conjugation Time-Reversal

ax(t) + by(t) x(t − t0 ) ejω0 t x(t) x∗ (t) x(−t)

Time- and Frequency-Scaling

x(at)

Convolution

x(t) ∗ y(t)

Multiplication

x(t)y(t)

aX(jω) + bY (jω) e−jωt0 X(jω) X(j(ω − ω0 )) X ∗ (−jω) X(−jω)   1 jω X |a| a X(jω)Y (jω) 1 X(jω) ∗ Y (jω) 2π

Differentiation in Time Integration

d x(t) Zdt t

x(t)dt

−∞

Differentiation in Frequency

tx(t)

Conjugate Symmetry for Real x(t) real Signals Symmetry for Real and Even x(t) real and even Signals Symmetry for Real and Odd x(t) real and odd Signals Even-Odd Decomposition for xe (t) = Ev{x(t)} [x(t) real] xo (t) = Od{x(t)} [x(t) real] Real Signals

jωX(jω) 1 X(jω) + πX(0)δ(ω) jω d j X(jω)  dω X(jω) = X ∗ (−jω)     ℜe{X(jω)} = ℜe{X(−jω)} ℑm{X(jω)} = −ℑm{X(−jω)}   |X(jω)| = |X(−jω)|    < ) X(jω) = −< ) X(−jω) X(jω) real and even

X(jω) purely imaginary and odd ℜe{X(jω)} jℑm{X(jω)}

Parseval’s Relation for Aperiodic Signals Z +∞ Z +∞ 1 2 |X(jω)|2 dω |x(t)| dt = 2π −∞ −∞

Table 4: Basic Continuous-Time Fourier Transform Pairs

Signal +∞ X

Fourier transform

ak ejkω0 t



+∞ X

ak δ(ω − kω0 )

k=−∞

k=−∞

jω0 t

e

2πδ(ω − ω0 )

cos ω0 t

π[δ(ω − ω0 ) + δ(ω + ω0 )]

sin ω0 t

π [δ(ω − ω0 ) − δ(ω + ω0 )] j

x(t) = 1

2πδ(ω)

Periodic  square wave 1, |t| < T1 x(t) = 0, T1 < |t| ≤ T2 and x(t + T ) = x(t) +∞ X δ(t − nT )

n=−∞ 

x(t)

1, |t| < T1 0, |t| > T1

sin W t πt δ(t) u(t) δ(t − t0 ) e−at u(t), ℜe{a} > 0 te−at u(t), ℜe{a} > 0 tn−1

−at u(t), (n−1)! e

ℜe{a} > 0

+∞ X 2 sin kω0 T1 δ(ω − kω0 ) k

k=−∞

  +∞ 2π X 2πk δ ω− T T k=−∞

2 sin ωT1 ω  1, |ω| < W X(jω) = 0, |ω| > W 1 1 + πδ(ω) jω e−jωt0 1 a + jω 1 (a + jω)2 1 (a + jω)n

Fourier series coefficients (if periodic) ak a1 = 1 ak = 0, otherwise a1 = a−1 = 12 ak = 0, otherwise 1 a1 = −a−1 = 2j ak = 0, otherwise a0 = 1, ak = 0, k 6= 0 ! this is the Fourier series representation for any choice of T >0 ω0 T 1 sinc π

ak =



kω0 T1 π

1 for all k T

— — — — — — — —



=

sin kω0 T1 kπ

Table 5: Properties of the Discrete-Time Fourier Transform Z 1 X(ejω )ejωn dω x[n] = 2π 2π jω

X(e ) =

+∞ X

x[n]e−jωn

n=−∞

Property

Aperiodic Signal

Fourier transform  Periodic with X(ejω ) period 2π Y (ejω ) jω aX(e ) + bY (ejω ) e−jωn0 X(ejω ) X(ej(ω−ω0 ) ) X ∗ (e−jω ) X(e−jω )

Convolution

x[n] y[n] ax[n] + by[n] x[n − n0 ] ejω0 n x[n] x∗ [n] x[−n]  x[n/k], if n = multiple of k x(k) [n] = 0, if n 6= multiple of k x[n] ∗ y[n]

Multiplication

x[n]y[n]

Differencing in Time

x[n] − x[n − 1] n X x[k]

Linearity Time-Shifting Frequency-Shifting Conjugation Time Reversal Time Expansions

Accumulation

k=−∞

X(ejkω ) X(eZjω )Y (ejω ) 1 X(ejθ )Y (ej(ω−θ) )dθ 2π 2π (1 − e−jω )X(ejω ) 1 X(ejω ) 1 − e−jω +∞ X j0 δ(ω − 2πk) +πX(e ) k=−∞

Differentiation in Frequency

Conjugate Symmetry Real Signals

nx[n]

for x[n] real

Symmetry for Real, Even x[n] real and even Signals Symmetry for Real, Odd x[n] real and odd Signals Even-odd Decomposition of xe [n] = Ev{x[n]} [x[n] real] xo [n] = Od{x[n]} [x[n] real] Real Signals

dX(ejω ) j dω  jω X(e ) = X ∗ (e−jω )     ℜe{X(ejω )} = ℜe{X(e−jω )} ℑm{X(ejω )} = −ℑm{X(e−jω )}   |X(ejω )| = |X(e−jω )|    < ) X(ejω ) = −< ) X(e−jω ) X(ejω ) real and even

X(ejω ) purely imaginary and odd

Parseval’s Relation for Aperiodic Signals Z +∞ X 1 2 |X(ejω )|2 dω |x[n]| = 2π 2π n=−∞

ℜe{X(ejω )} jℑm{X(ejω )}

Table 6: Basic Discrete-Time Fourier Transform Pairs

Signal X

Fourier transform jk(2π/N )n



ak e

+∞ X

k=−∞

k=hN i

ejω0 n



+∞ X



2πk ak δ ω − N



cos ω0 n

π

δ(ω − ω0 − 2πl)

{δ(ω − ω0 − 2πl) + δ(ω + ω0 − 2πl)}

l=−∞

+∞ π X {δ(ω − ω0 − 2πl) − δ(ω + ω0 − 2πl)} j

sin ω0 n

l=−∞

x[n] = 1



+∞ X

δ(ω − 2πl)

l=−∞

Periodic  square wave 1, |n| ≤ N1 x[n] = 0, N1 < |n| ≤ N/2 and x[n + N ] = x[n] +∞ X δ[n − kN ]

k=−∞

an u[n], x[n]



sin W n πn

1, 0,

|a| < 1

|n| ≤ N1 |n| > N1

=W π sinc 0
Wn π



δ[n] u[n] δ[n − n0 ] (n + 1)an u[n], |a| < 1 (n + r − 1)! n a u[n], |a| < 1 n!(r − 1)!



+∞ X

k=−∞



2πk ak δ ω − N

 +∞ 2π X δ ω− N k=−∞ 1 1 − ae−jω sin[ω(N1 + 12 )] sin(ω/2) 1, X(ejω ) = 0, X(ejω )periodic

2πk N





2πm = N 1, k = m, m ± N, m ± 2N, . . . ak = 0, otherwise 0 irrational ⇒ The signal is aperiodic (b) ω 2π 2πm (a) ω0 =  N 1 2 , k = ±m, ±m ± N, ±m ± 2N, . . . ak = 0, otherwise 0 irrational ⇒ The signal is aperiodic (b) ω 2π 2πr (a) ω0 =  N1  2j , k = r, r ± N, r ± 2N, . . . − 1 , k = −r, −r ± N, −r ± 2N, . . . ak =  2j 0, otherwise 0 irrational ⇒ The signal is aperiodic (b) ω 2π 1, k = 0, ±N, ±2N, . . . ak = 0, otherwise

1 + 1 − e−jω

ω0

ak ak

= =

ak =

— 0 ≤ |ω| ≤ W W < |ω| ≤ π with period 2π

— —

+∞ X

πδ(ω − 2πk)



k=−∞

1 (1 − ae−jω )2 1 (1 − ae−jω )r

sin[(2πk/N )(N1 + 12 )] , k 6= 0, ±N, ±2N, . . . N sin[2πk/2N ] 2N1 +1 , k = 0, ±N, ±2N, . . . N

1 for all k N



1

e−jωn0

ak (a)

l=−∞ +∞ X

Fourier series coefficients (if periodic)

— — —

Table 7: Properties of the Laplace Transform Property

Linearity

Signal

Transform

ROC

x(t)

X(s)

R

x1 (t)

X1 (s)

R1

x2 (t)

X2 (s)

R2

ax1 (t) + bx2 (t) aX1 (s) + bX2 (s) At least R1 ∩ R2

Time shifting

x(t − t0 )

e−st0 X(s)

R

Shifting in the s-Domain

es0 t x(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 s X |a| a

Conjugation

x∗ (t)

X ∗ (s∗ )

Convolution

x1 (t) ∗ x2 (t)

X1 (s)X2 (s)

Differentiation in the Time Domain

d x(t) dt

sX(s)

Differentiation in the s-Domain

−tx(t)

Integration in the Time Domain

Z

t

−∞

x(τ )d(τ )

d X(s) ds 1 X(s) s

“Scaled” ROC (i.e., s is in the ROC if (s/a) is in R) R At least R1 ∩ R2 At least R R 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. δ(t)

1

All s

2. u(t) 3. −u(−t) tn−1 u(t) (n − 1)! tn−1 5. − u(−t) (n − 1)! 4.

6. e−αt u(t) 7. −e−αt u(−t) tn−1 −αt e u(t) (n − 1)! tn−1 −αt 9. − e u(−t) (n − 1)! 8.

e−sT

10. δ(t − T ) 11. [cos ω0 t]u(t)

s + ω02 ω0 2 s + ω02 s+α (s + α)2 + ω02 ω0 (s + α)2 + ω02 s2

12. [sin ω0 t]u(t) 13. [e−αt cos ω0 t]u(t) 14. [e−αt sin ω0 t]u(t) 15. un (t) =

1 s 1 s 1 sn 1 sn 1 s+α 1 s+α 1 (s + α)n 1 (s + α)n

dn δ(t) dtn

16. u−n (t) = u(t) ∗ · · · ∗ u(t) {z } | n times

ℜe{s} > 0 ℜe{s} < 0 ℜe{s} > 0 ℜe{s} < 0 ℜe{s} > −ℜe{α} ℜe{s} < −ℜe{α} ℜe{s} > −ℜe{α} ℜe{s} < −ℜe{α} All s ℜe{s} > 0 ℜe{s} > 0 ℜe{s} > −ℜe{α} ℜe{s} > −ℜe{α}

sn

All s

1 sn

ℜe{s} > 0

Table 9: Properties of the z-Transform Property

Sequence

Transform

ROC

x[n] x1 [n] x2 [n]

X(z) X1 (z) X2 (z)

R R1 R2

Linearity

ax1 [n] + bx2 [n]

aX1 (z) + bX2 (z) At least the intersection of R1 and R2

Time shifting

x[n − n0 ]

z −n0 X(z)

R except for the possible addition or deletion of the origin

Scaling in the

ejω0 n x[n]

R

z-Domain

z0n x[n] n

a x[n]

−jω0 X(e   z) X zz0 X(a−1 z)

x[−n]

X(z −1 )

Inverted R (i.e., R−1 = the set of points z −1 where z is in R)

X(z k )

R1/k

Time reversal

Time expansion

x(k) [n] =



x[r], n = rk 0, n 6= rk for some integer r

z0 R Scaled version of R (i.e., |a|R = the set of points {|a|z} for z in R)

(i.e., the set of points z 1/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

1 X(z) 1−z −1

At least the intersection of R and |z| > 1

Differentiation in the z-Domain

nx[n]

−z dX(z) dz

R

k=−∞

x[k]

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. δ[n]

1

All z

2. u[n]

1 1−z −1

|z| > 1

3. u[−n − 1]

1 1−z −1

|z| < 1

4. δ[n − m]

z −m

All z except 0 (if m > 0) or ∞ (if m < 0)

5. αn u[n]

1 1−αz −1

|z| > |α|

6. −αn u[−n − 1]

1 1−αz −1

|z| < |α|

7. nαn u[n]

αz −1 (1−αz −1 )2

|z| > |α|

8. −nαn u[−n − 1]

αz −1 (1−αz −1 )2

|z| < |α|

9. [cos ω0 n]u[n]

1−[cos ω0 ]z −1 1−[2 cos ω0 ]z −1 +z −2

|z| > 1

10. [sin ω0 n]u[n]

[sin ω0 ]z −1 1−[2 cos ω0 ]z −1 +z −2

|z| > 1

11. [rn cos ω0 n]u[n]

1−[r cos ω0 ]z −1 1−[2r cos ω0 ]z −1 +r 2 z −2

|z| > r

12. [rn sin ω0 n]u[n]

[r sin ω0 ]z −1 1−[2r cos ω0 ]z −1 +r 2 z −2

|z| > r

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Table 1: Properties of the Continuous-Time Fourier Series x(t

Table 1: Properties of the Continuous-Time Fourier Series +∞ X x(t) = jkω0 t ak e k=−∞ 1 ak = T Property Periodic Convolution −jkω0 t x(t)e T...

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