ECE2111 Signals and systems Useful formulas
Sifting formula:
Convolution:
x[k]δ[n−k] foralln
x(τ)δ(t−τ)dτ forallt
(x1 ∗x2)[n] =
Infinite sum: If |α| < 1 then
Indefinite integrals:
(x1 ∗x2)(t) =
x1(τ)x2(t−τ) dτ
x1[k]x2[n−k]
αk = 1−α. k=0
xx k1k+1
e dx=e +c x dx=k+1x +c (fork̸=−1)
DTFT of sampled signal If x has continuous-time Fourier transform X then xs[n] = x(nT) for all n has discrete-time Fourier transform
X((ω−2πk)/T) forallω.
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Continuous-time Fourier series tables
If x is periodic with period T0 = 2π/ω0 (and x is sufficiently regular) then ∞ 1 T 0
x(t) = Xkejkω0t where Xk = T x(t)e−jkω0t dt k=−∞ 00
Time domain signal x(t) for −T0/2 ≤ t ≤ T0/2
cos(ω0 t) sin(ω0 t) 1
[u(t+T)−u(t−T)] δ(t)
Fourier series coefficients Xk
of periodic extension xperiodic(t)
12 (δ[k−1]+δ[k+1])
1 (δ[k−1]−δ[k+1]) 2j
δ[k] sin(kω0T)
Square wave Impulse train
with fundamental period T0 and fundamental frequency ω0 = 2π/T0 and the parameter T satisfies 0 < T < T0/2. The entry u(t + T ) − u(t − T ) should be interpreted as a periodic square wave that agrees with u(t + T) − u(t − T) for −T0/2 < t < T0/2. The entry δ should be interpreted as an impulse train with impulses at integer multiples of T0.
Table of Fourier series coefficients. In the table, all time domain signals are periodic
Time domain y(t) =
αx(t) + βz(t)
ejMω0tx(t)
T0 x(τ)z(t−τ)dτ 0
x(t)z(t) x(t)∗
Time domain x(t) is real
x(t) = x(−t)∗ 1 T 0
Frequency-domain Yk=
αXk + βZk e−jkω0τ −M T0XkZk
l=−∞ XlZk−l
Modulation
Periodic convolution Multiplication Conjugation
Conjugate symmetry in frequency Conjugate symmetry in time
Parseval’s relation
Frequency-domain Xk=X∗
−k Xk is real
T |x(t)|2 dt = |Xk|2
integer, and α and β are complex numbers. All time-domain signals are periodic with fundamental period T0 and fundamental frequency ω0 = 2π/T0. All integrals from 0 to T0 can be replaced with integrals from a to a + T0 for any real a.
Properties of continuous-time Fourier series. In the table, τ is a real number, M is an
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Discrete-time Fourier series tables
If x is periodic with period N0 = 2π/ω0 then N0−1
x[n]e−jkω0n.
x[n] = Xkejkω0n k=0
Time domain signal x[n] for −N0/2 ≤ n < N0/2
cos(ω0n) sin(ω0n) 1
u[n+N]−u[n−N] δ[n]
N0 n=0 Fourier series coefficients Xk
of periodic extension xperiodic[n]
12 (δ[k − 1] + δ[k + 1])
1 (δ[k−1]−δ[k+1]) 2j
δ[k] sin(kω0N)
Square wave Impulse train
periodic with fundamental period N0 and fundamental frequency ω0 = 2π/N0 and N is an integer with 0 < N < N0/2. All Fourier series coefficient sequences are also periodic with fundamental period N0. The time-domain entry u[n + N] − u[n − N] should be interpreted as an N0-periodic square wave that agrees with u[n + N] − u[n − N] for −N0/2 ≤ n < N0/2.
Table of discrete-time Fourier series coefficients. In the table, all time domain signals are
Time domain y[n] =
αx[n] + βz[n]
ejMω0nx[n]
N0−1 x[l]z[n − l] l=0
x[n]z[n] x[n]∗
Time domain x[n] is real
x[n] = x[−n]∗
Frequency-domain Yk =
αXk + βZk e−jkω0N −M
N0−1 XlZk−l l=0
Modulation
Periodic convolution Multiplication Conjugation
Conjugate symmetry in frequency Conjugate symmetry in time
Parseval’s relation
Frequency-domain Xk = X∗
−k Xk is real
|x[n]|2 dt = |Xk|2
N0 n=0 k=0
and β are complex numbers. Time-domain signals are periodic with fundamental period N0 and fundamental frequency ω0 = 2π/N0. Fourier series coefficients are periodic with fundamental period N0. Sums from 0 to N0 −1 can be replaced with sums from a to a+N0 −1 for any integer a.
Properties of discrete-time Fourier series. In the table, N and M are integers, and α
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Continuous-time Fourier transform tables
If x is an absolutely summable (and sufficiently regular) continuous-time signal then 1∞ ∞
Time domain signal
e−atu(t) (for Re(a) > 0)
u(t+T1)−u(t−T1) sin(W t)
X(ω)ejωt dω where X(ω) = x(t)e−jωt dt −∞
Continuous-time Fourier transform X(ω) =
2 sin(ωT1) ω
u(ω + W ) − u(ω − W ) 1
1 +πδ(ω) jω
δ(t−τ) e−jωτ
ejω0t 2πδ(ω − ω0)
te−atu(t) (for Re(a) > 0)
positive real number, the parameter a is a complex number, and the parameter τ is a real number.
Table of continuous-time Fourier transform pairs. In the table, the parameter W is a
Time domain y(t) =
αx(t) + βz(t)
∞ x(τ)z(t−τ)dτ −∞
x(t)z(t) x(t)∗ x(at)
Time domain x(t) is real
Frequency-domain Y(ω)=
αX(ω) + βZ(ω)
Linearity Delay Modulation Convolution Multiplication Conjugation Time-scaling
Conjugate symmetry in frequency Conjugate symmetry in time
Parseval’s relation
X(ω)Z(ω) 1∞
2π −∞ X(Ω)Z(ω − Ω) dΩ X(−ω)∗
1 X(ω/a) |a|
Frequency-domain X(ω) = X(−ω)∗
x(t) = x(−t)∗
X(ω) is real
|x(t)|2 dt = 2π |X(ω)|2 dω
Table 6: Properties of continuous-time Fourier transform. In the table, τ is a real number, ω0 is a real number, a is a non-zero real number, and α and β are complex numbers.
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Discrete-time Fourier transform tables
If x is an absolutely summable discrete-time signal then
Time domain signal
αnu[n] (for |α| < 1)
u[n+N]−u[n−N] sin(W n)
(n + 1)αnu[n] (for |α| < 1)
X(ω) = x[n]e−jωn
X(ω)ejωn dω
Discrete-time Fourier transform X(ω) =
sin(ω(N + 1/2)) sin(ω/2)
2π-periodic extension
of u(ω+W)−u(ω−W)
(1 − αe−jω)2
0 < W < π, the parameter α is a complex number, and the parameter n0 is an integer.
Table of discrete-time Fourier transform pairs. In the table, the parameter W satisfies
Time domain y[n] =
αx[n] + βz[n] x[n−N]
∞l=−∞ x[l]z[n − l] x[n]z[n]
Time domain
x[n] is real x[n] = x[−n]∗
Frequency-domain Y(ω)=
αX(ω) + βZ(ω) e−jωNX(ω) X(ω−ω0) X(ω)Z(ω)
Linearity Delay Modulation Convolution Multiplication Conjugation
Conjugate symmetry in frequency Conjugate symmetry in time
Parseval’s relation
1 2πX(Ω)Z(ω−Ω)dΩ 2π 0
X(−ω)∗ Frequency-domain
|x[n]|2 dt = 2π |X(ω)|2 dω
X(ω) = X(−ω)∗ X(ω) is real
Table 8: Properties of discrete-time Fourier transform. In the table, N is an integer, ω0 is a real number, and α and β are complex numbers. All discrete-time Fourier transforms are periodic with fundamental period 2π.
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Laplace transform tables
If x is a (sufficiently regular) continuous-time signal then ˆ ∞ −st
Region of convergence RoC(x)
Re(s) > −Re(a) Re(s) > −Re(a) C
Re(s) > −b
Re(s) > −b
Time domain signal Laplace transform x ( t ) = Xˆ ( s ) =
u(t) tu(t) e−at u(t) te−at u(t)
(s+a)2 δ(t−τ) e−sτ
e−bt cos(ω t)u(t) 0
e−bt sin(ω t)u(t) 0
( s + b ) 2 + ω 02 ω0
( s + b ) 2 + ω 02
and the parameters b and τ and ω0 are real numbers.
parameter a is a complex number,
Linearity Delay Modulation Convolution Conjugation Time-scaling
Integration Differentiation
Table of Laplace transform pairs. In the table, the
Time domain y(t) =
αx(t) + βz(t)
es0 t x(t)
∞ x(τ)z(t−τ)dτ −∞
x(t)∗ x(at)
t x(τ ) dτ −∞
Laplace transform Yˆ(s)=
α Xˆ ( s ) + β Zˆ ( s ) e−sτ Xˆ (s)
Xˆ(s−s0) Xˆ ( s ) Zˆ ( s ) Xˆ ( s ∗ ) ∗
1 Xˆ(s/a) |a|
Xˆ (s)/s s Xˆ ( s )
number, a is a non-zero real number, and α and β are complex numbers.
Properties of Laplace transform. In the table, τ is a real number, s0 is a complex
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Z-transform tables
If x is a discrete-time signal then
Time domain signal x [ n ] =
Xˆ(z)= x[n]z−n n=−∞
Z-transform Xˆ ( z ) =
Region of convergence RoC(x)
|z|>1 |z|>1 |z| > |a| |z| > |a| |z|>0 |z| > |a|
Table of Z-transform pairs. In the table, the parameter a is a complex number, and the
an cos[ω n]u[n] 0
z−1 (1−z−1)2
az−1 (1−az−1)2
an sin[ω n]u[n] 0
1−az−1 cos(ω0) 1−2az−1 cos(ω0)+a2z−2 az−1 sin(ω0) 1−2az−1 cos(ω0)+a2z−2
parameter ω0 is a real number.
Time domain y[n] =
αx[n] + βz[n] x[n−m]
∞k=−∞ x[k]z[n − k] x[n]∗
Properties of the Z-transform. In the table, m is
Z-transform Yˆ ( z ) =
α Xˆ ( z ) + β Zˆ ( z ) z − m Xˆ ( z )
Xˆ ( z ) Zˆ ( z )
Linearity Delay Modulation Convolution Conjugation Time-reversal
an integer, a, α and β are complex
Xˆ ( z ∗ ) ∗ Xˆ ( z − 1 )
Table 12: numbers.
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