DTFT Theorems and Properties
Property
Notation:
Linearity:
Time shifting:
Time reversal
Convolution:
Multiplication:
Correlation:
Time Domain
x(n)
x1 (n)
x2 (n)
a1 x1 (n) + a2 x2 (n)
x(n − k)
x(−n)
x1 (n) ∗ x2 (n)
x1 (n)x2 (n)
rx1 x2 (l) = x1 (l) ∗ x2 (−l)
Frequency Differentiation:
Wiener-Khintchine:
nx(n)
rxx (l) = x(l) ∗ x(−l)
Frequency Domain
X(ω)
X1 (ω)
X1 (ω)
a1 X1 (ω) + a2 X2 (ω)
e−jωk X(ω)
X(−ω)
X1R(ω)X2 (ω)
1
2π 2π X1 (λ)X2 (ω − λ)dλ
Sx1 x2 (ω) = X1 (ω)X2 (−ω)
= X1 (ω)X2∗ (ω) [if x2 (n) real]
j dX(ω)
dω
Sxx (ω) = |X(ω)|2
DTFT Symmetry Properties
Time Sequence
x(n)
x∗ (n)
x∗ (−n)
x(−n)
xR (n)
jxI (n)
x(n) real
x0e (n) = 12 [x(n) + x∗ (−n)]
x0o (n) = 21 [x(n) − x∗ (−n)]
DTFT
X(ω)
X ∗ (−ω)
X ∗ (ω)
X(−ω)
Xe (ω) = 21 [X(ω) + X ∗ (−ω)]
Xo (ω) = 21 [X(ω) − X ∗ (−ω)]
X(ω) = X ∗ (−ω)
XR (ω) = XR (−ω)
XI (ω) = −XI (−ω)
|X(ω)| = |X(−ω)|
∠X(ω) = −∠X(−ω)
XR (ω)
jXI (ω)
DFT Properties
Property
Notation:
Periodicity:
Linearity:
Time reversal
Circular time shift:
Circular frequency shift:
Complex conjugate:
Circular convolution:
Multiplication:
Parseval’s theorem:
Time Domain
x(n)
x(n) = x(n + N )
a1 x1 (n) + a2 x2 (n)
x(N − n)
x((n − l))N
x(n)ej2πln/N
x∗ (n)
x1 (n) ⊗ x2 (n)
x (n)x (n)
P1 N −1 2
∗
n=0 x(n)y (n)
1
Frequency Domain
X(k)
X(k) = X(k + N )
a1 X1 (k) + a2 X2 (k)
X(N − k)
X(k)e−j2πkl/N
X((k − l))N
X ∗ (N − k)
X1 (k)X2 (k)
1
1 (k) ⊗ X2 (k)
NX
1 PN −1
∗
k=0 X(k)Y (k)
N
Note: The following tables are courtesy of Professors Ashish Khisti and Ravi Adve and were
developed originally for ECE355. Please note that the notation used is different from that in
ECE455.
Fourier Properties
Property
DTFS
Synthesis
P
Analysis
1
N
Linearity
x[n] =
ak ejkΩ0 n
k=<N >
ak =
−jkΩ0 n
n=<N > x[n]e
P
αx[n] + βy[n] ↔
CTFS
Time Shifting
x[n − n0 ] ↔ ak e
x(t) =
ak ejkω0 t
P∞
1
2π
k=−∞
1
T
R
T
R
2π
αak + βbk
x(t − t0 ) ↔ ak e
x[n] =
X(ejΩ )ejΩn dΩ
1
2π
R ∞ x(t) = jωt
X(jω)e dω
−∞
=
R ∞ X(jω)−jωt
x(t)e
dt
−∞
αx[n] + βy[n] ↔
αx(t) + βy(t) ↔
αX(e
−jkω0 t0
CTFT
X(ejΩ ) =
P∞
−jΩn
−∞ x[n]e
ak =
x(t)e−jkω0 t dt
αx(t) + βy(t) ↔
αak + βbk
−j2πn0 k/N
DTFT
jΩ
) + βY (e
−jΩn0
x[n − n0 ] ↔ e
jΩ
)
X(e
αX(jω) + βY (jω)
jΩ
)
x(t − t0 ) ↔ e−jωt0 X(jω)
Frequency Shift
x[n]ej2πmn/N ↔ ak−m
x(t)ejmω0 t ↔ ak−m
x[n]ejΩ0 n ↔ X(ej(Ω−Ω0 )n )
x(t)ejω0 t ↔ X(j(ω − ω0 ))
Conjugation
x∗ [n] ↔ a∗−k
x∗ (t) ↔ a∗−k
x∗ [n] ↔ X ∗ (e−jΩ )
x∗ (t) ↔ X ∗ (−jω)
Time Reversal
x[−n] ↔ a−k
x(−t) ↔ a−k
x[−n] ↔ X(e−jΩ )
x(−t) ↔ X(−jω)
x[n] ∗ y[n] ↔ X(ejΩ )Y (ejΩ )
x(t) ∗ y(t) ↔ X(jω)Y (jω)
x[n]y[n] ↔
x(t)y(t) ↔
PN −1
Convolution
x[r]y[n − r]
↔ N a k bk
P −1
x[n]y[n] ↔ N
r=0 ar bk−r
r=0
R
T
x(τ )y(t − τ )dτ
↔ T ak bk
x(t)y(t) ↔ ak ∗ bk
Multiplication
1
2π
First Difference/
Derivative
Running Sum/
Integration
Parseval’s
Relation
x[n] − x[n − 1] ↔
(1 − e−j2πk/N )ak
Pn
k=−∞ x[k] ↔
ak
dx(t)
dt
Rt
−∞
↔ jkω0 ak
x(τ )dτ ↔
PN −1
|x[n]|2
Pn=0
N −1
= k=0 |ak |2
1
N
1
|x(t)|2 dt
TPT
∞
= k=−∞ |ak |2
R
2π
X(ejθ )Y (ej(Ω−θ) )dθ
x[n] − x[n − 1] ↔
(1 − e−jΩ )X(ejΩ )
ak
jkω0
1−e−j2πk/N
R
Pn
jΩ
X(e )
x[k] ↔ 1−e
−jΩ
j0
+πX(e )δ(Ω)
k=−∞
P∞
|x[n]|2
Rn=−∞ jΩ 2
1
= 2π 2π |X(e )| dΩ
Real and even
Real and even
signals
in frequency domain
Real and odd
Purely imaginary and odd
signals
in frequency domain
1
X(jω)
2π
dx(t)
dt
∗ Y (jω)
↔ jωX(jω)
Rt
x(τ )dτ ↔ X(jω)
jω
+πX(j0)δ(ω)
R∞
|x(t)|2 dt
−∞
R∞
1
= 2π −∞ |X(jω)|2 dω
−∞
Additional Property: A real-valued time-domain signal x(t) or x[n] will have a conjugate-symmetric Fourier
representation.
Notes:
1. For the CTFS, the signal x(t) has a period of T , fundamental frequency ω0 = 2π/T ; for the DTFS, the signal
x[n] has a period of N , fundamental frequency Ω0 = 2π/N . ak and bk denote the Fourier coefficients of x(t)
(or x[n]) and y(t) (or y[n]) respectively.
2. Periodic convolutions can be evaluated by summing or integrating over any single period, not just those
indicated above.
3. The “Running Sum” formula for the DTFT above is valid for Ω in the range −π < Ω ≤ π.
2
Fourier Pairs
Fourier Series Coefficients of Periodic Signals∗
Discrete-Time∗∗
Continuous-Time
Time Domain – x(t)
Frequency Domain – ak
Time Domain – x[n]
Frequency Domain – ak
Aejω0 t
a1 = A
ak = 0, k 6= 1
AejΩ0 n
a1 = A,
ak = 0, k 6= 1
A cos(ω0 t)
a1 = a−1 = A/2
ak = 0, k 6= 1
A cos(Ω0 n)
a1 = a−1 = A/2
ak = 0, k 6= 1
A sin(ω0 t)
A
a1 = a∗−1 = 2j
ak = 0, k 6= 1
A sin(Ω0 n)
A
a1 = a∗−1 = 2j
ak = 0, k 6= 1
x(t) = A
a0 = A, ak = 0 otherwise
x[n] = A
a0 = A, ak = 0 otherwise
P∞
n=−∞
δ(t − nT )
ak =
P∞
1
T
k=−∞
δ[n − kN ]
ak =
1
N
2T1
T
sin(kω0 T1 )
, k 6= 0
ak =
kπ
Periodic square wave
1
|t| < T1
x(t) =
0 T1 < |t| ≤ T2
a0 =
and x(t) = x(t + T )
Fourier Transform Pairs
Discrete-Time∗∗
Continuous-Time
Time Domain – x(t)
1, |t| < T1
x(t) =
0, |t| > T1
sin W t
πt
Frequency Domain – X(jω)
2 sin(ωT1 )
ω
1, |ω| < W
X(jω) =
0, otherwise
Time Domain – x[n]
1, |n| ≤ N1
x[n] =
0, |n| > N1
sin W n
πn
Frequency Domain – X(ejΩ )
sin(Ω(N1 + 1/2))
sin(Ω/2)
1, |Ω| ≤ W
jΩ
X(e ) =
0, otherwise
δ(t)
1
δ[n]
1
1
2πδ(ω)
1
+ πδ(ω)
jω
1
a + jω
1
(a + jω)n
1
2πδ(Ω)
1
+ πδ(Ω)
1 − e−jΩ
1
1 − ae−jΩ
1
(1 − ae−jΩ )r
u(t)
e−at u(t), Re(a) > 0
tn−1 −at
e u(t), Re(a) > 0
(n − 1)!
u[n]
an u[n], |a| < 1
(n + r − 1)! n
a u[n], |a| < 1
n!(r − 1)!
∗
In the Fourier series table, ω0 = 2π
and Ω0 = 2π
, where T and N are the periods of x(t) and x[n] respectively.
T
N
For the DTFS, ak is given only for k in the range −N/2 + 1 ≤ k ≤ N/2 for even N , −(N − 1)/2 ≤ k ≤ (N − 1)/2 for
odd N , and ak = ak+N ; for the DTFT X(ejΩ ) is given only for Ω in the range −π < Ω ≤ π, and X(ejΩ ) = X(ej(Ω+2π) ).
∗∗
Fourier Transform for Periodic Signals:
x(t) =
∞
X
k=−∞
x[n] =
X
∞
X
ak ejkω0 t ↔ X(jω) = 2π
ak δ(ω − kω0 )
k=−∞
ak ejkΩ0 n ↔ X(ejΩ ) = 2π
k=<N >
∞
X
k=−∞
3
ak δ(Ω − kΩ0 )
Common z-Transform Pairs
1
2
3
Signal, x(n)
δ(n)
u(n)
an u(n)
4
5
nan u(n)
−an u(−n − 1)
6
−nan u(−n − 1)
7
cos(ω0 n)u(n)
8
sin(ω0 n)u(n)
9
n
a cos(ω0 n)u(n)
10
an sin(ω0 n)u(n)
z-Transform, X(z)
1
1
1−z −1
1
1−az −1
az −1
(1−az −1 )2
1
1−az −1
az −1
(1−az −1 )2
1−z −1 cos ω0
1−2z −1 cos ω0 +z −2
z −1 sin ω0
1−2z −1 cos ω0 +z −2
1−az −1 cos ω0
1−2az −1 cos ω0 +a2 z −2
1−az −1 sin ω0
1−2az −1 cos ω0 +a2 z −2
ROC
All z
|z| > 1
|z| > |a|
|z| > |a|
|z| < |a|
|z| < |a|
|z| > 1
|z| > 1
|z| > |a|
|z| > |a|
z-Transform Properties
Property
Notation:
Linearity:
Time shifting:
Time Domain
x(n)
x1 (n)
x2 (n)
a1 x1 (n) + a2 x2 (n)
x(n − k)
z-Domain
X(z)
X1 (z)
X2 (z)
a1 X1 (z) + a2 X2 (z)
z −k X(z)
z-Scaling:
Time reversal
Conjugation:
z-Differentiation:
Convolution:
an x(n)
x(−n)
x∗ (n)
n x(n)
x1 (n) ∗ x2 (n)
X(a−1 z)
X(z −1 )
X ∗ (z ∗ )
−z dX(z)
dz
X1 (z)X2 (z)
4
ROC
ROC: r2 < |z| < r1
ROC1
ROC2
At least ROC1 ∩ ROC2
At least ROC, except
z = 0 (if k > 0)
and z = ∞ (if k < 0)
|a|r2 < |z| < |a|r1
1
< |z| < r12
r1
ROC
r2 < |z| < r1
At least ROC1 ∩ ROC2