Discrete-time Fourier series and transform - Continuous and Discrete Time Signals and Systems - page 483

Sequence Time-domain waveform Magnitude and phase spectra

x [ k ] =1

∞ ∞

(1) Constant
x [ k ] = 1

X ( W )

∑ ∑

( W −2 m p )

= 2π d

1

… …

m = −∞

… …

k

W −6 −4 −2 0 2 4 6

−6 p −4 p −2 p 2 p 4 p 6 p

0

x [ k ] =δ[ k ]

() 1

(2) Unit impulse
x [ k ] = δ [ k ]

X X W = = 1

1

1

k

W

−6 −4 −2 0

2 4 6

−3 p −2 p − p 0 p 2 p 3 p

x [ k ] = u [ k ]

∞

(3) Unit step
x [ k ] = u [ k ]

()

∑

( ( )

1

X Ω = π δ Ω Ω − − 2 m π + +

1 − e −jΩ

…

m = −∞

p

… …

k

−6 −4 − 0 6

W

− 2 p −3 p − p 0 p 2 p 3 p

k

1

x [ k ] = p u [ k ]

(4) Decaying exponential
x [ k ] = p k u [ k ]
with p < 1

() =

X W =

1 − p e −j W 1

p

1

…

p

2

1 − p

k

W

−6 −4 −2 0

2 4 6

−3 p −2 p − p 0 p 2 p 3 p

sin((2 N + 1) W /2)
sin( W /2)

1 k ≤ N

(5) Rectangular
x [ k ] =

() = () =

X X W

x [ k ] =

0 elsewhere

1 k ≤ N

1

2 N +1

0 elsewhere

k

W

− N 0

N

− −3 p −2 p − p p 2 p 3 p

0

k k

x x [ [ k k ] ] = = ( ( k k + + 1 1 p p u u [ [ k k ] ]

1

() (1 − p e −j W ) 2 (6) First-order time-rising

decaying exponential
x [ k ] = ( k + 1) p k u [ k ]
with p < 1

X W = =

1 1 1 2 p 2 p 2 p

1

3 p 2 3 p 2 3 p 2

…

(1 − p ) 2

k k k

W

−6 −6 −6 −4 −4 −4 −2 −2 −2 0 0 0

− − 3 p

−2 p − p 0 p 2 p 3 p

()
p

1 1 1 W ≤ W

(7) Sinc
x [ k ] = W
π

W Wk

()

x [ k ] = = sinc X W = = =

p

0 W < W ≤ p

Wk
π

1 1

sinc

… …

1

k

W

−6 −4 −2

0 2

−2π 0 2 p

j W k ∞

x [ [ k ] ] x [ [ k ] ] = = e

(8) Complex exponential
x [ k ] = e j k Ω 0

0

()

∑

( )

= 2 p

d W W p m

1

X Ω = 2 − − − − 2

0

… …

m =−∞

2 p

k

−6 −4 −2

0 2 4 6

W

−2 p 0

2 p

< x [ k ]

∞

(9) Cosine
x [ k ] = cos( Ω 0 k )

x [ k ] = cos Ω k

()

∑

[ )] ( ) ( X W = p W − + W − 2 m p + + d W − W − 2 m p

d

0

0 0

1 1

m =−∞

… …

p

k

W

−6 −4 −2

0 2

−2 −2 p 0 2 2 p